Classify sharp noncubic ranks of the H6 odd-Gram family

research/h6-odd-gram-rank/ADAPTATIONS.json

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+{
+  "assistance": "Separate OpenAI Codex agents; not independent human or formal-kernel review.",
+  "changes": [
+    "CLI/output paths made portable; generated evidence written outside package.",
+    "Producer no longer hashes four historical documentation files at runtime; source references remain in SOURCES.json and proof.",
+    "Independent comparison reads the freshly generated producer receipt and binds the live primary geometry source.",
+    "Terminology uses scalar factor return episodes; no number of physical helpers is inferred."
+  ],
+  "preserved_mathematical_function_asts": true,
+  "source_independent_sha256": "7903be2b98e89e844f3c364f0e2e8d9638f14d863af0b791b89d5409bbddbb1d",
+  "source_producer_sha256": "f0efdfa39b9c74ee3079883e1b4b0d3fbcf699adeb70437564769a5de9bb5281"
+}

research/h6-odd-gram-rank/ARTIFACTS.json

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+{
+  "files": {
+    "ADAPTATIONS.json": "2aedf56e56da3e7d045ff2fdccdb553c245ae033b49cbdebfe9446cbe116d2d9",
+    "PROOF.md": "00525fedbe8a5fe3173fb0966313540bb197f930c90cd968589475c43a527035",
+    "README.md": "2b23ddceb742f1d7bdf5c4a3a3bee047fe6739ab55982ba82842f0ad1e385f23",
+    "SOURCES.json": "265ab45303b68e190722ab5ac46bbc03b2e0f214cb4b6a8db2d6e8a2dbb248d1",
+    "TASK.json": "2014c5afebe53e92792b147770ca77b1e566e18b1dc2e701d9f7cbbe96415ec2",
+    "VERIFY.json": "50d0beca581aabb9d9d49e84f3966c15632aead65d41d92085c3dd8aede7ac93",
+    "derive.py": "cf01cfc364ab834edd69c05ae83a2d2a7af143f013287699c9dac80ee46404ec",
+    "independent.py": "e4c7eeb38afce6c2f191febc307d0730fcbbcc3c6667cef7ed5294b2a242b3db",
+    "verify.py": "a0b6cb76d6f52da5e42bacd83ce0804f27ddd91d24260b328e70293d3ad6f068"
+  }
+}

research/h6-odd-gram-rank/PROOF.md

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+# Odd-Gram rank theorem and its precise cost consequence
+
+Work over any characteristic-zero field F. All displayed rational constants
+embed in F; in particular the argument covers real and complex coefficients.
+Let E be the 240 signed E8 roots, represented by doubled integer coordinates:
+two nonzero entries ±2, or eight entries ±1 with an even number of negatives.
+Divide ordinary doubled-coordinate dot products by 4. Roots have norm 2 and
+pairings in {-2,-1,0,1,2}.
+
+The pinned primary `e8_cubed.py` constructs these roots in `e8_shells`, retains
+120 representatives whose first nonzero coordinate is positive, and constructs
+each two-block product in `construct_shell` as those 120 first roots times both
+signs of each second representative. Thus one H6 two-block shell contains the
+canonical 28,800 root-product ports, plus 2,160 single-block norm-four ports.
+This is a written binding to the exact pinned primary functions; the registered
+checks independently enumerate the roots, not the entire NumPy shell generator.
+
+Let K_f have entries f(a·c+b·d) on the canonical root-product ports. An odd
+extension to {-4,...,4} of the attainable Gram values is determined by its
+values at 1,2,3,4. The canonical Gram set excludes -4; its extension value
+f(-4)=-f(4) is determined by oddness, not an extra assumption. The Vandermonde matrix in their distinct nonzero squares is invertible
+in characteristic zero, so its unique odd interpolant has degree at most 7.
+Imposing f(2)=0 and f(4)=1 leaves
+
+```
+f(t)=(-1/12+64u+1280v)t +(1/48-20u-336v)t^3 +u t^5+v t^7.
+```
+
+Any odd formula with those same attainable values is covered, irrespective of
+how it is originally written. No claim is made about values outside this finite
+Gram set, even kernels, different normalization or characteristic two.
+
+## Exact root algebra
+
+On the 240 signed roots let S be the Gram matrix, R its entrywise square, B
+the antipodal permutation and J the all-ones matrix. Direct exact enumeration
+checks the second moment and every one of the 57,600 entries of R². They give
+
+```
+S²=60S, R²=24R+48J, SR=0, BS=-S, BR=R, SJ=0, RJ=120J.
+```
+
+Here SR=0 follows by pairing the intermediate root with its antipode; the
+other omitted products follow from B²=I and J²=240J. The five orthogonal
+idempotents summing to I are
+
+```
+P0=J/240                     rank 1
+P1=S/60                      rank 8
+P2=R/24-J/48                 rank 35
+P3=(I-B)/2-S/60              rank 112
+P4=(I+B)/2-R/24+J/60         rank 84.
+```
+
+Their ranks follow from traces and idempotence, which is valid over F.
+For positive odd j, the entrywise power S^[j] is
+`S+(2^j-2)(I-B)`; for positive even j it is
+`R+(2^j-4)(I+B)`. The zeroth entrywise power is J, including its diagonal.
+The binomial expansion of `(a·c+b·d)^j` is therefore diagonal on all tensor
+products of the P_i.
+
+An odd f is supported on the odd simultaneous-antipode subspace of the signed
+240×240 product. Restricting to one representative of each simultaneous
+antipodal pair divides its nonzero eigenvalues by 2. It preserves rank, and
+produces exactly the canonical 120×240 matrix above. Its six eigenspaces are
+the indicated tensor sector and its swap:
+
+| sectors | dimension | eigenvalue |
+|---|---:|---|
+| (1,0) and (0,1) |16| -195+151200u+3104640v |
+| (3,0) and (0,3) |224| 30-14400u-347760v |
+| (1,2) and (2,1) |560| 45-27360u-604800v |
+| (3,2) and (2,3) |7840| 1440u+25200v |
+| (1,4) and (4,1) |1344| 3600u+55440v |
+| (3,4) and (4,3) |18816| 5040v |
+
+The dimensions sum to 28,800. Both checkers derive these forms from the exact
+root algebra. Each also checks 18 actual 28,800-column operator actions at the
+cubic and two attaining parameters. The direct actions detect an omitted
+quotient factor and corrupted new-sector coefficients; they supplement the
+algebraic derivation rather than replacing its universal proof.
+
+## Exhaustive parameter cases and sharpness
+
+At u=v=0 only the first three sectors survive, giving rank 800.
+
+If v=0 and u≠0, the two new sectors of dimensions 7,840 and 1,344 survive.
+The three old sectors have distinct zero parameters, so at most the largest
+old sector, of dimension 560, can vanish. Hence
+`rank K_f >= 7840+1344+800-560 = 9424`.
+Equality holds at u=1/608, v=0.
+
+If v≠0, the 18,816 sector survives. When the 7,840 sector also survives their
+sum alone is 26,656, greater than 20,400. Otherwise u=-35v/2. The 1,344
+sector then has eigenvalue -7560v and survives. On that line the three old
+zeros occur at the distinct values v=13/30576, 1/3192, 1/2800. At most 560 old
+dimensions vanish, so
+`rank K_f >= 18816+1344+800-560 = 20400`.
+Equality holds at u=-1/160, v=1/2800. These alternatives exhaust F²; no real
+ordering, floating calculation or finite-sample extrapolation is used.
+
+These are sharp ranks of the root-product principal block. The whole H6
+matrix has rank at least that of its principal block. Attainment here does
+not assert the entire H6 matrix has that same rank.
+
+## Conditional full-width scalar-factor invoice
+
+Assume a realization expresses the full scalar K as a sum of scalar rank-one
+terms, each delivered by a paid FULL-to-zero return episode of width 17.
+Rank subadditivity forces at least rank(K) such terms/episodes and each pays
+17 rank units. Thus a genuinely quintic member forces at least
+`17*9424=160208`; a member with v≠0 forces at least `17*20400=346800`.
+Both exceed N=30960 before compensation. In the inherited deficit expression,
+for every positive replica count L and nonnegative other costs,
+`(L-1)N-L*C_K < 0`.
+
+The count is of rank-one return episodes, not simultaneous physical centres:
+reusing one centre repeatedly still counts its episodes. This implication is
+conditional on the stated scalar factorization and full-width return price.
+It does not bound arbitrary dirty helpers, nonisotropic kernels, operator
+factors, partial-frame networks, changed geometry or a different coverage law.
+No physical supplier, rounding/precision, owners/access/tape, joint outer
+transfer, new kappa or formal-kernel theorem is established here.

research/h6-odd-gram-rank/README.md

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+# Sharp noncubic ranks for the H6 odd-Gram family
+
+For every characteristic-zero odd Gram function with `f(2)=0` and `f(4)=1`,
+the canonical 28,800-port root-product principal block has rank at least
+**9,424** if its odd interpolation is genuinely quintic, and **20,400** if it
+has a nonzero septic coefficient. Both bounds are attained. The cubic member
+has root-block rank 800. These are exact ranks of this principal block, not
+claims about the rank of the entire 30,960-port H6 matrix.
+
+Under a **full-width 17 scalar rank-one return-episode** implementation, the
+noncubic ranks alone force paid rank at least 160,208 or 346,800, respectively.
+Both exceed all 30,960 ports before compensation. Reusing a physical centre
+still pays for its separate episodes. This is not a lower bound for arbitrary
+helpers, partial-frame networks, operator-valued factors, nonisotropic kernels
+or changed geometry, and it is not a new kappa claim.
+
+[PROOF.md](PROOF.md) gives the interpolation, five exact projectors, six
+eigenforms, complete parameter cases and conditional cost implication.
+`derive.py` and `independent.py` are separate standard-library implementations.
+Each freshly checks all 57,600 association-square entries and 18 exact actions
+with 28,800 columns apiece. Four producer mutations and one independent
+normalization mutation must fail. The all-parameter conclusion uses the written
+exhaustive case proof; it is not inferred from finitely sampled parameters.
+
+Run from the repository root with Python 3.12:
+
+```sh
+python3.12 -I -O -B -Werror scripts/research_package.py --package research/h6-odd-gram-rank
+python3.12 -I -O -B -Werror research/h6-odd-gram-rank/verify.py --out /tmp/fresh-odd-gram-result
+```
+
+The registered replay has a 120-second cap, requires no third-party modules,
+writes only fresh external temporary outputs and authenticates all inputs
+before and after. `e8_cubed.py` stays a live, exact-pinned source. The geometric
+binding to its `e8_shells` and `construct_shell` is a written source inspection;
+the NumPy shell generator and complete H6 supplier are not rerun. Exact source
+hashes and original Git blobs are in `SOURCES.json`; portable script
+adaptations are recorded in `ADAPTATIONS.json`.
+
+## Open next questions
+
+- Can a nonisotropic or operator-valued kernel improve the complete paid
+  invoice? First specify actual frames, factor maps and a restored word. No
+  source-bound realization or complete budget is provided here.
+- Can partial-frame returns avoid the full-width premise? First give one exact
+  dirty-input gather/scatter/restore mechanism with legal chronology. Matrix
+  rank alone does not price that mechanism.
+
+Credit Dominik Scholz's E8 geometry/factorization in PR606, DreamingOfClouds'
+H6/Walsh proposal in Discussion1167, and PR1830's complete compensation/encoder
+audit with its retained predecessor lineage. This adds the complete normalized
+odd-function classification beyond the cubic kernel; it claims no priority for
+E8 association algebra or generic rank subadditivity. Ross Richardson, with
+substantial OpenAI Codex assistance in derivation, separate audit and packaging.
+Separate AI implementations are not independent human review or formal-kernel
+verification. Apache-2.0.

research/h6-odd-gram-rank/SOURCES.json

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+{
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "path": "certificates/selected-result.json",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129",
+      "usage": "provenance"
+    },
+    {
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "path": "docs/research/agent-task-catalog.json",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0",
+      "usage": "provenance"
+    },
+    {
+      "git_blob": "d4444d142fb370498bcd9a0abec45e2f5ea0b0c5",
+      "path": "research/niemeier-lattices/code/e8_cubed.py",
+      "sha256": "ce24f397053c0ef5b26881bb7ccfab53fb0a3d762f2de04086fed8f7b41e8234",
+      "usage": "input"
+    },
+    {
+      "git_blob": "e7b1d7354b26c3c5d09c71a141c41414d0ec5545",
+      "path": "research/h6-independent-audit/spectral/README.md",
+      "sha256": "6f789925b5d3d86b51aa0e23ef8012e10ab54da51d89c767d5330ffb09707135",
+      "usage": "provenance"
+    },
+    {
+      "git_blob": "131c19ed91375aa2ad068cb14c02dd4fef971a8b",
+      "path": "research/h6-independent-audit/walsh/README.md",
+      "sha256": "1a74412eda67ffc0dcd0b4cbbbfcfda404a596fee1a54f39e09937130f24ca33",
+      "usage": "provenance"
+    },
+    {
+      "git_blob": "74832d898b6993b677a3c6fbfb30d03e509accaf",
+      "path": "research/h6-compensation-network/contract/THEOREM.md",
+      "sha256": "9752a525d22d815341b70143a06221ad4b3aec0ae8a6447786d88d0291fef45f",
+      "usage": "provenance"
+    },
+    {
+      "git_blob": "9b5bd0a67e327350016a622276306c094218dd0b",
+      "path": "research/h6-compensation-network/k_encoder/README.md",
+      "sha256": "949732ca1bc56b738bf26dcbf517a5d2e24ed9813bede0a0e30e620c9317bdce",
+      "usage": "provenance"
+    }
+  ],
+  "schema_version": 1
+}

research/h6-odd-gram-rank/TASK.json

@@ -0,0 +1,71 @@
+{
+  "actual_scope": "Sharp characteristic-zero ranks of the canonical H6 root-product odd-Gram kernel family, with a conditional full-width scalar factor return-episode cost consequence.",
+  "ai_assistance": "Substantial OpenAI Codex assistance in derivation, separate review and packaging; no human or formal-kernel verification claimed.",
+  "author": "Ross Richardson, with substantial OpenAI Codex assistance",
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "bottleneck_link": "The full-width scalar return bill already exceeds all 30,960 ports for every noncubic member, before compensation. This identifies which kernel/factor premise must change.",
+  "budget_minutes": 240,
+  "evidence": "PROOF.md and two independently implemented registered exact checks.",
+  "open_questions": [
+    {
+      "blocked_by": "No source-bound realization or global budget is supplied here.",
+      "first_step": "Specify actual source/target frames, scalar or operator factor maps and a complete restored word before pricing the full rank histogram.",
+      "question": "Can a nonisotropic or operator-valued kernel reduce the complete paid factor invoice?",
+      "why_it_matters": "Such kernels are outside the odd scalar Gram interpolation premise."
+    },
+    {
+      "blocked_by": "The theorem does not establish an arbitrary decoder floor.",
+      "first_step": "Give one exact dirty-input gather/scatter/restore mechanism and its legal source/target chronology.",
+      "question": "Can partial-frame returns realize a useful kernel without full-width scalar descents?",
+      "why_it_matters": "Matrix rank alone does not price arbitrary partial-frame networks."
+    }
+  ],
+  "original_card": {
+    "acceptance_gate": "An explicit replacement identity, verified on a small complete instance with dirty inputs, priced with ceiling.ceiling(unpaid=...). A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "allowed_variations": [
+      "One template change or one endpoint-contract change.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "bottleneck_link": "The deficit is one coordinate per family minus the centre charge; doubling unpaid coordinates per family scales it directly.",
+    "first_experiment": "Explain where the single unpaid coordinate per family comes from in the hosted template; propose one change and price it. Ideal budget: two unpaid coordinates per family is 3.77x (levers.py).",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "id": "coverage-law-change",
+    "methods": [
+      "State a replacement for the coverage identity (mass = 72W + 64N) and the contract change that realizes it, with its implementation cost.",
+      "Find an obstruction: show which data-endpoint obligation forces exactly one unpaid coordinate in the current template."
+    ],
+    "overlap_terms": [
+      "coverage identity",
+      "unpaid coordinate",
+      "deficit law",
+      "W - 64N"
+    ],
+    "question": "Can a template or call contract leave two or more unpaid coordinates per live family?",
+    "scope_change_triggers": [
+      "Counting a coordinate as unpaid while leaving it unadvanced in the data endpoint.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/idle-axis-hosting/PROOF.md",
+      "certificates/selected-result.json"
+    ]
+  },
+  "outcome": "partial",
+  "priority_revision": 6,
+  "schema_version": 1,
+  "scope_change_reason": "The card asks for a new coverage identity. This bounded alternative-kernel screen instead proves that noncubic odd Gram kernels cannot improve that coverage under explicit full-width scalar factor premises; it leaves partial-frame/operator-valued mechanisms open.",
+  "scope_status": "changed",
+  "stop_rule": "Stop at the complete two-parameter theorem and exact source-bound replay; no further family expansion or global supplier claim.",
+  "task_id": "coverage-law-change"
+}

research/h6-odd-gram-rank/VALIDATION.json

@@ -0,0 +1,23 @@
+{
+  "actions_per_checker": 18,
+  "all_parameter_scope": "Written characteristic-zero interpolation and exhaustive affine-case proof; finite exact evidence supports its coefficients.",
+  "association_entries_per_checker": 57600,
+  "columns_per_action": 28800,
+  "committed_export": "pending",
+  "cost_scope": "Full-width17 scalar rank-one return episodes only; not a universal physical-helper or arbitrary-decoder bound.",
+  "hosted_ci": "pending",
+  "independent_controls": 1,
+  "no_new_kappa": true,
+  "producer_controls": 4,
+  "python": "3.12.12 (main, Oct  9 2025, 11:07:00) [Clang 17.0.0 (clang-1700.6.3.2)]",
+  "rank_minima": [
+    800,
+    9424,
+    20400
+  ],
+  "registered_replay_log_sha256": "b7d88e8b57e014f2cc1a0bb60bf61bddd5f1c796fdd5f13408dfc0c8730f828c",
+  "seconds": 1.6231689999985974,
+  "stage": "precommit portable replay",
+  "status": "PASS_H6_ODD_GRAM_SHARP_RANKS",
+  "utc": "2026-10-18T12:53:44.028905+00:00"
+}

research/h6-odd-gram-rank/VERIFY.json

@@ -0,0 +1,23 @@
+{
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "-Werror",
+        "{package}/verify.py"
+      ],
+      "checks": "Two independent exact derivations: all 57,600 association-square entries, five idempotents/ranks, six eigenforms, complete parameter classification, eighteen 28,800-column actions each, four producer and one independent defect control.",
+      "timeout_seconds": 120
+    }
+  ],
+  "inputs": [
+    "research/niemeier-lattices/code/e8_cubed.py"
+  ],
+  "python": "3.12",
+  "requirements": [],
+  "schema_version": 1,
+  "status": "ready"
+}