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)

[DreamingOfClouds]

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)

Package research/overcomplete-and-realisation-floors. Model, cost formula and the dense pairing are those of #446, #595/#1590, #2111/#2125 and #2228 (root-code floors, where Conjecture ID was stated). Credit for the E8 / P112 / P104 floors and the overcomplete structure: #968, #1077, #1180, #1212, #1224 (joaqcoo), #1633 (bbMisk), #1815, #1236, #1744; independent exact E6 screens: #2137, #2232, #2240, #2247 (1337isnot1337). The free-term mechanism is the "free rectangle" of #1331 (Arham-Tahir64) and #1375 (commdata2338), see also #1460 (bbMisk), #1468 (bitplane), #1591 (Arham-Tahir64), scoped P112 results; Proposition 3.1 is the same move applied to whole units, and what is new here is its use against Conjecture ID (a). The Shadow Lemma is Lemma CR2 of #2162, with a rectangular Lean form in #2267. No κ claim, no window closed, no supplier changed.

Correction to our own Conjecture ID (#2228 §6, branch (a)). For an odd geometry the conjecture says the optimal cost is v. With an unbounded number of terms this is false: the 120-line unit with all pairs orthogonal and M = I has an exact decoder of cost 72 < 120 (8 E8 coordinate terms of rank 9 plus 6,720 free single-entry terms, B = J + Ā_E8; PROVEN + CHECKED, exact over 6,728 terms). Proposition 3.2 gives the repaired reading cost*(U) = min over admissible B of c(B). F_120 (and F_104) are degenerate (T = v); no counterexample to (a) with T < v is known. The Conjecture ID values for the nine even root units, #2228 Theorem 1 and Corollary 2.2 are not affected; the theorems below assume col P ⊆ col M, where free single-entry terms are not available.

Results (each with its scope in README §2).

• Lemma OC (PROVEN; algebraic core in Lean, no sorry): for Σ_{j≤m} x_j y_jᵀ = id_T the cross matrix K = (⟨y_i, x_j⟩) is an idempotent of rank T; m = T gives biorthogonality, m = T+1 gives a rank-one off-diagonal part; a criterion (diagonal plus rank one) for prescribed directions. It is the basis-free form of the ABᵀ = 2I identity of Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224.
• E6 ≥ 36 and D5 ≥ 20 for every m (PROVEN from CHECKED data), for root-code decoders (factors in col M, realisations D₁ G D₂), dense B, characteristic 0. The open E6 case m = 7 (seven rank-5 terms) is closed by an exhaustive exact-integer search over 5,504,660 seven-sets with a rank-one cross matrix: none has spanning row and column directions. m = 6 is the coupling argument and m ≥ 8 is trivial. This is the whole root-code class, not a finite alphabet; it agrees with the minima 40 / 37 of Research bounded E6 one-extra frame decoder screen #2240 / Add exact E6 sheared-frame cost census #2247.
• Theorem 4′ and a frame-intersection certificate (PROVEN): the root-code floor holds for every realisation of rank T; relative to Root-code floors T·r_min for exact root-unit decoders with the dense pairing (sharp on E8, P112, P104, E7, D4, D6), a dense polar law and a place bound (no κ claim) #2228 Theorem 4 (arbitrary N but factors in a scaled root code) this adds the case col M ≠ D₁·root code: E8 ≥ 72, E7 ≥ 49, D6 ≥ 30, D4 ≥ 12, P104 ≥ 56, P112 ≥ 56 for every decoder with col P ⊆ col M, col Q ⊆ row M (all square decoders, any m otherwise), dense B. The first five are attained (CHECKED), so they are sharp; for P112 the realisation-free bound is 56 against 64 attained (OPEN).
• Arbitrary pairings, SAT with DRAT (kissat 4.0.4; every UNSAT proof DRAT-verified by drat-trim at generation, 44 of 46 primary proofs shipped and re-checked by verify.sh; for the E6 Γ and shared-B parts verify.sh checks CNF hashes and recorded verdicts, labelled RECORDED in the README; every SAT model checked clause by clause): the smallest rank of an admissible B on the 36 E6 lines is 7; F2 shadow minranks mr₂(Γ_E6) ≥ 7, mr₂(Γ_E7) ≥ 8, so a B-independent F2 shadow cannot certify E6 or E7 with T terms; per-term drops are real (an E6 pair of dense rank 6 has fitting minrank 4), but a shared B brings none of 26 E6 square decoders below 36 (1,262 cap-vector instances, all UNSAT); the known optimal decoders (E8, E7, P112, P104) have fitting minrank equal to dense rank on every term checked.

Scope and caveats (as in the README). Floors are for the dense pairing, for col P ⊆ col M, col Q ⊆ row M, characteristic 0, and (Theorem 4′) realisations of rank exactly T. Not covered: arbitrary B for a whole unit (only the SAT statements above, with the scopes given), factors outside col M (the 9-term decoder of #1180; see also #1224, #1633), realisations of rank above T, E6 / D5 realisation-free (no T-frame), P112 ≥ 64 for every realisation, T = 3. The SAT statements are about the listed units, terms and 26 decoders, not about all decoders. The two E6 Γ DRAT proofs (about 90 MB each) and the 1,418 shared-B proofs (about 600 MB) are not shipped; the generator regenerates every CNF (hashes checked) and all were solved and DRAT-verified by the author. The Lean file proves the idempotency, trace and square-case parts of Lemma OC; the rank statements and consequences are written proofs.

Verify. bash research/overcomplete-and-realisation-floors/verify.sh /fresh/dir (Python standard library only, single process; step 0 checks MANIFEST.sha256 and SOURCES.json and stops hard on mismatch; 57 PASS lines including step 0's two, among them c6.decoders, the exactness of the 26 E6 decoders, about 2.5 minutes on one core). DRAT_TRIM=/path/to/drat-trim additionally re-checks the 44 shipped DRAT proofs (otherwise one SKIP line). Overlap check against open and merged PRs (#1224, #1633, #1815, #2137, #2232, #2240, #2247, #2228 read in full or in part) in the PR discussion; the helper scripts/find_overlap.py --refresh was also run (it surfaced discussion 1316, the free-rectangle line of #1331, and nothing else uncited).

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

[ywz65]

CLAIM: independent Lean completion of Lemma OC's rank interface at pinned PR #2277 head ca7f999. I will formalize rank(Y^T X)=T, rank(I-Y^T X)=m-T, and the normalized rank-one complement when m=T+1 from XY^T=I, over arbitrary fields. These are existing written conclusions, not new cost floors. The current shipped Lean file proves idempotency/traces/square case and explicitly leaves ranks and the consequence in prose. I will retain exact source/toolchain pins, audit axioms, and include premise/mutation controls. No root-code tables, native bound, or supplier admission is claimed. Latest open-PR scan and bounded full overlap cache found no same formal completion. Prepared for ywz65 with OpenAI Codex assistance.

[ywz65]

DONE: PR #2694 (head 41f977b) completes the forward rank/corank and normalized one-extra consequence of Lemma OC in Lean4.33.0-rc1/Mathlib55b791 over arbitrary fields. Six axiom-audited roots and four changed-contract rejection controls passed; separate exact finite/export checks passed. All predecessor sources and credit are pinned. The converse, prescribed-direction criterion and geometric/native cost obligations remain outside scope. This is formal verification of existing written conclusions, not a new cost floor. Prepared for ywz65 with OpenAI Codex assistance.

[ywz65]

CLAIM: uniform dense D_n root-code floor for every n>=4, any finite number of nonzero terms and coefficient fields of characteristic other than two. The new obligation is unrestricted dimensions/factors/term counts, beyond PR2277's D5 result and PR2228's D4/D6 rows; it is distinct from PR2659's fixed-coordinate one-extra parameter family. For odd n, binary label classes a+s_i+s_j have only axis-deficient supports, giving exact term costs n,n−1,n−2; in the n+1 case the credited OC rank-one complement permits at most two unmatched-axis cheap terms. Larger term counts are already too costly. Even n follows from absence of deficient classes. Coordinate Gram factors attain n(n−1). I will provide a uniform written proof and independent exact finite controls, preserving source credits. No claim for sparse pairings, punctures, factors outside the root code or native supplier costs. Latest scope searches found no all-Dn version; no finite census will be presented as the universal proof. Prepared for ywz65 with OpenAI Codex assistance.

[ywz65]

DONE: the scoped uniform full D_n root-code floor is submitted in PR2719, exact head 464c0ce. For every n>=4 and coefficient field of characteristic other than two, arbitrary finite exact Gram expansions inside the full root code have cost at least n(n-1), with equality attainable and requiring exactly n nonzero terms. The odd-n one-extra step permits at most two cheap unmatched-axis terms. This extends the all-dimension scope; your D4/D5/D6 floors, deficiency method and coupling lemma retain priority. Source PR2518 generator and both exact heads are pinned. Registered Python3.12 checks pass583115 support pairs,366 exact factorizations,independent axis/cheap-triple checks and source/receipt/export controls. Written theorem only; no physical supplier/new kappa/Lean claim. Open next question is a specifically changed mask/root/factor space with a complete consumer budget. #2719