Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa)

[joaqcoo]

Status: research note / scoped structural result. No κ claim, no supplier word. Only research/nine-term-decoder-normal-form-seed42/ is added. The selected record (PR1094), upstream/ and every existing file are unchanged.

What was learned

PR968 and PR1077 prove the punctured complex E8 copy-cost floor for exactly eight all-pairs terms. Nine or more terms remained open (PR968 §6, PR1071, PR1077, PR717 mixed-β). The live roots span R^8 throughout, which holds for p ≤ 56. The statements below hold over R or C with ordinary transpose:

• Theorem A (any m). D = RA + N1 and T = RB + N2 with RᵀN = 0. Exactness ⇔ ABᵀ = 2I, N1Bᵀ = 0, AN2ᵀ = 0, N1N2ᵀ = 0. The pairing ker A × ker B is nondegenerate.
• Corollary B (m = 9). At most one factor leaves the live root code. An exact ten-term decoder with outside-code columns on both factors is checked, so this is specific to nine terms.
• Corollary C. A one-sided nine-term decoder is the same as an overcomplete representation. Choose nine column normals b_k with one dependency z, and give each live line any coefficient vector with Σ d_ik b_k = 2r_i.
• Lemma D. Proportional column normals merge without increasing cost. So |supp z| ≤ 2 reduces to eight terms, where PR968/PR1077 give cst ≥ max(72−p, 56) for p ≤ 24.
• Propositions E–G.
  • Terms outside the dependency are rigid.
  • Each line avoids exactly one ratio class of masked terms.
  • A line avoiding two masked terms is orthogonal to a canonical nonzero vector.
  • On P120 all 511 codeword supports span R^8, so a one-sided nine-term decoder costs exactly 81 − Σ δ_k. Beating 72 needs Σ δ_k ≥ 10, which forces a masked row support of ≤ 36 lines.
  • A census of all 130,305 support pairs shows that 39,255 do not span. So the pairwise argument alone cannot close P120; this is recorded as open.

Residual class. For p ≤ 24, the floor can fail at nine terms only for decoders with one factor outside the root code and |supp z| ≥ 3. "Exactly one" uses the open PR1180 for the both-root-code case; it is cited, not re-verified.

Evidence, not proof. Exact nine-term decoders were built from weight frames of maximal subsystems. In the A8 frame every root is 2- or 3-sparse. The decoders cost 76–81 on P120. A bounded annealing search (27 runs, seeds 0–2, 3000 moves) found best ratio 67/111 = 0.6036, nothing below 64/112 or 7/13. This is a failed search over the stated domain, not an impossibility theorem.

Reproduce

python3 -I -B research/nine-term-decoder-normal-form-seed42/verify.py

Expected: PASS_NINE_TERM_NORMAL_FORM_S42 checks=80 pairs=130305 frames=5 A8_cost=81 (~35 s, stdlib only). It also passes under -O, from a relocated git archive copy, and alongside scripts/verify_selected_result.py. search.py reproduces SEARCH.json. Not run: hosted CI, repository-wide suites, Lean.

Scope, overlap, credit

• Not covered: a floor for the residual class, m ≥ 10, masked/E-graph-only decoders, p > 56, physical realization. All selected-record conditions are inherited.
• Base: 1b59b624d21924f8e3537cdc1ac52d3be6eda2fe. Sprint track 40–49 (structural decoder research), OS seed 42. The claim was recorded locally in CLAIM.md.
• Credit: DreamingOfClouds (PR717/595/446), themilkman216 (PR968), @woklerx (PR1077, PR1180).
• Concurrent and complementary: @mightyOil (PR1191/1197), @burntout76 (PR1212), @DavidAriasC (PR1213). @3xcalibur-tech's PR1171 (U_k hierarchy) is re-derived independently.
• The overlap scan covered open PRs through Independent bit construction: R439 frame509 at 20 ADDs with original H1-H5 (no new kappa) #1221.
• Prepared with substantial assistance from Claude Opus 5.5 (Anthropic) via Claude Code, which did the research, proofs, code and write-up. Submitted by @joaqcoo. No independent human review is claimed. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 00e6ed0810b8ebc7f6133e2c036d0f1c9a3a4162.

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

Reviewed scope: All-pairs decoder normal form, nondegenerate kernel pairing and one-sided nine-term reduction; dependency supports≤2 reduce to inherited eight-term floor. Overcomplete residual class remains open; bounded searches are not exclusions.

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.