Prove exactly three shared A2 rank-six equality fibers (no κ)

research/e8-a2-shared-equality/PROOF.md

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+# Exactly three A2 fibers share each rank-six equality support
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+## Statement and inherited premises
+
+Use the full E8 root-line labels R in F2^9 from PR3288, with leading
+coordinate one. For a 27-root projective A2 fiber F and a binary space V
+of dimension at most six, write
+
+```
+J_F(V) = |R intersect V| + |F intersect V|.
+```
+
+PR3262 and PR3288 supply the individual bound J_F(V)<=32. PR3288 also
+supplies its equality characterization and the count of 27 equality
+six-spaces for each of the 3,360 distinct fibers. These are cited premises.
+This package does not independently audit their proofs or model census.
+
+**Theorem.** If J_F(V)=32 for at least one fiber, then exactly three
+distinct A2 fibers satisfy J_F(V)=32 for this same V. Two distinct fibers
+share at most one equality six-space. Consequently there are exactly
+30,240 equality six-spaces in total, and every fiber shares equality with
+exactly 54 other fibers. A fourth distinct fiber cannot saturate the
+individual bound on that same support.
+
+The theorem concerns this finite full-E8 configuration. It does not say that
+pairwise compatibility of several fibers forces a common equality space.
+
+## A2 fibers in the quadratic model
+
+Write E=F2^8, with the nondegenerate plus quadratic form q and polar form
+B(x,y)=q(x+y)+q(x)+q(y). Root lines correspond to q(x)=1, with label
+(1,x). An A2 root plane gives an anisotropic binary two-space N. Its three
+nonzero vectors all have q=1. For p in N minus {0}, put
+
+```
+K = N-perp,
+H(N,p) = p+K,
+F(N,p) = {x in H(N,p):q(x)=1} minus {p}.
+```
+
+These are the 27-root fibers in the inherited model. To see the parity
+identification, choose two roots of an A2 plane. The mod-two evaluation
+against them has one zero class and three nonzero classes. Each nonzero
+class contains its unique A2 root p and the corresponding large
+projective evaluation fiber. The original real fibers have sizes
+36,27,27,27,1,1,1, as recorded in the pinned predecessor. Distinct pairs
+(N,p) give distinct affine sections and fibers.
+
+The Gauss sum on N is -2. Its orthogonal complement K has Gauss sum -8,
+since the ambient sum is +16. Translation by the orthogonal root p gives
+sum_(x in H(N,p)) (-1)^q(x)=+8. Thus H contains 28 roots, including p.
+The predecessor's seven-dimensional binary span is the homogenization of
+H. Its equality six-spaces have affine sections A=a+L of dimension five,
+contain 16 roots, and avoid p.
+
+## The singular radical of an equality section
+
+Fix one equality section A=a+L inside H(N,p). The hyperplane L in the
+nondegenerate six-space K has a one-dimensional radical <r>. Equivalently,
+L is the kernel of B(-,r) on K, for a unique nonzero r in K. Since the
+form is alternating, r belongs to L.
+
+The affine hyperplane A avoids p, so B(a-p,r)=1. As p is orthogonal to K,
+
+```
+B(a,r)=1.
+```
+
+The root count 16 says that the signed quadratic sum on A is zero. The
+quadratic restriction identity from PR3288, or its direct squared-sum
+calculation, gives
+
+```
+(sum_(x in A) (-1)^q(x))^2
+    = 32 [1+(-1)^(q(r)+B(a,r))].
+```
+
+It follows that q(r)=0. Hence
+
+```
+U = L-perp = N direct-sum <r>
+```
+
+has dimension three, radical <r>, and q(r)=0. Its quotient by the radical
+is the anisotropic plane N.
+
+## Four complementary planes, three admissible projections
+
+Every other fiber F(N',p') containing all 16 roots of this equality
+six-space has A contained in H(N',p'), by the inherited equality
+characterization. Therefore N' is contained in U. An anisotropic two-space
+in U is a complement to <r>. Conversely every complement is anisotropic,
+because q descends to the anisotropic quotient U/<r>.
+
+There are exactly four complements. Relative to U=N direct-sum <r>, they
+are the graphs
+
+```
+N_f = {u+f(u)r:u in N},       f in Hom(N,F2).
+```
+
+For each complement let p_f be the unique orthogonal projection of a
+onto N_f. The condition p_f=0 is that B(a,-) vanish on N_f. On a graph,
+
+```
+B(a,u+f(u)r) = B(a,u)+f(u),
+```
+
+because B(a,r)=1. Exactly one f makes this functional zero. For that
+complement p_f=0, which is not an A2 root and does not define a large
+fiber. For each of the remaining three complements, p_f is nonzero and
+q(p_f)=1.
+
+Each of those three affine sections H(N_f,p_f) contains A. Also p_f is
+not in A, since B(p_f,r)=0 whereas B(x,r)=1 for every x in A. Thus every
+root of A lies in F(N_f,p_f), giving joint score 32. Conversely the
+previous paragraph exhausts every possible fiber. This proves exactly
+three, including the initially chosen fiber.
+
+The 16 roots determine their affine five-space A. Otherwise they would
+fill an affine four-space on which q is constant one. Its direction would
+be a four-dimensional totally isotropic subspace of L. But L has radical
+dimension one and nondegenerate quotient dimension four, so its largest
+totally isotropic subspace has dimension three. This is impossible.
+Homogenizing shows that the roots determine V as well.
+
+## Pairwise uniqueness and the counts
+
+Distinct fibers have distinct seven-dimensional binary spans W_F.
+Indeed the leading-one section determines N-perp and then N, and its
+orthogonal projection determines p. If two distinct fibers share an
+equality V, then V is contained in the intersection of their two spans.
+That intersection has dimension at most six, so it is exactly V. Hence
+the shared equality space is unique.
+
+Double-counting the inherited 3,360 times 27 incidences now gives
+
+```
+number of equality spaces = 3360*27/3 = 30240.
+neighbors of each fiber = 27*(3-1) = 54.
+compatible unordered fiber pairs = 3360*54/2 = 90720.
+```
+
+Pairwise uniqueness is needed in the second count, to prevent counting a
+neighbor twice. These counts do not enumerate arbitrary cliques in the
+pairwise compatibility graph.
+
+## Explicit witness and verification boundary
+
+In the pinned simple-root encoding, one common equality support is
+
+```
+[3,11,27,31,39,47,59,63,71,79,83,87,103,111,123,127].
+```
+
+The three anisotropic normal planes, listed by their nonzero eight-bit
+vectors before homogenization, and their omitted singleton points are
+
+```
+N={84,128,212}, p=128;
+N={84,134,210}, p=210;
+N={82,134,212}, p=82.
+```
+
+The verifier uses q0(x)=x0*x1+x2*x3+x4*x5+x6*x7. The columns
+`[3,6,18,34,76,78,130,208]` give an explicit invertible transport to the
+pinned simple-root quadratic polynomial. All 256 values are compared.
+No P112 membership or caller compatibility is asserted for this witness.
+
+The primal enumeration visits the hyperplanes of each inherited fiber
+span and constructs the new cross-fiber incidence. The separate dual
+enumeration visits every U=N+<r> above, its four complementary planes and
+the four cosets of U-perp with B(a,r)=1. They agree on every support and
+every incident (N,p), not only totals. All 30,240 zero-projection exclusions
+are checked. The code does not import or run predecessor executable code.
+
+The new written proof and finite certificate are author-checked evidence.
+Independent human and formal proof-kernel review are unclaimed. The
+predecessor's individual bounds and E8 binding remain cited dependencies.
+No shortening threshold, physical circuit, paid moment, kappa or complete
+multiplication theorem is derived.
+
+Credit Nick Petalas / PR3288, PR3280 and PR3262, David Arias's quadratic
+model, DreamingOfClouds / PR3154, joaqcoo and the retained E8/P112 lineage.
+Quadratic-form restriction identities and binary symplectic linear algebra
+are classical. The new step is the simultaneous three-fiber classification
+and its incidence consequences.

research/e8-a2-shared-equality/README.md

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+# Exactly three A2 fibers share each rank-six equality support
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped written mathematical proof with an exact finite
+certificate. For the full E8 leading-one root labels, every binary
+rank-six support attaining the individual joint bound 32 for one 27-root
+A2 fiber attains it for exactly three distinct fibers. A fourth distinct
+fiber cannot attain 32 on that same support. Two distinct fibers share
+at most one equality six-space.
+
+There are 30,240 equality supports, each with three incidences. Every one
+of the 3,360 fibers has exactly 54 compatible neighbors. The individual
+bound, its 27 equality spaces per fiber and the E8 model remain credited
+premises from PR3288 and its predecessors. This contribution answers the
+separate simultaneous-equality opening, with a new three-projection proof
+and complete cross-fiber incidence.
+
+[PROOF.md](PROOF.md) gives the theorem, the exact hypotheses and a literal
+16-root attainer. The primal enumeration constructs equality hyperplanes
+inside fiber spans. The separate dual enumeration constructs singular
+radical three-spaces and their affine cosets. They compare every support
+and every incident fiber. They share the specified quadratic model, but
+neither imports predecessor executable code. This is separate
+combinatorial checking, not independent human review.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/e8-a2-shared-equality
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/e8-a2-shared-equality/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/e8-a2-shared-equality/verify.py
+```
+
+The registered replay runs both direct commands. Expected status is
+`PASS_A2_SHARED_EQUALITY`, with 30,240 supports, multiplicity three and
+54 neighbors per fiber. The complete canonical incidence SHA256 is
+`be65d72f039eaf568a318133a63a07166d6cf29e1d1d0ee5254f57741f393b62`.
+The optional `--receipt /absolute/path/outside/checkout.json` writes a
+generated receipt bound to package and input hashes. Receipt files and
+other run outputs belong outside the checkout.
+
+The dual enumeration covers 7,560 radical three-spaces, 30,240 affine
+cosets and every zero-projection exclusion. The primary checker also
+checks the explicit simple-root transport at all 256 points, the witness
+rank, a genuine balanced hyperplane containing its omitted singleton
+whose joint score is 31, and four invalid binary domains. Explicit
+exceptions preserve these checks under optimized Python.
+
+## Scope and source status
+
+The individual-fiber results are cited dependencies, not this deliverable's
+audit targets. [SOURCES.json](SOURCES.json) pins their proof and the exact
+simple-root convention. These inputs are authenticated as bytes, without
+executing their verifiers. Historical catalogue and selected-result pins
+are provenance. No unmerged source commit or network is needed at replay.
+
+This full-E8 theorem supplies no new P112 threshold, pairwise-clique
+classification, actual decoder or physical caller, native program,
+restoration implementation, paid moment, kappa or selected-frontier change.
+The five selected framework assumptions remain. Predecessor scientific
+replays, selected cold replay, Lean and Docker were not run. Independent
+human and formal proof-kernel review are unclaimed.
+
+This is a changed `complex-rank-shape-decoder` card and a scoped
+support-geometry side result. A real caller must specify its fibers and
+show that they require saturation on one common support before this
+obstruction can affect its budget. A finite compatibility improvement
+does not authorize a further construction campaign.
+
+## Implementation bounds
+
+All mathematical domains are fixed subsets of F2^8 or F2^9. The primal
+loops through 3,360 fibers and at most 127 hyperplanes of each. It retains
+30,240 support keys, 90,720 incidences and at most 90,720 compatibility
+pairs. The dual loops through at most 255 radical candidates and 1,120
+normal planes, then eight cosets for each of 7,560 three-spaces. Every
+point scan has 256 iterations. Basis construction has at most 512 input
+vectors and nine pivots. There is no recursion, subprocess, network,
+dynamic code evaluation or unbounded input search.
+
+Power of Ten rule 3 deviation: CPython allocates integers, tuples and
+dictionaries during these bounded finite enumerations. The fixed domains
+bound collection sizes; this is not a fixed-allocation implementation or
+a certified Python allocator bound. All functions are below sixty lines.
+Warning-strict normal and optimized execution are the available local
+diagnostics. No separately installed linter or formal proof checker is
+claimed.
+
+## Open next questions
+
+- Which pairwise-compatible triples actually share an equality space?
+  Start with the explicit compatibility graph and the support triples.
+  No clique classification is provided here.
+- Does a named current decoder or caller require four distinct fibers to
+  attain their individual maxima on the same rank-six support? First bind
+  its literal support and complete paid invoice. No such caller is supplied.
+
+Discovery stopped at the first verified simultaneous-compatibility result.
+
+## Credit and coordination
+
+Credit Nick Petalas / PR3288, PR3280 and PR3262, David Arias's quadratic
+model, DreamingOfClouds / PR3154, joaqcoo and the retained E8/P112 lineage.
+Quadratic restriction and binary symplectic linear algebra are classical.
+No global literature-priority claim is made. PR3302 concerns outside-span
+capacity 30 and supplies no simultaneous equality result used here.
+
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19371205)
+states the delivered question. Base main is
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision eight,
+selected PR6030. The overlap snapshot covered 522 open PRs, 6,536
+Discussion records and all 8,917 tracked local notes. Main/merged object
+and general-form searches, account history and current reservations were
+also checked. Arbitrary historical diffs and unpublished work remain
+coverage limits. Metadata and archive retention do not imply scientific
+acceptance or a selected-frontier update.

research/e8-a2-shared-equality/SOURCES.json

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+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
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