research: classify the full 4080-owner centre kernel using sheaves and integer fibres

.github/workflows/p112-common-slice.yml

@@ -15,9 +15,9 @@ jobs:
     if: github.repository == 'CrocSwap/integer-mult-bounds'
     runs-on: ubuntu-24.04
     timeout-minutes: 20
-    env:
-      TMPDIR: ${{ runner.temp }}/p112-common-slice
     steps:
+      - name: Initialise P112 work directory
+        run: echo "TMPDIR=$RUNNER_TEMP/p112-common-slice" >> "$GITHUB_ENV"
       - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           fetch-depth: 0

research/centre-sheaf-fibre/PROOF.md

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+# Exact full kernel on the 4,080-owner block
+
+Clear1nsight, with substantial OpenAI assistance. Apache-2.0.
+
+This is a scalar-information theorem for the inherited selected calls-0/4
+observation model. It is not a native programme, physical sharing construction,
+recursive tariff, or multiplication theorem. The finite premises below were
+reconstructed in fork-hosted CPython 3.12.15 runs at scientific head
+`403ef45de40e41baaf8b1f4cfddaf3ba1d57af1c`. The original source archive and
+predecessor implementations are unchanged.
+
+## 1. Fixed observation contract
+
+Let R be the selected 8-by-112 rational reader matrix and let rho_p be its
+112 involutory owner routes. The complete routes act on 73 binary coordinates;
+they preserve a 17-coordinate active space and fix 56 spectators. Computation
+on the active block is therefore faithful, not a projection that identifies
+otherwise different routes.
+
+Set a=0, b=6, c=85 and
+
+    H = <rho_a rho_b, rho_a rho_c>.
+
+Arrays x(g,p) are arbitrary rational scalars on H times the 112 ports, and
+are extended by zero to all other owners. The two observations are
+
+    Sx(g)  = sum_p R_p x(g,p),
+    4Tx(z) = sum_p R_p x(z rho_p,p).
+
+Here R_p is a column. Every touched output owner z is retained, including
+owners outside H. Dropping those output equations changes the problem.
+The inherited source slice and common-frame/constant-payload interpretation
+remain assumptions of its physical application. No statement about arbitrary
+704-field native states or the complete chronological word follows.
+
+The separately reconstructed finite premises are
+
+* |H|=4,080 and rank_Q(R)=8.
+* R_a-R_b-R_c=0, and these three columns have rank two.
+* The right cosets H rho_p partition the ports into the single class
+  {0,6,85} and 109 singleton classes. Every singleton column is nonzero.
+* The undirected compatibility graph on H, with right multiplications by
+  rho_a rho_b and rho_a rho_c, is connected. It has 8,160 edges and a
+  certified 4,079-edge spanning tree.
+
+The primary certificate checks all 112^2 ordered coset comparisons, using
+binary column matrices and rational ranks. The second implementation uses
+its own archive parser, forward coefficient propagation, binary row matrices,
+left-generated group closure, and nonzero minors modulo 101. Integrality of R
+is checked before the modular rank test. A nonzero minor modulo 101 proves
+the corresponding rational lower rank; the displayed exact relation gives
+the matching upper rank for the triple.
+
+## 2. Coset elimination theorem
+
+For involutions, H rho_p=H rho_q exactly when rho_p rho_q belongs to H.
+The coordinates contributing to an output z belong to a single such class.
+For a class C, an output equation has the form
+
+    R_C (x(z rho_p,p))_{p in C} = 0.
+
+If R_C has independent columns, all its incoming coordinates vanish. Every
+source coordinate x(g,p) appears at the output z=g rho_p, so this eliminates
+that entire port class, not just one output sample.
+
+For the selected H, 109 singleton classes consequently eliminate
+109*4,080=444,720 variables. The only possible nonzero coordinates are those
+on ports a,b,c. The source equations then force, uniquely at each g,
+
+    (x(g,a),x(g,b),x(g,c)) = alpha(g) (1,-1,-1).
+
+At z=g rho_a, the remaining output 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 and R_c forces all three amplitudes to agree. Connectivity
+makes alpha constant on H. Conversely, any constant amplitude satisfies every
+source and output equation. Thus, with v equal to (1,-1,-1) on these three
+ports at every owner of H and zero elsewhere,
+
+    ker(S) intersect ker(T) = Q v.
+
+This establishes the FULL 112-port kernel. The predecessor's existence of v
+and its single-triple census alone did not establish this equality. Merely
+checking which individual routes lie in <rho_a,rho_b,rho_c> is insufficient;
+the outgoing right-coset comparisons are the essential additional test.
+
+## 3. Cellular sheaf interpretation
+
+Take the bipartite graph of source owners and touched output owners, with a
+port edge from g to g rho_p. Give source vertices stalk ker(R), an output
+vertex in class C stalk ker(R_C), and every edge stalk Q. Restriction maps
+are the corresponding coordinate maps. A global section is exactly a choice
+of all x(g,p) satisfying both observation equations.
+
+The independent output stalks are zero. Eliminating their incident coordinates
+reduces every remaining source and output stalk to the same one-dimensional
+triple relation. Dividing each nonzero restriction by its relation coefficient
+makes edge compatibility ordinary equality. The reduced graph is connected,
+so its global sections form one copy of Q.
+
+Equivalently, eliminating output amplitudes leaves the constant sheaf on the
+4,080-vertex compatibility graph. Its zeroth cohomology has dimension one.
+This is a local-to-global calculation, not a claim that scalar consistency
+certifies chronological access, native phases, or a physical implementation.
+No assertion about higher cohomology of the unreduced sheaf is needed.
+
+## 4. Sharp information dimension and proper subblocks
+
+There are 112*4,080=456,960 scalar variables. S has rank 8*4,080=32,640, hence
+its kernel has dimension 424,320. The theorem gives
+
+    rank(T restricted to ker(S)) = 424,319,
+    rank(stack(S,T)) = 456,959.
+
+Suppose extra LINEAR observations L allow recovery of all T from (S,L), even
+by an otherwise arbitrary nonlinear decoder. Equal (S,L) observations must
+give equal T, so ker(S) intersect ker(L) is contained in ker(T). Rank-nullity
+on ker(S) therefore requires at least 424,319 additional scalar observations.
+A linear quotient of ker(S) by Qv attains that count abstractly. This existence
+statement supplies neither an efficient decoder nor a recursive cost.
+
+For every proper subset B of H, extend a supported common-kernel vector by
+zero to H. It must equal t v. Since v is nonzero at every owner of H and B
+omits at least one owner, t=0. Therefore all proper subblocks have zero common
+kernel and require 104|B| additional linear observations under the same
+whole-output contract. The result also holds on left translates of H.
+It does not decide equally sized arbitrary owner sets, unions of several
+cosets, other triples' subgroups, or the global support minimum.
+
+## 5. Primitive integer fibres and their limitation
+
+R is integral. Stack S and 4T into an integer matrix A. Since v has a coordinate
+one, every integer vector in its rational kernel is an integer multiple of v.
+Thus ker_Z(A)=Zv. Every nonempty nonnegative integer fibre is precisely
+
+    {n0+t v : t is an integer and lower <= t <= upper},
+    lower = max_{v_i=1}(-n0_i),
+    upper = min_{v_i=-1}(n0_i).
+
+All zero coordinates of v remain fixed. Unit moves by v and -v connect this
+interval, so {v} is a Markov basis, with both move directions permitted.
+For the polynomial-to-Laurent-monomial map specified by A, the toric ideal
+is principal, generated by z^(v+) - z^(v-). Indeed, every integer kernel
+relation is t v, and a difference of positive integer powers factors by that
+binomial. This is classical rank-one lattice-ideal reasoning.
+
+The primitive move has 4,080 positive and 8,160 negative unit entries, hence
+support 12,240 and sum(v)=-4,080. Adding a fixed total-count row therefore
+kills the kernel over Q and Z. More generally, an extra linear resource row
+w retains it exactly when w*v=0. This explicitly prevents interpreting an
+information-preserving coefficient move as a free legal scheduling move.
+No operation-count, ownership, or conservation constraint is silently dropped.
+These fibre moves preserve both observations at each move. A physical programme
+may temporarily leave the fibre and later restore its outputs; such programmes
+are not excluded by this integer statement.
+
+## 6. Decision, evidence, and credit
+
+The known witness spans all invisible directions on this fixed block. Its
+abstract information saving is exactly one out of 424,320 coordinates, not
+an unexplored large family of savings. No recursive child is shown removable.
+The decision is NARROW and COMPLETE for this information question. A larger
+construction requires a named changed block or consumer, actual returned-state
+contract, and a favourable fully charged cost screen before expansion.
+
+The source-dependent primary and independent implementations passed normally
+and under -O on GitHub's fork-hosted Ubuntu 24.04.5 / CPython 3.12.15 runner.
+The source archive, both relevant member hashes, query ordering, H=R/4,
+faithful active restriction, coset partition, and graph premises were checked.
+The independent direct residual calculation checks all 4,080 touched outputs
+of v and detects exactly three failures when one owner is removed. Eight
+small-group tests cover full matrices, all six nonempty proper subblocks,
+512 binary coefficient assignments, signs, routes, and the total-count row.
+These are separate same-author implementations, not independent human review.
+The committed receipt hook additionally compares typed results and rejects
+receipt/JSON mutations. Its final publication replay is recorded on the PR.
+
+Credit Matarener/#3656 for the subgroup and witness; David Charbonneau/#3147
+for the investigation; Ross Richardson/#3071 for the model and extractor;
+Sagar Thakrar/#3291 and Sudarsh Chaturvedi/#3168 for the corrected graph/census
+lineage; ywz65 and Aaron Wang/navinor for the observation/centre lineage; and
+all original selected-source notices. The earlier circle-elision proposals
+are not reopened; #3151 and #3194 retain their distinct obstruction credit.
+
+The mathematical formulation uses established theory from Jakob Hansen and
+Robert Ghrist, Toward a spectral theory of cellular sheaves, Journal of Applied
+and Computational Topology 3 (2019), 315-358, DOI 10.1007/s41468-019-00038-7;
+and Persi Diaconis and Bernd Sturmfels, Algebraic algorithms for sampling from
+conditional distributions, Annals of Statistics 26(1) (1998), 363-397,
+DOI 10.1214/aos/1030563990. No literature-wide novelty is claimed.

research/centre-sheaf-fibre/README.md

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+# Full centre kernel through sheaves and integer fibres
+
+A source-bound contribution to the #3147 investigation, answering the fixed
+4,080-owner full-kernel question in Matarener's #3656. Clear1nsight, with
+substantial OpenAI assistance. Apache-2.0.
+
+## Result
+
+For the exact subgroup H generated by the even route products on ports
+(0,6,85), the common kernel of the old and routed later observations on ALL
+112 ports is exactly one-dimensional. Its sole primitive integer generator
+is the already known (1,-1,-1) triple repeated at every owner. What is new is
+the full-kernel classification, the sharp information count for this block
+and all its proper subblocks, and the primitive integer-fibre description.
+
+The local-to-global reduction replaces a 456,960-variable kernel problem with
+109 independent outgoing port classes and one connected 4,080-vertex constant
+sheaf. It certifies 424,319 additional linear observations as necessary and
+sufficient ABSTRACTLY, given the old observations. Every proper subblock of H
+still requires 104 observations per owner. Arbitrary equally sized owner sets
+are outside this theorem.
+
+The integer fibre is a one-parameter interval connected by one primitive move.
+That move touches 12,240 coefficients and changes their sum by -4,080. An
+additional fixed-count constraint destroys it. Thus algebraic statistics does
+not turn this information identity into a free physical schedule.
+
+See PROOF.md for the full hypotheses and argument, and RESULT.json for the
+exact certificate. No scalar information count is called a recursive tariff.
+
+## Evidence and reproduction
+
+The unmodified mathematical kernels at head
+403ef45de40e41baaf8b1f4cfddaf3ba1d57af1c passed the actual registered workflow on
+Ubuntu 24.04.5 / CPython 3.12.15, including normal and optimised source-dependent
+primary and independent replays, all eight generic tests, and unchanged input
+checks. The run is
+https://github.com/Clear1nsight/integer-mult-bounds/actions/runs/39165978787 .
+The scientific head and its synthetic PR-merge checkout are distinguished in
+the run logs. Later publication checks are recorded on the fork PR.
+
+The primary reuses the pinned PR3071 reverse-covector source extractor and
+uses rational elimination and binary column products. The second imports no
+primary or predecessor executable, parses the archive itself, propagates
+coefficients forward, uses binary row products and modular minors, and directly
+checks all routed witness residuals. Both reconstruct identical geometry with
+SHA256 79515573d8231a59ece664a2ec421983989c9925e1ccab1ca4c71c7cfa3ea2be.
+
+From a repository checkout with the two pinned inputs present, run
+
+```sh
+python3.12 -I -B scripts/research_package.py --package research/centre-sheaf-fibre
+```
+
+The registered hook runs the eight small-group tests normally and under -O,
+then two fresh-source implementations and committed typed-receipt checks in
+both modes. It uses only the standard library. Direct source-dependent replay
+is also available through `receipt_check.py`. The explicit `--controls-only`
+option tests JSON/receipt fixtures and does NOT claim source verification.
+
+The direct primary and independent `--geometry` paths accept an explicitly
+labelled derived export; they do not pretend to re-extract the archive. The
+registered committed-receipt hook requires both actual archive paths instead.
+
+## Scope and stopping decision
+
+This is an exact rational scalar theorem under the inherited selected calls-0/4
+observation contract, not a native payload or full chronological programme.
+Physical frames, acquisition, dirty-state restoration, precision, ownership,
+hosting, and the complete supplier invoice remain unproved here. No paid child
+is deleted and the selected frontier is unchanged. No human or formal proof
+kernel has independently accepted this theorem. The two implementations share
+the same archived mathematical source and author.
+
+Decision NARROW / COMPLETE. The full kernel on this H is settled, so another
+search for more directions inside the same block is not justified. Any new
+physical investigation needs a changed contract and a complete cost gate.
+
+## Open next questions
+
+1. Can a union of distinct H cosets introduce additional dependent outgoing
+   classes and a materially larger kernel? First prove a changed coset
+   incidence and compute its compatibility rank. Do not infer it from this
+   one-block result, or count a scalar coordinate as a saved child.
+2. Does a named downstream consumer permit a weaker observation than the
+   whole later-query field? State every required output, allowed source read,
+   and restoration condition before constructing a quotient.
+3. Can a useful integer move preserve the actual resource rows of such a
+   caller? First specify those rows and test w*v=0. The fixed-count row fails
+   for the present move, so no scheduling claim is made.
+
+## Provenance, overlap, and publication status
+
+Canonical scientific base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority
+revision 6. Both live input pins are ancestral. Frozen sources are untouched.
+Relevant current PRs were read and targeted searches performed; complete
+Discussion and arbitrary-diff coverage is not established. Upstream claim and
+PR writes were rejected by the connected integration with HTTP 403. The public
+fork contribution is https://github.com/Clear1nsight/integer-mult-bounds/pull/1 .
+It is not an upstream submission. Neither upstream nor fork main was changed.
+
+Matarener/#3656 retains subgroup/witness credit; dcharb78/#3147 retains the
+research question; Ross Richardson/#3071 retains the observation model and
+extractor; Sagar Thakrar/#3291 and Sudarsh Chaturvedi/#3168 retain their corrected
+graph/census work. Credit ywz65, Aaron Wang/navinor, and the retained selected
+native, geometry, and compiler lineage. #3151/#3194's circle obstructions are
+not repeated as new. Hansen/Ghrist and Diaconis/Sturmfels retain classical
+sheaf and Markov-basis credit. Full references and qualifications are in
+PROOF.md. No separate agent, human, or formal-kernel review is claimed.

research/centre-sheaf-fibre/RESULT.json

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+{
+  "active_dimension": 17,
+  "additional_linear_observations_required": 424319,
+  "class_size_histogram": {
+    "1": 109,
+    "3": 1
+  },
+  "compatibility_edges": 8160,
+  "compatibility_vertices": 4080,
+  "constant_count_constraint_kills_kernel": true,
+  "deficient_coset_classes": [
+    {
+      "ports": [
+        0,
+        6,
+        85
+      ],
+      "rank": 2
+    }
+  ],
+  "full_joint_kernel_dimension": 1,
+  "full_variables": 456960,
+  "geometry_sha256": "79515573d8231a59ece664a2ec421983989c9925e1ccab1ca4c71c7cfa3ea2be",
+  "independent_dropped_owner_nonzero_outputs": 3,
+  "independent_nonzero_later_residuals": 0,
+  "independent_old_rank_mod101": 8,
+  "old_kernel_dimension": 424320,
+  "old_rank": 32640,
+  "owners": 4080,
+  "ports": 112,
+  "primitive_move_negative_entries": 8160,
+  "primitive_move_positive_entries": 4080,
+  "primitive_move_support": 12240,
+  "right_coset_class_count": 110,
+  "right_coset_classes_sha256": "af7b54230d4fd047858c9c6879fdf0d39249d725a61823a84fff8d6a73e982d3",
+  "schema_version": 1,
+  "spanning_tree_edges": 4079,
+  "spanning_tree_sha256": "0e821c4c81b3ae01dcee422e4c70e1ece3dbae908ae7ee9b8f77f82ef643e400",
+  "status": "PASS_CENTRE_SHEAF_COMMITTED_RECEIPT",
+  "subgroup_sha256": "edded9e0dcd065d7dfb9a5ece868f1777f9030ae14e3ea0adea060bccba90cc0"
+}

research/centre-sheaf-fibre/SOURCES.json

@@ -0,0 +1,10 @@
+{
+  "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"}
+  ]
+}