Decoder floors are tight in the frontier format: one F2^9 label table, factors in col M of a rank-8 realisation (P112 64, E8 72, P104 (#968's unit) 56; two-call batches 128) — no κ claim

[DreamingOfClouds]

Decoder floors are tight in the frontier format: one F2^9 label table, factors in col M of a rank-8 realisation (P112 64, E8 72, P104 (#968's unit) 56; two-call batches 128) — no κ claim

Status: research note, no κ claim (scoped negative result with explicit open items). The selected frontier is unchanged. Draft: overnight closures from the author's working cards are added below as RECORDED (not re-run by verify.sh); to be reviewed before marking ready.

Summary

This package pins down where the E8-family decoder (the copy cost of the P112 / E8 / P104 / E7 units, and of independent batches of them) can still be improved. In the format the frontier actually uses — one global F2^9 label table (the pairing of the whole unit has F2-rank ≤ 9; ≤ 7 for E7), factors in the column space of a rank-8 (minimal-rank) realisation (the square decoder of a rank-9 realisation is OPEN), any number of terms, any number of independent calls — the floors are P112 64, E8 72, P104 56 (P104 = #968's unit, F1 = H4; other P104 units are not claimed), E7 49, and 64k for k independent P112 calls (128 for two; symmetric / physical table for k ≥ 3, GL charts with k ≥ 3 are open), all attained. Every escape from that class is listed as OPEN with what it would have to achieve. Nothing here is a word-side result, and no κ is claimed.

Results (labels: PROVEN / CHECKED / RECORDED / OPEN; full proofs and the master table are in the README)

1. Theorem R, rigidity of the minimal-rank label table (CHECKED, 16 DRAT-verified UNSAT). On E8, P112, P104 (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's unit, F1 = H4, which lies in the 759 class, and one frame from each of the three invariant classes of disjoint second frames: three classes, four instances), the bit unit P111 (rank ≤ 9, general UVᵀ and symmetric PPᵀ) and on E7, E6 (rank ≤ 7), every fitting pairing of that rank is the dense pairing. Credit: the P111 row is prior work. Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620 (GamingPuzzled) found by SAT that the rank-9 fitting matrix of its archived P111 pattern is unique, and Research consolidation: checked P111 minrank and fitting-matrix uniqueness proofs #1097 (bitplane, merged) proof-checked it with LRAT in the general (asymmetric) model; Research consolidation: checked P111 minrank and fitting-matrix uniqueness proofs #1097's set is the archived Hamming order with 9 indices deleted, which is not Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's set (104 of 111 lines in common), so Theorem R re-derives it on Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's puncture in both models with DRAT. Derive one-sided P112 fitting rigidity from support spectra #1815 (jmbarrancoml, merged) proved a one-sided P112 fitting rigidity (fixed column spaces); Theorem R removes that hypothesis. New here: E8, P112, P104, E7, E6. Controls (10, SAT, dense) and negative controls (sub-units at h = 9; P112 at h = 10, SAT and non-dense) are replayed. So at h = 9 the "any B" slack of overcomplete decoders does not exist; non-dense tables start at h ≥ 10.
2. Theorem P (PROVEN from a CHECKED census): P112 ≥ 64 in every rank-8 realisation with factors in col M (every square decoder, any m otherwise) — Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277 had P112 ≥ 56 here and listed 64 as open. A support-propagation lemma certifies every label class except exactly the 8 (P112) / 16 (P104, Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's unit; the census was run on this unit only) Gram-deficient ones, which contain no 2-dimensional subspace and pair at rank 8 (P112) / ≥ 7 (P104). E8 72, E7 49, P104 (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's) 56 follow (also Dense-pairing floors for overcomplete root-code decoders (E6 36, D5 20, every m) and for every rank-T realisation (E8 72, E7 49, D6 30, D4 12, P104 56), with a correction to Conjecture ID (a) (no κ claim) #2277 Cor. 5.3). Attained by the decoders recomputed in c5.
3. ≤ 8 paid terms with factors anywhere ⇒ ≥ 64 (PROVEN); Structural proof: overcomplete factor-linear E8 decoders retain the puncture copy-cost floor #1180's nine-term example outside col M costs exactly 65 and its shape costs ≥ 64 for every admissible B (it is a split of a proportional pair; CHECKED); all 120 co-star dead sets span in every rank-8 realisation, so every nine-term masked co-star decoder (Discussion 1267) is a square decoder of a rank-9 realisation (CHECKED + PROVEN consequence).
4. Batching (PROVEN from CHECKED premises). k = 2 independent P112 calls on one F2^9 table cost ≥ 128 + 3t for every chart (O(9) and GL(9)), any rank-8 realisations, any m; k ≥ 3 (symmetric / physical table; GL charts with k ≥ 3 are open) ≥ 64k + 3t. Some O(9) charts make all 64 cross pairs rank 7, so the counting bound 120 is not the truth: the extra units come from a rank argument. Negative result: P112 two-call batch keeps the 128 copy floor under isometric cross-call label charts (rank-7 cross terms exist; PR1714 OQ3, isometric free-conversion case) #2294 (thakrarsagar) proves the k = 2 / O(9) / Gram core first and is credited; ours was derived independently, with GL(9), general rank-8 realisations and k ≥ 3 as additions.
5. Every admissible B on one coordinate block (CHECKED, 8 DRAT): rank_F2 B[N(f), N(f)] ≥ 8; and clique-hitting bounds on punctures (f(2) ≥ 14, f(6) ≥ 46, f(7) ≥ 52), with s/f(s) ≤ 1/7 on all computed values (s = 2..8; s = 3–5 RECORDED, s = 1 not computed).
6. RECORDED, not re-run here (the author's working cards; scopes quoted, labels not upgraded):
   • Bit windows of 13–14 totals: CLOSED (RECORDED). Word-priced P111 set-decoders with ≤ 14 totals cost ≥ 63 (STRAYS-BIT: n ≤ 13 PROVEN by exact rational Farkas certificates; n = 14 CHECKED, 735 integer-only classes kissat UNSAT + drat-trim VERIFIED); P110 with ≤ 12 totals ≥ 63 (its whole hosting window); the premise (P) is proven and no longer used; P111 / P110 rigidity at h = 9 drat-trim VERIFIED in the card (P111 rigidity credit: Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620, Research consolidation: checked P111 minrank and fitting-matrix uniqueness proofs #1097). n ≥ 15 is hosting-forbidden (R ≤ 4v), a note.
   • Projected-copy route to P110 = 62 (RECORDED pricing, not a theorem). SPEC-legal (a checker convention forbids it, not the SPEC; SPECCHECK), but on PR589's tuned chains the best term costs ≥ 3.57% c net including the +2.80% unit gain (BITCUT); ideal bound +0.27% c is not attainable with those chains. Not a supplier.
   • h = 10 root code (Gram realisation): CLOSED (RECORDED; card status CHECKED). Any m, any fitting B of rank ≤ 10: every term ≥ 8, P112 ≥ 64 (STRAYS-H10; 46/46 G-query DRATs verified, 688 term queries UNSAT with DRATs pending (1 verified; README rows 16a/19a)).
   • p = 45 closed for m ≥ 8 (RECORDED; card: PROVEN from CHECKED exact data): cost ≥ 40 > 39.71 (Theorem 45, dense B, column-space factors); Lemma G (PROVEN) for rank N = 8 (PUNCT4, PUNCT5).
   • Also recorded: E8 square root-code decoders ≥ 72 for every B (4022 pair classes, DRAT in the working card); D4 ≥ 12 for every decoder; E7 square maximal-normal ≥ 49 for every B; no helper-free nested delivery of Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's P104 decoder for any F2^9 table; PUNCT2 p = 25..44 for any physical table. None of rows 1–5 depends on any of this.
7. New tool (PROVEN, proof in README §3.3): the frame-module theorem. For every nine-term decoder of a unit with 8-frames (any realisation, any rank, factors anywhere) the frame cross matrix is J = I − νπᵀ/(π·ν) with π·ν ≠ 0, p(e) = Σ_f (M_ef/M_ff) p(f) + λ_e π, and the Schur defect is rank one; it is a realisation-free rank-9 Lemma 1.1 as a support calculus. It does not close the outside-col M cases (below).

OPEN (each leaves the class above; priced where a price exists)

• Label tables of rank h ≥ 10, beyond the closed h = 10 Gram root code. OPEN: non-Gram rank-8 realisations at h ≥ 10; h = 11 (per-term ≥ 7 except possibly at pair orbit 0 (N(f) × N(g), one split timed out) and at pair orbits not yet run (1, 2, 65–171); runs continue; a 112-port unit must reach cost ≤ 55 to tie); h ≥ 12 not attempted. Non-dense fitting tables exist at rank 10.
• Factors outside col M with m ≥ 9. m ≤ 8 is closed. At m = 9: (b) one outside factor with ρ = 8, (c) a square decoder of a rank-9 realisation (contains the philippeVV co-star families), (d) m ≥ 10. All three remain OPEN: the frame-module theorem gives a sound SAT model (three positive controls) but the decisive instances timed out under load, and a class-level rank-9 pair test is too weak (136-clique). Swarm results on one explicit rank-9 family, not the general case: Prove the sharp COPY cost of the fixed P112 two-column completion #2321, Research: P112's nine-term COPY floor holds for every nonzero coefficient #2385, Research: extra factors do not lower the fixed P112 COPY-power optimum #2398 (one fixed P112 completion, any number of terms with factors in its 9-dimensional spaces).
• Bit windows: 13–14 totals CLOSED (above, RECORDED). Note only: n ≥ 15 totals are hosting-forbidden; decoders outside the set-decoder model are priced above, not excluded.
• Puncture windows p ≥ 46, and p = 45 with m ≤ 7 (N-space nullity > 1). Residue stated in the cards: m = 8 with a d ≥ 3 root at 46 ≤ p ≤ 56; m ≥ 9 at p = 46 and wherever the count-relaxed census exceeds its allowance (p ≥ 50 not computed; p = 47–49 rest on an uncertified floating-point Q-census); 57 ≤ p ≤ 112 and any-B untouched.
• Batches with m ≥ 17 terms outside col M, GL(9) charts with k ≥ 3, h ≥ 10 joint tables, correlated demands, mixed units.
  Swarm: Research: a stronger P112 two-call COPY floor for arbitrary cross forms #2395 (stronger two-call P112 COPY floor for arbitrary cross forms) and Independent proof: P112 three-call COPY192 via a sharp block count #2342 (three-call P112 COPY 192) are cited from their PR statements, not re-derived or compared in scope here; compare with our Theorem Bk.

Scope and caveats

• The frontier-format hypothesis (one F2^9 table hosts the whole decoder) is an engineering fact of the word, not proved here; it is exactly what the OPEN h ≥ 10 item negates.
• Theorem R for P104 is checked for H4's frame (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's unit; H4 is in the 759 class) and one representative of each of the three invariant classes of disjoint frames (not proven to be single orbits). Not claimed: Theorem R or Theorem P for the other P104 units (all 1,260 frames were not finished; the census was run on H4 only), and the rank-9 square decoder (OPEN).
• Theorem P / the headline need the factors in col M of a rank-8 realisation; char 0 is the scope of the co-star consequence (Theorem P and Theorem B are field-independent).
• The pricing consequence of the puncture bounds (break-even s/f(s) > .1441, margin under 1%) uses the author's working price model and is RECORDED, not a theorem.
• Word-side overheads (helpers, transport, packing) are out of scope; non-E8 units are out of scope.

Verification

bash research/decoder-floors-frontier-format/verify.sh /fresh/dir (Python ≥ 3.11, standard library only, one core). Step 0 checks MANIFEST.sha256 and SOURCES.json. On a fresh git archive export: 74 PASS (2 in step 0), 0 SKIP, 0 FAIL with kissat and drat-trim (16 rigidity UNSAT proofs and 10 further drat-trim checks re-derived; about 7 minutes of CPU on a loaded machine), 70 PASS and 4 SKIP without them (about 20 s). HIT_FULL=1 additionally re-solves three slow clique-hitting instances. kissat 4.0.4 and drat-trim are optional (KISSAT=, DRAT_TRIM=); without them the lines that need them print SKIP. Overlap check against open and merged PRs (title and body searches; #2294, #2277, #2228, #2062, #2061, #1962, #620, #1097, #1815 read): the credits are in item 1 above and in README §7 (also #717, #1331, #1249; audits #2293 of our CR2/B2 and #2311 of our #2265 premises; the edge-form counterexamples #2334 / #2340 concern an edge-count reading of B2(iii), not the vertex form cited here).

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.
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