Validate sheaf reduction of the full 4080-owner centre kernel

research/centre-sheaf-fibre/PROOF.md

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+# Local-to-global reduction for a fixed subgroup block
+
+A general exact lemma, followed by a finite application that must be checked.
+All observations are scalar maps over Q. The same conclusions hold over
+characteristic-zero extensions. No physical caller or recursive cost is proved.
+
+## Coset elimination
+
+Let G contain involutions rho_p and a finite subgroup H. Let r_p be the columns
+of a full-row-rank d-by-n rational matrix R. Arrays x(g,p) are supported on H
+and extended by zero outside H. Set
+
+    Sx(g) = sum_p r_p x(g,p),
+    4Tx(z) = sum_p r_p x(z rho_p,p).
+
+Partition ports by H rho_p=H rho_q, equivalent to rho_p rho_q in H. For a class
+C and an output z in its coset, only C contributes. If R_C has independent
+columns, Tx(z)=0 forces each incoming coordinate to zero. Every source/port
+coordinate appears at z=g rho_p, so all coordinates in such classes vanish.
+
+Suppose the only deficient class is P={a,b,c}, rank two, with the unique
+relation r_a-r_b-r_c=0. The source equations force
+
+    x(g,a)=alpha(g), x(g,b)=x(g,c)=-alpha(g).
+
+At z=g rho_a, the later equation is
+
+    r_a alpha(g)-r_b alpha(g rho_a rho_b)-r_c alpha(g rho_a rho_c)=0.
+
+Independence of r_b,r_c forces equality of these three amplitudes. If the two
+even route products generate H, alpha is constant. Conversely every constant
+works. Hence the full common kernel is Qv, where v equals(1,-1,-1) on P at
+every owner of H and is zero on all other ports. This uses every port; absence
+of other individual routes from a larger group cannot replace the coset test.
+
+## Cellular sheaf
+
+Use the bipartite graph of source owners and touched output owners. Source
+vertex stalks are ker R; an output vertex for class C has stalk ker R_C. Each
+port edge has stalk Q, with restriction maps given by the corresponding source
+and output coordinates. Global sections are exactly ker S intersect ker T.
+Independent output stalks vanish and eliminate their incident coordinates.
+Remaining triple stalks have dimension one. Normalising by their nonzero
+coefficients makes every edge restriction equality. Connectivity therefore
+gives a one-dimensional global-section space. No higher-cohomology assertion
+about the unreduced sheaf is needed.
+
+## Information and proper subblocks
+
+S has rank dh. Thus rank(T restricted to ker S)=(n-d)h-1. If linear additional
+observations L permit recovery of T from(S,L), even by a nonlinear decoder,
+then ker S intersect ker L is contained in ker T. Consequently L needs at
+least(n-d)h-1 scalar coordinates; a linear quotient attains this abstractly.
+This does not give a native programme or tariff.
+
+Any kernel vector on a proper B subset H extends by zero to H, hence is a
+multiple of v. Because v is nonzero at every owner, that multiple must be
+zero. The proper-subblock requirement is therefore(n-d)|B|. Left translates
+of H have the same result, but arbitrary equally sized owner sets do not
+follow from it.
+
+## Primitive integer fibres
+
+When R is integral, stack S and4T into an integer matrix A. Since v has an
+entry one, ker_Z A=Zv. A nonempty nonnegative integer fibre equals
+
+    {n0+t v: t integer, lower<=t<=upper},
+    lower=max_{v_i=1}(-n0_i), upper=min_{v_i=-1}(n0_i).
+
+The single primitive move v in either direction connects every such fibre.
+The associated toric ideal is generated by the binomial for v-positive and
+v-negative. This is a coefficient-fibre statement, not an operation-scheduling
+model. In particular sum(v)=-h, so imposing a fixed total-count row kills the
+kernel. Extra resource invariants cannot be silently discarded.
+
+## Proposed finite application
+
+PR3071/3147/3656 give n=112,d=8,h=4080. If the full coset certificate passes,
+there are456960 variables, old rank32640, common kernel dimension one, and
+424319 additional linear observations are necessary and sufficient abstractly.
+The move has4080 positive and8160 negative entries. Source-dependent execution
+is pending in this draft. Physical phases, payload covariance, acquisition,
+chronology, dirty restoration, precision, hosting and a favourable complete
+supplier bill remain outside this scalar theorem.
+
+## References and credit
+
+Hansen and Ghrist, Toward a spectral theory of cellular sheaves, Journal of
+Applied and Computational Topology3 (2019),315-358, DOI10.1007/s41468-019-00038-7.
+Diaconis and Sturmfels, Algebraic algorithms for sampling from conditional
+distributions, Annals of Statistics26(1) (1998),363-397, DOI10.1214/aos/1030563990.
+These established theories motivate the formulation; they do not prove the
+source-specific premises.
+
+Credit Matarener/3656 for the4080-owner subgroup and witness, dcharb78/3147 for
+the programme, Ross Richardson/3071 for the observation model and extractor,
+Sagar Thakrar/3291 and Sudarsh Chaturvedi/3168 for the corrected graph/census
+lineage, and all selected source notices. Clear1nsight with substantial OpenAI
+assistance; Apache-2.0. No separate human or formal-kernel acceptance claimed.

research/centre-sheaf-fibre/README.md

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+# Centre sheaf and primitive integer fibre
+
+Draft source-bound investigation of PR3147's centre-sharing lineage, answering
+PR3656's full-kernel question on its explicit4080-owner subgroup. This does not
+supply a physical shared centre, a recursive tariff, or a new kappa.
+
+## Precise test
+
+The known triple(0,6,85) gives one common-kernel vector. Group all112 port
+routes by right cosets of its even subgroup H. Full-column-rank output classes
+force their input coordinates to zero. If the only deficient class is the
+known rank-two triple, remaining compatibility forces one constant amplitude
+across H. This would prove that the known vector spans the entire common
+kernel, even when all112 ports can vary.
+
+The conditional reduction is in PROOF.md. The actual coset certificate must
+pass before the application is claimed. The verifier deliberately rejects any
+additional deficient class instead of assuming it away.
+
+## Verification boundary
+
+Eight generic tests passed normally and with -O under local CPython3.13.5.
+They compare an independently assembled full matrix on a noncommutative small
+group, every proper support subset, and all binary nonnegative fibre points.
+They are not a substitute for the actual selected geometry.
+
+The container cannot fetch the selected archive. Source-dependent execution is
+registered for the repository's existing Python3.12 workflow. No successful
+source-dependent execution is claimed in this draft.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/centre-sheaf-fibre
+```
+
+`verify.py --emit-geometry` exports derived rational columns and binary routes
+after source authentication. `--geometry FILE` is explicitly arithmetic-only.
+Default commands preserve inputs. No predecessor heavy census is run.
+
+## Open next questions
+
+TASK.json records the exact dimension test and later physical-contract question.
+Stop after a useful source-bound result. Information dimension is not a native
+child rank, paid storage quantity, or permission to discard borrowed state.
+
+## Credit
+
+Matarener/PR3656 supplies the actual subgroup and witness. David Charbonneau/
+dcharb78/PR3147 supplies the dependent-column and route programme. Ross
+Richardson/PR3071 supplies the observation model and authenticated extractor
+reused unchanged. Sagar Thakrar/PR3291 supplies the corrected small-block floor;
+Sudarsh Chaturvedi/PR3168 supplies separate route checks. Aaron Wang/navinor/
+PR2391, ywz65, and the retained selected geometry/compiler lineage keep credit.
+
+Sheaf and integer-fibre methods are classical, not invented here. Prepared by
+Clear1nsight with substantial OpenAI assistance. No separate agent, independent
+human, proof-kernel or full multiplication verification is claimed. Apache-2.0.
+Existing sources and frontier are unchanged. The upstream coordination-comment
+attempt received HTTP403 and is not a published reservation.

research/centre-sheaf-fibre/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {"path": "certificates/selected-result.json", "usage": "provenance", "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81", "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"},
+    {"path": "docs/research/agent-task-catalog.json", "usage": "provenance", "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5", "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"},
+    {"path": "research/centre-block-repair-information/independent.py", "usage": "input", "git_blob": "d2db50ec7a595b1a1a87b7843fd5ca043c43fd25", "sha256": "b298ff3902c14db11e4ed057259785efc185e8ae3bb39c0d1c432e2340ceb4ea"},
+    {"path": "research/community-frontier1094-audit/selected-replay.tar.gz", "usage": "input", "git_blob": "c24daf067f9c543421d89ca2eeaa4eb78a57fb6f", "sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6"}
+  ]
+}

research/centre-sheaf-fibre/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "centre-path-sharing",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "original_card": {
+    "id": "centre-path-sharing",
+    "question": "Can the 64-per-call centre path (eight rank-8 COPY/ERASE totals through the private temporary) be shared across calls or shrunk?",
+    "starting_points": ["docs/research/research-family-status.md", "research/complex-local-word-ceiling/README.md", "research/split-centres-global/proofs/PR386-BIT-GEOMETRY.txt", "certificates/selected-result.json"],
+    "methods": ["Derive a fully charged favorable relaxation for sharing one centre path across k calls, including restoration, hosting and precision.", "Prove a lower bound on copied information per call for the selected decoder contract."],
+    "acceptance_gate": "Show the changed centre charge and copy profile per call with every added restoration or hosting cost charged, and its ceiling via ceiling.ceiling(centre=...). 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.",
+    "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."},
+    "first_experiment": "Write the per-call centre charge and copy children for sharing across k = 2 calls, charge the sharing overhead, and screen it; stop this route if even the ideal sharing gains under 10%. Ideal budgets: free 2.80x, shared by two 1.90x (levers.py).",
+    "bottleneck_link": "The centre path consumes 64 of about 99.6 unpaid coordinates per call, i.e. about 64% of the raw deficit.",
+    "allowed_variations": ["One sharing pattern across a fixed number of calls.", "One reduced copied-total contract for the existing decoder.", "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."],
+    "scope_change_triggers": ["Removing copies without a replacement contract for the decoder's copied totals.", "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."],
+    "overlap_terms": ["centre path", "copied totals", "private temporary", "shared centre", "batching"]
+  },
+  "scope_status": "changed",
+  "actual_scope": "User-assigned sheaf and integer-fibre approach to PR3147, refined to the complete112-port common observation kernel on PR3656's4080-owner subgroup. Test exact coset elimination and global compatibility before claiming a dimension.",
+  "scope_change_reason": "The simple circle route already has PR3151 and PR3194 obstructions. PR3656 supplies a concrete common-kernel witness but leaves its full kernel dimension open. This closes a different information obligation rather than repeating a circle cancellation or the single-triple census.",
+  "bottleneck_link": "Determines whether the first concrete large block has useful hidden information capacity or only its known one-dimensional perturbation. Information coordinates are not recursive tariffs.",
+  "budget_minutes": null,
+  "stop_rule": "Stop after one complete source-bound local-to-global kernel certificate or a counterexample to the proposed reduction. No physical supplier expansion without a changed favourable full boundary and cost ledger.",
+  "outcome": "partial",
+  "evidence": "Eight generic finite-group, matrix and nonnegative-fibre control tests pass normally and with -O on local CPython3.13.5. Actual archive-dependent subgroup and coset reduction await execution; draft status until verified.",
+  "author": "Clear1nsight",
+  "ai_assistance": "Substantial OpenAI assistance in proof, implementation, tests and writing. No separate agent, independent human or proof-kernel review is claimed.",
+  "open_questions": [
+    {"question": "Does exact outgoing-coset elimination reduce all112 ports on the named block to the known triple?", "why_it_matters": "It decides the full common-kernel dimension without an enormous dense matrix.", "first_step": "Run the authenticated source extractor and finite coset/rank certificate, then independently verify the derived geometry.", "blocked_by": "Archive-dependent execution not yet completed."},
+    {"question": "Can a different owner block or output contract yield an affordable physical sharing mechanism?", "why_it_matters": "A scalar kernel does not remove a paid recursive call.", "first_step": "Specify the actual source and return interfaces, chronology and all cost allowances before expanding search.", "blocked_by": "No native acquisition, restoration, hosting or complete supplier bill is supplied."}
+  ],
+  "coordination": {"assigned_pr": 3147, "specific_predecessor_pr": 3656, "specific_predecessor_head": "69e745992b966c647e10696b980cb4c96d7ebeed", "entry_canonical_main": "20ecf2891d80b5acf18d980b2164fb44ea7faa90", "overlap": "Targeted live PR searches for sheaf, centre, kernel and the named predecessors; current PR3656 had no comments. Not a complete Discussion or arbitrary-diff census.", "claim_attempt": "Upstream PR3656 claim comment rejected by integration with HTTP403. Not published.", "environment": "Container DNS unavailable; no local repository/selected archive. Fork branch is writable. Registered source-dependent CI, if available, will be distinguished from local generic tests."},
+  "family_gate": {"decision": "narrow", "reason": "One bounded information theorem and primitive fibre result only; no numerical saving, no full supplier or selected-frontier change."}
+}

research/centre-sheaf-fibre/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": ["research/centre-block-repair-information/independent.py", "research/community-frontier1094-audit/selected-replay.tar.gz"],
+  "commands": [
+    {"argv": ["{python}", "-I", "-B", "{package}/test_sheaf.py"], "timeout_seconds": 60, "checks": "Independent small matrix, proper-subblock and bounded nonnegative-fibre controls; not actual physical semantics."},
+    {"argv": ["{python}", "-I", "-B", "{package}/verify.py", "--emit-geometry"], "timeout_seconds": 300, "checks": "Authenticate original selected archive and credited source extractor, test all112 ports and emit derived geometry."},
+    {"argv": ["{python}", "-I", "-O", "-B", "{package}/verify.py"], "timeout_seconds": 300, "checks": "Repeat source-bound primary guards under optimization."},
+    {"argv": ["{python}", "-I", "-B", "{package}/independent.py"], "timeout_seconds": 120, "checks": "Separate raw-archive parser, sparse forward coefficient propagation, row-matrix routes, left group closure, modular minors and direct routed residuals."},
+    {"argv": ["{python}", "-I", "-O", "-B", "{package}/independent.py"], "timeout_seconds": 120, "checks": "Repeat the independent source and algebra audit with explicit checks under optimization."}
+  ]
+}