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)

[DreamingOfClouds]

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)

Package research/ideal-decoder-floors. Model, cost formula and exact decoders are those of the closed #446 and the merged #595/#1590 and of #2111/#2125; the E8, P112 and P104 floors are those of #620, #717 (D1), #1212, #968, #1077, #1180, #1236, #1744, #1714, re-derived here by one argument. This PR adds the lower-bound side for other units and a derivation of why the block rank is k + 1.

Place bound (PROVEN, every T). For every number field, prime 𝔭 | 2 and shift a: cost(B) ≥ |S_a|, the number of lines whose leading 2-adic terms do not cancel (Theorem 1). A tropical argument caps |S_a| by the largest pairwise orthogonal set, ≤ T for Gram decoders M = D₁GD₂ on every even root unit (Proposition 2). The 2-adic/tropical corollary and Lemma VCR2 of #2162 already contain Theorem 1's mechanism (all lines surviving, T = 3) and Proposition 2's cycle condition; no novelty is claimed for that part. CHECKED: frame bases attain it (8, 7, 4, 6 on E8, E7, D4, D6), so the ceiling is attained there and far below the optima (8 against 72 on E8, 7 against 49 on E7); a rank-of-sum witness of rank 44 shows that no certificate of the form rank_Q(G∘C) over admissible coverage C (bit model) proves more than 44 for the E8 value 72.

Dense polar law (PROVEN). For an odd-or-zero unit the dense pairing is B_ts = Ω(u_t, u_s) with u_t = (1, ℓ_t) and Ω = 1 ⊕ b on F2 ⊕ F2^k, so every term costs at most k + 1, and the rank of a block is an Ω-pairing rank (Theorem 3). This proves the upper bounds behind #968's rule "7 iff both normals are punctured and α·β is odd, else 8" (equalities CHECKED: E8, all 14,400 normal pairs: 9 / 8 / 8 / 7), and the rebuilt P104 frame-pair decoder costs 56 = 8 × 7 on 104 live lines (CHECKED).

Root-code floor (PROVEN + CHECKED). For exact decoders whose two factors lie in the live root code, for the dense pairing, over a field of characteristic 0 and with any number of terms: cost ≥ T·r_min (Theorem 4; r_min is a minimum over finitely many deficiency types). Sharp, with explicit exact decoders attaining it: E8 72, P112 64, P104 56, E7 49, E7 minus a 7-frame 42, D4 12, D6 30. For E7, #2125 found 49 as the first gain (an upper bound); here 49 is a lower bound in this class. For D5 and E6 the pairwise floors are 15 and 30; the forced form N ∝ Gram (CHECKED, N-space of dimension 1; for E6 this is #2137's kernel QG, which has priority) and a coupling lemma give square decoders ≥ 20 (D5) and ≥ 36 = v (E6), from CHECKED incidences (all 27 × 27 deficient E6 pairs are non-orthogonal).

Scope and caveats. The floor is for the dense pairing and for root-code factors only; it is not a bound over sparser admissible pairings, over factors outside the root code (cf. the 9-term decoder of #1180), over non-Gram geometries, or over overcomplete E6 (m = 7: 35 < 36 not excluded) and D5. Conjecture ID (cost* = min(v, T(k+1) − usable deficiencies)) is a conjecture; it matches the ten values on record (each in its stated scope). Forced triangular minors (#2176's tool) give only 7 per co-flat E8 term. Where the place bound is stated for every T it extends #2162's 2-adic/tropical corollary and the reduction-mod-2 idea of #1590/#2017; the new content is the count |S_a|, the ceiling and the root-code results. Open next questions (non-dense B, factors outside the root code, overcomplete E6 m = 7 / D5 m = 6, T = 3) are listed in README §6. No κ claim, no window closed, no supplier changed.

Verify. bash research/ideal-decoder-floors/verify.sh /fresh/dir (Python standard library only, single process, no SAT; step 0 checks MANIFEST.sha256 and SOURCES.json and stops hard on mismatch; 66 PASS lines including step 0).

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