[SHIN09-git]

Result

This answers #6691, open question 1. For any two positive port-block sizes, the generic support census equals the entire rational census of input-column and inverse-row block masks. The proof gives a signed integer witness for every realizable rational signature, a construction over every field with more than two elements, and necessary and sufficient binary exceptions.

Two blocks of three ports	Profiles up to relabeling	Labeled signatures
Generic, rational, or a field with more than two elements	188	33,727
Zero-one matrix entries, inverted over the rationals	179	32,287
Two-element field	167	30,547

A simultaneous pivot chart reduces the problem to two off-diagonal rectangles. Three binary obstruction classes account for 3,150, 15 and 15 missing labels. The counting formula uses O(pq) arithmetic operations. For balanced blocks, the fraction of ordinary signatures missing over the binary field decays as d^(3/2) (16/49)^d / (4√π). Proof · Exact counting formula

Price consequence. When enlarging either support cannot lower its message price, assigning each message to one block already attains the exact minimum. This holds even for label-dependent prices. A positive nonmonotone example instead costs 10 over the rationals and 11 over the two-element field. Exact separating prices and convex mixtures classify all 21 missing six-port profiles. For one common additive table, the three nested domains can have at most two distinct minima. Positive integer dependences exclude all 35 candidate pairs for two strict gaps. Both single gaps occur, for example at (10,10,11) and (8,9,9). Exact finite certificate. These are stipulated tariffs. The result covers square bases and does not supply physical decoder prices or a new κ. Worked matrix and prices

Reproduce

python3.12 -B scripts/research_package.py --package research/two-block-signatures

• Constructed and literally inverted every labeled six-port witness over Q, F3 and F2 where applicable. All 188 profile records include an explicit rational matrix, a binary matrix when possible, and exact finite-field multiplicities. Certificates
• An independent determinant/cofactor matching screen covered all 262,144 supports of a fixed six-port chart and all its message relabelings. It also recovered the prior 439 codes from all 65,536 four-port supports.
• The binary census compared all 714,240 complementary pairs. The ternary point-incidence census accounted for all 666,860,040 pairs by weighting 31 coordinate-symmetry orbits of the first subspace. Neither finite-field oracle imports the classification or witness constructor.
• Restricting coefficients to zero and one while inverting over Q gives 1,740 more signatures than the two-element field. A weighted census by maximal minors accounts for all 28,836,612,000 nonsingular six-port zero-one matrices and reproduces the full four-port histogram with literal cofactors. Restricted coefficient census
• Complete normal and optimized primary/census receipts agree. Twenty-three private-copy faults were rejected in both modes. The official source-history check and export passed in a separate full-history checkout. The read-only export, SHA-256 6592a61ad0508a0fe34d897b033721faac2c89b62cdc04dcaa07ae8ed356d7be, reran all seven commands without changing any file. Validation

Checks on the submitted head

GitHub passed planning and the registered package replay on this exact head. Hosted checks All seven hosted scientific receipts match the local ones in full. The hosted onboarding suite ran 711 tests with 24 skips, and the committed-input drift check passed.

On Ubuntu 24.04 WSL2 with CPython 3.12.3, the source-history and executable-mode checks and all seven registered commands passed. All 33 package files remained byte-identical. The upstream selection command picked only this package. The unmodified onboarding command ran 711 tests with 23 skips and no failures.

As an additional unbalanced check, the published binary space enumerator covered all 48,377,856 ordered complementary pairs for a 3+4 split and exactly matched the predicted 236,909 labeled binary signatures.

Repeat the additional 3+4 check from the package directory

python3.12 -B -c 'from binary_oracle import enumerate_pairs; from construction_check import labeled_codes; a,b,h = enumerate_pairs(7,3); print(sum(h.values()), len(h), set(h) == labeled_codes(3,4,True))'

Expected output is 48377856 236909 True.

Science boundary and credit

Gonzalo Mier supplied the prior zero-one rational and generic four-port screens. Nick Petalas supplied the four-port complementary-space characterization and six-port handoff. The prior matched-plane COPY result also covers overcomplete factorizations. Determinant matchings, finite Grassmannian counts and row-space incidence are classical. Three-block signatures, fine port-by-port joint patterns and physical caller admission are separate questions. The independent algorithms share one assisted author. Prepared for SHIN09-git with substantial OpenAI Codex assistance in derivation, implementation, testing and writing. Apache-2.0. #3990 · #3997 · #4365 · Source identities