[petalas]

Summary

Scoped written proof with exact finite corroboration. For fixed-support Eisenstein messages with one shared scalar quadratic form at each input port,

C >= N - sum_t rank(B_t)/2
coordinate-face example: 198
new product-affine bound: 199
actual example charge: 200

Here B_t is the alternating polar form of the used quadratic at port t. The proof determines the exact maximum dimension of a product of affine input spaces that linearizes every message. The dense family q=(sum u_i)(sum v_i) has coordinate restriction loss s and affine restriction loss one. This strengthens a nonlinear obstruction; it supplies no cheaper code or new kappa.

This contribution was produced by Nick Petalas. Credit ywz65 / #2607 for the open nonlinear-payload direction, Michael Fischler / #814 for the nonlinear message contract, Jose Manuel / #2017 for the cited block-linear floor, Jonathan Lam / #798 and Joel Pulikkan for geometry/pricing, Nick Petalas / #4895 for coordinate restrictions, and the retained lineage. Alternating-form decomposition and isotropic dimensions retain their classical status. No predecessor scientific verifier was audited or replayed.

Evidence

• Before: the exact coordinate-face obstruction gives 198 on the first strict fixture. After: the new polar certificate gives 199; a literal exact code costs 200.
• Normal and optimized checks agree on all 1,100 quadratic forms through five variables, 387,405 subspace restrictions, 69,677 translated restrictions, 64 vector maps, 1,600 product-subspace restrictions and 1,280 local decoder cases. Six malformed forms reject and four mathematical boundary controls pass.
• Both registered modes and direct warning-strict commands pass locally on CPython 3.12.15, macOS 27.0.1 arm64, standard library only. Direct elapsed times are 0.217/0.217 seconds, below the 120-second per-command bound.
• Metadata, ten committed package/input files and declared source history pass. Official export integrity and all four declared inputs pass. Both modes pass from a read-only Git-free export with empty PATH; source corruption rejects and all eleven exported files retain their bytes.
• Warning-strict source compilation and AST inspection pass. Ruff is unavailable locally.

From the repository root, reproduce with the exact local interpreter:

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

Other hosts may substitute CPython 3.12. Receipts are generated on stdout and bound to file hashes.

Limitations

The all-width argument invokes #2017's fixed-support linear theorem and the declared additive pairing price. Finite enumeration corroborates the new argument; it is not a proof kernel or independent human review. Only one shared scalar polar form per port is covered. Multiple independent forms at one port, cross-port quadratic products, adaptive queries, errors, private headers and correlated demands remain outside scope. The new bound need not be the optimal message cost.

Physical/native delivery and restoration, complete signed supplier costs, compiler, stock, precision/tape and analytic recovery remain open. Selected cold replay, Lean, Docker, independent human review and formal-kernel verification were not run. The selected frontier and its five assumptions are unchanged.

Open next questions

• Determine the largest common isotropic input space for two independent quadratic forms at one port. Start with literal small alternating matrices and a complete truth-table oracle; the single-form rank formula does not settle that dimension.
• Construct an exact nonlinear Eisenstein code below total demand, if possible, with every source/reader support, width and charge declared. No such construction or favorable physical invoice is supplied.

Related open work

The pre-DONE refresh inspected all sixteen open account PRs created since the advance CLAIM, including this PR. No other PR computes the Eisenstein product-affine restriction dimension or the displayed 198/199 bounds. No extra scientific computations were run to compare other contributions. The directly related #814, #2017, #2607 and #4895 are merged predecessors.

• Prove a sharp bounded-error threshold for frame cuts (no κ) #6266 proves a bounded-error outgoing-partition threshold for arbitrary nonlinear framed circuits. Its correctness premise, graph contract and quantity differ from this fixed-support zero-error Eisenstein pairing-price bound; no competing numerical value is supplied.

CI notes

Both hosted checks passed at exact head bbbe83e174392fc41a90b05a37bea1e7cd5fdb37 in run 41048507154: plan in 4m18s and replay (research/eisenstein-port-quadratic-floor) in 1m43s. The check list was read once after completion; run metadata separately confirms this exact head, completed status and success.

Hosted replay certifies execution of the registered normal and optimized finite checks. The local direct/registered, committed-byte/source-history and read-only export checks listed above also passed. No mathematical check was deferred solely to CI. These checks do not establish independent human or formal proof-kernel acceptance of the all-width written argument.

Advance CLAIM.

Merge Danger

Door: two-way. Blast Radius: isolated research package. Six new files; shared code, predecessor sources and selected-result metadata are unchanged.