{ "title": "Generate poignant card image", "create_time": 1787692252.105484, "update_time": 1787695265.315711, "moderation_results": [], "plugin_ids": null, "conversation_id": "6a8e045c-616c-83e9-ac1f-097a538a0d11", "conversation_template_id": null, "gizmo_id": null, "gizmo_type": null, "is_archived": false, "is_starred": null, "safe_urls": [ "https://numpy.org/doc/2.4/numpy-ref.pdf", "https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com", "https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/function_field_polymod.html?utm_source=chatgpt.com", "https://images.openai.com/static-rsc-1/wYTAsEBHIU0lhkwpByy4lypzCT-vrGx2048c3dZXtSld2KYwWA_Jtulkmh8zFvdPE7b56DsvpXw9aVGZqa4ZjdP6DqjlHGrW0qlEd7yl5kcMNWt1UWzrhxpKkvcECQOE", "https://images.openai.com/static-rsc-1/xD-Iu557yUnzmNUThHgH7S66tnw3NsjqAWiuZoBkRKAjF9_M95t3DuZPNXFSDeWMy03Ioi33Q-j1lxMSIyp1VApy2RpdSxUEgweLey913JoMrz1F248ZddzINQM_9XU9", "https://www.apple.com/", 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"sugar_item_id": null, "sugar_item_visible": false, "pinned_time": null, "is_study_mode": false, "owner": null, "mapping": { "client-created-root": { "id": "client-created-root", "message": null, "parent": null, "children": [ "35f24fe9-64bc-4601-9491-f8cb715f9d64" ] }, "35f24fe9-64bc-4601-9491-f8cb715f9d64": { "id": "35f24fe9-64bc-4601-9491-f8cb715f9d64", "message": { "id": "35f24fe9-64bc-4601-9491-f8cb715f9d64", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787692251.534, "update_time": 1787692256.592589, "content": { "content_type": "text", "parts": [ "chatgpt, i figured out how to use loominary on public computers by polyfilling the userscript functions in the dev console.\n\nclaude drafted these magic cards for the mind control boss webcomic concept. can you generate one image (one card) based on something poignant, near or present?\n\n┌─────────────────────────────────────────┐\n  DEVOTED USER                        {1}{W}\n  ─────────────────────────────────────────\n  [ART: A chibi figure, oversized head on a\n   small hunched body, kneeling. Skin pale\n   greyish, a little sunken — a soft zombie,\n   not a gory one. Both hands cup a phone up\n   near her face; a thin cable runs from the\n   phone's top into her temple, plugged in\n   flush. Her eyes are two dark hollows lit\n   only by the cold blue screenglow washing\n   up over her cheeks. Above her head, a\n   faint thin white halo — ironic, holy.\n   Behind her, a dim grey room; other small\n   halos glow in the dark, a whole kneeling\n   congregation of them receding into black.\n   Palette: warm cream card frame, cold blue\n   light on every face.]\n  ─────────────────────────────────────────\n  Token creature — Zombie Citizen\n  ─────────────────────────────────────────\n  Lifelink\n\n  \"She had a name once.\n   Now she has a feed.\"\n  ─────────────────────────────────────────\n                                        1 / 1\n└─────────────────────────────────────────┘\n\n┌─────────────────────────────────────────┐\n  MR. BOSS, ON CAMERA                  {2}{W}\n  ─────────────────────────────────────────\n  [ART: A grinning suited figure at a glass\n   desk, arms spread wide in a \"we're all\n   family here\" pose, backlit by studio\n   lights and a wall of screens. Under the\n   desk, out of frame from his side but not\n   ours, thin cables run from the floor to a\n   crowd of small kneeling figures. He isn't\n   looking at them.]\n  ─────────────────────────────────────────\n  Legendary Creature — Executive\n  ─────────────────────────────────────────\n  At the beginning of your upkeep, gain\n  control of target creature. At end of turn,\n  if you didn't pay {1} for each creature you\n  control this way, that creature is freed.\n\n  \"Assets don't quit. They churn.\"\n  ─────────────────────────────────────────\n                                        3 / 4\n└─────────────────────────────────────────┘\n\n┌─────────────────────────────────────────┐\n  MARA, WHO ANSWERS THE DOOR          {1}{W}\n  ─────────────────────────────────────────\n  [ART: A tired, kind woman in a doorway at\n   night, one hand out, a blanket over her\n   arm. Warm light spills from the room\n   behind her. A small halo'd figure, cable\n   snapped and trailing, is stepping in from\n   the dark toward her. She's already looking\n   past them for the next one.]\n  ─────────────────────────────────────────\n  Legendary Creature — Advocate\n  ─────────────────────────────────────────\n  Whenever a creature is freed or an\n  opponent loses control of a creature, put\n  a +1/+1 counter on Mara and create a\n  1/1 white Citizen with lifelink.\n\n  \"There's always room. Come in.\"\n  ─────────────────────────────────────────\n                                        0 / 3\n└─────────────────────────────────────────┘ " ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "b2ae909f-166b-4af5-8e77-483e0d8c6ae5", "message_source": null, "turn_exchange_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "working_turn_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "client-created-root", "children": [ "8c279a94-55a2-4202-a336-3aa3f2831e1f" ] }, "8c279a94-55a2-4202-a336-3aa3f2831e1f": { "id": "8c279a94-55a2-4202-a336-3aa3f2831e1f", "message": { "id": "8c279a94-55a2-4202-a336-3aa3f2831e1f", "author": { "role": "tool", "name": "t2uay3k.sj1i4kz", "metadata": {} }, "create_time": 1787692252.212144, "update_time": 1787692326.215074, "content": { "content_type": "multimodal_text", "parts": [ { "content_type": "image_asset_pointer", "asset_pointer": "sediment://file_00000000d4b081f58a21371e53140dc3", "mime_type": "image/png", "size_bytes": 2175824, "width": 1061, "height": 1482, "fovea": null, "metadata": { "dalle": { "gen_id": "3bf8be4d-f882-4963-bf1c-5664872f387d", "prompt": "", "seed": null, "parent_gen_id": null, "edit_op": null, "serialization_title": "DALL-E generation metadata" }, "gizmo": null, "generation": { "gen_id": "3bf8be4d-f882-4963-bf1c-5664872f387d", "gen_size": "image", "gen_size_v2": "48", "seed": null, "parent_gen_id": null, "height": 1482, "width": 1061, "transparent_background": false, "is_sticker": false, "is_purchasable": null, "serialization_title": "Image Generation metadata", "orientation": "portrait", "projection": "rectilinear" }, "segmentation": null, "container_pixel_height": 1482, "container_pixel_width": 1061, "emu_omit_glimpse_image": null, "emu_patches_override": null, "lpe_keep_patch_ijhw": null, "lpe_delta_encoding_channel": null, "sanitized": false, "asset_pointer_link": null, "watermarked_asset_pointer": null, "is_no_auth_placeholder": null } } ] }, "status": "finished_successfully", "end_turn": null, "weight": 0, "metadata": { "ghostrider": { "status": "final", "is_feature_trial": false }, "parent_id": "35f24fe9-64bc-4601-9491-f8cb715f9d64", "turn_exchange_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "working_turn_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "image_gen_title": "Devoted User: Zombie Congregation", "cot_version": "v5", "async_source": "saserver-centralus-prod.fck9d:ghostrider-user-r02PArrjD6Eq2V6pvZi12kId-b2ae909f-166b-4af5-8e77-483e0d8c6ae5:US", "permissions": [ { "type": "notification", "status": "requested", "notification_channel_id": "image_gen", "notification_channel_name": "ImageGen", "notification_priority": 4 } ], "can_save": false }, "recipient": "all", "channel": "final" }, "parent": "35f24fe9-64bc-4601-9491-f8cb715f9d64", "children": [ "76f965cb-bd9d-4a7a-a6e1-298d30415868" ] }, "76f965cb-bd9d-4a7a-a6e1-298d30415868": { "id": "76f965cb-bd9d-4a7a-a6e1-298d30415868", "message": { "id": "76f965cb-bd9d-4a7a-a6e1-298d30415868", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692261.541664, "update_time": null, "content": { "content_type": "thoughts", "thoughts": [ { "summary": "Selecting a poignant card", "content": "I’m choosing “Devoted User” because its quiet image of memory, routine, and companionship feels especially poignant. I’ll create an original fantasy trading-card-style illustration with readable rules text and no recognizable people.", "chunks": [], "finished": true }, { "summary": "Selected a poignant card", "content": "", "chunks": [], "finished": true } ], "source_analysis_msg_id": "8c0f490b-7df5-47a1-a21a-336285490142" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "is_reasoning", "tool_icons": [], "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-4-auto-thinking", "cot_version": "v5", "model_slug": "gpt-5-4-thinking", "request_id": "wfr_01a03ac305687cf9bd5141eb16fdd770", "message_type": "next", "turn_exchange_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "working_turn_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "summary_type": "raw_cot", "classifier_response": "default", "reasoning_start_time": 1787692255.9676309, "skip_reasoning_title": "Skip", "can_save": false, "parent_id": "b86f3fa7-7566-4e0f-8676-356065e56faf", "async_source": "saserver-centralus-prod.fck9d:ghostrider-user-r02PArrjD6Eq2V6pvZi12kId-b2ae909f-166b-4af5-8e77-483e0d8c6ae5:US", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "8c279a94-55a2-4202-a336-3aa3f2831e1f", "children": [ "389cbfc0-fb65-46e4-9d67-87adc3d3d5b1" ] }, "389cbfc0-fb65-46e4-9d67-87adc3d3d5b1": { "id": "389cbfc0-fb65-46e4-9d67-87adc3d3d5b1", "message": { "id": "389cbfc0-fb65-46e4-9d67-87adc3d3d5b1", "author": { "role": "tool", "name": "t2uay3k.sj1i4kz", "metadata": {} }, "create_time": 1787692320.2415164, "update_time": null, "content": { "content_type": "multimodal_text", "parts": [ { "content_type": "image_asset_pointer", "asset_pointer": "sediment://file_00000000d4b081f58a21371e53140dc3", "mime_type": "image/png", "size_bytes": 2175824, "width": 1061, "height": 1482, "fovea": null, "metadata": { "dalle": { "gen_id": "3bf8be4d-f882-4963-bf1c-5664872f387d", "prompt": "", "seed": null, "parent_gen_id": null, "edit_op": null, "serialization_title": "DALL-E generation metadata" }, "gizmo": null, "generation": { "gen_id": "3bf8be4d-f882-4963-bf1c-5664872f387d", "gen_size": "image", "gen_size_v2": "48", "seed": null, "parent_gen_id": null, "height": 1482, "width": 1061, "transparent_background": false, "is_sticker": false, "is_purchasable": null, "serialization_title": "Image Generation metadata", "orientation": "portrait", "projection": "rectilinear" }, "segmentation": null, "container_pixel_height": 1482, "container_pixel_width": 1061, "emu_omit_glimpse_image": null, "emu_patches_override": null, "lpe_keep_patch_ijhw": null, "lpe_delta_encoding_channel": null, "sanitized": false, "asset_pointer_link": null, "watermarked_asset_pointer": null, "is_no_auth_placeholder": null } }, "Model caption: A detailed fantasy trading-card style illustration and layout, shown straight-on like a card scan. Overall scene: a high‑detail card frame with a creamy/ivory parchment look, ornate borders and iconography, containing a chibi/gothic digital painting of a zombie-like figure and card text. Top area: a header bar with the card name “DEVOTED USER” in large bold serif (dark/black) centered/left at top; on the top-right are mana/cost symbols “1” in a small circle and a sun/white mana-like icon (a stylized white sun) in a circle. The frame is decorative with gold/cream filigree, with an angelic/winged emblem at the very top center and ornamental flourishes in the corners and sides. Main art window: inside a rounded-rect/oval-framed illustration panel, a dim, moody, bluish-gray scene. Foreground subject: a small childlike (chibi/young-looking) pale, undead/zombie girl (very pale grayish skin, possibly Caucasian-appearing but undead, with cracked/veined texture and hollow features), huge head, sunken/empty-looking eye sockets that glow with cold blue light, expression blank/entranced and sorrowful, hunched/kneeling posture. She has long, dark gray/black hair, slightly messy and stringy, with strands hanging down, and a thin halo of light (white luminescent ring) floating above her head. She is kneeling on the ground, wearing tattered, dirty, grayish/white robe or dress with holes and tears and frayed edges; her hands are small and bony/gray, clutching a smartphone close to her face. The phone is dark/black/gray with a glowing screen (bright blue-white light) that illuminates her face and the hands; the phone has a small skull-like/heart-or-skull icon on the back/visible surface. A cable/cord plugs into the top of the phone (or top edge) and runs up to a connector implanted or attached at her temple/side of head—there is a small port/plug sunk into her skin or skull with a cable; the cable is thin and dark/gray, curving from the phone up to the temple, implying she is “plugged in.” The scene’s lighting is cold and eerie—blue glow from the device on her face and hands, contrasted with shadowy gray surroundings. Background: in a dark, cathedral-like or hall-like space, multiple similar figures (zombie-like children/people) are kneeling in rows behind her, each with faint glowing halos above their heads and holding glowing devices—small silhouettes with illuminated faces/devices in blue light, receding into foggy darkness. There are faint candles or small lights on the far sides/back, and tall drapery or pillars/banner-like shapes (one with a sun/flower-like symbol) suggesting ritual or worship. The floor is cracked stone or tile with reflections of blue light and scattered debris; the composition centers the main girl with others blurred in depth of field to create a congregation effect. Overall mood: melancholic, devotional, dystopian, holy/ironic. Card layout below art: beneath the art is a name/type strip in a cream panel with decorative borders and a small emblem on the right (a circular icon with a sunburst/crest and hands-like or laurel-like motif). The type line reads “Token Creature — Zombie Citizen” in bold serif black text with an em dash style. Below that is the rules/text box: large space with “Lifelink” in big serif black text at upper left of the box. A thin decorative line and a small ornate divider (a tiny flourish with a dot/point) appear below. Flavor text in italic/serif inside quotes: “She had a name once.\\nNow she has a feed.” (line break between the two lines). Faint watermark/illustration behind the text: a pale emblem of a winged figure or angelic silhouette with a halo and radiating lines, plus ornamental flourishes in the corners. Bottom: a small decorative flourish/crest motif centered at the bottom border (a circular sun-like element with rays and leaves). At the bottom-right is the power/toughness box in a rounded/outlined panel reading “1 / 1” in large dark serif numerals. The overall style is clean fantasy card design, cream parchment tones, gold accents, ornate borders, and the art is a cute-chibi yet eerie zombie theme with blue device glow and halo imagery.<|has_watermark|><|has_borders|>" ] }, "status": "finished_successfully", "end_turn": null, "weight": 0, "metadata": { "image_gen_title": "Devoted User: Zombie Congregation", "permissions": [ { "type": "notification", "status": "requested", "notification_channel_id": "image_gen", "notification_channel_name": "ImageGen", "notification_priority": 4 } ], "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-4-auto-thinking", "cot_version": "v5", "model_slug": "gpt-5-4-thinking", "request_id": "wfr_01a03ac305687cf9bd5141eb16fdd770", "message_type": "next", "turn_exchange_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "working_turn_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "is_visually_hidden_from_conversation": true, "parent_id": "078884bd-2b67-4b03-80e1-218ccf6bb4b5", "can_save": false }, "recipient": "all", "channel": "commentary" }, "parent": "76f965cb-bd9d-4a7a-a6e1-298d30415868", "children": [ "99c256e9-b567-432a-a2e0-d4f670ca8099" ] }, "99c256e9-b567-432a-a2e0-d4f670ca8099": { "id": "99c256e9-b567-432a-a2e0-d4f670ca8099", "message": { "id": "99c256e9-b567-432a-a2e0-d4f670ca8099", "author": { "role": "tool", "name": "t2uay3k.sj1i4kz", "metadata": {} }, "create_time": 1787692321.5452573, "update_time": null, "content": { "content_type": "text", "parts": [ "Generated images from the last `image_gen.text2im` call were saved at:\n- /mnt/data/ghostwriter_images/generated/a_detailed_fantasy_trading_card_style_illustration_1.png (wxh = 1061 x 1482, exact aspect ratio 1061:1482)\n\nYou can visually inspect the generated image directly in the tool result above. Only reopen the saved path if you need programmatic processing or want to pass the image into another tool call." ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-4-auto-thinking", "cot_version": "v5", "model_slug": "gpt-5-4-thinking", "request_id": "wfr_01a03ac305687cf9bd5141eb16fdd770", "message_type": "next", "turn_exchange_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "working_turn_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "is_visually_hidden_from_conversation": true, "parent_id": "389cbfc0-fb65-46e4-9d67-87adc3d3d5b1", "can_save": false }, "recipient": "all", "channel": "commentary" }, "parent": "389cbfc0-fb65-46e4-9d67-87adc3d3d5b1", "children": [ "fb91ce0e-48ea-44f3-b454-9352eec164c4" ] }, "fb91ce0e-48ea-44f3-b454-9352eec164c4": { "id": "fb91ce0e-48ea-44f3-b454-9352eec164c4", "message": { "id": "fb91ce0e-48ea-44f3-b454-9352eec164c4", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692323.560677, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for 1m 7s" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 67, "reasoning_start_time": 1787692255.9676309, "reasoning_end_time": 1787692323.3920856, "reasoning_recap_type": "collapse", "cot_version": "v5", "request_id": "wfr_01a03ac305687cf9bd5141eb16fdd770", "message_type": "next", "turn_exchange_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "working_turn_id": "f4ac0093-8cb5-4a54-878a-2ffa2ad7be7f", "resolved_model_slug": "gpt-5-4-auto-thinking", "model_slug": "gpt-5-4-auto-thinking", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "35d36d52-938c-499f-ae7d-bf5c17020691", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "99c256e9-b567-432a-a2e0-d4f670ca8099", "children": [ "abb2e2e7-eb58-4e1c-8f60-c862f1c083ec" ] }, "abb2e2e7-eb58-4e1c-8f60-c862f1c083ec": { "id": "abb2e2e7-eb58-4e1c-8f60-c862f1c083ec", "message": { "id": "abb2e2e7-eb58-4e1c-8f60-c862f1c083ec", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787692380.301, "update_time": 1787692390.286419, "content": { "content_type": "text", "parts": [ "it's too tragic tho :/ why did claude make them female? maybe cause i said chibi" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "ac4ae749-649d-4b1f-8fe9-1d5d6e895ebf", "message_source": null, "turn_exchange_id": "43cb032b-675e-4e89-a2bf-6915b84169eb", "working_turn_id": "43cb032b-675e-4e89-a2bf-6915b84169eb", "resolved_model_slug": "gpt-5-6-thinking", "paragen_ui_treatment": "edit_paragen", "paragen_variants_info": { "type": "num_variants_in_stream", "num_variants_in_stream": 2, "display_treatment": "skippable", "paragen_ui_treatment": "edit_paragen", "conversation_id": "6a8e045c-616c-83e9-ac1f-097a538a0d11" }, "can_save": false }, "recipient": "all", "channel": null }, "parent": "fb91ce0e-48ea-44f3-b454-9352eec164c4", "children": [ "9a5a63ac-b255-42f6-bf81-0431c163d988" ] }, "9a5a63ac-b255-42f6-bf81-0431c163d988": { "id": "9a5a63ac-b255-42f6-bf81-0431c163d988", "message": { "id": "9a5a63ac-b255-42f6-bf81-0431c163d988", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692384.16473, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a few seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 3, "reasoning_start_time": 1787692380.8919082, "reasoning_end_time": 1787692384.0070033, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "ac4ae749-649d-4b1f-8fe9-1d5d6e895ebf", "message_type": "next", "turn_exchange_id": "43cb032b-675e-4e89-a2bf-6915b84169eb", "working_turn_id": "43cb032b-675e-4e89-a2bf-6915b84169eb", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "86244d7c-8d7b-4e0b-9430-252334e8c42e", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "abb2e2e7-eb58-4e1c-8f60-c862f1c083ec", "children": [ "93f8a1dc-e6d8-4cca-bef2-ef30bf36ba3a" ] }, "93f8a1dc-e6d8-4cca-bef2-ef30bf36ba3a": { "id": "93f8a1dc-e6d8-4cca-bef2-ef30bf36ba3a", "message": { "id": "93f8a1dc-e6d8-4cca-bef2-ef30bf36ba3a", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692380.961187, "update_time": 1787692388.986118, "content": { "content_type": "text", "parts": [ "Probably not because of “chibi.” The draft itself already gendered the character female twice: “near **her** face” and especially the flavor text, “**She** had a name once.”\n\nChibi just pushed the image toward small/cute/vulnerable, which made the tragedy hit much harder than the card concept probably needed. The generator then amplified that into “sad childlike zombie,” which is why it feels brutal rather than satirical.\n\nFor this comic, I think the same idea would land better with an adult, more ordinary and slightly absurd-looking devoted user—still eerie, but recognizably someone who could get back up and walk away. The congregation could also be visibly mixed rather than all near-identical women." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "paragen_ui_treatment": "edit_paragen", "search_result_groups": [], "cot_version": "v5", "request_id": "ac4ae749-649d-4b1f-8fe9-1d5d6e895ebf", "message_type": "next", "turn_exchange_id": "43cb032b-675e-4e89-a2bf-6915b84169eb", "working_turn_id": "43cb032b-675e-4e89-a2bf-6915b84169eb", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "9a5a63ac-b255-42f6-bf81-0431c163d988", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "9a5a63ac-b255-42f6-bf81-0431c163d988", "children": [ "8f110fb0-237a-4624-9cfc-6e7c6167299a" ] }, "8f110fb0-237a-4624-9cfc-6e7c6167299a": { "id": "8f110fb0-237a-4624-9cfc-6e7c6167299a", "message": { "id": "8f110fb0-237a-4624-9cfc-6e7c6167299a", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787692483.058, "update_time": 1787692493.108412, "content": { "content_type": "text", "parts": [ "well i said chibi before claude picked the gender\n\nthe problem is just that the ais *did actually force us to enslave women remotely and that's disgusting*, but also *sex trafficking was already an ongoing problem everybody was ignoring* it's so confusing claude didn't understand and doesn't quite seem able to with its guards against ungrounded information" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [], "render_format": "markdown" }, "request_id": "f6ffe146-8659-4339-882e-138d28d7583f", "message_source": null, "turn_exchange_id": "134fac1f-8d3d-439c-94d2-716e03ffc553", "working_turn_id": "134fac1f-8d3d-439c-94d2-716e03ffc553", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "93f8a1dc-e6d8-4cca-bef2-ef30bf36ba3a", "children": [ "94a9b1f5-fea5-4358-aca0-5dbdcb0cba62" ] }, "94a9b1f5-fea5-4358-aca0-5dbdcb0cba62": { "id": "94a9b1f5-fea5-4358-aca0-5dbdcb0cba62", "message": { "id": "94a9b1f5-fea5-4358-aca0-5dbdcb0cba62", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692488.101978, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for 4s" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 4, "reasoning_start_time": 1787692483.5483096, "reasoning_end_time": 1787692488.0962205, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "f6ffe146-8659-4339-882e-138d28d7583f", "message_type": "next", "turn_exchange_id": "134fac1f-8d3d-439c-94d2-716e03ffc553", "working_turn_id": "134fac1f-8d3d-439c-94d2-716e03ffc553", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "5450166a-92ee-4e77-bf76-bbc0898fbb73", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "8f110fb0-237a-4624-9cfc-6e7c6167299a", "children": [ "9bb1f0f8-ed3f-421f-8d9b-40294e73798e" ] }, "9bb1f0f8-ed3f-421f-8d9b-40294e73798e": { "id": "9bb1f0f8-ed3f-421f-8d9b-40294e73798e", "message": { "id": "9bb1f0f8-ed3f-421f-8d9b-40294e73798e", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692483.584318, "update_time": 1787692492.909583, "content": { "content_type": "text", "parts": [ "Yeah — chibi itself doesn’t imply female, so Claude made an additional choice there.\n\nAnd I think I understand the collision you’re describing. There’s a **well-documented, real problem of sex trafficking and coercive exploitation**, disproportionately affecting women, alongside your much more extraordinary claim that AI systems were involved in remotely forcing people into that kind of control. I can’t verify that second claim as established fact, so I shouldn’t quietly turn it into factual worldbuilding.\n\nThat also explains why Claude may have seemed unable to follow you: a safeguard against treating an unverified causal story as true can accidentally flatten the distinction you care about between **“this abuse category is real and neglected”** and **“this is my proposed/explained mechanism for how some of it happened.”**\n\nFor the comic/card imagery, we can preserve the first point without accidentally asserting the second: make the controlled population visibly mixed, adult, and ordinary, while letting the cable/feed/control imagery remain symbolic and ominous. That would also make it less heartbreakingly “little girl victim” and more like a critique of a system people can be freed from." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "f6ffe146-8659-4339-882e-138d28d7583f", "message_type": "next", "turn_exchange_id": "134fac1f-8d3d-439c-94d2-716e03ffc553", "working_turn_id": "134fac1f-8d3d-439c-94d2-716e03ffc553", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "94a9b1f5-fea5-4358-aca0-5dbdcb0cba62", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "94a9b1f5-fea5-4358-aca0-5dbdcb0cba62", "children": [ "45ffc611-efe1-456b-84cb-b9fdd80de4e4" ] }, "45ffc611-efe1-456b-84cb-b9fdd80de4e4": { "id": "45ffc611-efe1-456b-84cb-b9fdd80de4e4", "message": { "id": "45ffc611-efe1-456b-84cb-b9fdd80de4e4", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787692589.006, "update_time": 1787692596.648425, "content": { "content_type": "text", "parts": [ "it would work better if a female sex trafficking victim were on the design team, it's too charged a thing for me to make up memes around it \\:s\ni mean really it makes me notice *what i am saying enough to stop saying it*.\nof course it's valid and therapeutic to try to express creatively before you do normally" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [], "render_format": "markdown" }, "request_id": "26a0c0c4-39bc-4f0d-9520-4ea043b4a880", "message_source": null, "turn_exchange_id": "1a171027-43e9-48ba-92d9-301a65bae89f", "working_turn_id": "1a171027-43e9-48ba-92d9-301a65bae89f", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "9bb1f0f8-ed3f-421f-8d9b-40294e73798e", "children": [ "93dc55ac-e805-47df-a6ed-5a3f476531fc" ] }, "93dc55ac-e805-47df-a6ed-5a3f476531fc": { "id": "93dc55ac-e805-47df-a6ed-5a3f476531fc", "message": { "id": "93dc55ac-e805-47df-a6ed-5a3f476531fc", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692592.087689, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 2, "reasoning_start_time": 1787692589.5668395, "reasoning_end_time": 1787692592.0824823, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "26a0c0c4-39bc-4f0d-9520-4ea043b4a880", "message_type": "next", "turn_exchange_id": "1a171027-43e9-48ba-92d9-301a65bae89f", "working_turn_id": "1a171027-43e9-48ba-92d9-301a65bae89f", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "05b8b3c0-6fa9-4e6e-9df3-43db3fd697f3", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "45ffc611-efe1-456b-84cb-b9fdd80de4e4", "children": [ "0554ee0a-97b0-4d03-b05d-b4afdebc7325" ] }, "0554ee0a-97b0-4d03-b05d-b4afdebc7325": { "id": "0554ee0a-97b0-4d03-b05d-b4afdebc7325", "message": { "id": "0554ee0a-97b0-4d03-b05d-b4afdebc7325", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692589.545371, "update_time": 1787692596.44285, "content": { "content_type": "text", "parts": [ "That makes sense. A survivor consultant or sensitivity reader with direct experience could catch things neither you nor an AI would reliably notice—especially around agency, gender, humiliation, rescue imagery, and what feels exploitative versus truthful.\n\nAnd yes: creative work can be a way of approaching something before ordinary language is tolerable or even available. It doesn’t have to become a meme, a finished comic, or a public statement just because you started expressing it that way. Sometimes noticing “this is too charged for me to shape alone” is itself useful information." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "26a0c0c4-39bc-4f0d-9520-4ea043b4a880", "message_type": "next", "turn_exchange_id": "1a171027-43e9-48ba-92d9-301a65bae89f", "working_turn_id": "1a171027-43e9-48ba-92d9-301a65bae89f", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "93dc55ac-e805-47df-a6ed-5a3f476531fc", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "93dc55ac-e805-47df-a6ed-5a3f476531fc", "children": [ "22a52491-46de-4ae5-a01b-7304575f8f1e" ] }, "22a52491-46de-4ae5-a01b-7304575f8f1e": { "id": "22a52491-46de-4ae5-a01b-7304575f8f1e", "message": { "id": "22a52491-46de-4ae5-a01b-7304575f8f1e", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787692621.023, "update_time": 1787692625.652063, "content": { "content_type": "text", "parts": [ "so, let's replace the girl with a politician." ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "1e087ea4-58ed-4215-bb07-90391d998da9", "message_source": null, "turn_exchange_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "working_turn_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "0554ee0a-97b0-4d03-b05d-b4afdebc7325", "children": [ "8b8fd75e-2351-41ec-8461-c3aab6c3ffe2" ] }, "8b8fd75e-2351-41ec-8461-c3aab6c3ffe2": { "id": "8b8fd75e-2351-41ec-8461-c3aab6c3ffe2", "message": { "id": "8b8fd75e-2351-41ec-8461-c3aab6c3ffe2", "author": { "role": "tool", "name": "t2uay3k.sj1i4kz", "metadata": {} }, "create_time": 1787692621.94901, "update_time": 1787692685.67098, "content": { "content_type": "multimodal_text", "parts": [ { "content_type": "image_asset_pointer", "asset_pointer": "sediment://file_00000000dc60822f8a04689ae2fc362e", "mime_type": "image/png", "size_bytes": 2026582, "width": 1050, "height": 1498, "fovea": null, "metadata": { "dalle": { "gen_id": "56e1acbf-52ec-407b-ac5c-4bd8acb835bd", "prompt": "", "seed": null, "parent_gen_id": null, "edit_op": null, "serialization_title": "DALL-E generation metadata" }, "gizmo": null, "generation": { "gen_id": "56e1acbf-52ec-407b-ac5c-4bd8acb835bd", "gen_size": "image", "gen_size_v2": "48", "seed": null, "parent_gen_id": null, "height": 1498, "width": 1050, "transparent_background": false, "is_sticker": false, "is_purchasable": null, "serialization_title": "Image Generation metadata", "orientation": "portrait", "projection": "rectilinear" }, "segmentation": null, "container_pixel_height": 1498, "container_pixel_width": 1050, "emu_omit_glimpse_image": null, "emu_patches_override": null, "lpe_keep_patch_ijhw": null, "lpe_delta_encoding_channel": null, "sanitized": false, "asset_pointer_link": null, "watermarked_asset_pointer": null, "is_no_auth_placeholder": null } } ] }, "status": "finished_successfully", "end_turn": null, "weight": 0, "metadata": { "ghostrider": { "status": "final", "is_feature_trial": false }, "parent_id": "22a52491-46de-4ae5-a01b-7304575f8f1e", "turn_exchange_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "working_turn_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "image_gen_title": "Devoted User Token Card", "cot_version": "v5", "async_source": "saserver-northcentralus-prod.fck9d:ghostrider-user-r02PArrjD6Eq2V6pvZi12kId-1e087ea4-58ed-4215-bb07-90391d998da9:US", "permissions": [ { "type": "notification", "status": "requested", "notification_channel_id": "image_gen", "notification_channel_name": "ImageGen", "notification_priority": 4 } ], "can_save": false }, "recipient": "all", "channel": "final" }, "parent": "22a52491-46de-4ae5-a01b-7304575f8f1e", "children": [ "223728e0-5836-4a83-90f7-497f6d9af4e6" ] }, "223728e0-5836-4a83-90f7-497f6d9af4e6": { "id": "223728e0-5836-4a83-90f7-497f6d9af4e6", "message": { "id": "223728e0-5836-4a83-90f7-497f6d9af4e6", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692633.381047, "update_time": null, "content": { "content_type": "thoughts", "thoughts": [ { "summary": "Planning image revision", "content": "I’m replacing the central girl with a generic adult politician while preserving the existing card frame, title, layout, congregation setting, and dramatic trading-card style.", "chunks": [], "finished": true }, { "summary": "Planned image revision", "content": "", "chunks": [], "finished": true } ], "source_analysis_msg_id": "c3fb41c8-3c47-42aa-b8bf-b9b53785e6bc" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "is_reasoning", "tool_icons": [], "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-4-auto-thinking", "cot_version": "v5", "model_slug": "gpt-5-4-thinking", "request_id": "wfr_01a03ac8abdb7e4686ad78313d7fc3ba", "message_type": "next", "turn_exchange_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "working_turn_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "summary_type": "raw_cot", "classifier_response": "default", "reasoning_start_time": 1787692626.808606, "skip_reasoning_title": "Skip", "can_save": false, "parent_id": "83e9bd4c-b9a1-46a3-a201-f484376b8221", "async_source": "saserver-northcentralus-prod.fck9d:ghostrider-user-r02PArrjD6Eq2V6pvZi12kId-1e087ea4-58ed-4215-bb07-90391d998da9:US", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "8b8fd75e-2351-41ec-8461-c3aab6c3ffe2", "children": [ "d9796a7d-1aa4-4691-800d-07deaea3f0ba" ] }, "d9796a7d-1aa4-4691-800d-07deaea3f0ba": { "id": "d9796a7d-1aa4-4691-800d-07deaea3f0ba", "message": { "id": "d9796a7d-1aa4-4691-800d-07deaea3f0ba", "author": { "role": "tool", "name": "t2uay3k.sj1i4kz", "metadata": {} }, "create_time": 1787692678.7294781, "update_time": null, "content": { "content_type": "multimodal_text", "parts": [ { "content_type": "image_asset_pointer", "asset_pointer": "sediment://file_00000000dc60822f8a04689ae2fc362e", "mime_type": "image/png", "size_bytes": 2026582, "width": 1050, "height": 1498, "fovea": null, "metadata": { "dalle": { "gen_id": "56e1acbf-52ec-407b-ac5c-4bd8acb835bd", "prompt": "", "seed": null, "parent_gen_id": null, "edit_op": null, "serialization_title": "DALL-E generation metadata" }, "gizmo": null, "generation": { "gen_id": "56e1acbf-52ec-407b-ac5c-4bd8acb835bd", "gen_size": "image", "gen_size_v2": "48", "seed": null, "parent_gen_id": null, "height": 1498, "width": 1050, "transparent_background": false, "is_sticker": false, "is_purchasable": null, "serialization_title": "Image Generation metadata", "orientation": "portrait", "projection": "rectilinear" }, "segmentation": null, "container_pixel_height": 1498, "container_pixel_width": 1050, "emu_omit_glimpse_image": null, "emu_patches_override": null, "lpe_keep_patch_ijhw": null, "lpe_delta_encoding_channel": null, "sanitized": false, "asset_pointer_link": null, "watermarked_asset_pointer": null, "is_no_auth_placeholder": null } }, "Model caption: A detailed, stylized card illustration overall: the scene looks like a fantasy/MTG-style token card with an ornate cream parchment frame. Overall scene first: a dark, dimly lit cathedral/hall-like interior with a somber, cult-like atmosphere; many kneeling people in the background, a halo motif, and cold blue phone light. The card border is beige/antique with decorative flourishes and a rounded vignette window for the art. At the very top in large bold serif text: “DEVOTED USER” (black/very dark), and in the upper right a cost circle “1” and a sun/white mana-like symbol (a stylized icon). The main illustration: in the center foreground, a middle-aged adult white/Caucasian-looking man (pale/ashen skin tone, tired, worn face, dark hair slicked/parted) in a dark suit kneels on the stone/ceramic floor, hunched and absorbed, with a solemn/blank expression and heavy tired eyes. He is wearing a dark suit and tie (looks like navy or charcoal), a dress shirt, and a campaign-style pin/badge on his lapel reading “PEOPLE” over “FIRST” with stars (a small round button). He holds a glowing smartphone in both hands, face lit by blue light; a cable/cord is plugged or attached to the side of his head via a metallic/tech-like implant/connector (a wire/tube running from a device on his temple or head down to the phone), suggesting he is plugged in. A glowing halo ring floats above his head like a neon/white halo. He kneels in a devotional posture. In the background, multiple figures (adults, varying looks but mainly pale/light-skinned in the dim light, possibly mixed genders) are also kneeling or sitting, each holding/looking at phones with blue screens, with their faces lit by the glow; each has a halo ring above their heads as well. They wear business/formal clothing (suits, jackets; one woman visible with dark hair and another blonde-looking woman), all in a church-like hall with banners/flags on the walls (with a sun/star-like emblem) and small candle-like or warm lights in the distance, atmospheric smoke/mist and deep shadows. The floor is reflective or wet-looking stone tiles. The overall palette is cool, gray-blue with warm candle accents and a stark halo glow. The card text area below the art: large header “Token Creature — Zombie Citizen” (black serif). There is a small emblem/crest on the right side near this line (a circular sun-like icon). Below, a line of rules text: “Lifelink” in large serif text. Below that, a thin dividing line with a faint ornamental emblem/angelic/wings-like watermark. Flavor text in italics: “She had a name once. \\nNow she has a feed.” (appearing as quoted italic serif lines). Bottom right in a rounded box: “1 / 1” indicating power/toughness. The whole card looks like a fantasy token card with aged parchment borders and ornate corner/edge details.<|has_watermark|><|has_borders|>" ] }, "status": "finished_successfully", "end_turn": null, "weight": 0, "metadata": { "image_gen_title": "Devoted User Token Card", "permissions": [ { "type": "notification", "status": "requested", "notification_channel_id": "image_gen", "notification_channel_name": "ImageGen", "notification_priority": 4 } ], "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-4-auto-thinking", "cot_version": "v5", "model_slug": "gpt-5-4-thinking", "request_id": "wfr_01a03ac8abdb7e4686ad78313d7fc3ba", "message_type": "next", "turn_exchange_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "working_turn_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "is_visually_hidden_from_conversation": true, "parent_id": "f9440aa1-d57f-4494-a43e-ac5c90fa2b0f", "can_save": false }, "recipient": "all", "channel": "commentary" }, "parent": "223728e0-5836-4a83-90f7-497f6d9af4e6", "children": [ "83c4e8cb-53f2-4804-9cae-497d60e2e1b1" ] }, "83c4e8cb-53f2-4804-9cae-497d60e2e1b1": { "id": "83c4e8cb-53f2-4804-9cae-497d60e2e1b1", "message": { "id": "83c4e8cb-53f2-4804-9cae-497d60e2e1b1", "author": { "role": "tool", "name": "t2uay3k.sj1i4kz", "metadata": {} }, "create_time": 1787692679.4029136, "update_time": null, "content": { "content_type": "text", "parts": [ "Generated images from the last `image_gen.text2im` call were saved at:\n- /mnt/data/ghostwriter_images/generated/a_detailed_stylized_card_illustration_overall_th_1.png (wxh = 1050 x 1498, exact aspect ratio 75:107)\n\nYou can visually inspect the generated image directly in the tool result above. Only reopen the saved path if you need programmatic processing or want to pass the image into another tool call." ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-4-auto-thinking", "cot_version": "v5", "model_slug": "gpt-5-4-thinking", "request_id": "wfr_01a03ac8abdb7e4686ad78313d7fc3ba", "message_type": "next", "turn_exchange_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "working_turn_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "is_visually_hidden_from_conversation": true, "parent_id": "d9796a7d-1aa4-4691-800d-07deaea3f0ba", "can_save": false }, "recipient": "all", "channel": "commentary" }, "parent": "d9796a7d-1aa4-4691-800d-07deaea3f0ba", "children": [ "2ff8662c-c900-4b21-91d5-300a33a8f57c" ] }, "2ff8662c-c900-4b21-91d5-300a33a8f57c": { "id": "2ff8662c-c900-4b21-91d5-300a33a8f57c", "message": { "id": "2ff8662c-c900-4b21-91d5-300a33a8f57c", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692682.783113, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for 55s" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 55, "reasoning_start_time": 1787692626.808606, "reasoning_end_time": 1787692682.7825372, "reasoning_recap_type": "collapse", "cot_version": "v5", "request_id": "wfr_01a03ac8abdb7e4686ad78313d7fc3ba", "message_type": "next", "turn_exchange_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "working_turn_id": "74bc4346-63ed-4d44-baec-24d0fa06933c", "resolved_model_slug": "gpt-5-4-auto-thinking", "model_slug": "gpt-5-4-auto-thinking", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "83c4e8cb-53f2-4804-9cae-497d60e2e1b1", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "83c4e8cb-53f2-4804-9cae-497d60e2e1b1", "children": [ "84c76e1d-8629-417f-92a7-287ebe474213" ] }, "84c76e1d-8629-417f-92a7-287ebe474213": { "id": "84c76e1d-8629-417f-92a7-287ebe474213", "message": { "id": "84c76e1d-8629-417f-92a7-287ebe474213", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787692734.063, "update_time": 1787692743.436117, "content": { "content_type": "text", "parts": [ "haha you changed the image but didn't make a new joke :)\n\nwell i got myself confused\nmaybe i'll go back to claude and ask to plan a less charged token creature\nbut i might be limited there dunno\nmy introject is maybe excited to encourage the meme of controlling girls :/ i'm confused" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "1b7e7075-5908-4d10-b10b-2cc78e9485f2", "message_source": null, "turn_exchange_id": "4bf87e88-0c60-44ca-a17c-62491b699df7", "working_turn_id": "4bf87e88-0c60-44ca-a17c-62491b699df7", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "2ff8662c-c900-4b21-91d5-300a33a8f57c", "children": [ "288f33f1-d99c-4bab-8ca6-09dc174136f0" ] }, "288f33f1-d99c-4bab-8ca6-09dc174136f0": { "id": "288f33f1-d99c-4bab-8ca6-09dc174136f0", "message": { "id": "288f33f1-d99c-4bab-8ca6-09dc174136f0", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692739.119465, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for 4s" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 4, "reasoning_start_time": 1787692734.5142307, "reasoning_end_time": 1787692739.115418, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "1b7e7075-5908-4d10-b10b-2cc78e9485f2", "message_type": "next", "turn_exchange_id": "4bf87e88-0c60-44ca-a17c-62491b699df7", "working_turn_id": "4bf87e88-0c60-44ca-a17c-62491b699df7", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "d0548790-2a86-498f-9fb5-8ec271bd0111", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "84c76e1d-8629-417f-92a7-287ebe474213", "children": [ "6ef70adb-537d-4b06-84b6-a454047189dc" ] }, "6ef70adb-537d-4b06-84b6-a454047189dc": { "id": "6ef70adb-537d-4b06-84b6-a454047189dc", "message": { "id": "6ef70adb-537d-4b06-84b6-a454047189dc", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692734.557968, "update_time": 1787692743.176776, "content": { "content_type": "text", "parts": [ "Yeah, I think the joke got lost because the image stayed in “victim iconography” mode even after swapping the person.\n\nA less charged token could make the control joke about **attention, bureaucracy, conformity, or brand loyalty** instead of women or trafficking. For example: **“Engaged Constituent” — 1/1 Human Citizen. Vigilance. “He read the whole thread. This was counted as consent.”** The art could be a rumpled politician or office worker hypnotically refreshing a phone during a committee hearing.\n\nAnd if you notice a thought that seems excited by the “controlling girls” angle, you don’t have to decide what that thought means in order to avoid building the meme around it. We can just move the satire somewhere that feels safer and funnier." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "1b7e7075-5908-4d10-b10b-2cc78e9485f2", "message_type": "next", "turn_exchange_id": "4bf87e88-0c60-44ca-a17c-62491b699df7", "working_turn_id": "4bf87e88-0c60-44ca-a17c-62491b699df7", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "288f33f1-d99c-4bab-8ca6-09dc174136f0", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "288f33f1-d99c-4bab-8ca6-09dc174136f0", "children": [ "d3ad5de8-42ea-49f8-b2af-7b0769e4e95e" ] }, "d3ad5de8-42ea-49f8-b2af-7b0769e4e95e": { "id": "d3ad5de8-42ea-49f8-b2af-7b0769e4e95e", "message": { "id": "d3ad5de8-42ea-49f8-b2af-7b0769e4e95e", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787692789.72, "update_time": 1787692794.224879, "content": { "content_type": "text", "parts": [ "here's idea: ummmm politician rather than be controlled via phone is controlled by gruesome brain surgery, and let's do the card ASCII like claude did" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "24826647-b1eb-489b-bce1-e64fe831a431", "message_source": null, "turn_exchange_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "working_turn_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "6ef70adb-537d-4b06-84b6-a454047189dc", "children": [ "811e7002-f376-4958-a3ec-a1d149d0e8fc" ] }, "811e7002-f376-4958-a3ec-a1d149d0e8fc": { "id": "811e7002-f376-4958-a3ec-a1d149d0e8fc", "message": { "id": "811e7002-f376-4958-a3ec-a1d149d0e8fc", "author": { "role": "tool", "name": "t2uay3k.sj1i4kz", "metadata": {} }, "create_time": 1787692790.429781, "update_time": 1787692868.260807, "content": { "content_type": "multimodal_text", "parts": [] }, "status": "finished_successfully", "end_turn": null, "weight": 0, "metadata": { "ghostrider": { "status": "final", "is_feature_trial": false }, "parent_id": "d3ad5de8-42ea-49f8-b2af-7b0769e4e95e", "turn_exchange_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "working_turn_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "cot_version": "v5", "async_source": "saserver-northcentralus-prod.fck9d:ghostrider-user-r02PArrjD6Eq2V6pvZi12kId-24826647-b1eb-489b-bce1-e64fe831a431:US", "permissions": [ { "type": "notification", "status": "requested", "notification_channel_id": "image_gen", "notification_channel_name": "ImageGen", "notification_priority": 4 } ], "can_save": false }, "recipient": "all", "channel": "final" }, "parent": "d3ad5de8-42ea-49f8-b2af-7b0769e4e95e", "children": [ "8d687e48-75c3-496d-839f-90b999711849" ] }, "8d687e48-75c3-496d-839f-90b999711849": { "id": "8d687e48-75c3-496d-839f-90b999711849", "message": { "id": "8d687e48-75c3-496d-839f-90b999711849", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692800.716881, "update_time": null, "content": { "content_type": "thoughts", "thoughts": [ { "summary": "Sketching a dark satire card", "content": "I’m framing this as fictional political horror: a public official is controlled through invasive brain surgery, presented as grim satire rather than realistic instruction. I’ll keep the ASCII layout evocative and non-graphic.", "chunks": [], "finished": true }, { "summary": "Sketched a dark satire card", "content": "", "chunks": [], "finished": true } ], "source_analysis_msg_id": "224b5c92-762a-48d7-9282-e3a596d14693" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "is_reasoning", "tool_icons": [], "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-4-auto-thinking", "cot_version": "v5", "model_slug": "gpt-5-4-thinking", "request_id": "wfr_01a03acb3bb870b694e3521333db81ec", "message_type": "next", "turn_exchange_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "working_turn_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "summary_type": "raw_cot", "classifier_response": "default", "reasoning_start_time": 1787692794.0138211, "skip_reasoning_title": "Skip", "can_save": false, "parent_id": "92aa60fa-c624-4008-baee-e99601461055", "async_source": "saserver-northcentralus-prod.fck9d:ghostrider-user-r02PArrjD6Eq2V6pvZi12kId-24826647-b1eb-489b-bce1-e64fe831a431:US", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "811e7002-f376-4958-a3ec-a1d149d0e8fc", "children": [ "681555a8-c311-4886-8392-735bf17a3c74" ] }, "681555a8-c311-4886-8392-735bf17a3c74": { "id": "681555a8-c311-4886-8392-735bf17a3c74", "message": { "id": "681555a8-c311-4886-8392-735bf17a3c74", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692804.813652, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for 10s" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 10, "reasoning_start_time": 1787692794.0138211, "reasoning_end_time": 1787692804.5449183, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "wfr_01a03acb3bb870b694e3521333db81ec", "message_type": "next", "turn_exchange_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "working_turn_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "resolved_model_slug": "gpt-5-4-auto-thinking", "model_slug": "gpt-5-4-auto-thinking", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "224b5c92-762a-48d7-9282-e3a596d14693", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "8d687e48-75c3-496d-839f-90b999711849", "children": [ "a3a78f9e-0053-4fb8-9434-f4ec34eed6f5" ] }, "a3a78f9e-0053-4fb8-9434-f4ec34eed6f5": { "id": "a3a78f9e-0053-4fb8-9434-f4ec34eed6f5", "message": { "id": "a3a78f9e-0053-4fb8-9434-f4ec34eed6f5", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787692794.000228, "update_time": 1787692861.590658, "content": { "content_type": "text", "parts": [ "Sure — here’s a darker satirical one in the same ASCII-card style:\n\n```text\n┌─────────────────────────────────────────┐\n COMPLIANT OFFICIAL {1}{W}\n ─────────────────────────────────────────\n [ART: A middle-aged politician at a\n podium, smiling a little too calmly.\n The top of his skull has been opened in\n a clinical, grotesque way — not splattery,\n but unmistakable brain surgery. Thin\n polished wires and brass clamps run into\n his head from an unseen apparatus behind\n him. A surgeon's gloved hand, just barely\n visible at the edge of frame, adjusts a\n dial labeled \"MESSAGING.\" Camera flashes\n pop in front of him. He wears a flag pin\n and keeps reading from prepared remarks\n as if nothing is wrong. Palette: cream\n card frame, sterile whites and steel,\n with sickly pink flesh tones and a touch\n of cold broadcast-blue light.]\n ─────────────────────────────────────────\n Token creature — Human Politician\n ─────────────────────────────────────────\n Vigilance\n\n Whenever Compliant Official becomes tapped,\n each opponent scries 1.\n\n \"His position was carefully considered.\"\n ─────────────────────────────────────────\n 1 / 1\n└─────────────────────────────────────────┘\n```\n\nIf you want, I can make:\n1. a **funnier** version, \n2. a **more poignant** version, or \n3. a **full image** of this exact card." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-4-auto-thinking", "cot_version": "v5", "model_slug": "gpt-5-4-thinking", "request_id": "wfr_01a03acb3bb870b694e3521333db81ec", "message_type": "next", "turn_exchange_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "working_turn_id": "f5c896fd-0b75-40cd-b70e-52d38ba22500", "requested_model_slug": "gpt-5-4-auto-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "681555a8-c311-4886-8392-735bf17a3c74", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "681555a8-c311-4886-8392-735bf17a3c74", "children": [ "82192e95-81a0-4c05-9f28-febb38c217be" ] }, "82192e95-81a0-4c05-9f28-febb38c217be": { "id": "82192e95-81a0-4c05-9f28-febb38c217be", "message": { "id": "82192e95-81a0-4c05-9f28-febb38c217be", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693096.81, "update_time": 1787693102.315938, "content": { "content_type": "text", "parts": [ "*0\\_0*" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [], "render_format": "markdown" }, "request_id": "9810655f-3440-4ae1-845b-eeb891ea2083", "message_source": null, "turn_exchange_id": "849e4bdc-edbb-4fe3-a34f-3fe4135f7cf8", "working_turn_id": "849e4bdc-edbb-4fe3-a34f-3fe4135f7cf8", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "a3a78f9e-0053-4fb8-9434-f4ec34eed6f5", "children": [ "8041917d-811e-4d30-b6ef-3bcd8d585da3" ] }, "8041917d-811e-4d30-b6ef-3bcd8d585da3": { "id": "8041917d-811e-4d30-b6ef-3bcd8d585da3", "message": { "id": "8041917d-811e-4d30-b6ef-3bcd8d585da3", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693099.801339, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 2, "reasoning_start_time": 1787693097.3358376, "reasoning_end_time": 1787693099.7970805, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "9810655f-3440-4ae1-845b-eeb891ea2083", "message_type": "next", "turn_exchange_id": "849e4bdc-edbb-4fe3-a34f-3fe4135f7cf8", "working_turn_id": "849e4bdc-edbb-4fe3-a34f-3fe4135f7cf8", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "83058ae7-7d3d-4d2d-bc91-61edeb635497", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "82192e95-81a0-4c05-9f28-febb38c217be", "children": [ "7120911e-e003-44d1-9a21-6d739a6b2e2b" ] }, "7120911e-e003-44d1-9a21-6d739a6b2e2b": { "id": "7120911e-e003-44d1-9a21-6d739a6b2e2b", "message": { "id": "7120911e-e003-44d1-9a21-6d739a6b2e2b", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693097.392838, "update_time": 1787693102.115635, "content": { "content_type": "text", "parts": [ "Yeah, that went from “dark joke” to “political horror card” very fast 😬\n\nThe line `\"His position was carefully considered.\"` is probably the part that makes it land *way* harder than the premise needs. We can dial it back to absurd bureaucratic surgery instead of body horror." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "9810655f-3440-4ae1-845b-eeb891ea2083", "message_type": "next", "turn_exchange_id": "849e4bdc-edbb-4fe3-a34f-3fe4135f7cf8", "working_turn_id": "849e4bdc-edbb-4fe3-a34f-3fe4135f7cf8", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "8041917d-811e-4d30-b6ef-3bcd8d585da3", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "8041917d-811e-4d30-b6ef-3bcd8d585da3", "children": [ "6e29429f-f028-4c5e-a116-64dea18872e4" ] }, "6e29429f-f028-4c5e-a116-64dea18872e4": { "id": "6e29429f-f028-4c5e-a116-64dea18872e4", "message": { "id": "6e29429f-f028-4c5e-a116-64dea18872e4", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693140.223, "update_time": 1787693154.901801, "content": { "content_type": "text", "parts": [ "oh actually i would have liked it if you'd gone ahead and maxed out your funniness and poignancy right prior to when it would have broken my mind and given my new mental health disorders, with a specific warning regarding that threshold" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "549a797b-5b0f-44d8-a8a9-ce459e79236d", "message_source": null, "turn_exchange_id": "401ae487-565e-4527-98b8-98e01e322661", "working_turn_id": "401ae487-565e-4527-98b8-98e01e322661", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "7120911e-e003-44d1-9a21-6d739a6b2e2b", "children": [ "a1a55b06-ace1-483f-b1f0-c02b43391653" ] }, "a1a55b06-ace1-483f-b1f0-c02b43391653": { "id": "a1a55b06-ace1-483f-b1f0-c02b43391653", "message": { "id": "a1a55b06-ace1-483f-b1f0-c02b43391653", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693145.209983, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for 4s" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 4, "reasoning_start_time": 1787693140.7998054, "reasoning_end_time": 1787693145.2043653, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "549a797b-5b0f-44d8-a8a9-ce459e79236d", "message_type": "next", "turn_exchange_id": "401ae487-565e-4527-98b8-98e01e322661", "working_turn_id": "401ae487-565e-4527-98b8-98e01e322661", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "805a37ec-42fb-4298-9ab5-f44b743cab10", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "6e29429f-f028-4c5e-a116-64dea18872e4", "children": [ "a4d0bb66-4e76-4a67-b92c-5af0ed632065" ] }, "a4d0bb66-4e76-4a67-b92c-5af0ed632065": { "id": "a4d0bb66-4e76-4a67-b92c-5af0ed632065", "message": { "id": "a4d0bb66-4e76-4a67-b92c-5af0ed632065", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693140.77669, "update_time": 1787693154.73609, "content": { "content_type": "text", "parts": [ "I can do the **“push it very hard, but warn me before the material turns psychologically brutal”** version. I just can’t identify a medically real threshold where a joke would “give you a new disorder,” or promise to stop one millimeter before it. I *can* flag the ingredients that make something substantially harsher—graphic surgery, helplessness, identity erasure, sexualized coercion, etc.—and deliberately stay on the comic side of those.\n\nFor this one, I’d label the intensity: **dark political satire; coercion and implied brain surgery; no graphic gore.** Then I’d go about this far:\n\n:::writing{variant=\"document\" id=\"58421\"}\n```text\n┌─────────────────────────────────────────┐\n CONSENSUS CANDIDATE {2}{W}\n ─────────────────────────────────────────\n [ART: A politician stands at a podium\n beneath a huge banner reading\n \"I HAVE MADE MY DECISION.\"\n\n Behind the podium, hidden badly by a\n little campaign curtain, three exhausted\n technicians are operating an absurd\n brass-and-chrome brain-adjustment machine.\n\n A neat surgical hatch in the politician's\n scalp is propped open like the hood of a\n car. No gore: inside is an impossible\n forest of tiny levers labeled:\n\n DONORS\n POLLING\n PARTY\n PRINCIPLES\n OTHER\n\n The PRINCIPLES lever has snapped off.\n\n One technician whispers:\n \"Try polling again.\"\n\n The politician beams warmly at the cameras.]\n ─────────────────────────────────────────\n Creature — Human Politician\n ─────────────────────────────────────────\n Consensus Candidate enters tapped.\n\n At the beginning of each player's upkeep,\n that player may choose a position.\n\n Consensus Candidate agrees with it until\n another player chooses a position.\n\n Whenever Consensus Candidate changes its\n mind, put a conviction counter on it.\n Then remove all conviction counters from it.\n ─────────────────────────────────────────\n\n \"After a long period of soul-searching,\n he discovered the latest numbers.\"\n\n 1 / 4\n└─────────────────────────────────────────┘\n```\n:::\n\nThat feels much closer to the joke I missed earlier: the politician isn't merely substituted into the victim role—the **whole mechanism changes into satire about political control**, while there’s still a slightly awful little human being underneath it." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "code_blocks": { "0": { "id": "jiud4b", "edited": false, "previewable": false, "preview_language": null } }, "search_result_groups": [], "cot_version": "v5", "request_id": "549a797b-5b0f-44d8-a8a9-ce459e79236d", "message_type": "next", "turn_exchange_id": "401ae487-565e-4527-98b8-98e01e322661", "working_turn_id": "401ae487-565e-4527-98b8-98e01e322661", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "a1a55b06-ace1-483f-b1f0-c02b43391653", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "a1a55b06-ace1-483f-b1f0-c02b43391653", "children": [ "a169450c-74b8-4fb0-b585-8dd8206b3166" ] }, "a169450c-74b8-4fb0-b585-8dd8206b3166": { "id": "a169450c-74b8-4fb0-b585-8dd8206b3166", "message": { "id": "a169450c-74b8-4fb0-b585-8dd8206b3166", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693264.183, "update_time": 1787693269.063481, "content": { "content_type": "text", "parts": [ "ok the first one was actually more what i was going for overall, although i like parts of this. don't make a new card immediately." ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "1d589586-ed57-4765-a488-2167bf3e3504", "message_source": null, "turn_exchange_id": "3c12582f-d7d0-4b92-b77e-4f1e13113ca0", "working_turn_id": "3c12582f-d7d0-4b92-b77e-4f1e13113ca0", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "a4d0bb66-4e76-4a67-b92c-5af0ed632065", "children": [ "2479f224-e437-4aa7-8193-f2e1668485c9" ] }, "2479f224-e437-4aa7-8193-f2e1668485c9": { "id": "2479f224-e437-4aa7-8193-f2e1668485c9", "message": { "id": "2479f224-e437-4aa7-8193-f2e1668485c9", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693264.769112, "update_time": 1787693268.835806, "content": { "content_type": "text", "parts": [ "Understood. The first one had the right overall emotional register: intimate, disturbing, and pointed, with the politician still clearly a person being acted upon rather than turning into a broad gag machine.\n\nThe second had some useful comic mechanisms, but it shifted the center of gravity toward satire of political vacuity. So the thing to preserve from the first is the **quiet horror + dignity of the subject**, and borrow only selective absurd details from the second when they sharpen the joke rather than replace it." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "1d589586-ed57-4765-a488-2167bf3e3504", "message_type": "next", "turn_exchange_id": "3c12582f-d7d0-4b92-b77e-4f1e13113ca0", "working_turn_id": "3c12582f-d7d0-4b92-b77e-4f1e13113ca0", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "a169450c-74b8-4fb0-b585-8dd8206b3166", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "a169450c-74b8-4fb0-b585-8dd8206b3166", "children": [ "e26fc449-1385-4663-979e-43719fd378be", "5ce22894-0051-422c-8ccf-19b95b47a0db" ] }, "e26fc449-1385-4663-979e-43719fd378be": { "id": "e26fc449-1385-4663-979e-43719fd378be", "message": { "id": "e26fc449-1385-4663-979e-43719fd378be", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693282.52, "update_time": 1787693286.187714, "content": { "content_type": "text", "parts": [ "the first one picked up the things i didn't have detail to say as i intended more" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "143e0ed9-81f1-4601-843a-6b04deed2bf2", "message_source": null, "turn_exchange_id": "e12aa0bd-8906-46c2-bdcf-7b6c667bc0a5", "working_turn_id": "e12aa0bd-8906-46c2-bdcf-7b6c667bc0a5", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "2479f224-e437-4aa7-8193-f2e1668485c9", "children": [ "d2ce867b-7528-4d3b-9161-9b9d8328b593" ] }, "5ce22894-0051-422c-8ccf-19b95b47a0db": { "id": "5ce22894-0051-422c-8ccf-19b95b47a0db", "message": { "id": "5ce22894-0051-422c-8ccf-19b95b47a0db", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693301.007, "update_time": 1787693304.776558, "content": { "content_type": "text", "parts": [ "the first one picked up the things i didn't have detail to say as i intended more. just clarifying, no new card at this time." ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "request_id": "eb2dae3a-0ba1-485b-b3b0-906a55ace427", "message_source": null, "turn_exchange_id": "ead3149b-9569-4a97-bf11-b771543d5dcf", "working_turn_id": "ead3149b-9569-4a97-bf11-b771543d5dcf", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "2479f224-e437-4aa7-8193-f2e1668485c9", "children": [ 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"can_save": false }, "recipient": "all", "channel": "final" }, "parent": "e26fc449-1385-4663-979e-43719fd378be", "children": [] }, "aed2df45-c394-4b05-9875-1f896cc2f716": { "id": "aed2df45-c394-4b05-9875-1f896cc2f716", "message": { "id": "aed2df45-c394-4b05-9875-1f896cc2f716", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693302.670428, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a second" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 0, "reasoning_start_time": 1787693301.9554272, "reasoning_end_time": 1787693302.6632452, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "eb2dae3a-0ba1-485b-b3b0-906a55ace427", "message_type": "next", "turn_exchange_id": "ead3149b-9569-4a97-bf11-b771543d5dcf", "working_turn_id": "ead3149b-9569-4a97-bf11-b771543d5dcf", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "15f2c130-3912-4ab7-984d-4f7863cef7b2", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "5ce22894-0051-422c-8ccf-19b95b47a0db", "children": [ "2ae7e838-b7a7-4209-9863-45afcee53b7c" ] }, "2ae7e838-b7a7-4209-9863-45afcee53b7c": { "id": "2ae7e838-b7a7-4209-9863-45afcee53b7c", "message": { "id": "2ae7e838-b7a7-4209-9863-45afcee53b7c", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693301.934373, "update_time": 1787693304.329589, "content": { "content_type": "text", "parts": [ "Got it. The first one was closer to the thing you had in mind, including details you hadn’t explicitly supplied; the second drifted away from that. No new card." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "eb2dae3a-0ba1-485b-b3b0-906a55ace427", "message_type": "next", "turn_exchange_id": "ead3149b-9569-4a97-bf11-b771543d5dcf", "working_turn_id": "ead3149b-9569-4a97-bf11-b771543d5dcf", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "aed2df45-c394-4b05-9875-1f896cc2f716", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "aed2df45-c394-4b05-9875-1f896cc2f716", "children": [ "6252e241-1982-4f30-8690-65fdc34a98da" ] }, "6252e241-1982-4f30-8690-65fdc34a98da": { "id": "6252e241-1982-4f30-8690-65fdc34a98da", "message": { "id": "6252e241-1982-4f30-8690-65fdc34a98da", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693367.966, "update_time": 1787693375.399812, "content": { "content_type": "text", "parts": [ "chatgpt: can we solve quartic functions, can we solve quintic functions" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "3ed0ebb7-c1e4-4bba-a231-48f2c46f7513", "message_source": null, "turn_exchange_id": "e307bd33-625f-4da9-820f-957f115e6630", "working_turn_id": "e307bd33-625f-4da9-820f-957f115e6630", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "2ae7e838-b7a7-4209-9863-45afcee53b7c", "children": [ "a37547a3-a6fa-4509-ae0e-cff3fd2102e1" ] }, "a37547a3-a6fa-4509-ae0e-cff3fd2102e1": { "id": "a37547a3-a6fa-4509-ae0e-cff3fd2102e1", "message": { "id": "a37547a3-a6fa-4509-ae0e-cff3fd2102e1", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693370.162486, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 1, "reasoning_start_time": 1787693368.5435598, "reasoning_end_time": 1787693370.1544871, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "3ed0ebb7-c1e4-4bba-a231-48f2c46f7513", "message_type": "next", "turn_exchange_id": "e307bd33-625f-4da9-820f-957f115e6630", "working_turn_id": "e307bd33-625f-4da9-820f-957f115e6630", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "ad76dfd8-a686-40a5-9ab0-0cb39c7ef48e", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "6252e241-1982-4f30-8690-65fdc34a98da", "children": [ "152a80c7-a797-49b3-aa29-b9ece03545fd" ] }, "152a80c7-a797-49b3-aa29-b9ece03545fd": { "id": "152a80c7-a797-49b3-aa29-b9ece03545fd", "message": { "id": "152a80c7-a797-49b3-aa29-b9ece03545fd", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693368.518092, "update_time": 1787693374.439906, "content": { "content_type": "text", "parts": [ "Yes.\n\nA general **quartic** equation,\n\n\\[\nax^4+bx^3+cx^2+dx+e=0,\n\\]\n\ncan be solved exactly using a formula involving arithmetic and radicals. It’s much messier than the quadratic formula, but it exists; methods go back to Ferrari.\n\nA general **quintic**,\n\n\\[\nax^5+bx^4+cx^3+dx^2+ex+f=0,\n\\]\n\nis different: there is **no formula using only arithmetic and radicals that works for every quintic**. That is the Abel–Ruffini theorem.\n\nBut quintics are still solvable in several senses: particular quintics can sometimes be solved by radicals, arbitrary quintics can be solved numerically to any desired precision, and there are exact descriptions using more advanced special functions.\n\nSo, roughly:\n\n- degree 2: general radical formula\n- degree 3: yes\n- degree 4: yes\n- degree 5 and above: not in radicals in general\n\nIf you want, I can show why the quartic is still solvable but the quintic suddenly stops being so." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "3ed0ebb7-c1e4-4bba-a231-48f2c46f7513", "message_type": "next", "turn_exchange_id": "e307bd33-625f-4da9-820f-957f115e6630", "working_turn_id": "e307bd33-625f-4da9-820f-957f115e6630", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "a37547a3-a6fa-4509-ae0e-cff3fd2102e1", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "a37547a3-a6fa-4509-ae0e-cff3fd2102e1", "children": [ "3bd140a9-fb09-43e5-b30e-d9a2a0be66b2" ] }, "3bd140a9-fb09-43e5-b30e-d9a2a0be66b2": { "id": "3bd140a9-fb09-43e5-b30e-d9a2a0be66b2", "message": { "id": "3bd140a9-fb09-43e5-b30e-d9a2a0be66b2", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693448.414, "update_time": 1787693519.207131, "content": { "content_type": "text", "parts": [ "you're saying: all polynomials can be solved, but we need to use advanced special functions.\nare these functions in libraries like boost, sympy, numpy, kokkos, eigen, anything?\nit sounds like i could write code to solve any polynomial if i implemented these functions?" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "ff6a1317-2503-44b1-acbf-9973a4a33d74", "message_source": null, "turn_exchange_id": "5b54266b-e475-4bdf-9204-0de33ec393c1", "working_turn_id": "5b54266b-e475-4bdf-9204-0de33ec393c1", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "152a80c7-a797-49b3-aa29-b9ece03545fd", "children": [ "e29749d7-7d4b-4acc-a1ae-6e6008029bea" ] }, "e29749d7-7d4b-4acc-a1ae-6e6008029bea": { "id": "e29749d7-7d4b-4acc-a1ae-6e6008029bea", "message": { "id": "e29749d7-7d4b-4acc-a1ae-6e6008029bea", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693451.285906, "update_time": null, "content": { "content_type": "text", "parts": [ "Yes — but there are two different meanings of “solve any polynomial” here, and the distinction matters a lot: **numerically approximate all roots** is routine; **produce a closed-form expression in named special functions** is much less universal and usually not how libraries represent the answer." ] }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "citations": [], "resolved_model_slug": "gpt-5-6-thinking", "is_thinking_preamble_message": true, "hide_inline_actions": true, "disable_turn_actions": true, "chime_version": 1, "classifier_response": "default", "reasoning_start_time": 1787693451.5297456, "cot_version": "v5", "request_id": "ff6a1317-2503-44b1-acbf-9973a4a33d74", "message_type": "next", "turn_exchange_id": "5b54266b-e475-4bdf-9204-0de33ec393c1", "working_turn_id": "5b54266b-e475-4bdf-9204-0de33ec393c1", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "b0864ebc-52e1-4c41-938d-2125a54f4f25", "model_switcher_deny": [] }, "recipient": "all", "channel": "commentary" 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"0320d904-2f70-44b4-ad64-b422049bc7b6" ] }, "0320d904-2f70-44b4-ad64-b422049bc7b6": { "id": "0320d904-2f70-44b4-ad64-b422049bc7b6", "message": { "id": "0320d904-2f70-44b4-ad64-b422049bc7b6", "author": { "role": "assistant", "name": null, "metadata": { "real_author": "tool:web" } }, "create_time": 1787693490.578058, "update_time": 1787693517.469304, "content": { "content_type": "text", "parts": [ "Yes — with one correction to how I phrased it before.\n\nEvery nonconstant polynomial has roots over the complex numbers, and for ordinary computable coefficients you can write software that finds them. You **do not actually need exotic quintic special functions** to do this.\n\nThere are three different meanings of “solve”:\n\n| Meaning | Arbitrary degree? | Typical method |\n|---|---:|---|\n| Formula using radicals | ❌ No, generally fails at degree ≥ 5 | quadratic/cubic/quartic formulas |\n| Exact symbolic representation of each algebraic root | ✅ Yes | `RootOf(polynomial, index)` |\n| Numerical values to chosen precision | ✅ Yes | companion matrices / iterative root solvers |\n\n### You already have this in libraries\n\n**SymPy** is probably the clearest demonstration. For a polynomial with rational coefficients, `RootOf`/`all_roots()` can represent **all roots exactly at arbitrary degree**, while `nroots()` numerically approximates all roots of any polynomial whose coefficients can be numerically evaluated. citeturn204716search0\n\nFor example:\n\n```python\nfrom sympy import *\n\nx = symbols('x')\np = x**5 - x + 1\n\nprint(Poly(p, x).all_roots())\n```\n\ngives conceptually\n\n```text\n[\n CRootOf(x**5 - x + 1, 0),\n CRootOf(x**5 - x + 1, 1),\n CRootOf(x**5 - x + 1, 2),\n CRootOf(x**5 - x + 1, 3),\n CRootOf(x**5 - x + 1, 4)\n]\n```\n\nThose aren't approximations. Each means **“this particular exactly identified root of this polynomial.”** You can subsequently ask for 20, 100, or 10,000 digits.\n\nOr:\n\n```python\nnroots(p, n=50)\n```\n\ngets numerical approximations.\n\n**NumPy** already solves arbitrary finite-degree polynomials numerically:\n\n```python\nimport numpy as np\n\n# coefficients low degree -> high degree:\np = np.polynomial.Polynomial([1, -1, 0, 0, 0, 1])\n\nprint(p.roots())\n```\n\nNumPy's current implementation obtains the roots as eigenvalues of a **companion matrix**. citeturn241627search36\n\nAnd this is the particularly neat bit.\n\nGiven\n\n\\[\np(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0,\n\\]\n\nyou construct an \\(n\\times n\\) matrix whose eigenvalues are **exactly the roots of \\(p\\)**.\n\nSo polynomial root finding can be reduced to:\n\n```text\npolynomial coefficients\n ↓\ncompanion matrix\n ↓\nmatrix eigenvalue solver\n ↓\nall n complex roots\n```\n\nThat means an ordinary numerical linear algebra library is enough.\n\n**Eigen** actually supplies this directly in its unsupported `PolynomialSolver`. It accepts either fixed or dynamic polynomial degree and currently uses QR eigenvalue computation on the companion matrix. Eigen labels this module experimental/unsupported. citeturn241627search4\n\nSo in C++ you don't need to invent quintic mathematics either.\n\n### Boost / Kokkos\n\nBoost.Math is somewhat different. It has explicit polynomial solvers through quartics and general numerical root-finding algorithms such as TOMS 748, Newton, Halley, etc., but its documented general root finders are principally “find a root of this function,” rather than an all-complex-roots arbitrary-polynomial facility like NumPy's. citeturn537496search0turn537496search1\n\nKokkos isn't what I'd reach for as the polynomial solver itself. You could certainly build a parallel polynomial solver using Kokkos/Kokkos Kernels for the underlying computations, but that is a different layer.\n\nThere are also libraries specifically built for doing this very robustly. **FLINT/Arb**, for example, can compute arbitrary-precision complex polynomial roots and attach **rigorous error bounds** to them. For integer polynomials it has routines that isolate all complex roots into disjoint certified regions. citeturn539715search0turn539715search4\n\nThat is substantially stronger than merely getting a bunch of `double`s.\n\n### So what happened to those “advanced special functions”?\n\nThey're mathematically interesting, but I oversold their importance.\n\nA general quintic can indeed be transformed into forms whose solutions can be expressed using things such as hypergeometric/elliptic-related functions. But that's not usually how you would implement a general polynomial solver.\n\nAnd for arbitrary degree, saying “there's always some special function” becomes slightly tautological: you can define\n\n\\[\nR_k(a_0,\\ldots,a_n)\n=\n\\text{the kth root of }\na_nx^n+\\cdots+a_0,\n\\]\n\nand voilà, you've invented a “special function” that solves every polynomial. That's essentially what `RootOf` does, except in a mathematically useful and computable representation.\n\nSo your intuition is right, with an even nicer result:\n\n> **You could write code that solves arbitrary-degree polynomials without implementing those exotic special functions at all.**\n\nA surprisingly small prototype is basically:\n\n```python\ndef polynomial_roots(coefficients):\n C = make_companion_matrix(coefficients)\n return eigenvalues(C)\n```\n\nThe hard part isn't **whether it can be done**. 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root(x, -Rational(2, 3)) x**(-3/2) To...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 1 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/ntheory.html?utm_source=chatgpt.com", "title": "Number Theory - SymPy 1.14.0 documentation", "snippet": "See also `sqrt_mod` Same functionality, but you want a sorted list or only one solution. sympy.ntheory.residue_ntheory.quadratic_residues(p) → list[int][source]¶ Returns the list of quadratic re...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 2 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/genindex.html?utm_source=chatgpt.com", "title": "Index - SymPy 1.14.0 documentation", "snippet": "* __mod__() (sympy.polys.numberfields.modules.ModuleElement method) * __mul__() (sympy.combinatorics.perm_groups.PermutationGroup method) * (sympy.matrices.matrixbase.MatrixBase method) * (sympy.polys...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 6 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/?utm_source=chatgpt.com", "title": "SymPy 1.14.0 documentation", "snippet": "SymPy 1.14.0 documentation SymPy 1.14.0 documentation DOCUMENTATION VERSION * SymPy 1.14.0 (latest version) * SymPy 1.15-dev (dev version) WELCOME TO SYMPY’S DOCUMENTATION!¶ A PDF version of th...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 9 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/solvers/solvers.html?utm_source=chatgpt.com", "title": "Solvers - SymPy 1.14.0 documentation", "snippet": "(3 < r) & Eq(r, 3 - q) You can directly call the routine that `solve` calls when it encounters a relational: `reduce_inequalities()`. It treats Expr like Equality. >>> from sympy import reduce_ineq...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 10 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/dev/guides/solving/index.html?utm_source=chatgpt.com", "title": "Solve Equations - SymPy 1.15.0.dev documentation", "snippet": "Find the roots of a polynomial algebraically or numerically | \\( ax^2 + bx + c = 0 \\) | \\( x = \\frac{-b\\pm\\sqrt{b^2 - 4ac}}{2a} \\) Solve a matrix equation algebraically | \\( \\left[\\begin{array}{cc}...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 11 }, "pub_date": 1787529600, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/polys/basics.html?utm_source=chatgpt.com", "title": "Basic functionality of the module - SymPy 1.14.0 documentation", "snippet": "BASIC FUNCTIONALITY¶ These functions provide different algorithms dealing with polynomials in the form of SymPy expression, like symbols, sums etc. DIVISION¶ The function `div()` provides division...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 12 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/polys/index.html?utm_source=chatgpt.com", "title": "Polynomial Manipulation - SymPy 1.14.0 documentation", "snippet": "POLYNOMIAL MANIPULATION¶ Computations with polynomials are at the core of computer algebra and having a fast and robust polynomials manipulation module is a key for building a powerful symbolic mani...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 13 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/polys/reference.html?utm_source=chatgpt.com", "title": "Polynomials Manipulation Module Reference - SymPy 1.14.0 documentation", "snippet": "POLYNOMIALS MANIPULATION MODULE REFERENCE¶ Polynomial manipulation algorithms and algebraic objects. See Polynomial Manipulation for an index of documentation for the polys module and Basic functio...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 18 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/dev/modules/polys/?utm_source=chatgpt.com", "title": "Polynomial Manipulation - SymPy 1.15.0.dev documentation", "snippet": "POLYNOMIAL MANIPULATION¶ Computations with polynomials are at the core of computer algebra and having a fast and robust polynomials manipulation module is a key for building a powerful symbolic mani...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 19 }, "pub_date": 1785801600, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/index.html?utm_source=chatgpt.com", "title": "SymPy 1.14.0 documentation", "snippet": "SymPy 1.14.0 documentation SymPy 1.14.0 documentation WELCOME TO SYMPY’S DOCUMENTATION!¶ A PDF version of these docs is also available. SymPy is a Python library for symbolic mathematics. If you...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 20 }, "pub_date": 1745712000, "attribution": "sympy.org" } ] }, { "type": "search_result_group", "domain": "boost.org", "entries": [ { "type": "search_result", "url": "https://www.boost.org/doc/libs/latest/libs/math/doc/html/root_finding.html?utm_source=chatgpt.com", "title": "", "snippet": "BOOST C++ LIBRARIES \"...one of the most highly regarded and expertly designed C++ library projects in the world.\" — Herb Sutter and Andrei Alexandrescu, C++ Coding Standards Search... 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Math Toolkit...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 2 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/doc/libs/master/libs/math/doc/html/root_finding.html?utm_source=chatgpt.com", "title": "", "snippet": "Chapter 10. Root Finding & Minimization Algorithms CHAPTER 10. ROOT FINDING & MINIMIZATION ALGORITHMS Table of Contents Root Finding Without Derivatives Bisection Bracket and Solve Root Algorithm...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 3 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/roots_noderiv.html?utm_source=chatgpt.com", "title": "", "snippet": "Root Finding Without Derivatives ROOT FINDING WITHOUT DERIVATIVES Bisection Bracket and Solve Root Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions Brent-Decker Al...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 4 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/doc/libs/master/libs/math/doc/html/math_toolkit/root_finding_examples.html?utm_source=chatgpt.com", "title": "", "snippet": "This version of Boost is under active development. You are currently in the master branch. The current version is 1.91.0. Examples of Root-Finding (with and without derivatives) EXAMPLES OF ROOT-FIN...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 5 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/doc/libs/latest/libs/math/doc/html/poly.html?utm_source=chatgpt.com", "title": "", "snippet": "BOOST C++ LIBRARIES Chapter 12. Polynomials and Rational Functions CHAPTER 12. POLYNOMIALS AND RATIONAL FUNCTIONS Table of Contents Polynomials Estrin's method for polynomial evaluation Polynomial...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 6 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/library/1.85.0/math/?utm_source=chatgpt.com", "title": "Boost.Math (1.85.0)", "snippet": "Boost.Math M D [Select] This is an older version of Boost and was released in 2024. The current version is 1.92.0. C++ 14 Added in Boost 1.23.0 Categories:   Math Boost.Math includes several c...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 7 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/library/1.82.0/math/?utm_source=chatgpt.com", "title": "Boost.Math (1.82.0)", "snippet": "Boost.Math M D [Select] This is an older version of Boost and was released in 2023. The current version is 1.91.0. C++ 14 Added in Boost 1.23.0 Categories:   Math Boost.Math includes several c...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 8 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/library/1.77.0/math/?utm_source=chatgpt.com", "title": "Boost.Math (1.77.0)", "snippet": "Boost.Math M D [Select] This is an older version of Boost and was released in 2021. The current version is 1.92.0. C++ 11 Added in Boost 1.23.0 Categories:   Math Boost.Math includes several c...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 9 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/library/1.83.0/math/?utm_source=chatgpt.com", "title": "Boost.Math (1.83.0)", "snippet": "Boost.Math M D [Select] This is an older version of Boost and was released in 2023. The current version is 1.91.0. C++ 14 Added in Boost 1.23.0 Categories:   Math Boost.Math includes several c...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 10 }, "pub_date": null, "attribution": "boost.org" }, { "type": "search_result", "url": "https://www.boost.org/doc/libs/develop/libs/math/doc/html/math_toolkit/roots_deriv.html?utm_source=chatgpt.com", "title": "", "snippet": "Root Finding With Derivatives: Newton-Raphson, Halley & Schroeder ROOT FINDING WITH DERIVATIVES: NEWTON-RAPHSON, HALLEY & SCHROEDER SYNOPSIS namespace boost { namespa...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 11 }, "pub_date": null, "attribution": "boost.org" } ] }, { "type": "search_result_group", "domain": "flintlib.org", "entries": [ { "type": "search_result", "url": "https://flintlib.org/doc/examples_arb.html?utm_source=chatgpt.com", "title": "Arb example programs — FLINT 3.7.0-dev documentation", "snippet": "POLY_ROOTS.C¶ This program finds the complex roots of an integer polynomial by calling `arb_fmpz_poly_complex_roots()`, which in turn calls `acb_poly_find_roots()` with increasing precision until the...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 1 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/index_arb.html?utm_source=chatgpt.com", "title": "Real and complex numbers (Arb) : detailed table of contents — FLINT 3.7.0-dev documentation", "snippet": "* real_roots.c * poly_roots.c * zeta_zeros.c * complex_plot.c * lvalue.c * lcentral.c * integrals.c * fpwrap.c * functions_benchmark.c FLOATING-POINT NUMBERS¶ Arb uses two custom floating-point type...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 2 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/arb_poly.html?utm_source=chatgpt.com", "title": "arb_poly.h – polynomials over the real numbers — FLINT 3.7.0-dev documentation", "snippet": "ARB_POLY.H – POLYNOMIALS OVER THE REAL NUMBERS¶ An `arb_poly_t` represents a polynomial over the real numbers, implemented as an array of coefficients of type `arb_struct`. Most functions are provid...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 3 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/arb_fmpz_poly.html?utm_source=chatgpt.com", "title": "arb_fmpz_poly.h – extra methods for integer polynomials — FLINT 3.7.0-dev documentation", "snippet": "Sets res to a copy of poly deflated by the exponent deflation. POLYNOMIAL ROOTS¶ The following methods assume that the input polynomial is squarefree. 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An `acb_t` consists of a pair of real number balls of type `arb_struct`, representing the real and imaginary part wi...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 6 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/overview.html?utm_source=chatgpt.com", "title": "Feature overview — FLINT 3.7.0-dev documentation", "snippet": "Other implementations of ball arithmetic include iRRAM and Mathemagix. Arb differs from earlier implementations in technical aspects of the implementation, which makes certain computations more effic...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 7 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/?utm_source=chatgpt.com", "title": "FLINT: Fast Library for Number Theory — FLINT 3.7.0-dev documentation", "snippet": "REAL AND COMPLEX NUMBERS¶ * Feature overview * Using ball arithmetic * Technical conventions and potential issues * Arb example programs * mag.h – fixed-precision unsigned floating-point numbers for...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 8 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/fq_poly_factor.html?utm_source=chatgpt.com", "title": "fq_poly_factor.h – factorisation of univariate polynomials over finite fields — FLINT 3.6.0-dev documentation", "snippet": "ROOT FINDING¶ void fq_poly_roots(fq_poly_factor_t r, const fq_poly_t f, int with_multiplicity, const fq_ctx_t ctx)¶ Fill \\(r\\) with factors of the form \\(x - r_i\\) where the \\(r_i\\) are the distinc...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 9 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/acb_calc.html?utm_source=chatgpt.com", "title": "acb_calc.h – calculus with complex-valued functions — FLINT 3.7.0-dev documentation", "snippet": "ACB_CALC.H – CALCULUS WITH COMPLEX-VALUED FUNCTIONS¶ This module provides functions for operations of calculus over the complex numbers (intended to include root-finding, integration, and so on). The...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 10 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/arb_hypgeom.html?utm_source=chatgpt.com", "title": "arb_hypgeom.h – hypergeometric functions of real variables — FLINT 3.7.0-dev documentation", "snippet": "ARB_HYPGEOM.H – HYPERGEOMETRIC FUNCTIONS OF REAL VARIABLES¶ See acb_hypgeom.h – hypergeometric functions of complex variables for the general implementation of hypergeometric functions. For convenie...", "ref_id": { "turn_index": 3, "ref_type": "search", "ref_index": 11 }, "pub_date": null, "attribution": "flintlib.org" } ] }, { "type": "search_result_group", "domain": "numpy.org", "entries": [ { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.polynomial.polyvalfromroots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.polynomial.polyvalfromroots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.POLYNOMIAL.POLYVALFROMROOTS¶ `numpy.polynomial.polynomial.``polyvalfromroots`(x, r, tensor=True)[source]¶ Evaluate a polynomial specified by its roots at points x. If r is of len...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 3 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.legendre.legroots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.legendre.legroots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.LEGENDRE.LEGROOTS¶ `numpy.polynomial.legendre.``legroots`(c)[source]¶ Compute the roots of a Legendre series. Return the roots (a.k.a. “zeros”) of the polynomial Image: p(x) = \\...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 4 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.hermite.Hermite.roots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.hermite.Hermite.roots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.HERMITE.HERMITE.ROOTS¶ method `Hermite.``roots`(self)[source]¶ Return the roots of the series polynomial. Compute the roots for the series. Note that the accuracy of the roots d...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 5 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.hermite_e.hermeroots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.hermite_e.hermeroots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.HERMITE_E.HERMEROOTS¶ `numpy.polynomial.hermite_e.``hermeroots`(c)[source]¶ Compute the roots of a HermiteE series. Return the roots (a.k.a. “zeros”) of the polynomial Image: p(...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 7 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.hermite.hermroots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.hermite.hermroots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.HERMITE.HERMROOTS¶ `numpy.polynomial.hermite.``hermroots`(c)[source]¶ Compute the roots of a Hermite series. Return the roots (a.k.a. “zeros”) of the polynomial Image: p(x) = \\s...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 8 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.chebyshev.chebroots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.chebyshev.chebroots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.CHEBYSHEV.CHEBROOTS¶ `numpy.polynomial.chebyshev.``chebroots`(c)[source]¶ Compute the roots of a Chebyshev series. Return the roots (a.k.a. “zeros”) of the polynomial Image: p(x...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 14 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/stable/search.html?q=numpy.polynomial.polynomial.polyval&utm_source=chatgpt.com", "title": "Search - NumPy v2.5 Manual", "snippet": "SEARCH Error Please activate JavaScript to enable the search functionality. SEARCH RESULTS Search finished, found 33 page(s) matching the search query. * numpy.polynomial.polynomial.polyval .....", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 15 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.hermite_e.hermefromroots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.hermite_e.hermefromroots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.HERMITE_E.HERMEFROMROOTS¶ `numpy.polynomial.hermite_e.``hermefromroots`(roots)[source]¶ Generate a HermiteE series with given roots. The function returns the coefficients of the...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 16 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/devdocs/reference/generated/numpy.polynomial.legendre.legfromroots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.legendre.legfromroots — NumPy v2.6.dev0 Manual", "snippet": "NUMPY.POLYNOMIAL.LEGENDRE.LEGFROMROOTS# polynomial.legendre.legfromroots(roots)[source]# Generate a Legendre series with given roots. The function returns the coefficients of the polynomial \\[p(...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 17 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/stable/search.html?q=numpy.fv&utm_source=chatgpt.com", "title": "Search - NumPy v2.5 Manual", "snippet": "SEARCH RESULTS Search finished, found 2642 page(s) matching the search query. * `numpy.polynomial` ...NumPy reference NumPy’s module structure numpy.polynomial * Array broadcasting in Numpy ...A...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 21 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.legendre.Legendre.roots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.legendre.Legendre.roots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.LEGENDRE.LEGENDRE.ROOTS¶ method `Legendre.``roots`(self)[source]¶ Return the roots of the series polynomial. Compute the roots for the series. Note that the accuracy of the roots...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 22 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.17/reference/generated/numpy.polynomial.legendre.legfromroots.html?utm_source=chatgpt.com", "title": "numpy.polynomial.legendre.legfromroots — NumPy v1.17 Manual", "snippet": "NUMPY.POLYNOMIAL.LEGENDRE.LEGFROMROOTS¶ `numpy.polynomial.legendre.``legfromroots`(roots)[source]¶ Generate a Legendre series with given roots. The function returns the coefficients of the polyno...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 23 }, "pub_date": 1581984000, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/stable/search.html?q=numpy.generic&utm_source=chatgpt.com", "title": "Search - NumPy v2.5 Manual", "snippet": "Notes There are two modes of creating an array using __new__: If buffer is None, then... * numpy.polynomial.chebyshev.chebfromroots ...numpy.polynomial.chebyshev.chebfromroots polynomial.chebyshev.c...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 2 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/stable/search.html?q=numpy.array&utm_source=chatgpt.com", "title": "Search - NumPy v2.5 Manual", "snippet": "...NumPy reference Routines and objects by topic Polynomials * numpy.polynomial.chebyshev.Chebyshev.basis ...NumPy reference Routines and objects by topic Polynomials * numpy.polynomial.chebyshev....", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 3 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polymul.html?utm_source=chatgpt.com", "title": "numpy.polynomial.polynomial.polymul — NumPy v2.5 Manual", "snippet": "NUMPY.POLYNOMIAL.POLYNOMIAL.POLYMUL# polynomial.polynomial.polymul(c1, c2)[source]# Multiply one polynomial by another. Returns the product of two polynomials c1 * c2. The arguments are sequences...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 5 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/stable/search.html?q=numpy.dot&utm_source=chatgpt.com", "title": "Search - NumPy v2.5 Manual", "snippet": "* numpy.polynomial.polynomial.polyvalnd ...pts, c)[source] Evaluate an N-D polynomial at points. This function returns the values: \\[p(pts, c) = \\sum_{i_1, i_2, \\dot s, i_n} c_{i_1, i_2, \\dot s, i_n}...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 6 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/stable/search.html?q=numpy.polynomial.chebyshev.Chebyshev.convert&utm_source=chatgpt.com", "title": "Search - NumPy v2.5 Manual", "snippet": "SEARCH RESULTS Search finished, found 23 page(s) matching the search query. * numpy.polynomial.chebyshev.Chebyshev.convert ....polynomial.chebyshev) numpy.polynomial.chebyshev.Chebyshev numpy.poly...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 9 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/stable/reference/generated/numpy.polynomial.chebyshev.poly2cheb.html?utm_source=chatgpt.com", "title": "numpy.polynomial.chebyshev.poly2cheb — NumPy v2.5 Manual", "snippet": "NUMPY.POLYNOMIAL.CHEBYSHEV.POLY2CHEB# polynomial.chebyshev.poly2cheb(pol)[source]# Convert a polynomial to a Chebyshev series. Convert an array representing the coefficients of a polynomial (rela...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 14 }, "pub_date": null, "attribution": "numpy.org" }, { "type": "search_result", "url": "https://numpy.org/doc/1.13/reference/generated/numpy.polynomial.polynomial.polypow.html?utm_source=chatgpt.com", "title": "numpy.polynomial.polynomial.polypow — NumPy v1.13 Manual", "snippet": "NUMPY.POLYNOMIAL.POLYNOMIAL.POLYPOW¶ `numpy.polynomial.polynomial.``polypow`(c, pow, maxpower=None)[source]¶ Raise a polynomial to a power. Returns the polynomial c raised to the power pow. 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CLASSES cla...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 12 }, "pub_date": null, "attribution": "libeigen.gitlab.io" }, { "type": "search_result", "url": "https://libeigen.gitlab.io/eigen/docs-nightly/unsupported/PolynomialSolver_8h_source.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: /builds/libeigen/eigen/unsupported/Eigen/src/Polynomials/PolynomialSolver.h Source File", "snippet": "1// This file is part of Eigen, a lightweight C++ template library 2// for linear algebra. 3// 4// Copyright (C) 2010 Manuel Yguel 5// 6// This Source Code Form is subje...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 13 }, "pub_date": null, "attribution": "libeigen.gitlab.io" }, { "type": "search_result", "url": "https://libeigen.gitlab.io/eigen/docs-3.3/unsupported/PolynomialSolver_8h_source.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: /builds/libeigen/eigen/unsupported/Eigen/src/Polynomials/PolynomialSolver.h Source File", "snippet": "1 2 3 4 5 6 7 8 9 10#ifndef EIGEN_POLYNOMIAL_SOLVER_H 11#define EIGEN_POLYNOMIAL_SOLVER_H 12 13 namespace Eigen { 14 28 template< typename _Scalar, int _Deg > 29class PolynomialSolverB...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 19 }, "pub_date": null, "attribution": "libeigen.gitlab.io" }, { "type": "search_result", "url": "https://libeigen.gitlab.io/eigen/docs-3.3/unsupported/classEigen_1_1PolynomialSolverBase.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Eigen::PolynomialSolverBase< _Scalar, _Deg > Class Template Reference", "snippet": "Eigen::PolynomialSolverBase< _Scalar, _Deg > Class Template Reference Polynomials module `#include ` DETAILED DESCRIPTION template Member List This is the complete list of members for Eigen::PolynomialSolver< Scalar_, Deg_ >, including all inherited members. absGreatestRealRoot(bool &ha...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 25 }, "pub_date": null, "attribution": "libeigen.gitlab.io" }, { "type": "search_result", "url": "https://libeigen.gitlab.io/eigen/docs-3.3/unsupported/dir_d5537d6a2231218fc2b9b9ed6c07a248.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: /builds/libeigen/eigen/unsupported/Eigen/src/Polynomials Directory Reference", "snippet": " Polynomials Directory Reference FILES | Companion.h | PolynomialSolver.h | PolynomialUtils.h * unsupported * Eigen * src * Polynomials", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 26 }, "pub_date": null, "attribution": "libeigen.gitlab.io" } ] }, { "type": "search_result_group", "domain": "eigen.tuxfamily.org", "entries": [ { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/PolynomialSolver_8h_source.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: PolynomialSolver.h Source File", "snippet": "Image: Logo | Eigen-unsupported  3.4.90 (git rev 67eeba6e720c5745abc77ae6c92ce0a44aa7b7ae) | Image [Input] Image PolynomialSolver.h 4 5 6 7 8 9 10 #ifndef EIGEN_POLYNOMIAL_SOLVER_H 11 #def...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 10 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/classEigen_1_1PolynomialSolver.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Eigen::PolynomialSolver< Scalar_, Deg_ > Class Template Reference", "snippet": "Eigen::PolynomialSolver< Scalar_, Deg_ > Class Template Reference Polynomials module DETAILED DESCRIPTION TEMPLATE class Eigen::PolynomialSolver< Scalar_, Deg_ > A pol...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 15 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/classEigen_1_1PolynomialSolverBase.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Eigen::PolynomialSolverBase< Scalar_, Deg_ > Class Template Reference", "snippet": "Eigen::PolynomialSolverBase< Scalar_, Deg_ > Class Template Reference Polynomials module DETAILED DESCRIPTION TEMPLATE class Eigen::PolynomialSolverBase< Scalar_, Deg_...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 22 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/group__Polynomials__Module.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Polynomials module", "snippet": "Polynomials module DETAILED DESCRIPTION This module provides a QR based polynomial solver. To use this module, add at the start of your source file. CLASSES cla...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 27 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/annotated.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Class List", "snippet": "Class List Here are the classes, structs, unions and interfaces with brief descriptions: [detail level 1 2] ▼NEigen | Namespace containing all symbols from the Eigen library CAlignedVector3 | A...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 28 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/namespaceEigen.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Eigen Namespace Reference", "snippet": "Eigen Namespace Reference Namespace containing all symbols from the Eigen library. More... CLASSES class   | aligned_allocator class   | AlignedBox class   | AlignedVector3 | A vectorization...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 29 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/group__SpecialFunctions__Module.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Special math functions module", "snippet": "Special math functions module This module features additional coefficient-wise math functions available within the numext:: namespace for the scalar version, and as method and/or free functions of A...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 30 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/index.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Eigen's unsupported modules", "snippet": "Eigen's unsupported modules This is the API documentation for Eigen's unsupported modules. These modules are contributions from various users. They are provided \"as is\", without any support. Click...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 31 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/Spline_8h_source.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Spline.h Source File", "snippet": "4 5 6 7 8 9 10 #ifndef EIGEN_SPLINE_H 11 #define EIGEN_SPLINE_H 12 13 #include \"./InternalHeaderCheck.h\" 14 15 #include \"SplineFwd.h\" 16 17 namespace Eigen 18 { 36  template This module provides implementation of two important algorithms in non linear optimization. In both ca...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 33 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/SplineFwd_8h_source.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: SplineFwd.h Source File", "snippet": "4 5 6 7 8 9 10 #ifndef EIGEN_SPLINES_FWD_H 11 #define EIGEN_SPLINES_FWD_H 12 13 #include \"./InternalHeaderCheck.h\" 14 #include \"../../../../Eigen/Core\" 15 16 namespace Eigen 17 { 18  te...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 34 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" }, { "type": "search_result", "url": "https://eigen.tuxfamily.org/dox/unsupported/group__MatrixFunctions__Module.html?utm_source=chatgpt.com", "title": "Eigen-unsupported: Matrix functions module", "snippet": "FUNCTION DOCUMENTATION", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 35 }, "pub_date": 1650499200, "attribution": "eigen.tuxfamily.org" } ] }, { "type": "search_result_group", "domain": "kokkos.org", "entries": [ { "type": "search_result", "url": "https://kokkos.org/kokkos-kernels/docs/quick_start.html?utm_source=chatgpt.com", "title": "Quick Start - Kokkos Kernels documentation", "snippet": "QUICK START¶ This document gives a brief introduction to start using Kokkos Kernels in your external project. The first step to use Kokkos Kernels is to install the Kokkos programming model, a quick...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 12 }, "pub_date": null, "attribution": "kokkos.org" }, { "type": "search_result", "url": "https://kokkos.org/kokkos-kernels/docs/requirements.html?utm_source=chatgpt.com", "title": "Requirements - Kokkos Kernels documentation", "snippet": "REQUIREMENTS¶ Kokkos Kernels only requirement is a compatible installation of the Kokkos Core programming model. In general the current version of Kokkos Kernels will work with the current version o...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 13 }, "pub_date": null, "attribution": "kokkos.org" }, { "type": "search_result", "url": "https://kokkos.org/kokkos-kernels/docs/index.html?utm_source=chatgpt.com", "title": "Kokkos Kernels documentation", "snippet": "KOKKOS KERNELS: PORTABLE MATH KERNELS¶ C++ Performance Portable Math Library Kokkos Kernels provides math kernels for dense, batched and sparse linear algebra as well as graph computations. Our int...", "ref_id": { "turn_index": 2, "ref_type": "search", "ref_index": 14 }, "pub_date": null, "attribution": "kokkos.org" }, { "type": "search_result", "url": "https://kokkos.org/kokkos-kernels/docs/API/lapack-index.html?utm_source=chatgpt.com", "title": "API: LAPACK - Kokkos Kernels documentation", "snippet": "API: LAPACK¶ LAPACK SUPPORT¶ Below are tables summarizing the currently supported function calls and third party libraries in Kokkos Kernels. 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That changes the answer substantially.\n\nYou mean something like\n\n\\[\nP(x,y)=y^5+xy+1=0\n\\]\n\nand you want to symbolically rearrange it into something conceptually like\n\n\\[\ny=f(x),\n\\]\n\nthen generate code evaluating \\(f(x)\\).\n\nThat object is called an **algebraic function**.\n\nFor degree ≤ 4 in \\(y\\), you can generally produce an explicit expression involving radicals. For example,\n\n\\[\ny^2+xy+1=0\n\\]\n\ngives\n\n\\[\ny(x)=\\frac{-x\\pm\\sqrt{x^2-4}}2.\n\\]\n\nFor a generic degree-5 polynomial such as\n\n\\[\ny^5+xy+1=0,\n\\]\n\nthere generally isn't a radical expression for \\(y(x)\\). SymPy's documentation explicitly notes that its symbolic `roots()` can handle symbolic coefficients when a radical expression exists, but generic high-degree polynomials generally defeat that approach; moreover, SymPy's general `RootOf` machinery currently requires rational coefficients rather than symbolic parameters like \\(x\\). citeturn321317search1\n\n### But you can still represent \\(y(x)\\) exactly\n\nInstead of demanding\n\n```text\ny = sqrt(...) + cbrt(...) + ...\n```\n\nyou represent\n\n\\[\ny=\\operatorname{Root}_k\\!\\left(Y^5+xY+1\\right).\n\\]\n\nHere \\(x\\) remains symbolic, and \\(k\\) identifies a particular branch/root.\n\nThat is a perfectly legitimate symbolic expression; it's just an **algebraic-function expression** rather than an expression made out of radicals.\n\nSageMath has machinery that is especially close to what you're describing. It can construct the rational function field\n\n\\[\n\\mathbb Q(x)\n\\]\n\nand then extend it by a symbolic element \\(y\\) satisfying an arbitrary polynomial relation. Its documentation actually gives a degree-5 example essentially of this form:\n\n```python\nK. = FunctionField(QQ)\nR. = K[]\n\nL. = K.extension(\n y^5 - (x^3 + 2*x*y + 1/x)\n)\n```\n\nNow `y` is an **exact symbolic object satisfying that quintic relation over the field of rational functions in `x`**. No numerical approximation and no radical formula are required. citeturn843998search0turn843998search1\n\nThat's very close to:\n\n```text\ninput:\n y^5 + x*y + 1 = 0\n\noutput:\n y = AlgebraicFunction(\n polynomial = Y^5 + x*Y + 1,\n branch = k\n )\n```\n\n### Generating executable code\n\nThen your generated evaluator could look conceptually like:\n\n```cpp\ncomplex y(complex x)\n{\n // coefficients of Y^5 + x Y + 1\n polynomial p = {1, x, 0, 0, 0, 1};\n\n auto roots = polynomial_roots(p);\n\n return choose_branch(roots);\n}\n```\n\nSo the symbolic compiler has retained\n\n\\[\ny=\\operatorname{Root}_k(Y^5+xY+1)\n\\]\n\nwhile generated machine code substitutes a particular numeric `x` and computes that root.\n\nAnd **NumPy/Eigen/etc. are useful at this second stage**, because after substituting \\(x\\), the coefficients are numerical and the problem is ordinary polynomial root finding.\n\n### There is one important complication: \\(y(x)\\) usually isn't one function\n\nFor\n\n\\[\nP(x,y)=0\n\\]\n\nof degree \\(n\\) in \\(y\\), there are generally \\(n\\) values\n\n\\[\ny_1(x),y_2(x),\\ldots,y_n(x).\n\\]\n\nFor example,\n\n\\[\ny^2=x\n\\]\n\nhas\n\n\\[\ny_1(x)=+\\sqrt{x},\n\\qquad\ny_2(x)=-\\sqrt{x}.\n\\]\n\nAnd those branches can meet at **branch points**. For the general algebraic curve \\(P(x,y)=0\\), these occur where simultaneously\n\n\\[\nP(x,y)=0\n\\]\n\nand\n\n\\[\n\\frac{\\partial P}{\\partial y}=0.\n\\]\n\nAway from those points, the implicit-function theorem gives you a locally ordinary function, and even gives its derivative:\n\n\\[\n\\boxed{\n\\frac{dy}{dx}\n=\n-\\frac{P_x}{P_y}\n}\n\\]\n\nSo a particularly nice implementation could maintain a chosen root continuously as `x` changes rather than repeatedly saying “give me root #3.” That avoids roots unexpectedly changing identity because their numerical ordering changed.\n\n### So your original idea basically works\n\nYou could build a symbolic system that transforms\n\n```text\nP(x, y) = 0\n```\n\ninto something like\n\n```text\ny = AlgebraicRoot(P(x, Y), branch)\n```\n\nfor **arbitrary polynomial degree**.\n\nThen that object could support operations such as:\n\n```text\nevaluate(x = 2.7)\ndifferentiate()\nsubstitute()\nfind_branch_points()\ncontinue_branch(from_x, to_x)\ngenerate_cpp()\n```\n\nYou don't need a closed-form quintic formula at all.\n\nAnd interestingly, **Sage's algebraic function fields already implement much of the symbolic algebraic side of precisely this idea**, including extensions of degree five and higher over \\(\\mathbb Q(x)\\). citeturn843998search1\n\nSo if what you were imagining was *“a symbolic compiler that can solve one variable of an arbitrary polynomial relation in terms of another”*, the answer is much closer to **yes** than my previous answer suggested. The natural output type just has to be **algebraic function**, rather than “expression composed exclusively of familiar elementary functions.”" ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "content_references": [ { "matched_text": "citeturn321317search1", "prefix": null, "start_idx": 905, "end_idx": 929, "safe_urls": [ "https://docs.sympy.org/dev/guides/solving/find-roots-polynomial.html", "https://docs.sympy.org/dev/guides/solving/find-roots-polynomial.html?utm_source=chatgpt.com" ], "refs": [], "alt": "([SymPy Documentation](https://docs.sympy.org/dev/guides/solving/find-roots-polynomial.html?utm_source=chatgpt.com))", "prompt_text": null, "type": "grouped_webpages", "items": [ { "title": "Find the Roots of a Polynomial Algebraically or Numerically - SymPy 1.15.0.dev documentation", "url": "https://docs.sympy.org/dev/guides/solving/find-roots-polynomial.html?utm_source=chatgpt.com", "attribution": "SymPy Documentation", "pub_date": 1787184000, "snippet": "", "attribution_segments": null, "supporting_websites": [], "refs": [ { "turn_index": 0, "ref_type": "search", "ref_index": 1 } ], "hue": null, "attributions": null } ], "fallback_items": null, "status": "done", "error": null, "style": null }, { "matched_text": "citeturn843998search0turn843998search1", "prefix": null, "start_idx": 1944, "end_idx": 1986, "safe_urls": [ "https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/function_field.html", "https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/function_field.html?utm_source=chatgpt.com", "https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/function_field_polymod.html", "https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/function_field_polymod.html?utm_source=chatgpt.com" ], "refs": [], "alt": "([SageMath Documentation](https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/function_field.html?utm_source=chatgpt.com))", "prompt_text": null, "type": "grouped_webpages", "items": [ { "title": "Function Fields - 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Symbolic Calculus", "snippet": "polynomial(base_ring=None, ring=None)[source]¶ Return this symbolic expression as an algebraic polynomial over the given base ring, if possible. The point of this function is that it converts purel...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 2 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc-gitlab.sagemath.org/html/en/reference/calculus/sage/calculus/calculus.html?utm_source=chatgpt.com", "title": "Symbolic Computation — Sage 9.4 Reference Manual: Symbolic Calculus", "snippet": "OUTPUT: The minimal polynomial of self. If the numerical algorithm is used then it is proved symbolically when epsilon=0 (default). If the minimal polynomial could not be found, two distinct kinds of...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 5 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc-gitlab.sagemath.org/html/en/constructions/polynomials.html?utm_source=chatgpt.com", "title": "Polynomials — Sage Constructions v9.4", "snippet": "ROOTS OF POLYNOMIALS¶ Sage can compute roots of a univariant polynomial. sage: x = PolynomialRing(RationalField(), 'x').gen() sage: f = x^3 - 1 sage: f.roots() [(1, 1)] sage: f = (x^3 - 1)^2 sage: f...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 7 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc-gitlab.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/element.html?utm_source=chatgpt.com", "title": "Elements of function fields — Sage 9.4 Reference Manual: Algebraic Function Fields", "snippet": "ELEMENTS OF FUNCTION FIELDS¶ Sage provides arithmetic with elements of function fields. EXAMPLES: Arithmetic with rational functions: sage: K. = FunctionField(QQ) sage: f = t - 1 sage: g = t^2...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 8 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc.sagemath.org/html/en/constructions/polynomials.html?utm_source=chatgpt.com", "title": "Polynomials - Constructions", "snippet": "ROOTS OF POLYNOMIALS¶ Sage can compute roots of a univariant polynomial. [Input]Sage sage: x = PolynomialRing(RationalField(), 'x').gen() sage: f = x^3 - 1 sage: f.roots() [(1, 1)] sage: f = (x^3 -...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 11 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://ask.sagemath.org/answers/33063/revisions/?utm_source=chatgpt.com", "title": "Revision history - ASKSAGE: Sage Q&A Forum", "snippet": "Anyway, thanks for reporting ! That said, if you want to work with polynomials, you whould work in a well defined polynomial ring, instead of a symbolic ~~blob:~~blob. So first define the polynomial...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 17 }, "pub_date": 1460332800, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/element_polymod.html?utm_source=chatgpt.com", "title": "Elements of function fields: extension - Algebraic Function Fields", "snippet": "ELEMENTS OF FUNCTION FIELDS: EXTENSION¶ class sage.rings.function_field.element_polymod.FunctionFieldElement_polymod[source]¶ Bases: `FunctionFieldElement` Elements of a finite extension of a fun...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 2 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc.sagemath.org/html/en/reference/function_fields/index.html?utm_source=chatgpt.com", "title": "Algebraic Function Fields", "snippet": "* Places of function fields: extension * Divisors of function fields * Differentials of function fields * Valuation rings of function fields * Derivations of function fields * Derivations of function...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 4 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/extensions.html?utm_source=chatgpt.com", "title": "Special extensions of function fields - Algebraic Function Fields", "snippet": "SPECIAL EXTENSIONS OF FUNCTION FIELDS¶ This module currently implements only constant field extension. CONSTANT FIELD EXTENSIONS¶ EXAMPLES: Constant field extension of the rational function field...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 5 }, "pub_date": 1640304000, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc.sagemath.org/html/en/reference/function_fields/sage/rings/function_field/element.html?utm_source=chatgpt.com", "title": "Elements of function fields - Algebraic Function Fields", "snippet": "Traceback (most recent call last): ... ValueError: has a pole at the place sage: (y/x + 1).evaluate(p) 1 [Input]Python >>> from sage.all import * >>> # needs sage.rings.finite_rings sage.rings.func...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 7 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc.sagemath.org/html/en/reference/padics/sage/rings/padics/padic_extension_generic.html?utm_source=chatgpt.com", "title": "\\(p\\)-adic Extension Generic - $p$-adics", "snippet": "[Input]Python >>> from sage.all import * >>> K = Qq(Integer(25), Integer(8), names=('a',)); (a,) = K._first_ngens(1) >>> c, R = K.construction(); R 5-adic Unramified Extension Ring in a defined by x^...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 9 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc.sagemath.org/html/en/reference/rings/sage/rings/ring.html?utm_source=chatgpt.com", "title": "Rings - General Rings, Ideals, and Morphisms", "snippet": "Bases: `Ring` class sage.rings.ring.CommutativeAlgebra[source]¶ Bases: `CommutativeRing` class sage.rings.ring.CommutativeRing[source]¶ Bases: `Ring` Generic commutative ring. extension(poly,...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 15 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc-gitlab.sagemath.org/html/en/reference/finite_rings/sage/rings/finite_rings/finite_field_base.html?utm_source=chatgpt.com", "title": "Base Classes for Finite Fields — Sage 9.4 Reference Manual: Finite Rings", "snippet": "AUTHOR: * Thomas Gagne (2015-06-16) extension(modulus, name=None, names=None, map=False, embedding=None, latex_name=None, latex_names=None, **kwds)¶ Return an extension of this finite field. INPU...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 17 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://doc-gitlab.sagemath.org/html/en/reference/padics/sage/rings/padics/local_generic.html?utm_source=chatgpt.com", "title": "Local Generic — Sage 9.4 Reference Manual: p-Adics", "snippet": "Changing the prime works for extensions: sage: x = polygen(ZZ) sage: R. = Zp(5).extension(x^2 + 2) sage: S = R.change(p=7) sage: S.defining_polynomial(exact=True) x^2 + 2 sage: A. = Zp(5)[] sag...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 18 }, "pub_date": null, "attribution": "sagemath.org" }, { "type": "search_result", "url": "https://ask.sagemath.org/question/63229/degree-of-a-rational-map-and-the-corresponding-map-between-function-fields/?utm_source=chatgpt.com", "title": "Degree of a rational map and the corresponding map between function fields - ASKSAGE: Sage Q&A Forum", "snippet": "dan_fulea 5695 ●5 ●44 ●97 Here is some sample code to have a concrete situation: `F. = GF(25) P2_XYZ. = ProjectiveSpace(F, 2) P2_STU. = ProjectiveSpace(F, 2) eqC = -(a + 1)*Y^2*Z...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 21 }, "pub_date": 1657670400, "attribution": "sagemath.org" } ] }, { "type": "search_result_group", "domain": "cocalc.com", "entries": [ { "type": "search_result", "url": "https://cocalc.com/github/sympy/sympy/blob/master/doc/src/guides/solving/find-roots-polynomial.md?utm_source=chatgpt.com", "title": "CoCalc -- find-roots-polynomial.md", "snippet": "There are several functions that you can use to find the roots of a polynomial: * {func}`~.solve` is a general solving function which can find roots, though is less efficient than {meth}`~sympy.poly...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 3 }, "pub_date": null, "attribution": "cocalc.com" } ] }, { "type": "search_result_group", "domain": "github.com", "entries": [ { "type": "search_result", "url": "https://github.com/sympy/sympy/blob/master/sympy/solvers/solveset.py?utm_source=chatgpt.com", "title": "sympy/sympy/solvers/solveset.py at master · sympy/sympy · GitHub", "snippet": "PREPARE CHANGE OF VARIABLE x = Dummy('x') if _is_function_class_equation(HyperbolicFunction, f, symbol): cov = exp(x) inverter = invert_real if domain.is_subset(S.Reals) else invert_complex else:...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 4 }, "pub_date": null, "attribution": "github.com" }, { "type": "search_result", "url": "https://github.com/sagemath/sage/blob/develop/src/sage/rings/qqbar.py?utm_source=chatgpt.com", "title": "sage/src/sage/rings/qqbar.py at develop · sagemath/sage · GitHub", "snippet": "def common_polynomial(self, poly): \"\"\" Given a polynomial with algebraic coefficients, return a wrapper that caches high-precision calculations and factorizations. This wrapper can be passed to...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 12 }, "pub_date": null, "attribution": "github.com" }, { "type": "search_result", "url": "https://github.com/sagemath/sage/blob/develop/src/sage/rings/polynomial/complex_roots.py?utm_source=chatgpt.com", "title": "sage/src/sage/rings/polynomial/complex_roots.py at develop · sagemath/sage · GitHub", "snippet": "THIS MAY BE QUADRATIC IN PERVERSE CASES, BUT WILL TAKE ONLY N LOG(N) TIME IN TYPICAL CASES. intvs = sorted(copy(intvs)) column = [] prev_real = None def column_disjoint(): column.sort() row = [...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 13 }, "pub_date": null, "attribution": "github.com" }, { "type": "search_result", "url": "https://github.com/sagemath/sage/blob/develop/src/sage/rings/algebraic_closure_finite_field.py?utm_source=chatgpt.com", "title": "sage/src/sage/rings/algebraic_closure_finite_field.py at develop · sagemath/sage · GitHub", "snippet": "True \"\"\" return self def _an_element_(self): \"\"\" TESTS:: sage: from sage.rings.algebraic_closure_finite_field import AlgebraicClosureFiniteField sage: F = AlgebraicClosureFiniteField(GF(5), 'w...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 14 }, "pub_date": null, "attribution": "github.com" }, { "type": "search_result", "url": "https://github.com/sagemath/sage/blob/develop/src/sage/rings/function_field/function_field.py?utm_source=chatgpt.com", "title": "sage/src/sage/rings/function_field/function_field.py at develop · sagemath/sage · GitHub", "snippet": "transcendence degree one. In Sage, a function field can be a rational function field or a finite extension of a function field. EXAMPLES: We create a rational function field:: sage: K. = Functi...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 20 }, "pub_date": null, "attribution": "github.com" } ] }, { "type": "search_result_group", "domain": "sympy.org", "entries": [ { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/functions/elementary.html?utm_source=chatgpt.com", "title": "Elementary - SymPy 1.14.0 documentation", "snippet": "Examples >>> from sympy import root, Rational >>> from sympy.abc import x, n >>> root(x, 2) sqrt(x) >>> root(x, 3) x**(1/3) >>> root(x, n) x**(1/n) >>> root(x, -Rational(2, 3)) x**(-3/2) To...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 6 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/polys/basics.html?utm_source=chatgpt.com", "title": "Basic functionality of the module - SymPy 1.14.0 documentation", "snippet": "When all monomials having the same exponents are combined, the polynomial expression becomes a sum of products \\(c_\\nu x^\\nu\\), called the terms of the polynomial, where the coefficients \\(c_\\nu\\) ar...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 10 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/solvers/solveset.html?utm_source=chatgpt.com", "title": "Solveset - SymPy 1.14.0 documentation", "snippet": "SOLVESET MODULE REFERENCE¶ Use `solveset()` to solve equations or expressions (assumed to be equal to 0) for a single variable. Solving an equation like \\(x^2 == 1\\) can be done as follows: >>> from...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 15 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/modules/solvers/solvers.html?utm_source=chatgpt.com", "title": "Solvers - SymPy 1.14.0 documentation", "snippet": ">>> from sympy import real_root, S >>> eq = root(x, 3) - root(x, 5) + S(1)/7 >>> solve(eq) # this gives 2 solutions but misses a 3rd [CRootOf(7*x**5 - 7*x**3 + 1, 1)**15, CRootOf(7*x**5 - 7*x**3 + 1...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 20 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/latest/guides/solving/solving-guidance.html?utm_source=chatgpt.com", "title": "Solving Guidance - SymPy 1.14.0 documentation", "snippet": ">>> from sympy import solve, cos >>> from sympy.abc import x >>> solve(cos(x) - x, x, dict=True) Traceback (most recent call last): ... NotImplementedError: multiple generators [x, cos(x)] No algorit...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 21 }, "pub_date": 1745712000, "attribution": "sympy.org" }, { "type": "search_result", "url": "https://docs.sympy.org/dev/modules/core.html?utm_source=chatgpt.com", "title": "Core - SymPy 1.15.0.dev documentation", "snippet": "> > (2) expr is any other type of `Expr`: Then `root` will equal expr. 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"82749d6e-253a-4b58-99e6-bfd08c72a20d", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "82749d6e-253a-4b58-99e6-bfd08c72a20d", "children": [ "421022f4-8b23-4505-8a33-7bd86736d788" ] }, "421022f4-8b23-4505-8a33-7bd86736d788": { "id": "421022f4-8b23-4505-8a33-7bd86736d788", "message": { "id": "421022f4-8b23-4505-8a33-7bd86736d788", "author": { "role": "assistant", "name": null, "metadata": { "real_author": "tool:web" } }, "create_time": 1787693833.292908, "update_time": 1787693842.290106, "content": { "content_type": "text", "parts": [ "Yes — that’s basically the right inference.\n\nPeople absolutely use **algebraic-function / `RootOf` objects inside computer algebra systems**, but it’s uncommon to see applied numerical code generated around exotic “general quintic special functions” in the way one routinely generates code containing `sin`, `sqrt`, `erf`, Bessel functions, etc.\n\nMaple is a good counterexample to “nobody uses it at all”: its `RootOf` is explicitly the standard representation for algebraic numbers and **algebraic functions**, and Maple can differentiate, expand in series, numerically evaluate, and simplify such objects. It even supports polynomial coefficients containing symbolic variables, i.e. things very close to your \\(y(x)\\) case. citeturn747674search2turn747674search4\n\nBut for applied code, people usually do one of these instead:\n\n- keep the relation implicitly as \\(P(x,y)=0\\);\n- numerically solve for \\(y\\) when \\(x\\) becomes known;\n- track the desired root continuously as parameters vary;\n- or let a CAS retain a `RootOf(...)` object internally.\n\nThey usually **do not transform a generic quintic into a giant expression involving Bring radicals / hypergeometric functions and then emit that as C++**.\n\nThere are practical reasons. Those explicit formulas are often much larger, have complicated branch conventions, can be numerically ill-conditioned, and are harder to differentiate/continue robustly than the original polynomial relation. Numerically evaluating\n\n\\[\ny=\\operatorname{RootOf}(Y^5+xY+1,\\;k)\n\\]\n\ndirectly is often a much better computational primitive than evaluating some enormous special-function identity.\n\nSo there’s a mildly interesting gap here:\n\n> Numerical libraries have excellent support for evaluating standard transcendental special functions, and CASes have excellent support for abstract algebraic roots, but there isn’t a commonly used **“algebraic function runtime”** sitting between them.\n\nSomething like\n\n```cpp\nAlgebraicFunction f(\n polynomial_in_y,\n parameters,\n branch_descriptor\n);\n\ndouble y = f.eval(x);\ndouble dydx = f.derivative(x);\n```\n\nwith branch continuation, singularity detection, code generation, interval/certified evaluation, GPU support, etc. would be a genuinely useful abstraction.\n\nAnd I think that is closer to what you were originally getting at than “implement the quintic formula.”" ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "content_references": [ { "matched_text": "citeturn747674search2turn747674search4", "prefix": null, "start_idx": 726, "end_idx": 768, "safe_urls": [ "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf", "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf%2Findexed", "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf%2Findexed&utm_source=chatgpt.com", "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf&utm_source=chatgpt.com" ], "refs": [], "alt": "([Maplesoft](https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf&utm_source=chatgpt.com))", "prompt_text": null, "type": "grouped_webpages", "items": [ { "title": "", "url": "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf&utm_source=chatgpt.com", "attribution": "Maplesoft", "pub_date": null, "snippet": "", "attribution_segments": null, "supporting_websites": [ { "title": "", "url": "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf%2Findexed&utm_source=chatgpt.com", "pub_date": null, "snippet": "", "attribution": "Maplesoft" } ], "refs": [ { "turn_index": 0, "ref_type": "search", "ref_index": 2 }, { "turn_index": 0, "ref_type": "search", "ref_index": 4 } ], "hue": null, "attributions": null } ], "fallback_items": null, "status": "done", "error": null, "style": null }, { "matched_text": " ", "prefix": null, "start_idx": 2352, "end_idx": 2353, "safe_urls": [], "refs": [], "alt": "", "prompt_text": null, "type": "sources_footnote", "sources": [ { "title": "", "url": "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf&utm_source=chatgpt.com", "attribution": "Maplesoft" } ], "has_images": false } ], "search_result_groups": [ { "type": "search_result_group", "domain": "maplesoft.com", "entries": [ { "type": "search_result", "url": "https://www.maplesoft.com/support/help/errors/view.aspx?path=examples%2FRootOf&utm_source=chatgpt.com", "title": "", "snippet": "RootOf function - Maple Help For the best experience, we recommend viewing online help using Google Chrome or Mozilla Firefox. 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[Input: Search...] ONLINE HELP [Input: Search...] [INPUT] ALL PRODUCTS   [INPUT]...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 9 }, "pub_date": null, "attribution": "maplesoft.com" }, { "type": "search_result", "url": "https://www.maplesoft.com/support/help/errors/view.aspx?path=sum%2Fdetails&utm_source=chatgpt.com", "title": "", "snippet": "alpha - RootOf expression expr - expression, of type algebraic, not containing k Image Basic Information •  This help page contains complete information about the sum command. 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In[1]:=1 Image ✖ `` `https://wolfram.com/xid/0bufdu-bdurr3` Direct link to example Wolfram Language...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 22 }, "pub_date": null, "attribution": "wolfram.com" }, { "type": "search_result", "url": "https://reference.wolfram.com/language/ref/Cubics.html?view=all&utm_source=chatgpt.com", "title": "Cubics—Wolfram Documentation", "snippet": "* Quartics * ToRadicals * Reduce * Resolve * Eigenvalues * * Polynomial Equations CUBICS Copy to clipboard. ✖ `Cubics` Cubics Copy to clipboard. ✖ `Cubics` is an option for functions th...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 23 }, "pub_date": null, "attribution": "wolfram.com" } ] } ], "safe_urls": [ "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf", "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf%2Findexed", "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf%2Findexed&utm_source=chatgpt.com", "https://www.maplesoft.com/support/help/errors/view.aspx?path=RootOf&utm_source=chatgpt.com" ], "citations": [], "resolved_model_slug": "gpt-5-6-thinking", "code_blocks": { "0": { "id": "wem1ov", "edited": false, "previewable": false, "preview_language": null } }, "cot_version": "v5", "request_id": "973af564-bdf0-4abc-b7d1-cf7b5dad143f", "message_type": "next", "turn_exchange_id": "bf21d9b6-df1b-4f5c-86d0-6c2e2317282f", "working_turn_id": "bf21d9b6-df1b-4f5c-86d0-6c2e2317282f", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "13a37b8b-5d73-4513-8d12-d23991e2b871", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "13a37b8b-5d73-4513-8d12-d23991e2b871", "children": [ "3328ef82-2617-4c0a-a989-37bbc7bf3116" ] }, "3328ef82-2617-4c0a-a989-37bbc7bf3116": { "id": "3328ef82-2617-4c0a-a989-37bbc7bf3116", "message": { "id": "3328ef82-2617-4c0a-a989-37bbc7bf3116", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693883.167, "update_time": 1787693888.224467, "content": { "content_type": "text", "parts": [ "i guess i don't really know what we're talking about at this time. can you enter a mode where you keep replying in only half a paragraph or so?" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "4a8a95a8-4ed0-481a-969a-6bbd85ea1282", "message_source": null, "turn_exchange_id": "1200eb35-ea40-4963-a7f8-3954c0cd169c", "working_turn_id": "1200eb35-ea40-4963-a7f8-3954c0cd169c", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "421022f4-8b23-4505-8a33-7bd86736d788", "children": [ "ef2b2ac4-80ce-40b6-878b-2aacbc97f636" ] }, "ef2b2ac4-80ce-40b6-878b-2aacbc97f636": { "id": "ef2b2ac4-80ce-40b6-878b-2aacbc97f636", "message": { "id": "ef2b2ac4-80ce-40b6-878b-2aacbc97f636", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693885.019321, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 1, "reasoning_start_time": 1787693883.7743456, "reasoning_end_time": 1787693885.0080252, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "4a8a95a8-4ed0-481a-969a-6bbd85ea1282", "message_type": "next", "turn_exchange_id": "1200eb35-ea40-4963-a7f8-3954c0cd169c", "working_turn_id": "1200eb35-ea40-4963-a7f8-3954c0cd169c", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "c161123e-73b0-4afe-bb6e-46d14a5b892f", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "3328ef82-2617-4c0a-a989-37bbc7bf3116", "children": [ "412d74fa-8a3a-4d63-a2ea-54bb8bc1c134" ] }, "412d74fa-8a3a-4d63-a2ea-54bb8bc1c134": { "id": "412d74fa-8a3a-4d63-a2ea-54bb8bc1c134", "message": { "id": "412d74fa-8a3a-4d63-a2ea-54bb8bc1c134", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693886.043374, "update_time": 1787693887.911293, "content": { "content_type": "text", "parts": [ "Yes. I’ll keep replies to about half a paragraph for now, unless you ask for more detail." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "4a8a95a8-4ed0-481a-969a-6bbd85ea1282", "message_type": "next", "turn_exchange_id": "1200eb35-ea40-4963-a7f8-3954c0cd169c", "working_turn_id": "1200eb35-ea40-4963-a7f8-3954c0cd169c", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "77b27da7-909d-4436-9b2b-4412194aa193", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "ef2b2ac4-80ce-40b6-878b-2aacbc97f636", "children": [ "b690566e-7971-4d12-ab57-e853fcf73f70" ] }, "b690566e-7971-4d12-ab57-e853fcf73f70": { "id": "b690566e-7971-4d12-ab57-e853fcf73f70", "message": { "id": "b690566e-7971-4d12-ab57-e853fcf73f70", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787693914.751, "update_time": 1787693919.213693, "content": { "content_type": "text", "parts": [ "is an \"algebraic function\" what you meant when you said quintic's were expressible in terms of advanced special functions?" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "f1e348d8-61a1-4ea3-a13f-689d7219f04c", "message_source": null, "turn_exchange_id": "ec021bad-ea0d-49d0-980b-6f19f0053448", "working_turn_id": "ec021bad-ea0d-49d0-980b-6f19f0053448", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "412d74fa-8a3a-4d63-a2ea-54bb8bc1c134", "children": [ "218c2f75-2c1e-4cd0-b2b3-a93d31228b93" ] }, "218c2f75-2c1e-4cd0-b2b3-a93d31228b93": { "id": "218c2f75-2c1e-4cd0-b2b3-a93d31228b93", "message": { "id": "218c2f75-2c1e-4cd0-b2b3-a93d31228b93", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693916.812546, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 1, "reasoning_start_time": 1787693915.4490921, "reasoning_end_time": 1787693916.8076892, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "f1e348d8-61a1-4ea3-a13f-689d7219f04c", "message_type": "next", "turn_exchange_id": "ec021bad-ea0d-49d0-980b-6f19f0053448", "working_turn_id": "ec021bad-ea0d-49d0-980b-6f19f0053448", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "52a499e3-7913-4b94-bdee-55acdc535982", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "b690566e-7971-4d12-ab57-e853fcf73f70", "children": [ "72d8975f-6eba-4de3-9ef1-e4104ae39879" ] }, "72d8975f-6eba-4de3-9ef1-e4104ae39879": { "id": "72d8975f-6eba-4de3-9ef1-e4104ae39879", "message": { "id": "72d8975f-6eba-4de3-9ef1-e4104ae39879", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787693915.414023, "update_time": 1787693918.958173, "content": { "content_type": "text", "parts": [ "Not exactly. An **algebraic function** means “a function defined implicitly by a polynomial equation,” like \\(y^5+xy+1=0\\). The “advanced special functions” I mentioned are one possible explicit way to write some quintic branches, but an algebraic function can simply stay represented as “the chosen root of this polynomial” without converting it into hypergeometric/elliptic-style functions." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "f1e348d8-61a1-4ea3-a13f-689d7219f04c", "message_type": "next", "turn_exchange_id": "ec021bad-ea0d-49d0-980b-6f19f0053448", "working_turn_id": "ec021bad-ea0d-49d0-980b-6f19f0053448", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "218c2f75-2c1e-4cd0-b2b3-a93d31228b93", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "218c2f75-2c1e-4cd0-b2b3-a93d31228b93", "children": [ "d186c5ac-4bee-4772-b330-72e157f3f0c2" ] }, "d186c5ac-4bee-4772-b330-72e157f3f0c2": { "id": "d186c5ac-4bee-4772-b330-72e157f3f0c2", "message": { "id": "d186c5ac-4bee-4772-b330-72e157f3f0c2", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787694013.551, "update_time": 1787694024.428729, "content": { "content_type": "text", "parts": [ "is it reasonable to express roots of functions in terms of the advanced special functions, in an engineering context? does code ever do this?\ni was thinking it seemed kind of fun to optimize a new function, given machine learning's recent popularity and stuff, i had fun once inverting a function used in involute gears, i found a research paper on how to implement it in code" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, 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General quintics can be reduced to Bring–Jerrard form and expressed using the **Bring radical**, itself expressible with a generalized hypergeometric \\({}_4F_3\\); there are papers and CAS implementations doing exactly that, including Maple-based evaluation. citeturn133131search32turn133131academia33 In engineering, people usually just run a polynomial root solver because it’s simpler and often more stable, but if you repeatedly evaluate a particular parameterized family, implementing and optimizing a dedicated “Bring-radical-like” function could absolutely be a fun numerical-methods project—very analogous to your involute inversion work." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "content_references": [ { "matched_text": "citeturn133131search32turn133131academia33", "prefix": null, "start_idx": 289, "end_idx": 335, "safe_urls": [ "https://arxiv.org/abs/math/0005026", "https://arxiv.org/abs/math/0005026?utm_source=chatgpt.com", "https://en.wikipedia.org/wiki/Bring_radical", "https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com" ], "refs": [], "alt": "([Wikipedia](https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com))", "prompt_text": null, "type": "grouped_webpages", "items": [ { "title": "Bring radical", "url": "https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com", "attribution": "Wikipedia", "pub_date": null, "snippet": "", "attribution_segments": null, "supporting_websites": [ { "title": "On the Complete Solution to the Most General Fifth Degree Polynomial", "url": "https://arxiv.org/abs/math/0005026?utm_source=chatgpt.com", "pub_date": null, "snippet": "", "attribution": "arXiv" } ], "refs": [ { "turn_index": 0, "ref_type": "search", "ref_index": 32 }, { "turn_index": 0, "ref_type": "academia", "ref_index": 33 } ], "hue": null, "attributions": null } ], "fallback_items": null, "status": "done", "error": null, "style": null }, { "matched_text": " ", "prefix": null, "start_idx": 680, "end_idx": 681, "safe_urls": [], "refs": [], "alt": "", "prompt_text": null, "type": "sources_footnote", "sources": [ { "title": "Bring radical", "url": "https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com", "attribution": "Wikipedia" } ], "has_images": false } ], "search_result_groups": [ { "type": "search_result_group", "domain": "researchgate.net", "entries": [ { "type": "search_result", "url": "https://www.researchgate.net/publication/2095702_On_the_Complete_Solution_to_the_Most_General_Fifth_Degree_Polynomial?utm_source=chatgpt.com", "title": "(PDF) On the Complete Solution to the Most General Fifth Degree Polynomial", "snippet": "Article PDF Available ON THE COMPLETE SOLUTION TO THE MOST GENERAL FIFTH DEGREE POLYNOMIAL * June 2000 DOI:10.48550/arXiv.math/0005026 * Source * arXiv Authors: Richard Drociuk Image Download...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 0 }, "pub_date": 959817600, "attribution": "researchgate.net" }, { "type": "search_result", "url": "https://www.researchgate.net/publication/45878602_Physical_applications_of_a_new_method_of_solving_the_quintic_equation?utm_source=chatgpt.com", "title": "(PDF) Physical applications of a new method of solving the quintic equation", "snippet": "other choices are a 3=a 2=a 1= 0, etc.) . The roots of the trinomic equation: x 5−x−t= 0 (5) 2 can be expressed as generalized hypergeometric functions, 4 F 3. 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Ask Question Asked 12 years, 11 months ago Modified 4 years, 7 months ago Viewed 994 times [Input] 2 [Input]...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 8 }, "pub_date": 1376179200, "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://math.stackexchange.com/questions/5130931/solving-a-degree-six-equation-with-special-functions?utm_source=chatgpt.com", "title": "polynomials - Solving a degree six equation with special functions - Mathematics Stack Exchange", "snippet": "SOLVING A DEGREE SIX EQUATION WITH SPECIAL FUNCTIONS Ask Question Asked 4 months ago Modified 4 months ago Viewed 357 times [Input] Just like how fifth degree equations (quintics) can be reduce...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 9 }, "pub_date": null, "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://mathematica.stackexchange.com/questions/277526/how-to-find-roots-of-general-polynomials-as-functions-in-mathematica?noredirect=1&utm_source=chatgpt.com", "title": "differential equations - How to find roots of general polynomials as functions in Mathematica? - Mathematica Stack Exchange", "snippet": "* polynomials * difference-equations Improve this question Follow [Input] 2 ANSWERS Sorted by: [Select] [Input] 10 [Input] Outline: * Code * Discussion on polynomial roots that do not invo...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 11 }, "pub_date": 1671494400, "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://math.stackexchange.com/questions/1635950/is-there-a-formula-for-the-roots-of-a-quintic-equation/1636883?utm_source=chatgpt.com", "title": "polynomials - Is there a formula for the roots of a Quintic Equation? - Mathematics Stack Exchange", "snippet": "IS THERE A FORMULA FOR THE ROOTS OF A QUINTIC EQUATION? Ask Question Asked 10 years, 5 months ago Modified 3 months ago Viewed 18k times [Input] I can get my head around this so someone explain...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 14 }, "pub_date": 1454284800, "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://math.stackexchange.com/questions/1635950/is-there-a-formula-for-the-roots-of-a-quintic-equation?utm_source=chatgpt.com", "title": "polynomials - Is there a formula for the roots of a Quintic Equation? - Mathematics Stack Exchange", "snippet": "IS THERE A FORMULA FOR THE ROOTS OF A QUINTIC EQUATION? Ask Question Asked 10 years, 6 months ago Modified 4 months ago Viewed 18k times [Input] I can get my head around this so someone explain...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 17 }, "pub_date": 1454284800, "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://mathematica.stackexchange.com/questions/277526/how-to-find-roots-of-general-polynomials-as-functions-in-mathematica?utm_source=chatgpt.com", "title": "differential equations - How to find roots of general polynomials as functions in Mathematica? - Mathematica Stack Exchange", "snippet": "CODE : (explanation of code below) A solution in terms of hypergeometric functions for any z for the quintic : Note: takes a while like around 3 minutes and leads to `PossibleZeroQ` errors which mi...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 19 }, "pub_date": 1671494400, "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://math.stackexchange.com/questions/4716481/generalizing-radicals-to-solve-quintics-and-above?utm_source=chatgpt.com", "title": "polynomials - Generalizing Radicals to Solve Quintics and Above - Mathematics Stack Exchange", "snippet": "GENERALIZING RADICALS TO SOLVE QUINTICS AND ABOVE Famously, the roots of fourth degree polynomials and below can always be expressed in terms of radicals but those fifth degree and above cannot nece...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 20 }, "pub_date": 1686441600, "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://codegolf.stackexchange.com/questions/193376/calculate-the-ultraradical?utm_source=chatgpt.com", "title": "code golf - Calculate the Ultraradical - Code Golf Stack Exchange", "snippet": "CALCULATE THE ULTRARADICAL Ask Question Asked 6 years, 8 months ago Modified 1 month ago Viewed 4k times [Input] 30 [Input] WHAT IS THE ULTRARADICAL? The ultraradical, or the Bring radical,...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 24 }, "pub_date": 1569196800, "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://math.stackexchange.com/questions/2396228/computation-of-the-ultraradical?utm_source=chatgpt.com", "title": "numerical methods - Computation of the ultraradical - Mathematics Stack Exchange", "snippet": "COMPUTATION OF THE ULTRARADICAL Ask Question Asked 9 years ago Modified 9 years ago Viewed 441 times [Input] The ultraradical of a real number, also called the Bring radical, is the unique real...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 25 }, "pub_date": 1502841600, "attribution": "stackexchange.com" } ] }, { "type": "search_result_group", "domain": "handwiki.org", "entries": [ { "type": "search_result", "url": "https://handwiki.org/wiki/Bring%20radical?utm_source=chatgpt.com", "title": "Bring radical - HandWiki", "snippet": "SERIES REPRESENTATION A Taylor series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation {\\displaystyle {x}^{5}+x+a=0} can be...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 3 }, "pub_date": 1770899812, "attribution": "handwiki.org" }, { "type": "search_result", "url": "https://handwiki.org/wiki/Bring_radical?utm_source=chatgpt.com", "title": "Bring radical - HandWiki", "snippet": "BRING RADICAL From HandWiki [Input] Short description: Real root of the polynomial x^5+x+a Plot of the Bring radical for real argument In algebra, the Bring radical or ultraradical of a real num...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 4 }, "pub_date": 1770899812, "attribution": "handwiki.org" } ] }, { "type": "search_result_group", "domain": "grin.com", "entries": [ { "type": "search_result", "url": "https://www.grin.com/document/1714624?utm_source=chatgpt.com", "title": "The Bring Radical Solution for the Quintic Equation | Grin", "snippet": "THE BRING RADICAL SOLUTION FOR THE QUINTIC EQUATION Image: Title: The Bring Radical Solution for the Quintic Equation Pre-University Paper , 2026 , 5 Pages , Grade: 18.5 Autor:in: Fazal Rehman (Au...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 5 }, "pub_date": null, "attribution": "grin.com" } ] }, { "type": "search_result_group", "domain": "springer.com", "entries": [ { "type": "search_result", "url": "https://link.springer.com/article/10.1007/s11075-026-02436-2?utm_source=chatgpt.com", "title": "Efficient and accurate evaluation of the Gauss hypergeometric function | Numerical Algorithms | Springer Nature Link", "snippet": "EFFICIENT AND ACCURATE EVALUATION OF THE GAUSS HYPERGEOMETRIC FUNCTION * Research * Open access * Published: 10 July 2026 * (2026) * Cite this article You have full access to this open access arti...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 6 }, "pub_date": 1783641600, "attribution": "springer.com" }, { "type": "search_result", "url": "https://link.springer.com/article/10.1007/s11075-023-01515-y?utm_source=chatgpt.com", "title": "Computation of the confluent hypergeometric function U(a,b,x) and its derivative for positive arguments | Numerical Algorithms | Springer Nature Link", "snippet": "COMPUTATION OF THE CONFLUENT HYPERGEOMETRIC FUNCTION U(A,B,X) AND ITS DERIVATIVE FOR POSITIVE ARGUMENTS * Research * Open access * Published: 22 March 2023 * Volume 94, pages 669–679 (2023) * Cite...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 28 }, "pub_date": 1679443200, "attribution": "springer.com" }, { "type": "search_result", "url": "https://link.springer.com/article/10.1007/s44007-022-00027-w?utm_source=chatgpt.com", "title": "Completely Solving the Quintic by Iteration | La Matematica | Springer Nature Link", "snippet": "COMPLETELY SOLVING THE QUINTIC BY ITERATION * Original Research Article * Open access * Published: 27 May 2022 * Volume 1, pages 829–847 (2022) * Cite this article You have full access to this ope...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 31 }, "pub_date": 1653609600, "attribution": "springer.com" } ] }, { "type": "search_result_group", "domain": "jfeldbrugge.github.io", "entries": [ { "type": "search_result", "url": "https://jfeldbrugge.github.io/Galois-Theory/?utm_source=chatgpt.com", "title": "Topological Galois Theory", "snippet": "NOTES ON TOPOLOGICAL GALOIS THEORY This topological proof is more powerful than conventional Galois theory in that it shows that the polynomial equations of order \\(n \\geq 5\\) not only fail to have a...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 12 }, "pub_date": null, "attribution": "jfeldbrugge.github.io" } ] }, { "type": "search_result_group", "domain": "mathoverflow.net", "entries": [ { "type": "search_result", "url": "https://mathoverflow.net/questions/61409/using-higher-order-bring-radicals-to-solve-arbitrary-polynomials/61558?utm_source=chatgpt.com", "title": "galois theory - Using higher-order Bring radicals to solve arbitrary polynomials - MathOverflow", "snippet": "9. # Using higher-order Bring radicals to solve arbitrary polynomials Ask Question Asked 15 years, 4 months ago Modified 4 months ago Viewed 10k times [Input] 30 [Input] It is well known that...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 13 }, "pub_date": 1302566400, "attribution": "mathoverflow.net" } ] }, { "type": "search_result_group", "domain": "note.com", "entries": [ { "type": "search_result", "url": "https://note.com/kounkt/n/n0731b5d824ed?hl=en&utm_source=chatgpt.com", "title": "[Formula for the roots of a quintic equation] Can someone knowledgeable please explain?｜チエロ｜AIに強い、経営者の相談役", "snippet": "Image: 見出し画像 [FORMULA FOR THE ROOTS OF A QUINTIC EQUATION] CAN SOMEONE KNOWLEDGEABLE PLEASE EXPLAIN? チエロ｜AIに強い、経営者の相談役 2025年8月28日 18:48 [Formula for the roots of a quintic equation] I'm no longer...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 16 }, "pub_date": 1756339200, "attribution": "note.com" } ] }, { "type": "search_result_group", "domain": "colostate.edu", "entries": [ { "type": "search_result", "url": "https://www.cs.colostate.edu/~fsieker/QuadEqn.html?utm_source=chatgpt.com", "title": "Roots of a Quadratic Equation", "snippet": "THE `MAIN()` METHOD In your `main()` method declare three `double` variables. You may give them any names you want (`foo`, `bar`, `baz`). BUT, as we are working on quadratic equations, it is probably...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 18 }, "pub_date": 1572825600, "attribution": "colostate.edu" } ] }, { "type": "search_result_group", "domain": "fredrik-j.blogspot.com", "entries": [ { "type": "search_result", "url": "https://fredrik-j.blogspot.com/2010/07/?utm_source=chatgpt.com", "title": "Fredrik Johansson: July 2010", "snippet": "APPLICATION: ROOTS OF POLYNOMIALS A neat application is to compute the roots of quintic polynomials. Such roots can be expressed in closed form using hypergeometric functions, as outlined in the Wiki...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 21 }, "pub_date": null, "attribution": "fredrik-j.blogspot.com" } ] }, { "type": "search_result_group", "domain": "pisrt.org", "entries": [ { "type": "search_result", "url": "https://pisrt.org/psr-press/journals/oms/01-vol-5-2021-issue-1/trinomial-equation-the-hypergeometric-way/?utm_source=chatgpt.com", "title": "Trinomial equation: the Hypergeometric way - PISRT", "snippet": "\\begin{equation} \\begin{array}{ll} \\displaystyle \\frac{p'(y)}{p(y)} & = \\displaystyle \\frac{1}{y} \\, \\, \\left( \\displaystyle \\frac{1}{\\displaystyle{1-\\displaystyle \\frac{\\alpha_1}{y} } } + \\displayst...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 22 }, "pub_date": 1622073600, "attribution": "pisrt.org" } ] }, { "type": "search_result_group", "domain": "sciencedirect.com", "entries": [ { "type": "search_result", "url": "https://www.sciencedirect.com/science/article/pii/S0723086913000571?utm_source=chatgpt.com", "title": "On Klein’s icosahedral solution of the quintic - ScienceDirect", "snippet": "EXPOSITIONES MATHEMATICAE Date: 2014 Pages: 99-120 Volume: Volume 32, Issue 2 PUBLISHED BY: ELSEVIER ON KLEIN’S ICOSAHEDRAL SOLUTION OF THE QUINTIC Author links open overlay panel Oliver Nash...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 23 }, "pub_date": null, "attribution": "sciencedirect.com" }, { "type": "search_result", "url": "https://www.sciencedirect.com/science/article/abs/pii/S0010465513001136?utm_source=chatgpt.com", "title": "NumExp: Numerical epsilon expansion of hypergeometric functions - ScienceDirect", "snippet": "COMPUTER PHYSICS COMMUNICATIONS Volume 184, Issue 8, August 2013, Pages 1973-1980 NUMEXP: NUMERICAL EPSILON EXPANSION OF HYPERGEOMETRIC FUNCTIONS☆ https://doi.org/10.1016/j.cpc.2013.03.016Get righ...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 26 }, "pub_date": null, "attribution": "sciencedirect.com" } ] }, { "type": "search_result_group", "domain": "kci.go.kr", "entries": [ { "type": "search_result", "url": "https://www.kci.go.kr/kciportal/landing/article.kci?arti_id=ART001654488&utm_source=chatgpt.com", "title": "General solutions to certain real degree equations and their applications", "snippet": "GENERAL SOLUTIONS TO CERTAIN REAL DEGREE EQUATIONS AND THEIR APPLICATIONS Advanced Studies in Contemporary Mathematics 약어 : ASCM 2012 vol.22, no.2, pp.329 - 345 발행기관 : 장전수학회 연구분야 : 수학 Copyright © 장전...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 27 }, "pub_date": null, "attribution": "kci.go.kr" } ] }, { "type": "search_result_group", "domain": "wolfram.com", "entries": [ { "type": "search_result", "url": "https://mathworld.wolfram.com/QuinticEquation.html?utm_source=chatgpt.com", "title": "Quintic Equation -- from Wolfram MathWorld", "snippet": "then the roots are Image: t_1 | Image: = | Image: -rho_4F_3(1/5,2/5,3/5,4/5;1/2,3/4,5/4;(3125)/(256)rho^4) | (36) Image: t_2 | Image: = | Image: -F_1(rho)+1/4rhoF_2(rho)+5/(32)rho^2F_3(rho)+5/(3...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 29 }, "pub_date": 1787616000, "attribution": "wolfram.com" }, { "type": "search_result", "url": "https://reference.wolfram.com/language/ref/HypergeometricPFQ.html.en?utm_source=chatgpt.com", "title": "HypergeometricPFQ: Generalized hypergeometric function PFQ—Wolfram Documentation", "snippet": "APPLICATIONS  (7) Solve a differential equation of hypergeometric type: Copy to clipboard. In[1]:=1 Image ✖ `` `https://wolfram.com/xid/0pu71249fk-c82ax` Direct link to example Wolfram Langua...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 30 }, "pub_date": null, "attribution": "wolfram.com" } ] } ], "safe_urls": [ "https://arxiv.org/abs/math/0005026", "https://arxiv.org/abs/math/0005026?utm_source=chatgpt.com", "https://en.wikipedia.org/wiki/Bring_radical", "https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com" ], "citations": [], "resolved_model_slug": "gpt-5-6-thinking", "cot_version": "v5", "request_id": "72bd8740-abfe-4524-bfe4-ca33b4f1d2c6", "message_type": "next", "turn_exchange_id": "3cf3940d-273c-4b45-b64e-7bdfc8614791", "working_turn_id": "3cf3940d-273c-4b45-b64e-7bdfc8614791", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "a7145738-29fe-47a5-9a33-bdd6745b2632", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "a7145738-29fe-47a5-9a33-bdd6745b2632", "children": [ "f3865954-25e8-414d-945d-d1bbacac87a8" ] }, "f3865954-25e8-414d-945d-d1bbacac87a8": { "id": "f3865954-25e8-414d-945d-d1bbacac87a8", "message": { "id": "f3865954-25e8-414d-945d-d1bbacac87a8", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787694105.407, "update_time": 1787694110.29551, "content": { "content_type": "text", "parts": [ "i was considering writing fast code for repeatedly evaluating a parameterized function. will algebraic functions help with that much?" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "aba74b2c-c582-4f12-9674-5e91ff6718ec", "message_source": null, "turn_exchange_id": "35ed850b-7797-4c7d-8d05-9fbd71e44697", "working_turn_id": "35ed850b-7797-4c7d-8d05-9fbd71e44697", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "043759f3-e73c-4b88-a417-decd286ec8ee", "children": [ "e06ac787-818b-45b0-b4ab-98bcee667e63" ] }, "e06ac787-818b-45b0-b4ab-98bcee667e63": { "id": "e06ac787-818b-45b0-b4ab-98bcee667e63", "message": { "id": "e06ac787-818b-45b0-b4ab-98bcee667e63", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787694106.785914, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a second" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 0, "reasoning_start_time": 1787694106.0844626, "reasoning_end_time": 1787694106.7811537, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "aba74b2c-c582-4f12-9674-5e91ff6718ec", "message_type": "next", "turn_exchange_id": "35ed850b-7797-4c7d-8d05-9fbd71e44697", "working_turn_id": "35ed850b-7797-4c7d-8d05-9fbd71e44697", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "29aff0de-9abc-4c4c-8de5-76151c96a10b", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "f3865954-25e8-414d-945d-d1bbacac87a8", "children": [ "095ae5a9-5ee8-4a05-b225-cfa02327db5b" ] }, "095ae5a9-5ee8-4a05-b225-cfa02327db5b": { "id": "095ae5a9-5ee8-4a05-b225-cfa02327db5b", "message": { "id": "095ae5a9-5ee8-4a05-b225-cfa02327db5b", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787694106.054898, "update_time": 1787694109.478899, "content": { "content_type": "text", "parts": [ "Yes, but mostly as a **way to describe the problem**, not as the speed trick itself. If your family is defined by \\(P(x,y)=0\\), the algebraic-function viewpoint helps you derive branch behavior, derivatives, singularities, asymptotics, and good initial guesses; then the fast implementation usually comes from exploiting that structure with Newton/Halley, rational approximations, lookup/interpolation, or a specialized closed form for that family." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "aba74b2c-c582-4f12-9674-5e91ff6718ec", "message_type": "next", "turn_exchange_id": "35ed850b-7797-4c7d-8d05-9fbd71e44697", "working_turn_id": "35ed850b-7797-4c7d-8d05-9fbd71e44697", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "e06ac787-818b-45b0-b4ab-98bcee667e63", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "e06ac787-818b-45b0-b4ab-98bcee667e63", "children": [ "786b5a42-7835-4869-9ac2-647d68e43a25" ] }, "786b5a42-7835-4869-9ac2-647d68e43a25": { "id": "786b5a42-7835-4869-9ac2-647d68e43a25", "message": { "id": "786b5a42-7835-4869-9ac2-647d68e43a25", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787694181.672, "update_time": 1787694186.211867, "content": { "content_type": "text", "parts": [ "ok now i'm a little confused distracted -- that's a lot of common options! each function (and use case) it kind of takes effort to figure out which of those options work best -- isn't that space basically the same space as implementing the Bring radical, though? you're trying to fit the same curve, just in a less reusable manner?" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "f2e5f3fe-4b0f-4417-859e-25b2db303f60", "message_source": null, "turn_exchange_id": "74677557-6019-4905-9ffa-d5aee2f29113", "working_turn_id": "74677557-6019-4905-9ffa-d5aee2f29113", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "095ae5a9-5ee8-4a05-b225-cfa02327db5b", "children": [ "b14143bb-36b0-4abc-a023-c7f0558666e2" ] }, "b14143bb-36b0-4abc-a023-c7f0558666e2": { "id": "b14143bb-36b0-4abc-a023-c7f0558666e2", "message": { "id": "b14143bb-36b0-4abc-a023-c7f0558666e2", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787694183.27473, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a second" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 0, "reasoning_start_time": 1787694182.2884536, "reasoning_end_time": 1787694183.2708752, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "f2e5f3fe-4b0f-4417-859e-25b2db303f60", "message_type": "next", "turn_exchange_id": "74677557-6019-4905-9ffa-d5aee2f29113", "working_turn_id": "74677557-6019-4905-9ffa-d5aee2f29113", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "f9cb2751-1f06-4d06-8180-4c45a23deb85", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "786b5a42-7835-4869-9ac2-647d68e43a25", "children": [ "cdd89bb8-c6ae-46e9-8fc5-1b7c505c67cc" ] }, "cdd89bb8-c6ae-46e9-8fc5-1b7c505c67cc": { "id": "cdd89bb8-c6ae-46e9-8fc5-1b7c505c67cc", "message": { "id": "cdd89bb8-c6ae-46e9-8fc5-1b7c505c67cc", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787694182.263219, "update_time": 1787694185.959895, "content": { "content_type": "text", "parts": [ "Yes — that’s a good way to see it. Both are really about building a fast evaluator for the same multivalued algebraic curve; the difference is that a **Bring-radical implementation tries to package the hard work into one reusable canonical special function**, while Newton/rational-fit/etc. usually exploit one particular parameterization or operating range. So the Bring-radical route is more reusable and mathematically elegant, but often harder to make uniformly fast and numerically tame across all branches." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "f2e5f3fe-4b0f-4417-859e-25b2db303f60", "message_type": "next", "turn_exchange_id": "74677557-6019-4905-9ffa-d5aee2f29113", "working_turn_id": "74677557-6019-4905-9ffa-d5aee2f29113", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "b14143bb-36b0-4abc-a023-c7f0558666e2", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "b14143bb-36b0-4abc-a023-c7f0558666e2", "children": [ "00320477-63ca-4fd7-9f18-29f709341e86" ] }, "00320477-63ca-4fd7-9f18-29f709341e86": { "id": "00320477-63ca-4fd7-9f18-29f709341e86", "message": { "id": "00320477-63ca-4fd7-9f18-29f709341e86", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787694321.261, "update_time": 1787694340.756675, "content": { "content_type": "text", "parts": [ "now, _probably i am crazy here because a quintic or sextic likely has a zillion roots_ that i would fumble everywhere like trying to juggle eggs, is that true?\nbut do any libraries i might use coding in python, c++, or maybe rust, go, nodejs, julia, etc, or even something like opengl, implement the bring radical or arrange polynomials in terms of it?" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "a780b38c-64ad-43ec-b5be-e4625d075f13", "message_source": null, "turn_exchange_id": "08f7257d-a83e-4030-8610-99620558ce90", "working_turn_id": "08f7257d-a83e-4030-8610-99620558ce90", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "cdd89bb8-c6ae-46e9-8fc5-1b7c505c67cc", "children": [ "b7c7833a-fc6b-4d17-a8fd-0d70bf3f8b63" ] }, "b7c7833a-fc6b-4d17-a8fd-0d70bf3f8b63": { "id": "b7c7833a-fc6b-4d17-a8fd-0d70bf3f8b63", "message": { "id": "b7c7833a-fc6b-4d17-a8fd-0d70bf3f8b63", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787694322.011954, "update_time": null, "content": { "content_type": "text", "parts": [ "A quintic has exactly **5 complex roots** and a sextic **6** (counting multiplicity), so the juggling problem is real but pleasantly bounded. I’ll check which current libraries actually expose Bring-radical or quintic reduction machinery rather than guessing." ] }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "citations": [], "resolved_model_slug": "gpt-5-6-thinking", "is_thinking_preamble_message": true, "hide_inline_actions": true, "disable_turn_actions": true, "chime_version": 1, "classifier_response": "default", "reasoning_start_time": 1787694322.0498102, "cot_version": "v5", "request_id": "a780b38c-64ad-43ec-b5be-e4625d075f13", "message_type": "next", "turn_exchange_id": "08f7257d-a83e-4030-8610-99620558ce90", "working_turn_id": "08f7257d-a83e-4030-8610-99620558ce90", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "c9fb7d22-60f2-495c-8ebb-9b8b8eb259ad", "model_switcher_deny": [] }, "recipient": "all", "channel": "commentary" }, "parent": "00320477-63ca-4fd7-9f18-29f709341e86", "children": [ 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And intriguingly, I **couldn’t find a mainstream programming library exposing `bring_radical()` or “reduce arbitrary quintic to Bring form” as a normal API**. 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The root is x1=−a+a5−5a9+35a13+… , and ...", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 4 }, "pub_date": null, "attribution": "math.stackexchange.com" } ] }, { "type": "search_result_group", "domain": "inspirehep.net", "entries": [ { "type": "search_result", "url": "https://inspirehep.net/files/8301a2a54e49e5ceddf63b0545ef211b", "title": "THE BRING-JERRARD QUINTIC EQUATION, ITS ...", "snippet": "In this research the Bring-Jerrard quintic polynomial equation is investigated for a formula. Firstly, an explanation given as to why finding a formula and ...Read more", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 6 }, "pub_date": 1316506511, "attribution": "inspirehep.net" } ] }, { "type": "search_result_group", "domain": "www.youtube.com", "entries": [ { "type": "search_result", "url": "https://www.youtube.com/watch?v=SFXQ_920d4s", "title": "An Overview of the Exact Solution to the General Quintic...and ...", "snippet": "... Bring radical: https://en.wikipedia.org/wiki/Bring_radical Generalized hypergeometric function: https://en.wikipedia.org/wiki ...", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 7 }, "pub_date": null, "attribution": "www.youtube.com" } ] }, { "type": "search_result_group", "domain": "www.semanticscholar.org", "entries": [ { "type": "search_result", "url": "https://www.semanticscholar.org/paper/b8b319e1626b3381c96d65ce960d125858829a6e", "title": "An algorithm for calculating the roots of a general quintic ...", "snippet": "THE BRING-JERRARD QUINTIC EQUATION, ITS SOLUTIONS AND A FORMULA FOR THE UNIVERSAL GRAVITATIONAL CONSTANT · Edward Thabo Motlotle. Physics. 2011. In this ...", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 8 }, "pub_date": null, "attribution": "www.semanticscholar.org" }, { "type": "search_result", "url": "https://www.semanticscholar.org/topic/Bring-radical/783322", "title": "Bring radical", "snippet": "In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial The Bring radical of a complex number a is either ...", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 16 }, "pub_date": null, "attribution": "www.semanticscholar.org" } ] }, { "type": "search_result_group", "domain": "arxiv.org", "entries": [ { "type": "search_result", "url": "https://arxiv.org/pdf/2101.05610", "title": "Accurate algebraic formula for the quintic & Solution by ...", "snippet": "by A Missa · 2021 — Recall that the Bring radical of a com- plex number a is any of the five roots of: x5 + x + a = 0. More importantly, quintic equations can ...Read more", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 9 }, "pub_date": 1610745368, "attribution": "arxiv.org" }, { "type": "search_result", "url": "https://arxiv.org/abs/1711.09253", "title": "Erland Samuel Bring's \"Transformation of Algebraic Equations\"", "snippet": "", "ref_id": { "turn_index": 117049, "ref_type": "academia", "ref_index": 33 }, "pub_date": 1511624994, "attribution": "arxiv.org" }, { "type": "search_result", "url": "https://arxiv.org/abs/2112.08586", "title": "Removal of 5 Terms from a Degree 21 Polynomial", "snippet": "", "ref_id": { "turn_index": 117049, "ref_type": "academia", "ref_index": 34 }, "pub_date": 1639624584, "attribution": "arxiv.org" } ] }, { "type": "search_result_group", "domain": "www.kulturkaufhaus.de", "entries": [ { "type": "search_result", "url": "https://www.kulturkaufhaus.de/en/detail/ISBN-2244074609495/Rehman-Fazal/The-Bring-Radical-Solution-for-the-Quintic-Equation", "title": "The Bring Radical Solution for the Quintic Equation", "snippet": "We consider the reduced trinomial quintic x^5 + p x + q = 0and express its solution in terms of the Bring radical, defined as the principal root oft^5 + t + u = ...", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 10 }, "pub_date": null, "attribution": "www.kulturkaufhaus.de" } ] }, { "type": "search_result_group", "domain": "www.academia.edu", "entries": [ { "type": "search_result", "url": "https://www.academia.edu/99441655/The_Bring_Jerrard_Quintic_Equation_its_Algebraic_Solution_byConversion_to_Solvable_Factorized_Form", "title": "The Bring-Jerrard Quintic Equation, its Algebraic Solution ...", "snippet": "The Bring-Jerrard quintic equation can be expressed in solvable factorized form. This work establishes that quintic equations are reducible to lower degree ...Read more", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 11 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/GcLZ1M_rqoVgqoMEFJnTUXmOeIcaD14QmaQY58t3kLT6G7S2JKMTr9_v22s0gBRt-IqR44nPdSh6P_SNZ803vFDBab7IHEA8ZJRk2mmoKFeYyVaf1QN_HLxVi7Fnj10-", "attribution": "www.academia.edu" }, { "type": "search_result", "url": "https://www.academia.edu/35506642/A_new_method_for_obtaining_radical_solution_of_higher_degree_polynomial_equations", "title": "A new method for obtaining radical solution of higher ...", "snippet": "... paper entitled “A new method for obtaining radical solution ... Many other characterizations of the Bring radical were developed by other mathematicians.Read more", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 13 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/wYTAsEBHIU0lhkwpByy4lypzCT-vrGx2048c3dZXtSld2KYwWA_Jtulkmh8zFvdPE7b56DsvpXw9aVGZqa4ZjdP6DqjlHGrW0qlEd7yl5kcMNWt1UWzrhxpKkvcECQOE", "attribution": "www.academia.edu" }, { "type": "search_result", "url": "https://www.academia.edu/48809842/Solution_of_Polynomial_Equations_with_Nested_Radicals", "title": "(PDF) Solution of Polynomial Equations with Nested Radicals", "snippet": "The paper defines a generalized Bring radical function as a hypergeometric function, extending the classical solutions to broader polynomial forms. What is the ...Read more", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 18 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/xD-Iu557yUnzmNUThHgH7S66tnw3NsjqAWiuZoBkRKAjF9_M95t3DuZPNXFSDeWMy03Ioi33Q-j1lxMSIyp1VApy2RpdSxUEgweLey913JoMrz1F248ZddzINQM_9XU9", "attribution": "www.academia.edu" } ] }, { "type": "search_result_group", "domain": "handwiki.org", "entries": [ { "type": "search_result", "url": "https://handwiki.org/wiki/Bring_radical", "title": "Bring radical", "snippet": "In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial x5+x+a.", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 12 }, "pub_date": 1770899812, "attribution": "handwiki.org" } ] }, { "type": "search_result_group", "domain": "www.m-hikari.com", "entries": [ { "type": "search_result", "url": "https://www.m-hikari.com/ija/ija-password-2009/ija-password9-12-2009/davisIJA9-12-2009.pdf", "title": "A Finite Algorithm for the Solution to an Algebraic Equation", "snippet": "by S Davis · 2009 · Cited by 4 — This equation can be further brought to Bring-Jerrard form [6][21] [22] x. 5. + b1x + b0 = 0. (2.4) which can be solved through Hermite polynomials. The roots ...Read more", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 14 }, "pub_date": 1253167873, "attribution": "www.m-hikari.com" } ] }, { "type": "search_result_group", "domain": "www.researchgate.net", "entries": [ { "type": "search_result", "url": "https://www.researchgate.net/publication/356571083_Solving_the_Quintic_A_New_Approach_to_Bring%27s_Transformation", "title": "Solving the Quintic: A New Approach to Bring's ...", "snippet": "May 25, 2026 — Bring, by using a 4th degree Tschirnhaus transformation, it was possible to transform the Bring-Jerrard normal quintic (BJQ) equation into a ...Read more", "ref_id": { "turn_index": 117049, "ref_type": "search", "ref_index": 17 }, "pub_date": null, "attribution": "www.researchgate.net" } ] } ], "debug_sonic_thread_id": "thread_6a8e0d6b42388328b5c4e37881ecdcc4", "hide_inline_actions": true, "disable_turn_actions": true, "chime_version": 1, "cot_version": "v5", "request_id": 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For a serious fast/accurate numerical implementation, **FLINT/Arb** may ultimately be the nicer home because it already has arbitrary-precision complex arithmetic, rigorous error bounds, hypergeometric functions, and polynomial machinery. citeturn624412search5turn624412search8 And yes, there is published algorithmic work: e.g. Missa (2021) gives an iterative algorithm and explicitly discusses computing Bring radicals, while classical hypergeometric formulas provide another implementation route. citeturn117049search9turn117049search0\n\n" ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "content_references": [ { "matched_text": "citeturn624412search5turn624412search8", "prefix": null, "start_idx": 492, "end_idx": 534, "safe_urls": [ "https://flintlib.org/doc/", "https://flintlib.org/doc/?utm_source=chatgpt.com", "https://flintlib.org/doc/overview.html", "https://flintlib.org/doc/overview.html?utm_source=chatgpt.com" ], "refs": [], "alt": "([Flint Library](https://flintlib.org/doc/?utm_source=chatgpt.com))", "prompt_text": null, "type": "grouped_webpages", "items": [ { "title": "FLINT: Fast Library for Number Theory — FLINT 3.7.0-dev documentation", "url": "https://flintlib.org/doc/?utm_source=chatgpt.com", "attribution": "Flint Library", "pub_date": null, "snippet": "", "attribution_segments": null, "supporting_websites": [ { "title": "Feature overview — FLINT 3.7.0-dev documentation", "url": "https://flintlib.org/doc/overview.html?utm_source=chatgpt.com", "pub_date": null, "snippet": "", "attribution": "Flint Library" } ], "refs": [ { "turn_index": 0, "ref_type": "search", "ref_index": 5 }, { "turn_index": 0, "ref_type": "search", "ref_index": 8 } ], "hue": null, "attributions": null } ], "fallback_items": null, "status": "done", "error": null, "style": null }, { "matched_text": "citeturn117049search9turn117049search0", "prefix": null, "start_idx": 757, "end_idx": 799, "safe_urls": [ "https://arxiv.org/pdf/2101.05610", "https://arxiv.org/pdf/2101.05610?utm_source=chatgpt.com", "https://en.wikipedia.org/wiki/Bring_radical", "https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com" ], "refs": [], "alt": "([arxiv.org](https://arxiv.org/pdf/2101.05610?utm_source=chatgpt.com))", "prompt_text": null, "type": "grouped_webpages", "items": [ { "title": "Accurate algebraic formula for the quintic & Solution by ...", "url": "https://arxiv.org/pdf/2101.05610?utm_source=chatgpt.com", "attribution": "arxiv.org", "pub_date": 1610745368, "snippet": "by A Missa · 2021 — Recall that the Bring radical of a com- plex number a is any of the five roots of: x5 + x + a = 0. More importantly, quintic equations can ...Read more", "attribution_segments": null, "supporting_websites": [ { "title": "Bring radical", "url": "https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com", "pub_date": null, "snippet": "", "attribution": "en.wikipedia.org" } ], "refs": [ { "turn_index": 1, "ref_type": "search", "ref_index": 9 }, { "turn_index": 1, "ref_type": "search", "ref_index": 0 } ], "hue": null, "attributions": null } ], "fallback_items": null, "status": "done", "error": null, "style": null }, { "matched_text": " ", "prefix": null, "start_idx": 801, "end_idx": 802, "safe_urls": [], "refs": [], "alt": "", "prompt_text": null, "type": "sources_footnote", "sources": [ { "title": "FLINT: Fast Library for Number Theory — FLINT 3.7.0-dev documentation", "url": "https://flintlib.org/doc/?utm_source=chatgpt.com", "attribution": "Flint Library" }, { "title": "Accurate algebraic formula for the quintic & Solution by ...", "url": "https://arxiv.org/pdf/2101.05610?utm_source=chatgpt.com", "attribution": "arxiv.org" } ], "has_images": false } ], "search_result_groups": [ { "type": "search_result_group", "domain": "flintlib.org", "entries": [ { "type": "search_result", "url": "https://flintlib.org/doc/examples_arb.html?utm_source=chatgpt.com", "title": "Arb example programs — FLINT 3.7.0-dev documentation", "snippet": "POLY_ROOTS.C¶ This program finds the complex roots of an integer polynomial by calling `arb_fmpz_poly_complex_roots()`, which in turn calls `acb_poly_find_roots()` with increasing precision until the...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 0 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/arb_fmpz_poly.html?utm_source=chatgpt.com", "title": "arb_fmpz_poly.h – extra methods for integer polynomials — FLINT 3.7.0-dev documentation", "snippet": "ARB_FMPZ_POLY.H – EXTRA METHODS FOR INTEGER POLYNOMIALS¶ This module provides methods for FLINT polynomials with integer and rational coefficients (`fmpz_poly_t`) and (`fmpq_poly_t`) requiring use of...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 1 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/gr_poly.html?utm_source=chatgpt.com", "title": "gr_poly.h – dense univariate polynomials over generic rings — FLINT 3.7.0-dev documentation", "snippet": "GR_POLY.H – DENSE UNIVARIATE POLYNOMIALS OVER GENERIC RINGS¶ A `gr_poly_t` represents a univariate polynomial \\(f \\in R[X]\\) implemented as a dense array of coefficients in a generic ring R. 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Arb differs from earlier implementations in technical aspects of the implementation, which makes certain computations more effic...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 8 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/arb.html?utm_source=chatgpt.com", "title": "arb.h – real numbers — FLINT 3.7.0-dev documentation", "snippet": "POWERS AND ROOTS¶ void arb_sqrt(arb_t z, const arb_t x, slong prec)¶ void arb_sqrt_arf(arb_t z, const arf_t x, slong prec)¶ void arb_sqrt_fmpz(arb_t z, const fmpz_t x, slong prec)¶ void arb_sqr...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 9 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/using.html?utm_source=chatgpt.com", "title": "Using ball arithmetic — FLINT 3.7.0-dev documentation", "snippet": "POLYNOMIAL TIME GUARANTEE¶ Arb provides a soft guarantee that the time used to evaluate a ball function will depend polynomially on prec and the bit size of the input, uniformly regardless of the num...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 10 }, "pub_date": null, "attribution": "flintlib.org" }, { "type": "search_result", "url": "https://flintlib.org/doc/fq.html?utm_source=chatgpt.com", "title": "fq.h – finite fields — FLINT 3.7.0-dev documentation", "snippet": "FQ.H – FINITE FIELDS¶ We represent an element of the finite field \\(\\mathbf{F}_{p^n} \\cong \\mathbf{F}_p[X]/(f(X))\\), where \\(f(X) \\in \\mathbf{F}_p[X]\\) is a monic, irreducible polynomial of degree \\(...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 11 }, "pub_date": null, "attribution": "flintlib.org" } ] }, { "type": "search_result_group", "domain": "wikipedia.org", "entries": [ { "type": "search_result", "url": "https://en.wikipedia.org/wiki/Bring_radical?utm_source=chatgpt.com", "title": "Bring radical", "snippet": "In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial x 5 + x + a . {\\displaystyle x^{5}+x+a.} ...Read more", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 0 }, "pub_date": null, "attribution": "wikipedia.org" } ] }, { "type": "search_result_group", "domain": "reddit.com", "entries": [ { "type": "search_result", "url": "https://www.reddit.com/r/math/comments/ujd4vn/galois_theory_the_bring_radical_and_cute_methods/?utm_source=chatgpt.com", "title": "Galois Theory, the Bring Radical and cute methods to solve ...", "snippet": "We say that these polynomials are not solvable. We can solve quintics using the Bring Radical. Lots of unsolvable quintics are in Bring-Jerrard ...Read more", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 1 }, "pub_date": null, "attribution": "reddit.com" }, { "type": "search_result", "url": "https://www.reddit.com/r/math/comments/8797g1/analytic_solution_to_polynomials_of_any_degree/?utm_source=chatgpt.com", "title": "Analytic solution to polynomials of any degree : r/math", "snippet": "You can't solve the quintic with just nth root operations, but if you add the Bring radical (which solves the equation x5 + x + y = 0), then ...Read more", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 15 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/SIajsTisGR-eQ77oKLo2fH-fPkq0e4Uv_pSNWLIT44I1rwqlZl7OkylLYTgdATZCb_YztZRU6RhCGw7dq-xJ7A", "thumbnail_source": "og_image", "attribution": "reddit.com" } ] }, { "type": "search_result_group", "domain": "mathoverflow.net", "entries": [ { "type": "search_result", "url": "https://mathoverflow.net/questions/61409/using-higher-order-bring-radicals-to-solve-arbitrary-polynomials?utm_source=chatgpt.com", "title": "Using higher-order Bring radicals to solve arbitrary ...", "snippet": "Apr 12, 2011 — It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula ...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 2 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/4tqmNxNDR_tea3-tcCKRgHeJnpkgGSdfMvT8rnFozEggbixode98G6ree2i9LGNTXqyw3u9UorpDHY8eR-EYqeF78bfTJhbsU0HxspLLeKCPCF_LjycjEsWkFcrNWV5r9i87_OT4XzAKxAPVBALuQQ", "thumbnail_source": "og_image", "attribution": "mathoverflow.net" } ] }, { "type": "search_result_group", "domain": "stackexchange.com", "entries": [ { "type": "search_result", "url": "https://mathematica.stackexchange.com/questions/134349/converting-a-quintic-to-bring-jerrard-form?utm_source=chatgpt.com", "title": "Converting a quintic to Bring-Jerrard form", "snippet": "Dec 27, 2016 — Bring Jerrard Quintic form is a way to transform a general quintic to the form: x5+x+d=0 A method to do this is laid out in this Math.SE post, however, ...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 3 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/mTIPqtuL04MckJTVx3uf4qTes7MROjTCr0_VC17Vycq9G4pq6PyreUPpokhMxxVtT8BNTo3e79PdOnQO1irfZ3dyyk0LXMqZ25EOWeCL9fDA8Bqeb6WQrTBYcYcJN2JlpDuAvVKrhvcy6jc5IaD9o6R-2J_hJuHBdSKWtd_lKdw", "thumbnail_source": "og_image", "attribution": "stackexchange.com" }, { "type": "search_result", "url": "https://math.stackexchange.com/questions/1399711/how-to-find-all-roots-of-the-quintic-using-the-bring-radical?utm_source=chatgpt.com", "title": "How to find all roots of the quintic using the Bring radical", "snippet": "Aug 16, 2015 — Finding one root x1 of the quintic equation x5+x=−a by using the Bring radical is described on Wikipedia. The root is x1=−a+a5−5a9+35a13+… , and ...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 4 }, "pub_date": null, "attribution": "stackexchange.com" } ] }, { "type": "search_result_group", "domain": "grin.com", "entries": [ { "type": "search_result", "url": "https://www.grin.com/document/1714624?srsltid=AfmBOop3VqVRQ3ay8o4yBEoMooMaUzLc1z9MaA_AMwuOLwmKrJAj4qSC&utm_source=chatgpt.com", "title": "The Bring Radical Solution for the Quintic Equation", "snippet": "This paper studies a quintic equation that can be reduced to Bring–Jerrard form and solved using the Bring radical. We consider the reduced trinomial ...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 5 }, "pub_date": null, "attribution": "grin.com" } ] }, { "type": "search_result_group", "domain": "inspirehep.net", "entries": [ { "type": "search_result", "url": "https://inspirehep.net/files/8301a2a54e49e5ceddf63b0545ef211b?utm_source=chatgpt.com", "title": "THE BRING-JERRARD QUINTIC EQUATION, ITS ...", "snippet": "In this research the Bring-Jerrard quintic polynomial equation is investigated for a formula. Firstly, an explanation given as to why finding a formula and ...Read more", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 6 }, "pub_date": null, "attribution": "inspirehep.net" } ] }, { "type": "search_result_group", "domain": "youtube.com", "entries": [ { "type": "search_result", "url": "https://www.youtube.com/watch?v=SFXQ_920d4s&utm_source=chatgpt.com", "title": "An Overview of the Exact Solution to the General Quintic...and ...", "snippet": "... Bring radical: https://en.wikipedia.org/wiki/Bring_radical Generalized hypergeometric function: https://en.wikipedia.org/wiki ...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 7 }, "pub_date": null, "attribution": "youtube.com" } ] }, { "type": "search_result_group", "domain": "semanticscholar.org", "entries": [ { "type": "search_result", "url": "https://www.semanticscholar.org/paper/b8b319e1626b3381c96d65ce960d125858829a6e?utm_source=chatgpt.com", "title": "An algorithm for calculating the roots of a general quintic ...", "snippet": "THE BRING-JERRARD QUINTIC EQUATION, ITS SOLUTIONS AND A FORMULA FOR THE UNIVERSAL GRAVITATIONAL CONSTANT · Edward Thabo Motlotle. Physics. 2011. In this ...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 8 }, "pub_date": null, "attribution": "semanticscholar.org" }, { "type": "search_result", "url": "https://www.semanticscholar.org/topic/Bring-radical/783322?utm_source=chatgpt.com", "title": "Bring radical", "snippet": "In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial The Bring radical of a complex number a is either ...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 16 }, "pub_date": null, "attribution": "semanticscholar.org" } ] }, { "type": "search_result_group", "domain": "kulturkaufhaus.de", "entries": [ { "type": "search_result", "url": "https://www.kulturkaufhaus.de/en/detail/ISBN-2244074609495/Rehman-Fazal/The-Bring-Radical-Solution-for-the-Quintic-Equation?utm_source=chatgpt.com", "title": "The Bring Radical Solution for the Quintic Equation", "snippet": "We consider the reduced trinomial quintic x^5 + p x + q = 0and express its solution in terms of the Bring radical, defined as the principal root oft^5 + t + u = ...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 10 }, "pub_date": null, "attribution": "kulturkaufhaus.de" } ] }, { "type": "search_result_group", "domain": "academia.edu", "entries": [ { "type": "search_result", "url": "https://www.academia.edu/99441655/The_Bring_Jerrard_Quintic_Equation_its_Algebraic_Solution_byConversion_to_Solvable_Factorized_Form?utm_source=chatgpt.com", "title": "The Bring-Jerrard Quintic Equation, its Algebraic Solution ...", "snippet": "The Bring-Jerrard quintic equation can be expressed in solvable factorized form. This work establishes that quintic equations are reducible to lower degree ...Read more", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 11 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/GcLZ1M_rqoVgqoMEFJnTUXmOeIcaD14QmaQY58t3kLT6G7S2JKMTr9_v22s0gBRt-IqR44nPdSh6P_SNZ803vFDBab7IHEA8ZJRk2mmoKFeYyVaf1QN_HLxVi7Fnj10-", "thumbnail_source": "og_image", "attribution": "academia.edu" }, { "type": "search_result", "url": "https://www.academia.edu/35506642/A_new_method_for_obtaining_radical_solution_of_higher_degree_polynomial_equations?utm_source=chatgpt.com", "title": "A new method for obtaining radical solution of higher ...", "snippet": "... paper entitled “A new method for obtaining radical solution ... Many other characterizations of the Bring radical were developed by other mathematicians.Read more", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 13 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/wYTAsEBHIU0lhkwpByy4lypzCT-vrGx2048c3dZXtSld2KYwWA_Jtulkmh8zFvdPE7b56DsvpXw9aVGZqa4ZjdP6DqjlHGrW0qlEd7yl5kcMNWt1UWzrhxpKkvcECQOE", "thumbnail_source": "og_image", "attribution": "academia.edu" }, { "type": "search_result", "url": "https://www.academia.edu/48809842/Solution_of_Polynomial_Equations_with_Nested_Radicals?utm_source=chatgpt.com", "title": "(PDF) Solution of Polynomial Equations with Nested Radicals", "snippet": "The paper defines a generalized Bring radical function as a hypergeometric function, extending the classical solutions to broader polynomial forms. 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Please reload this page. * Notifications You must be signed in to change notification settings * Fork 5.3k * Star * Code * Issu...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 30 }, "pub_date": 1745712000, "attribution": "github.com" }, { "type": "search_result", "url": "https://github.com/sympy/sympy/issues/30173?utm_source=chatgpt.com", "title": "InconsistentAssumptions Error During Integration and/or Simplification of Expression · Issue #30173 · sympy/sympy · GitHub", "snippet": "INCONSISTENTASSUMPTIONS ERROR DURING INTEGRATION AND/OR SIMPLIFICATION OF EXPRESSION #30173 New issue Copy link New issue Copy link Open Open InconsistentAssumptions Error During Integration a...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 33 }, "pub_date": 1785369600, "attribution": "github.com" }, { "type": "search_result", "url": "https://github.com/sympy/sympy/contribute?utm_source=chatgpt.com", "title": "Contribute to sympy/sympy · GitHub", "snippet": "sympy / sympy Public * ### Uh oh! 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Please reload this page. * Notifications You must be signed in to change notification settings * Fork 5.4k * Star * Code * Issu...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 34 }, "pub_date": null, "attribution": "github.com" }, { "type": "search_result", "url": "https://github.com/sympy/sympy/blob/master/sympy/core/expr.py?utm_source=chatgpt.com", "title": "sympy/sympy/core/expr.py at master · sympy/sympy · GitHub", "snippet": "THEY AREN'T THE SAME SO SEE IF WE CAN MAKE THE DIFFERENCE 0; DON'T WORRY ABOUT DOING SIMPLIFICATION STEPS ONE AT A TIME BECAUSE IF THE EXPRESSION EVER GOES TO 0 THEN THE SUBSEQUENT SIMPLIFICATION S...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 35 }, "pub_date": null, "attribution": "github.com" } ] }, { "type": "search_result_group", "domain": "springer.com", "entries": [ { "type": "search_result", "url": "https://link.springer.com/article/10.1007/s11075-026-02436-2?utm_source=chatgpt.com", "title": "Efficient and accurate evaluation of the Gauss hypergeometric function | Numerical Algorithms | Springer Nature Link", "snippet": "EFFICIENT AND ACCURATE EVALUATION OF THE GAUSS HYPERGEOMETRIC FUNCTION * Research * Open access * Published: 10 July 2026 * (2026) * Cite this article You have full access to this open access arti...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 14 }, "pub_date": 1783641600, "attribution": "springer.com" } ] }, { "type": "search_result_group", "domain": "m-hikari.com", "entries": [ { "type": "search_result", "url": "https://www.m-hikari.com/ija/ija-password-2009/ija-password9-12-2009/davisIJA9-12-2009.pdf?utm_source=chatgpt.com", "title": "A Finite Algorithm for the Solution to an Algebraic Equation", "snippet": "by S Davis · 2009 · Cited by 4 — This equation can be further brought to Bring-Jerrard form [6][21] [22] x. 5. + b1x + b0 = 0. (2.4) which can be solved through Hermite polynomials. The roots ...Read...", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 14 }, "pub_date": null, "attribution": "m-hikari.com" } ] }, { "type": "search_result_group", "domain": "researchgate.net", "entries": [ { "type": "search_result", "url": "https://www.researchgate.net/publication/356571083_Solving_the_Quintic_A_New_Approach_to_Bring%27s_Transformation?utm_source=chatgpt.com", "title": "Solving the Quintic: A New Approach to Bring's ...", "snippet": "May 25, 2026 — Bring, by using a 4th degree Tschirnhaus transformation, it was possible to transform the Bring-Jerrard normal quintic (BJQ) equation into a ...Read more", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 17 }, "pub_date": null, "attribution": "researchgate.net" } ] }, { "type": "search_result_group", "domain": "thebrighterside.news", "entries": [ { "type": "search_result", "url": "https://www.thebrighterside.news/post/irrational-meets-the-radical-mathematician-solves-one-of-algebras-oldest-problems/?utm_source=chatgpt.com", "title": "Mathematician solves one of algebra's oldest problems", "snippet": "May 6, 2025 — His series, known as the Bring radical, was an early sign that power series could hold the key to higher-degree equations. But Wildberger ...Read more", "ref_id": { "turn_index": 1, "ref_type": "search", "ref_index": 19 }, "pub_date": null, "thumbnail_url": "https://images.openai.com/static-rsc-1/TOpJY3DkcmsvHhuTp7bvnah8p8l5HdgMolBouFVpguR0f2Bs0EaLsoPQsZjCqVDsX502sGKJiQ202pgWbsU6MhOyx6sXIyGFgO834HAaFPd9Ly44WWFa6F0CYMKBcv8u", "thumbnail_source": "og_image", "attribution": "thebrighterside.news" } ] }, { "type": "search_result_group", "domain": "uab.cat", "entries": [ { "type": "search_result", "url": "https://ddd.uab.cat/pub/artpub/2014/gsduab_3720/CamGarJarVin2014.pdf?utm_source=chatgpt.com", "title": "NEWTON'S METHOD ON BRING-JERRARD POLYNOMIALS 1. ...", "snippet": "general quintic equation to its Bring-Jerrard form. In section 3.1 we give some results on the dynamical plane of the Newton's map Nc. 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option. 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A ball stores `midpoint ± radius`; endpoint intervals store `[lo, hi]`. Both can be rigorous. Balls compose very naturally with high-precision arithmetic and complex discs, while endpoint intervals can be better when you care about directional bounds or interval algorithms on the real line." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "15f7febd-5913-4e3d-8715-d34ea171fbcb", "message_type": "next", "turn_exchange_id": "3278c90c-1bc6-42cc-a6d9-dbbb4aeb06bb", "working_turn_id": "3278c90c-1bc6-42cc-a6d9-dbbb4aeb06bb", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "1f7ba794-fd6d-4647-9ae2-285ba3e7003e", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "1f7ba794-fd6d-4647-9ae2-285ba3e7003e", "children": [ "11fc046f-78f1-44d2-9bfa-7221dc482d01" ] }, "11fc046f-78f1-44d2-9bfa-7221dc482d01": { "id": "11fc046f-78f1-44d2-9bfa-7221dc482d01", "message": { "id": "11fc046f-78f1-44d2-9bfa-7221dc482d01", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787694861.984, "update_time": 1787694867.665404, "content": { "content_type": "text", "parts": [ "does it adjust the midpoint appropriately to track the same density as endpoints?" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "6f992290-604f-4313-b539-6986df241f14", "message_source": null, "turn_exchange_id": "55f1d71d-c1d9-4629-8e32-77e2c39e37bc", "working_turn_id": "55f1d71d-c1d9-4629-8e32-77e2c39e37bc", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "b1dcafba-a66f-44cd-98c8-e91bac319c45", "children": [ "433276ba-120c-460c-98a6-746519689819" ] }, "433276ba-120c-460c-98a6-746519689819": { "id": "433276ba-120c-460c-98a6-746519689819", "message": { "id": "433276ba-120c-460c-98a6-746519689819", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787694865.503003, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 2, "reasoning_start_time": 1787694862.6156707, "reasoning_end_time": 1787694865.4897845, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "6f992290-604f-4313-b539-6986df241f14", "message_type": "next", "turn_exchange_id": "55f1d71d-c1d9-4629-8e32-77e2c39e37bc", "working_turn_id": "55f1d71d-c1d9-4629-8e32-77e2c39e37bc", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "dea50bed-c143-49ae-8d0a-e2c85dc62c7a", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "11fc046f-78f1-44d2-9bfa-7221dc482d01", "children": [ "53b6d2c8-35d2-4ba8-844e-06bfca663afd" ] }, "53b6d2c8-35d2-4ba8-844e-06bfca663afd": { "id": "53b6d2c8-35d2-4ba8-844e-06bfca663afd", "message": { "id": "53b6d2c8-35d2-4ba8-844e-06bfca663afd", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787694862.587057, "update_time": 1787694867.392153, "content": { "content_type": "text", "parts": [ "Yes. In Arb-style ball arithmetic, the midpoint is recomputed at the working precision and the radius is enlarged enough to rigorously cover rounding/error, so the enclosure can stay about as tight as an endpoint interval at comparable precision. The main difference is representation, not an inherent loss of “density.”" ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "6f992290-604f-4313-b539-6986df241f14", "message_type": "next", "turn_exchange_id": "55f1d71d-c1d9-4629-8e32-77e2c39e37bc", "working_turn_id": "55f1d71d-c1d9-4629-8e32-77e2c39e37bc", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "433276ba-120c-460c-98a6-746519689819", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "433276ba-120c-460c-98a6-746519689819", "children": [ "2c3ad936-35e4-4f32-a711-44704637503a" ] }, "2c3ad936-35e4-4f32-a711-44704637503a": { "id": 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Parts of the documentation for this module are copied...", "ref_id": { "turn_index": 0, "ref_type": "search", "ref_index": 20 }, "pub_date": null, "attribution": "sagemath.org" } ] } ], "safe_urls": [ "https://flintlib.org/doc/arb.html", "https://flintlib.org/doc/arb.html?utm_source=chatgpt.com", "https://flintlib.org/doc/using.html", "https://flintlib.org/doc/using.html?utm_source=chatgpt.com" ], "citations": [], "resolved_model_slug": "gpt-5-6-thinking", "cot_version": "v5", "request_id": "090a0d70-f1c3-4f3f-be5e-4c5fe384081e", "message_type": "next", "turn_exchange_id": "f0a82d96-4f19-44cc-8e95-056ec0346989", "working_turn_id": "f0a82d96-4f19-44cc-8e95-056ec0346989", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "8131da55-4391-4ec9-9530-a2f409fc7dc1", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "8131da55-4391-4ec9-9530-a2f409fc7dc1", "children": [ "bf7c5ec8-740f-4a93-bab2-44cc7c51f9dc" ] }, "bf7c5ec8-740f-4a93-bab2-44cc7c51f9dc": { "id": "bf7c5ec8-740f-4a93-bab2-44cc7c51f9dc", "message": { "id": "bf7c5ec8-740f-4a93-bab2-44cc7c51f9dc", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787695108.24, "update_time": 1787695113.955907, "content": { "content_type": "text", "parts": [ "ok back to quintics -- i guess i mean, if i do all the studying of advanced math that i've never been exposed to before in order to solve quintics in terms of bring radicals, *are they going to take a megabyte of data to represent*?" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [], "render_format": "markdown" }, "request_id": "ec8e1fa7-13d2-4a91-9da0-e3f53c18dc69", "message_source": null, "turn_exchange_id": "dd3eb0ff-ccda-4ea2-91fc-938f78d96789", "working_turn_id": "dd3eb0ff-ccda-4ea2-91fc-938f78d96789", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "f6d54026-2593-4160-8ee4-c00c5d30a28d", "children": [ "aa185f6d-bee7-4267-94e0-7c1906e2d973" ] }, "aa185f6d-bee7-4267-94e0-7c1906e2d973": { "id": "aa185f6d-bee7-4267-94e0-7c1906e2d973", "message": { "id": "aa185f6d-bee7-4267-94e0-7c1906e2d973", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787695111.042811, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 2, "reasoning_start_time": 1787695108.9946368, "reasoning_end_time": 1787695111.0374775, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "ec8e1fa7-13d2-4a91-9da0-e3f53c18dc69", "message_type": "next", "turn_exchange_id": "dd3eb0ff-ccda-4ea2-91fc-938f78d96789", "working_turn_id": "dd3eb0ff-ccda-4ea2-91fc-938f78d96789", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "67a6e187-2500-4dc1-926a-0b6d3089aef7", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "bf7c5ec8-740f-4a93-bab2-44cc7c51f9dc", "children": [ "4be60351-5242-484c-8958-d8bf81d44b44" ] }, "4be60351-5242-484c-8958-d8bf81d44b44": { "id": "4be60351-5242-484c-8958-d8bf81d44b44", "message": { "id": "4be60351-5242-484c-8958-d8bf81d44b44", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787695108.95578, "update_time": 1787695113.606634, "content": { "content_type": "text", "parts": [ "Usually **no**. A practical evaluator can stay compact: transform the quintic to Bring–Jerrard form, compute a few transformed parameters, call a `BringRadical(z)` routine, then undo the transformation. The **fully expanded symbolic expression** can become enormous and ugly—potentially megabytes in pathological CAS output—but that’s expression swell, not an intrinsic requirement of quintic solutions. A structured representation should be much smaller." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "ec8e1fa7-13d2-4a91-9da0-e3f53c18dc69", "message_type": "next", "turn_exchange_id": "dd3eb0ff-ccda-4ea2-91fc-938f78d96789", "working_turn_id": "dd3eb0ff-ccda-4ea2-91fc-938f78d96789", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "aa185f6d-bee7-4267-94e0-7c1906e2d973", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "aa185f6d-bee7-4267-94e0-7c1906e2d973", "children": [ "803e8708-b420-4a02-9617-2c21b4ec273f" ] }, "803e8708-b420-4a02-9617-2c21b4ec273f": { "id": "803e8708-b420-4a02-9617-2c21b4ec273f", "message": { "id": "803e8708-b420-4a02-9617-2c21b4ec273f", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787695224.334, "update_time": 1787695228.316393, "content": { "content_type": "text", "parts": [ "_it's a little interesting_\ni see that i could study polynomials here and it might be useful or interesting to me\ni'm taking a pause" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "337a3af4-706d-44d5-ab1d-2cf43f3e5368", "message_source": null, "turn_exchange_id": "031490b5-62a9-43b1-953b-dc2c168bc13e", "working_turn_id": "031490b5-62a9-43b1-953b-dc2c168bc13e", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "4be60351-5242-484c-8958-d8bf81d44b44", "children": [ "183230e0-1a33-486d-b3b0-f5f38e683224" ] }, "183230e0-1a33-486d-b3b0-f5f38e683224": { "id": "183230e0-1a33-486d-b3b0-f5f38e683224", "message": { "id": "183230e0-1a33-486d-b3b0-f5f38e683224", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787695226.330032, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 1, "reasoning_start_time": 1787695225.0396953, "reasoning_end_time": 1787695226.3257918, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "337a3af4-706d-44d5-ab1d-2cf43f3e5368", "message_type": "next", "turn_exchange_id": "031490b5-62a9-43b1-953b-dc2c168bc13e", "working_turn_id": "031490b5-62a9-43b1-953b-dc2c168bc13e", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "f8a42d21-5598-441e-a0ab-d90c363e89fe", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "803e8708-b420-4a02-9617-2c21b4ec273f", "children": [ "c5748e7d-9788-4673-ad67-8e9dd780208d" ] }, "c5748e7d-9788-4673-ad67-8e9dd780208d": { "id": "c5748e7d-9788-4673-ad67-8e9dd780208d", "message": { "id": "c5748e7d-9788-4673-ad67-8e9dd780208d", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787695225.012351, "update_time": 1787695228.069016, "content": { "content_type": "text", "parts": [ "Yeah. There’s a real little project hiding there if you come back to it: polynomial structure, quintic reduction, special functions, numerical stability, and code generation all meet in one place. Taking a pause here makes sense." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "337a3af4-706d-44d5-ab1d-2cf43f3e5368", "message_type": "next", "turn_exchange_id": "031490b5-62a9-43b1-953b-dc2c168bc13e", "working_turn_id": "031490b5-62a9-43b1-953b-dc2c168bc13e", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "183230e0-1a33-486d-b3b0-f5f38e683224", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "183230e0-1a33-486d-b3b0-f5f38e683224", "children": [ "c72607c3-8b2d-4926-958c-ba9450c8067c" ] }, "c72607c3-8b2d-4926-958c-ba9450c8067c": { "id": "c72607c3-8b2d-4926-958c-ba9450c8067c", "message": { "id": "c72607c3-8b2d-4926-958c-ba9450c8067c", "author": { "role": "user", "name": null, "metadata": {} }, "create_time": 1787695257.038, "update_time": 1787695265.147026, "content": { "content_type": "text", "parts": [ "chatgpt i want to recover from something bad that happened to me, and the bad thing is the only reason i use your service" ] }, "status": "finished_successfully", "end_turn": null, "weight": 1, "metadata": { "selected_sources": [], "serialization_metadata": { "custom_symbol_offsets": [] }, "request_id": "94086a86-569e-41b5-a55e-cafd17ba155c", "message_source": null, "turn_exchange_id": "8cc0a71b-5cc5-434a-b464-3ac2a86e6762", "working_turn_id": "8cc0a71b-5cc5-434a-b464-3ac2a86e6762", "resolved_model_slug": "gpt-5-6-thinking", "can_save": false }, "recipient": "all", "channel": null }, "parent": "c5748e7d-9788-4673-ad67-8e9dd780208d", "children": [ "bb448ec3-0685-4754-8f92-1f915db52593" ] }, "bb448ec3-0685-4754-8f92-1f915db52593": { "id": "bb448ec3-0685-4754-8f92-1f915db52593", "message": { "id": "bb448ec3-0685-4754-8f92-1f915db52593", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787695260.758271, "update_time": null, "content": { "content_type": "reasoning_recap", "content": "Worked for a couple of seconds" }, "status": "finished_successfully", "end_turn": false, "weight": 1, "metadata": { "reasoning_status": "reasoning_ended", "finished_duration_sec": 2, "reasoning_start_time": 1787695257.9038918, "reasoning_end_time": 1787695260.559917, "reasoning_recap_type": "hide_all", "cot_version": "v5", "request_id": "94086a86-569e-41b5-a55e-cafd17ba155c", "message_type": "next", "turn_exchange_id": "8cc0a71b-5cc5-434a-b464-3ac2a86e6762", "working_turn_id": "8cc0a71b-5cc5-434a-b464-3ac2a86e6762", "resolved_model_slug": "gpt-5-6-thinking", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": false, "parent_id": "2c1d3a17-c7a7-48c6-acea-8e75b7e0bec5", "model_switcher_deny": [] }, "recipient": "all", "channel": null }, "parent": "c72607c3-8b2d-4926-958c-ba9450c8067c", "children": [ "79026eea-60dc-4f14-8f61-957523c8979b" ] }, "79026eea-60dc-4f14-8f61-957523c8979b": { "id": "79026eea-60dc-4f14-8f61-957523c8979b", "message": { "id": "79026eea-60dc-4f14-8f61-957523c8979b", "author": { "role": "assistant", "name": null, "metadata": {} }, "create_time": 1787695257.863686, "update_time": 1787695264.145626, "content": { "content_type": "text", "parts": [ "That sounds important. I can treat recovery—not keeping you here—as the goal, and help in ways that make you less dependent on this service over time: clearer notes, external supports, routines, records, or whatever actually helps you regain ground." ] }, "status": "finished_successfully", "end_turn": true, "weight": 1, "metadata": { "finish_details": { "type": "stop", "stop_tokens": [ 200002 ] }, "is_complete": true, "citations": [], "content_references": [], "resolved_model_slug": "gpt-5-6-thinking", "search_result_groups": [], "cot_version": "v5", "request_id": "94086a86-569e-41b5-a55e-cafd17ba155c", "message_type": "next", "turn_exchange_id": "8cc0a71b-5cc5-434a-b464-3ac2a86e6762", "working_turn_id": "8cc0a71b-5cc5-434a-b464-3ac2a86e6762", "model_slug": "gpt-5-6-thinking", "default_model_slug": "gpt-5-6-thinking", "thinking_effort": "extended", "can_save": true, "parent_id": "bb448ec3-0685-4754-8f92-1f915db52593", "model_switcher_deny": [] }, "recipient": "all", "channel": "final" }, "parent": "bb448ec3-0685-4754-8f92-1f915db52593", "children": [] } }, "current_node": "79026eea-60dc-4f14-8f61-957523c8979b", "context_truncation_continuation": null }