Prove exact unequal query capacity for every odd cycle

research/odd-cycle-query-capacity/PROOF.md

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+# Every unequal odd-cycle query mask
+
+JoyboyBrian, substantial OpenAI Codex assistance; Apache-2.0. Credit Tekalign
+PR3362's C5 proof/interface, Nick Petalas PR4909's bipartite interface, and
+classical finite linear-programming duality and capacitated matching methods.
+No literature-priority claim is made for odd-cycle coding or matching.
+
+## Contract and exact answer
+
+Let n=2k+1, k>=2, and d_i be any nonnegative integers, S=sum d_i. Over any
+field F take an S by S fitting matrix M with M_ii=I_(d_i), and M_ij=0 for
+distinct nonadjacent vertices of C_n. The two allowed ordered blocks at every
+edge are arbitrary rectangular matrices. They need not be symmetric, invertible,
+commuting, or simultaneously diagonalizable. A zero-width block is empty.
+Let alpha be the largest sum of d_i on an independent vertex set. Then
+
+    min rank(M) = max(alpha, ceil(S/2)).                 (1)
+
+For linear query factorizations M=AB this is the exact minimum intermediate
+field-coordinate dimension. It is not a native call count, physical COPY cost,
+Gaussian rank, admitted supplier, or multiplication exponent. No assertion is
+made about nonlinear encoders, changed zero masks, or a physical embedding of
+these cycles inside the P112 geometry.
+
+## Lower bound for arbitrary directed blocks
+
+Factor M=AB through r=rank(M) dimensions. Let B_i be its i-th column block,
+A_i its i-th row block, and U_i=im B_i. Because A_i B_i=I, B_i is injective,
+dim U_i=d_i, and A_i is injective on U_i. Because both ordered forbidden blocks
+are zero, A_i annihilates the sum of all nonneighbor U_j. Thus
+
+    U_i intersect sum_(j nonadjacent to i, j!=i) U_j = 0. (2)
+
+For every independent set the U_i form a direct sum, by applying A_i to a
+putative relation. Therefore r>=alpha.
+
+Put t_i=dim(U_i intersect U_(i+1)). The two subspaces U_i intersect U_(i-1)
+and U_i intersect U_(i+1) lie in U_i and have zero intersection: i-1 and i+1
+are nonadjacent when n>=5. Hence
+
+    t_(i-1)+t_i <= d_i;  2 sum_i t_i <= S.               (3)
+
+For each i take J_i={i,i+1,i+3,i+5,...,i+2k-1} modulo n. The subgraph on
+J_i consists of exactly the edge {i,i+1} and k-1 isolated vertices. Applying
+each isolated A_j kills all the other subspaces. Their sum is therefore direct
+with U_i+U_(i+1), so
+
+    r >= sum_(j in J_i) d_j - t_i.
+
+Every vertex lies in exactly k+1 of these n rotated sets. Sum the inequalities
+and use (3):
+
+    n r >= (k+1) S - sum_i t_i >= (k+1/2) S = n S/2.
+
+Integer dimensions give r>=ceil(S/2), in every characteristic. This proves the
+entire lower bound, including arbitrary noncommuting payload blocks. Finite
+matrix tests are corroboration of this argument rather than its justification.
+
+## Exact integral sharing
+
+An edge count x_i>=0 reserves x_i distinct coordinates at both endpoints.
+Private count p_i=d_i-x_(i-1)-x_i must be nonnegative. One shared coordinate
+uses the rank-one all-ones two-by-two block; one private coordinate uses a
+scalar identity. These disjoint groups fit the full ordered zero mask over every
+field and have rank sum_i(x_i+p_i)=S-sum_i x_i.
+
+For completeness the needed cycle matching fact is proved here. Consider the
+finite LP max sum_i x_i with x_i>=0 and x_(i-1)+x_i<=d_i. Its dual is min
+sum_i d_i y_i with y_i>=0 and y_i+y_(i+1)>=1. Truncating y_i at one preserves
+feasibility and cannot increase the objective. An extreme dual point is either
+an integral vertex cover or all y_i=1/2. Indeed a proper component of fractional
+vertices connected by active edge equalities is a path, whose alternating
+plus/minus perturbation preserves all active equalities for a sufficiently small
+amount; an endpoint cannot be pinned by an adjacent integral value without
+becoming integral itself. Such a path is not extreme. If the component is the
+whole odd cycle, its alternating equations force all values to one half.
+Finite LP strong duality therefore gives the fractional optimum
+
+    min(S-alpha, S/2).                                 (4)
+
+Here integral covers have minimum weight S-alpha because their complements
+are independent sets. To pass to integers, choose an extreme optimal primal
+point. Any positive-support component that is a proper path has integer edge
+values: a full-rank active system is solved from a zero edge/end or a tight
+endpoint constraint and then successively along integer vertex capacities.
+Equivalently the path incidence matrix has every nonsingular square minor
+0 or +/-1 by removing a leaf. Fractionality can consequently occur only when
+all n edges are positive and every vertex constraint is tight. The cyclic
+system x_(i-1)+x_i=d_i has determinant two; any fractional solution has all
+x_i integer-plus-one-half. Round x_0,x_2,...,x_(n-1) down and the intervening
+edges up. Every vertex keeps its total except vertex 0, whose total drops by
+one. Nonnegativity remains valid, and the objective loses exactly one half.
+Thus the maximum *integral* sharing is
+
+    min(S-alpha, floor(S/2)).                          (5)
+
+When (4) is integer, the fractional extreme case cannot be optimal: the sum of
+an odd number of half-integers is a noninteger. This also covers zero widths
+and zero total. Equations (5) and the channel construction attain (1).
+The LP argument is a classical matching method, not a new general LP theorem.
+
+## Compressed deterministic construction
+
+The implementation does not call an LP solver or expand d_i coordinates.
+Fix the closing edge x_(n-1)=t with 0<=t<=D=min(d_0,d_(n-1)). The remaining
+capacities on endpoints are reduced by t. A path maximum matching is obtained
+by saturating the leftmost remaining edge as far as possible and continuing.
+An exchange proves this greedy rule: if an optimum uses fewer copies at the
+left edge, an unused copy at its right endpoint can be taken directly, or one
+copy on the next edge can be moved left without reducing the number of edges.
+After fixing the left edge repeat on the shorter path.
+
+Let g(t)=t+the optimum path sharing. The capacitated path matching/cover dual
+is integral, either by the leaf argument above or ordinary weighted bipartite
+matching duality. Thus for integer t it equals the minimum of the path-cover
+affine functions
+
+    cover_weight + t*(1 - number_of_endpoints_in_cover).
+
+Those slopes are -1,0,1. The same LP expression on real t is concave, so g on
+integers is discretely concave. Binary search for the first index satisfying
+g(t)>=g(t+1) finds a maximum, including an arbitrary long flat plateau and
+D=0. It uses at most 2 ceil(log2(D+1))+1 greedy path evaluations; each takes
+O(n) integer operations and O(n) compressed integers. The maximum arithmetic
+operand bit length is O(log(S+1)); source/target coordinate expansion and any
+physical index/bit-operation costs are separate. `validate` checks every
+coverage equation and exact answer, and emits a weighted independent witness using linear-space backpointer
+reconstruction, without quadratic prefix-tuple copying.
+
+## Discriminating cases and stop
+
+Seven width-two vertices need seven query coordinates, rather than twice the
+scalar optimum four. For d=(2,1,2,1,2,1,1), alpha=6 exceeds S/2=5; the actual
+answer is six. For seven unit vertices alpha=3 but ceil(S/2)=4. Both terms
+are essential. The C5 predecessor is recovered exactly, not claimed again.
+
+The proof closes this cycle-mask dimension contract. It does not fund copying,
+dirty compensation/restoration, owner access, source coefficient preparation,
+frames, hosting, stock, finite precision, physical tapes, compiler or outer
+transfer. A new physical construction needs those contracts before another
+implementation campaign. No selected frontier is changed.

research/odd-cycle-query-capacity/README.md

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+# Exact query capacity for every unequal odd cycle
+
+For every C_(2k+1), k>=2, arbitrary nonnegative payload widths d_i and every
+field, the exact fitting rank is
+
+    max(maximum independent-set payload weight, ceil(total payload/2)).
+
+Allowed ordered edge blocks may be rectangular, nonsymmetric and noncommuting.
+[PROOF.md](PROOF.md) proves the all-field lower bound and an attaining integral
+edge/private-channel factorization. The compressed deterministic constructor
+uses O(n log(max(d)+2)) integer operations, without expanding payload coordinates.
+This is a query-dimension contract, not a physical supplier or new kappa.
+
+Seven width-two payloads need seven coordinates, while two separate scalar
+rank-four solutions would need eight. The unequal C7 (2,1,2,1,2,1,1) needs six
+coordinates: its weight-six independent set is stronger than half-total five.
+Seven unit payloads instead need four, stronger than independent-set weight three.
+
+## Scope and family decision
+
+This extends Tekalign #3362's C5 interface to all odd cycle lengths, alongside
+Nick Petalas #4909's bipartite-mask result. It changes the mask class, retaining
+the same complete ordered zero and identity-block contracts. All rotated
+one-edge-plus-isolates subspaces establish the new general lower-bound step.
+Classical matching, vertex-cover/LP and odd-cycle index-coding methods retain
+credit; no literature priority or new general matching algorithm is claimed.
+The bounded finite census does not establish the universal theorem by itself.
+
+Family decision: EXPAND once to close this exact mask contract, then STOP at
+that result. No source-bound P112 realization is asserted. Physical reads,
+copies, dirty restoration, frames, ownership, hosting, preparation, numerical
+stock/precision, fixed tapes, compiler and outer recovery remain UNKNOWN.
+A dimension optimum does not price them as free or improve the selected result.
+
+## Reproduction
+
+Use Python 3.12 and only its standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/odd-cycle-query-capacity
+python3.12 -I -O -B research/odd-cycle-query-capacity/verify.py
+```
+
+The read-only replay checks every width vector in {0,1,2,3}^5, {0,1,2}^7
+and {0,1}^9 against independently enumerated coordinate sharing and independent
+sets. It checks all attaining matrix entries and ranks over GF(2), GF(3) and
+GF(101), every directed binary scalar fitting matrix on C5/C7, malformed inputs,
+underpaid/incorrect channel recipes and a forbidden ordered block. An eleven-
+vertex fixture uses 4097-bit widths while retaining only O(n) integers. Exact
+counts and the deterministic recipe digest are in [RESULT.json](RESULT.json).
+An independent column-span implementation checks 288 directed rectangular-block
+fitting matrices, all rotated subspace inequalities and a sharp seven-width-two
+attainer with noncommuting allowed blocks. Source/receipt corruption must fail. Hosted CI and independent mathematical
+acceptance are not asserted here.
+
+Canonical base bee95ce88f9f19802f0613f92838895cd0e14a1f, priority revision 7,
+selected conditional kappa 0.00302015 unchanged. Initial overlap covered 310
+open PRs, 4096 Discussion records and all 6690 tracked package notes, with
+reported complete inventories; absence of a match is not proof of novelty.
+The canonical investigation is Discussion #1587. Exact predecessor bytes and
+the original card are retained in SOURCES.json and TASK.json.
+
+## Open next questions
+
+1. Determine a named triangle-free mask with overlapping odd cycles for which
+   arbitrary fitting blocks improve on integral coordinate pairing. First
+   compare its rank-factor subspace constraints with its complete edge-sharing
+   LP; no physical decoder should be assumed.
+2. Exhibit an actual recursive caller whose admitted ordered read mask is one
+   of these longer odd cycles, with all query/source/copy and cleanup costs.
+   A geometric/matching realization and complete consumer tariff are blockers.
+
+Credit Tekalign #3362, Sagar Thakrar #2484, ywz65 #2471, Nick Petalas #4909,
+Cinna Davis, Adam Gall, ctoth, Philippe Van Velzen and the retained P112/native
+lineage. New scoped proof, constructor and checks: JoyboyBrian, with substantial
+OpenAI Codex assistance. Apache-2.0. No human or proof-kernel review claimed.