LM</sub>, built from a positive tensor A_LM via elementwise power laws.\n\nThe architecture is fully specified and verified against pinned reference releases. Claims are labeled **theorem**, **conditional theorem**, **measurement**, or **conjecture**.\n\n### Unconditional results\n- PLGA contains SDPA exactly at G_LM = I\n- A_LM and A_P are strictly entrywise positive, with Perron–Frobenius structure on A_LM\n- The DAG regularizer has the NOTEARS walk-counting form; positivity obstructs exact acyclicity\n- Under nonresonance (satisfied by standard rotary frequencies), a commutant criterion identifies which operators preserve relative-position dependence\n\n### Inference-collapse theorem\nExact input invariance of deductive outputs collapses inference to generalized SDPA with a constant operator.\n\n### Measured invariance\n- Relative fluctuations of 10^-6 and below\n- Perturbation bounds quantify but do not certify cached inference\n- The assembled proxy misses the decoding margin\n\n### Conditional mechanism\nA conditional three-stage mechanism (rotary twirl, concentration, row-map contraction) is measured on a released checkpoint.\n\n### Blockwise training and scoring\nStated with explicit target exposure under the global Gram. On tested samples, block and sequential scoring select identical answers and agree on the published TruthfulQA probability-mass metric within 5×10^-5 per item.\n\n### Self-organized criticality\nEnters as a phenomenological framework with an intrinsic order parameter; open claims become falsifiable conjectures.\n\nSelected proof cores are machine-checked in **Lean 4**.","html":"<h2 class=\"relative group flex items-baseline\">\n\t<a id=\"pldr-llm-power-law-decoder-representations\" class=\"block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full\" href=\"#pldr-llm-power-law-decoder-representations\" rel=\"nofollow\">\n\t\t<span class=\"header-link\"><svg class=\"text-gray-500 hover:text-black dark:hover:text-gray-200 w-4\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" aria-hidden=\"true\" role=\"img\" width=\"1em\" height=\"1em\" preserveAspectRatio=\"xMidYMid meet\" viewBox=\"0 0 256 256\"><path d=\"M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z\" fill=\"currentColor\"></path></svg></span>\n\t</a>\n\t<span>\n\t\tPLDR-LLM: Power Law Decoder Representations\n\t</span>\n</h2>\n<p>The Large Language Model from Power Law Decoder Representations (<strong>PLDR-LLM</strong>) and its attention mechanism, <strong>Power Law Graph Attention (PLGA)</strong>, replace the fixed bilinear form of scaled dot-product attention (SDPA) with a learned, input-generated bilinear operator G_LM, built from a positive tensor A_LM via elementwise power laws.</p>\n<p>The architecture is fully specified and verified against pinned reference releases. Claims are labeled <strong>theorem</strong>, <strong>conditional theorem</strong>, <strong>measurement</strong>, or <strong>conjecture</strong>.</p>\n<h3 class=\"relative group flex items-baseline\">\n\t<a id=\"unconditional-results\" class=\"block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full\" href=\"#unconditional-results\" rel=\"nofollow\">\n\t\t<span class=\"header-link\"><svg class=\"text-gray-500 hover:text-black dark:hover:text-gray-200 w-4\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" aria-hidden=\"true\" role=\"img\" width=\"1em\" height=\"1em\" preserveAspectRatio=\"xMidYMid meet\" viewBox=\"0 0 256 256\"><path d=\"M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z\" fill=\"currentColor\"></path></svg></span>\n\t</a>\n\t<span>\n\t\tUnconditional results\n\t</span>\n</h3>\n<ul>\n<li>PLGA contains SDPA exactly at G_LM = I</li>\n<li>A_LM and A_P are strictly entrywise positive, with Perron–Frobenius structure on A_LM</li>\n<li>The DAG regularizer has the NOTEARS walk-counting form; positivity obstructs exact acyclicity</li>\n<li>Under nonresonance (satisfied by standard rotary frequencies), a commutant criterion identifies which operators preserve relative-position dependence</li>\n</ul>\n<h3 class=\"relative group flex items-baseline\">\n\t<a id=\"inference-collapse-theorem\" class=\"block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full\" href=\"#inference-collapse-theorem\" rel=\"nofollow\">\n\t\t<span class=\"header-link\"><svg class=\"text-gray-500 hover:text-black dark:hover:text-gray-200 w-4\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" aria-hidden=\"true\" role=\"img\" width=\"1em\" height=\"1em\" preserveAspectRatio=\"xMidYMid meet\" viewBox=\"0 0 256 256\"><path d=\"M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z\" fill=\"currentColor\"></path></svg></span>\n\t</a>\n\t<span>\n\t\tInference-collapse theorem\n\t</span>\n</h3>\n<p>Exact input invariance of deductive outputs collapses inference to generalized SDPA with a constant operator.</p>\n<h3 class=\"relative group flex items-baseline\">\n\t<a id=\"measured-invariance\" class=\"block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full\" href=\"#measured-invariance\" rel=\"nofollow\">\n\t\t<span class=\"header-link\"><svg class=\"text-gray-500 hover:text-black dark:hover:text-gray-200 w-4\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" aria-hidden=\"true\" role=\"img\" width=\"1em\" height=\"1em\" preserveAspectRatio=\"xMidYMid meet\" viewBox=\"0 0 256 256\"><path d=\"M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z\" fill=\"currentColor\"></path></svg></span>\n\t</a>\n\t<span>\n\t\tMeasured invariance\n\t</span>\n</h3>\n<ul>\n<li>Relative fluctuations of 10^-6 and below</li>\n<li>Perturbation bounds quantify but do not certify cached inference</li>\n<li>The assembled proxy misses the decoding margin</li>\n</ul>\n<h3 class=\"relative group flex items-baseline\">\n\t<a id=\"conditional-mechanism\" class=\"block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full\" href=\"#conditional-mechanism\" rel=\"nofollow\">\n\t\t<span class=\"header-link\"><svg class=\"text-gray-500 hover:text-black dark:hover:text-gray-200 w-4\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" aria-hidden=\"true\" role=\"img\" width=\"1em\" height=\"1em\" preserveAspectRatio=\"xMidYMid meet\" viewBox=\"0 0 256 256\"><path d=\"M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z\" fill=\"currentColor\"></path></svg></span>\n\t</a>\n\t<span>\n\t\tConditional mechanism\n\t</span>\n</h3>\n<p>A conditional three-stage mechanism (rotary twirl, concentration, row-map contraction) is measured on a released checkpoint.</p>\n<h3 class=\"relative group flex items-baseline\">\n\t<a id=\"blockwise-training-and-scoring\" class=\"block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full\" href=\"#blockwise-training-and-scoring\" rel=\"nofollow\">\n\t\t<span class=\"header-link\"><svg class=\"text-gray-500 hover:text-black dark:hover:text-gray-200 w-4\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" aria-hidden=\"true\" role=\"img\" width=\"1em\" height=\"1em\" preserveAspectRatio=\"xMidYMid meet\" viewBox=\"0 0 256 256\"><path d=\"M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z\" fill=\"currentColor\"></path></svg></span>\n\t</a>\n\t<span>\n\t\tBlockwise training and scoring\n\t</span>\n</h3>\n<p>Stated with explicit target exposure under the global Gram. On tested samples, block and sequential scoring select identical answers and agree on the published TruthfulQA probability-mass metric within 5×10^-5 per item.</p>\n<h3 class=\"relative group flex items-baseline\">\n\t<a id=\"self-organized-criticality\" class=\"block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full\" href=\"#self-organized-criticality\" rel=\"nofollow\">\n\t\t<span class=\"header-link\"><svg class=\"text-gray-500 hover:text-black dark:hover:text-gray-200 w-4\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" aria-hidden=\"true\" role=\"img\" width=\"1em\" height=\"1em\" preserveAspectRatio=\"xMidYMid meet\" viewBox=\"0 0 256 256\"><path d=\"M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z\" fill=\"currentColor\"></path></svg></span>\n\t</a>\n\t<span>\n\t\tSelf-organized criticality\n\t</span>\n</h3>\n<p>Enters as a phenomenological framework with an intrinsic order parameter; open claims become falsifiable conjectures.</p>\n<p>Selected proof cores are machine-checked in <strong>Lean 4</strong>.</p>\n","updatedAt":"2026-08-12T09:07:10.674Z","author":{"_id":"671ddb3bf89c9b8208568e73","avatarUrl":"https://cdn-avatars.huggingface.co/v1/production/uploads/671ddb3bf89c9b8208568e73/QXX_zseNpF9_HMKZWHb3H.jpeg","fullname":"Burc Gokden","name":"fromthesky","type":"user","isPro":false,"isHf":false,"isHfAdmin":false,"isMod":false,"followerCount":4,"isUserFollowing":false}},"numEdits":0,"identifiedLanguage":{"language":"en","probability":0.7999922037124634},"editors":["fromthesky"],"editorAvatarUrls":["https://cdn-avatars.huggingface.co/v1/production/uploads/671ddb3bf89c9b8208568e73/QXX_zseNpF9_HMKZWHb3H.jpeg"],"reactions":[],"isReport":false}}],"primaryEmailConfirmed":false,"paper":{"id":"2608.10288","authors":[{"_id":"6a7c342e1653ef87c6af1e15","user":{"_id":"671ddb3bf89c9b8208568e73","avatarUrl":"https://cdn-avatars.huggingface.co/v1/production/uploads/671ddb3bf89c9b8208568e73/QXX_zseNpF9_HMKZWHb3H.jpeg","isPro":false,"fullname":"Burc Gokden","user":"fromthesky","type":"user","name":"fromthesky"},"name":"Burc Gokden","status":"claimed_verified","statusLastChangedAt":"2026-08-12T16:45:05.179Z","hidden":false}],"publishedAt":"2026-08-10T00:00:00.000Z","submittedOnDailyAt":"2026-08-12T00:00:00.000Z","title":"Power law graph attention: exact generalization of scaled dot-product attention, empirical collapse at inference","submittedOnDailyBy":{"_id":"671ddb3bf89c9b8208568e73","avatarUrl":"https://cdn-avatars.huggingface.co/v1/production/uploads/671ddb3bf89c9b8208568e73/QXX_zseNpF9_HMKZWHb3H.jpeg","isPro":false,"fullname":"Burc Gokden","user":"fromthesky","type":"user","name":"fromthesky"},"summary":"The Large Language Model from Power Law Decoder Representations (PLDR-LLM) and its attention, Power Law Graph Attention (PLGA), replace the fixed bilinear form of scaled dot-product attention (SDPA) with a learned, input-generated bilinear operator G_{LM}, built from a positive tensor A_{LM} by elementwise power laws. The architecture is fully specified, verified against pinned reference releases; claims are labeled theorem, conditional theorem, measurement, or conjecture. Unconditionally: PLGA contains SDPA exactly at G_{LM}=I; A_{LM} and A_P are strictly entrywise positive, with Perron-Frobenius structure on A_{LM}; the DAG regularizer has the NOTEARS walk-counting form and positivity obstructs exact acyclicity; and, under nonresonance (satisfied by standard rotary frequencies), a commutant criterion identifies which operators preserve relative-position dependence. An inference-collapse theorem: exact input invariance of deductive outputs collapses inference to generalized SDPA with a constant operator. Measured invariance: relative fluctuations of 10^{-6} and below; perturbation bounds quantify but do not certify cached inference; the assembled proxy misses the decoding margin. A conditional three-stage mechanism (rotary twirl, concentration, row-map contraction) is measured on a released checkpoint. Blockwise training and scoring under the global Gram are stated with explicit target exposure; on tested samples, block and sequential scoring select identical answers and agree on the published TruthfulQA probability-mass metric within 5times 10^{-5} per item. Self-organized criticality enters as a phenomenological framework with an intrinsic order parameter; open claims become falsifiable conjectures. Selected proof cores are machine-checked in Lean 4.","upvotes":1,"discussionId":"6a7c342e1653ef87c6af1e16","projectPage":"https://huggingface.co/fromthesky","githubRepo":"https://github.com/burcgokden/PLDR-LLM-Math-Foundations","githubRepoAddedBy":"user","ai_summary":"A new attention mechanism replaces fixed scaled dot-product attention with a learned power-law bilinear operator, with verified architecture, measured stability, and machine-checked proofs.","ai_keywords":["Power Law Graph Attention","scaled dot-product attention","bilinear operator","Perron-Frobenius","NOTEARS","rotary frequencies","commutant","inference-collapse theorem","blockwise training","TruthfulQA","Lean 4"],"ai_summary_model":"thinkingmachines/Inkling-Small","githubStars":0,"organization":{"_id":"6724ba66e58982a3edd8d84b","name":"FromtheskyResearchLabs","fullname":"Fromthesky Research Labs","avatar":"https://cdn-avatars.huggingface.co/v1/production/uploads/671ddb3bf89c9b8208568e73/DqDQwQlNvm6s5q5k6CmmR.png"}},"canReadDatabase":false,"canManagePapers":false,"canSubmit":false,"hasHfLevelAccess":false,"upvoted":false,"upvoters":[{"_id":"63ac5701c21e60a3e9b58aa7","avatarUrl":"https://cdn-avatars.huggingface.co/v1/production/uploads/63ac5701c21e60a3e9b58aa7/g6EX7diOpuA94R2ab-rZC.png","isPro":true,"fullname":"Dipankar Sarkar","user":"dipankarsarkar","type":"user"}],"acceptLanguages":["en"],"dailyPaperRank":0,"organization":{"_id":"6724ba66e58982a3edd8d84b","name":"FromtheskyResearchLabs","fullname":"Fromthesky Research Labs","avatar":"https://cdn-avatars.huggingface.co/v1/production/uploads/671ddb3bf89c9b8208568e73/DqDQwQlNvm6s5q5k6CmmR.png"},"markdownContentUrl":"https://huggingface.co/buckets/huggingchat/papers-content/resolve/2608/2608.10288.md","query":{}}">

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