[petalas]

Summary

Scoped proof and exact implementation of compact universal certificates for dirty linear circuits with independent rectangular operators. Each register has a fixed payload type. A nonzero path of length k has a literal zero/one counterexample using one coordinate per visit to each type, totaling k+1 coordinates. The degree-six example uses dimensions 4 and 3, instead of padding both types to seven. This is a constructive bound, not a minimum-dimension theorem or a new kappa.

compile each typed program into r+q dependency states
certify initial span, arrow closure and zero complete endpoint projections
or return a typed coefficient path and rectangular witness matrices

The new compiler, checker, proof and examples are isolated in research/rectangular-dirty-certificates. The equal-space method is zxv's #5235. Expanded typed block identities are already in Philipp / mightyOil's #1294 following Charles Bethin #1278. Krzysztof Wójt's #5262 covers named inverses in a common space. Classical weighted-automata/PIT methods retain their Kiefer and Raz–Shpilka credit, with the retained dirty-echo, phase and tensor lineage listed in the package. This contribution was produced by Nick Petalas.

Evidence

• Before: Add compact universal certificates for dirty linear circuits (no new κ) #5235's compact bridge explicitly excludes typed rectangular operators. Its predecessor theorem is cited, not audited.
• After: the written typed bridge preserves r+q states per program and supplies per-type finite witnesses. The checker validates closure equations without elimination and shares the compiler and arithmetic with the producer.
• Both registered warning-strict normal/optimized checks pass locally on CPython 3.12.15. Each takes under one second. Their generated source-hash receipts agree exactly.
• Exact finite coverage: 160 explicit-word differential pairs, 86 equal and 74 different; all 74 differences have literal rectangular witnesses. There are 64 dimension vectors, 352 full-column or zero-input echo cases, six dirty-product path witnesses, and fourteen named rejecting/discrepancy controls.
• Metadata, committed bytes/modes and source ancestry pass at 454fa2f43e2465518946ae51fc6f64bd275c877d. Official export and offline integrity checking pass for sixteen payload files. Normal and optimized read-only Git-free replays pass with every exported file's bytes and modes unchanged. Both CLI examples and refusal to overwrite an existing output pass.

From the worktree root, reproduce with the exact local interpreter path:

PYTHON=/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12
"$PYTHON" -B scripts/research_package.py --package research/rectangular-dirty-certificates
"$PYTHON" -I -W error -B research/rectangular-dirty-certificates/verify.py
"$PYTHON" -I -O -W error -B research/rectangular-dirty-certificates/verify.py
"$PYTHON" -B scripts/check_research_commit.py --package research/rectangular-dirty-certificates --check-source-history

Any CPython 3.12 interpreter can replace that installation-specific path. The package uses the standard library only.

Limitations and unrun obligations

The written algebraic theorem is over arbitrary fields; the executable uses exact rational arithmetic with explicit input and certificate size caps. The checker trusts its shared compiler, arithmetic and path reconstruction. The separate explicit-word and literal-matrix finite oracles do not constitute independent human or formal proof-kernel review. No new general PIT algorithm or global literature priority is claimed.

Arrow inverse/group/commutation relations, external aliases, nonlinear operations, native implementation, rounding, precision, physical tapes, an actual supplier adapter and the complete signed invoice remain outside this proof. No selected cold replay, proof-assistant build or physical admission was run. All five selected framework assumptions remain, and the selected record is unchanged.

Open next questions

• Bind one literal mixed-width supplier boundary, retaining every dirty input and returned role. Extra native relations and paid delivery are the first applicability gate.
• Characterize when scalar spaces already separate all paths in a type graph. Begin with acyclic graphs and multiple-cycle obstructions. This classification is not supplied here.

Related open work

The completion refresh read this account's open PRs created since the CLAIM at 2026-10-29T13:06:25Z: #6172 through #6185, including this PR. No other submission in that list studies compact equivalence certificates for typed rectangular independent-operator programs. The nearest certificate packages #5235 and #5262 are merged and credited above. Adjacent open work has different contracts or quantities:

• Add exact P104 ADD-coefficient fault localization (no new κ) #6175 localizes ADD-coefficient faults for a fixed rational scalar endpoint; this package certifies complete program identities with independent typed operators and computes no coefficient repair set.
• Prove functional decoder pairings cost at least v over every field #6178 constructs rank-preserving lifts of functional decoder pairings; its submatrix-rank theorem differs from the closure certificate and path-witness dimension bound here.
• Prove stable joint-bank Gram spectral obstruction (no new κ) #6181 bounds a joint-bank Gram spectral correction under explicit dyadic relations; those operator relations are outside this independent-arrow grammar.
• Prove a sharp 1/4 error floor for contractive P104 mode deletion (no κ) #6183 bounds coefficient error for contractive P104 mode deletion; this package proves exact endpoint equality or supplies a counterexample and computes no approximation optimum.
• Prove exact arithmetic chirp cut spectra and robust bond floors (no κ) #6184 computes arithmetic-phase cut spectra and transfer-dimension obstructions; it does not compute this program-equivalence certificate or per-type path witness.
• Construct a paid arbitrary-bit Sandwich boundary (no new κ) #6185 constructs a paid arbitrary-bit Sandwich boundary under a completed-child contract; its nonlinear bit/tape interface is not certified by this linear-algebra package.

No extra computations were run to check those contributions. None reports a competing value for the quantities computed here.

CI notes

Both hosted checks passed for exact head 454fa2f43e2465518946ae51fc6f64bd275c877d in run 40881001159: plan in 5m21s and replay (research/rectangular-dirty-certificates) in 1m32s. The completed check list was read once and both entries report pass.

Hosted replay certifies execution of the registered normal and optimized package checks. The recorded local scientific, publication bytes/modes/source-history, export-integrity and read-only Git-free checks also passed. The written proof, independent-arrow hypothesis, shared-compiler trust and unrun physical/full-supplier obligations retain their stated limits. Independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way. This adds one self-contained research package.

Blast Radius: package. Shared code, pinned predecessor files and the selected frontier are unchanged.