Prove exact full-mask rank with separated-port excess (no new κ)

research/full-mask-separated-excess/PROOF.md

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+# Full-mask attainment with separated-port excess
+
+This is a scoped algebraic proof over every field. The finite checks corroborate
+the new constructions. No physical tariff or multiplication theorem is proved.
+
+## Contract and credited premises
+
+Use the 112 ports, matching and strict ordered mask of Sagar Thakrar's
+PR2484, retained in `paired-plane-block-rank`. If port i lies in plane E_i,
+a fitting matrix has diagonal block I_(n_i), and its ordered (i,j) block is
+zero for i != j unless E_i and E_j are orthogonal. Permitted rectangular
+blocks are arbitrary. They need not be symmetric or commute.
+
+The credited premises are:
+
+1. PR2484's 49 supports partition the ports, and distinct ports within a
+   support are permitted pairs.
+2. Its 22-plane set Q is pairwise nonorthogonal and nonorthogonal to the
+   five planes C=(13,23,35,16,34). The induced permitted graph on C is C5.
+   Both ports within a plane have forced off-diagonal zeros.
+3. The all-field C5 block lemma in `paired-plane-block-rank` gives rank at
+   least 5d when each cycle plane has total width 2d.
+
+These are predecessor results, cited as dependencies. This package does not
+claim to discover their geometry, partition or uniform rank 49d, and does
+not replay their verification suites. Source bytes and proofs are pinned.
+Tekalign's PR3362 extends the C5 lower bound to unequal widths and explicitly
+leaves full unequal-mask attainment open. We settle one restricted family.
+
+## Theorem
+
+Let T be the 44 ports in the planes of Q. Choose an integer d>=1 and
+arbitrary integers e_p>=0 for p in T. Set
+
+    n_p = d+e_p for p in T, and n_p=d otherwise.
+    E = sum_(p in T) e_p.
+
+The exact minimum rank of a fitting matrix on the complete 112-port mask is
+
+    49d + E.
+
+Equivalently this is the exact minimum intermediate field-coordinate
+dimension for a linear factorization M=A B of such a fitting matrix.
+This is a full-mask result for this family, not just a principal-mask bound.
+
+## Lower bound
+
+Restrict any fitting matrix to the 44 Q ports and the ten C ports. All
+ordered cross blocks between those two sets vanish. The Q part is the
+identity of dimension 44d+E. The cycle part, grouping the two ports within
+each plane, has five diagonal blocks I_(2d) and forbidden nonneighbor
+blocks. The credited C5 lemma bounds its rank below by 5d over every field.
+The principal submatrix therefore has rank at least 49d+E, as does the
+full matrix. This applies to all arbitrary ordered free blocks.
+
+## Attaining construction
+
+No two distinct Q ports can occur in the same partition support: Q ports
+are mutually forbidden, whereas each support consists of permitted pairs.
+Thus exactly 44 of the 49 supports contain a Q port, each containing one.
+
+For each support S allocate a query vector of dimension
+
+    c_S = max_(p in S) n_p.
+
+At port p in S, embed its n_p coordinates as the first n_p coordinates
+of this vector. Sum these embeddings to encode the query. Decode at p by
+projection onto its first n_p coordinates. The self-block is I_(n_p).
+Blocks between distinct supports vanish. Blocks within a support are
+rectangular initial-coordinate overlaps, which the mask permits. This
+constructs an actual full-mask factorization over every field using only
+zero and one coefficients.
+
+For a support containing Q port p, c_S=d+e_p. For the other five supports,
+c_S=d. Consequently total query dimension is exactly 49d+E, attaining
+the lower bound. The construction is independent of the characteristic.
+
+One can also see its exact rank directly. For each support and local
+coordinate, the associated expanded matrix is a nonempty all-ones block
+on the ports that possess that coordinate. These blocks use disjoint row
+and column sets. Each has rank one over every field, so their total rank
+is sum_S c_S. Arbitrary independent invertible port charts preserve the
+identity/zero mask and rank, though they do not reduce the universal floor.
+
+## First instance and scope
+
+Set d=1 and add one coordinate at Q port 0. The full matrix has 113 rows
+and columns and exact minimum rank 50. No assertion is made that every
+unequal vector satisfies a formula based only on total excess. Allowing
+excess at other ports changes the family and can change both bounds.
+
+The compressed recipe stores 112 widths and 49 capacities. It does not
+expand enormous payload matrices or implement the queries on a physical
+machine. Native copies, access, frame changes, dirty restoration, numerical
+precision, hosting and complete supplier costs remain outside the theorem.
+
+This contribution is produced by Nick Petalas. Credit Sagar Thakrar/PR2484,
+Tekalign/PR3362, ywz65/PR2541 and PR2471, Cinna Davis/PR1825,
+Adam Gall/PR1889, ctoth, Philippe Van Velzen and the retained P112 lineage.
+The support embedding and rank arguments use elementary linear algebra;
+no claim of literature priority for those methods is made. Apache-2.0.

research/full-mask-separated-excess/README.md

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+# Exact full-mask rank for separated-port excess payloads
+
+The complete fixed PR2484 112-port mask has minimum fitting rank
+
+    49d + sum_(p in T) e_p
+
+when every port has width d>=1 and only the 44 ports T of the retained
+separated 22-plane set Q receive arbitrary excess widths e_p>=0.
+The theorem holds over every field, with arbitrary rectangular ordered free
+blocks, and the bound is attained on the full mask. This is a scoped proof
+with finite corroboration. No new kappa or supplier admission is claimed.
+
+This answers a restricted family of Tekalign's [PR3362](https://github.com/CrocSwap/integer-mult-bounds/pull/3362)
+Open next question 1, which asks for full unequal-width 112-port attainment.
+The credited uniform 49d theorem and Q/C5 lower bound are dependencies.
+The new result is the full-mask lift for all 44 independently increased
+separated widths. The first example has d=1, width two at port 0, all other
+widths one, and exact minimum rank 50 in a 113-by-113 matrix.
+
+[PROOF.md](PROOF.md) gives the lower bound and explicit support-wise encoder
+and decoder. It also separates the result from the general unequal-width
+problem. [SOURCES.json](SOURCES.json) pins the predecessor proofs, partition,
+matching and label archive at canonical base
+`624de9dc66858af0945e5f8a0942488d62280d43`. No predecessor executable is imported
+or run. The card's scope is explicitly changed to this algebraic side result;
+this does not supply a new paid profile or satisfy the physical mechanism gate.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/full-mask-separated-excess
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/full-mask-separated-excess/verify.py
+```
+
+Expected direct output:
+
+```text
+PASS_FULL_MASK_SEPARATED_EXCESS matrices=50 field_ranks=150 controls=10
+```
+
+The registered hook runs normal and optimized isolated checks. The direct
+verifier works without Git when the declared input files retain their paths.
+`--emit` prints a newly generated receipt; acceptance compares the complete
+canonical JSON to [RESULT.json](RESULT.json). The receipt binds package and
+proof file hashes and the source manifest. It is generated, not hand-written.
+
+The 50 full-mask matrices are all new nonuniform payloads: one single-unit
+increase at each of the 44 Q ports, then three larger patterns both with and
+without independent invertible port charts. Each is checked against every
+forced identity and ordered zero block and ranked by literal elimination over
+F2, F3 and F101. The patterns include d=2 and simultaneously unequal excesses.
+A 201-bit base-width recipe is checked without expanding its matrix. These
+finite instances corroborate the universal written proof; they do not enumerate
+all fitting matrices or prove the universal lower bound by search.
+
+Ten controls cover invalid base widths, malformed/negative/Boolean excesses,
+a one-direction forbidden leak, a changed diagonal identity, an omitted
+extra-coordinate charge, and refusal to expand beyond 512 coordinates.
+The archive and source files are authenticated before use. All acceptance
+checks remain enabled under optimized Python.
+
+## Limits and implementation checks
+
+This is an exact linear query-capacity theorem. It does not price native
+copies, access, frames, dirty restoration, precision, hosting or a complete
+consumer. No complete multiplication theorem, selected-frontier change,
+independent human review or formal proof-kernel acceptance is claimed.
+
+The verifier uses managed Python allocation, a documented Power of Ten
+memory-rule deviation. Expanded matrices are capped at 512 coordinates;
+there are 50 fixed fixtures and three fixed finite fields. Compressed input
+integers have at most 4096 bits, and compressed arrays have fixed sizes
+112 and 49. Pinned files are capped at 16 MiB and the extracted JSON member
+is capped at 16 MiB. Tar/JSON/Python runtime allocation is not statically
+controlled. Functions stay below 60 lines and there is no recursion. The
+small callback set in rejection controls is a documented indirect-call
+exception, restricted to the ten immediate controls. The source hash bounds
+the exact runtime input. `-W error` is part of both registered commands; no
+additional project Python linter configuration was found. The finite expansion
+cap is exercised by a rejection control. A future larger replay must revise
+these bounds before expansion. No asynchronous or persistent work is started.
+
+## Open next questions
+
+- Can excess widths on the ten C5 ports, or on the other 58 ports, also
+  attain Tekalign's principal lower bound on the full mask? First increase
+  one such width, compare the support capacity with that bound, and give
+  a full fitting factorization or a stronger obstruction. The present
+  theorem leaves those changes open.
+- Can one of these unequal query layers improve a fully paid native
+  implementation? First choose source/target charts and price every read,
+  copy, frame change and restoration. Those costs remain unknown here.
+
+## Credit
+
+This contribution is produced by Nick Petalas. Credit Sagar Thakrar/PR2484
+for the scalar partition and Q/C5 witness; ywz65/PR2541 and PR2471 for the
+all-field uniform block theorem and earlier mask work; Tekalign/PR3362
+for the unequal principal theorem and full-mask question; Cinna Davis/PR1825,
+Adam Gall/PR1889, ctoth, Philippe Van Velzen and the retained E8/P112 lineage.
+No predecessor credit is changed. Elementary support embeddings and linear
+rank inequalities are classical methods; no literature-priority claim is made.
+New material is Apache-2.0. Independent human and formal proof-kernel review
+are unclaimed.