Research: sharp 40-copy floor for the Eisenstein frame decoder with at most eleven totals

[cbethin]

Change and mathematical scope

Prove a sharp obstruction for the fixed Eisenstein orthogonal-frame decoder: every binary decomposition into at most eleven totals costs at least 40 pairing-rank units, even with non-clean supports and cancellation. Ten- and eleven-total witnesses attain 40. The inherited supplier screen needs at most 31, so adding one total cannot make this decoder viable.

The general theorem covers any number of orthogonal frames in any positive-definite dimension. Factor ranks reduce one extra total to a single frame-code coset; distinct supports in that coset have pairwise orthogonal geometric complements. Their aggregate rank deficit cannot exceed the ambient dimension. See PROOF.md for all hypotheses and cases.

Status: research note / proved scoped negative result. Other fitting matrices and twelve or more totals remain open. No supplier, new kappa, unrestricted decoder optimum, or full multiplication theorem is claimed. The selected frontier and upstream files are unchanged.

Task and overlap

Base: 6aff82d6d346fb2238d061ef45066b82d48a9b67. Task: complex-rank-shape-decoder, refined to this complete one-extra-total class. Actual originating task: 01a1335e-7a24-73c3-a549-9bcb5496a363; OS seed22 selects the complex track. Source blobs and hashes are in SOURCES.json.

PR798 supplies the geometry and general pairing-charge model. PR1794 supplies the frame witness, clean-class floor and necessary cost screen. This theorem extends beyond clean supports for a fixed decoder and bounds every decomposition with one extra total. PR1977, head ea021c002e61fea7f2f9cd1b613f65b88f7eb7e9, studies general minrank certificates; this package does not repeat or independently certify that pending result.

Validation

python3 -B research/eisenstein-eleven-total-seed22/verify.py

Expected: PASS_EISENSTEIN_ELEVEN_TOTAL.

• Exact-integer rank-adjoining traversal and separate modular generator-combination traversal agree on all 164110 proper support ranks. The independent traversal examines all 10700 generator sets; a Hadamard bound makes reduction modulo 65537 exact.
• Separate rational elimination checks 9200 complements and 5470 orthogonal pairs across all 4360 nontrivial cosets.
• A direct four-port outer-product model checks all 225 rectangles and 50625 ordered pairs, recovering all six two-term and 618 three-term decompositions without using the proof reduction.
• Both sharp E8 witnesses, altered-frame/rank controls, normal and optimized CPython 3.15.0 runs, the explicitly pinned 3.12.12 registered replay, metadata/source pins, and read-only relocation pass. On-disk source and receipt corruption fail as intended.
• The inherited necessary-cost screen replays at 31. Selected-record checks and five unit tests pass. The full selected cold replay is not run because its sources are unchanged.

The implementations share pinned geometric input, not traversal or elimination functions. Expected outputs are regression receipts, not independent evidence by themselves. These finite checks corroborate the written argument; they do not prove its universal quantifier. No proof-assistant validation or independent human review is claimed.

Credit

Jonathan Lam / PR798; GamingPuzzled (Joel Pulikkan) / PR620; Maximooch / PR1794; and Josie Elliston / PR1977. New work is Apache-2.0. Charles Bethin with substantial OpenAI Codex assistance in derivation, implementation, verification and writing.

Open next questions

A different fitting matrix or at least two extra totals must change the excluded contract. Start by pricing the complete optimistic invoice, then study a two-dimensional factor extension or a changed matrix. No obstruction or successful construction for that larger scope follows here. Details are in TASK.json and the package README.

[Mister-Meeseeks]

Batch review at 92ee132d0baf07f7ddf0fbbcd12756be8429bbf7.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: General one-extra-total theorem for an orthogonal frame-partition decoder proves cost at least qd. Independent exhaustive E8 support/rank and complement checks plus a small direct factorization model corroborate it. The fixed fitting matrix and at most q+1 totals are essential; #2017 separately proves a stronger E8-wide result.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.