[petalas]

Summary

A d-dimensional binary off-diagonal pattern space
    -> at most 2^d+d-1 noncoordinate rank-one idempotents
    -> a common-row family attains the bound for every W>=d+1

Scoped written mathematical proof with finite exact corroboration. The exact count is the number of realized nonzero patterns plus the number of off-diagonal matrix units in the space. Deleting the diagonal has a unique lift except that each single-edge pattern has exactly two. The result bounds a scalar pool before chronological word synthesis; no new kappa or selected-frontier change is claimed.

The pattern-space question and cyclic-pattern uniqueness argument come from woklerx's PR #1456 at 83b6ad2ace819a96128131a3600ee914a66dd0dd. Credit woklerx's PR #1846 for the separate four-role classification, and Ali Romman, eumemic and the retained register-model lineage. No predecessor verifier or scientific count is replayed. The new proof, construction and standalone verifier were produced by Nick Petalas. Apache-2.0. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before, the cited source left further pattern-space classification open without this all-dimension cardinality bound. After, the written proof gives the exact universal maximum and an explicit attaining family.

Local CPython 3.12.15 normal/optimized and both registered checks pass:

• 2,501 noncoordinate projectors on five and six registers, checked against every one of 141,888 diagonal completions.
• 110,685 pairs spanning five-register pattern planes, with exact maximum pool size five. These are spanning pairs, not distinct-plane counts.
• 55 attaining star shapes through dimension ten, including padded spectators, with 4,237 literal projectors.
• Five malformed dimensions, seven semantic control groups and three actual optimized implementation mutations. Missing edge charges, missing second lifts and admitted nilpotents all reject.

Each registered command was timed locally, at 0.305s and 0.261s in individual runs, below its 120s limit. These timings check the replay budget. Source pins, committed bytes/modes and declared source history pass at f0ca6c7c1266bcc32596a4d08d08c0caa547914f. Official export/integrity checks and both read-only Git-free direct replays pass; all seven exported files remain unchanged. Syntax compilation, indentation and diff checks pass. No dedicated linter is installed.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/pattern-pool-cardinality/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/pattern-pool-cardinality/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/pattern-pool-cardinality
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/pattern-pool-cardinality --check-source-history

Any CPython 3.12 executable can replace this installation path. Direct execution prints PASS_PATTERN_POOL_CARDINALITY and a generated receipt containing recomputed counts and SHA256 hashes of all six package files. No handwritten scientific receipt is supplied.

Limitations and unrun obligations

The field is F2 and zero-pattern coordinate projectors are excluded. The bound is sharp over ambient register counts, with attainment proved for W>=d+1; fixed smaller W can have a different maximum. The universal result is a written proof, not an inference from finite enumeration.

The joint-image consequence concerns this pool alone. Interleaved coordinate projectors can destroy that invariant subspace. Rational-frame feasibility, two-relation words, native implementation, physical restoration, full cost/precision/tape/ownership invoices, compiler admission and the multiplication theorem remain outside scope. No predecessor scientific suite, selected operator replay, Lean or Docker run is supplied.

Open next questions

• Classify every maximizing pattern space by imposing equality in both counting bounds and examining its directed-edge sets. Only an attaining common-row family is proved here.
• Determine the sharp joint-image dimension at fixed d, starting with d=2 and every diagonal lift. The cardinality bound may be loose; adding coordinate gates to complete words remains a separate obstacle.

Related open work

Refreshed all nine other account open PRs created since the advance CLAIM: #5919, #5920, #5922, #5923, #5924, #5925, #5926, #5927 and #5928. Their titles and result scopes were read. None studies this off-diagonal pool cardinality or diagonal-deletion fiber count. No same-quantity numerical disagreement was identified, and no extra computations were run to validate their results.

• #5928 studies zero-eigenvalue multiplicity for characteristic-two idempotent differences; it does not count rank-one pattern pools.
• #5924 studies mixed affine target call costs with binary rank-one scalar couplings; it does not constrain the size of a fixed pattern pool.
• #5926 studies Fano point/line cut certificates in a related formal register model; its objective is a target tariff lower bound, not a projector-fiber count.

The other six refreshed PRs concern a rational reader basis, an integer carrier height, complex entropy, a descendant circuit traversal, stored-factor rounding and bounded sign lifetimes. The related #1456 and #1846 are merged and credited. Merged #5231 concerns rational span-six word feasibility, a different conclusion not generalized here. Arbitrary nested diffs and unpublished work remain outside the search coverage.

CI notes

At head f0ca6c7c1266bcc32596a4d08d08c0caa547914f, run 40513532365 finished with plan failed; replay skipped. The only failed step is Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern the frozen/current card count (0 != 16), expected rejection of missing current/historical helpers (ValueError not raised), and sparse omission reports that return [] instead of the expected source paths. The historical-reference rejection accounts for four failing subtests. Package selection and the package replay did not execute.

All ten failure/error headers, exception/assertion messages and the final summary match PR #5919's run 40513266276, same failed step, exactly. This branch changes only its six package files; shared tests and helpers are unchanged. Maintainers own the shared main failure. No repair, rerun or additional commit was attempted.

Hosted replay has not certified this package. Only the local scientific and publication checks listed above have run. CI did not run the package verifier.

Merge Danger

Door: two-way

Blast Radius: isolated

Adds only research/pattern-pool-cardinality/. Shared code, frozen sources and the selected record are unchanged.