Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim)

[DreamingOfClouds]

Research note, no κ claim. Adds research/decoder-floors-reduction/ only; bash verify.sh re-runs every check (about 6 min, ≤ 3 processes, numpy only, no SAT solver).

Target

philippeVV's gain law (#1587, follow-up 6): gain(L) = |L| − C(L) is monotone and superadditive, so "decoder optimisation is dead" for every E8 line set is the single statement C(P120) ≥ 72 (gain ≤ 48).
It implies P112 ≥ 64, P111 ≥ 63 and P104 ≥ 56. The converse does not follow (P112 ≥ 64 does not imply P120 ≥ 72). philippeVV's follow-up 8 closes all lift cases and names R1–R4.

New results in this PR

1. Universality (PROVEN, any E8 line set with the dense table). Every decoder of every masked realisation becomes a decoder of the single matrix N₀ = G/2 at the same cost, by adding cost-0 single-entry terms on non-orthogonal pairs. So C(L) is the minimum over decoders of N₀(L). The inside / one-sided / two-sided split of N₀ is exhaustive, and the rank of a realisation is not an invariant. This is a full-class statement, so it survives the gain-law padding.
2. Structure (PROVEN, any line set). The outside factors of a two-sided decoder are free. Sylvester gives m ≥ 10, and m = 10 forces (s,t) = (1,1). In the Lemma-D normal form |J_Q| ≥ t + 2.
3. P112 two-sided decoders.
   • Projection transfer: cost(P,Q) = cost(P,Q_H) − Σ_{J_Q} ε^col with (P,Q_H) one-sided. Only outside corrections are new.
   • Pivot bound (CONDITIONAL on Decoder floors, case (b) for every number of terms: one-sided P112 decoders cost ≥ 64 without corrections (unique optimum) and with ≤ 2 non-full corrections (no κ claim) #2816 R4).
   • m = 10 with a 2-element circuit is closed (via Decoder floors, case (b): nine terms outside col M (P112 ≥ 64, #968 P104 ≥ 56; no κ claim) #2724 and 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 #2338).
   • CHECKED catalogue of outside-column corrections (511 classes × 1200 directions).
   • For t = 1: pair lemma, low-column collinearity, at most two weight-7 columns, and an exact saving count. The disjoint (1,1), m = 10 case is closed (E_Q, E_P ≤ 8, so E_Q + E_P ≤ 16 < 17; idempotent-rank and Ω-orthonormality argument, CHECKED d4.py, omega.py).
   • Items 3.x depend on the open PRs Decoder floors, case (b): nine terms outside col M (P112 ≥ 64, #968 P104 ≥ 56; no κ claim) #2724 / Decoder floors, case (b) for every number of terms: one-sided P112 decoders cost ≥ 64 without corrections (unique optimum) and with ≤ 2 non-full corrections (no κ claim) #2816 and are labelled CONDITIONAL on them.
4. Batching reduces to the single-call floor.
   • Multiplicity inequality and converse.
   • Twin-fold theorem TF (proportional shared terms never help).
   • Transport lemma: twins are the only cost-monotone line-wise folds.
   • Fold barrier: rank-(2,2) multi-diagonal terms survive every fold.
   • Relative-rank lemma and free dangling.
   • Covering ceiling of 9 per block.
   • Lemma K′ and Lemma E: the GL(9) digon clique costs ≥ 64k + 7.
   • Theorems C, D and M.
   • Dangling-proof witness bound: one shared term with square coordinate restrictions is closed unconditionally in 277/512 (pair, chart) cases (CHECKED on 8 charts).

Map

Slice (P112 unless stated)	Status	PR
Inside (both factor families in the root code), any m	≥ 64	#2338 (merged)
Rank-8 rigidity	proven	#1212 (merged)
One side outside, m = 9	≥ 64; P104 ≥ 56	#2724
One side outside, every m, ≤ 2 non-full corrections	≥ 64, unique optimum	#2816
… ≤ 3 non-full corrections	≥ 64 except six multisets	#2826 (draft)
Rank-9 realisations of E8	B-form / A2 family	#2819
A2 family	≥ 64 / 56 / 72	#2784 (+ B-locus follow-up)
h = 10, 11	every term ≥ 8 / ≥ 7	#2639
Punctured units (Model C)	p ≤ 46, 48–56, p = 47 with m ≥ 8	#2639, #2806
D_n units	floors	#2719
Every realisation ⇒ decoders of N₀; outside factors free	PROVEN	this PR
Two-sided ⇒ one-sided + outside corrections; pivot bound; small circuits; catalogue; t = 1 lemmas	PROVEN / CHECKED, CONDITIONAL on #2724 / #2816	this PR
Disjoint two-sided (1,1), m = 10	≥ 64, CONDITIONAL on #2816	this PR
Batching ⇒ single call except rank-(2,2) multi-diagonal terms; 277/512 one-shared-term cases	PROVEN / CHECKED	this PR

Conjectures (the remaining gaps)

Decoder optimisation is dead on every slice above, modulo the open PRs cited. On P112, K1 ∧ K2 ⇒ C(P112) ≥ 64 (with #2338; CONDITIONAL on #2816). On P120 the master is only restated on N₀: the route needs #2338's inside floor on P120, K4 and a two-sided P120 analogue of K1.

• K-P120 (master). Every decoder of N₀(P120) costs ≥ 72. By universality this is C(P120) ≥ 72, restated rather than reduced.
• K1. Every two-sided decoder of N₀(P112) costs ≥ 64. Equivalently, outside corrections never exceed the excess of the one-sided projection. K1 ∧ K2 ⇒ C(P112) ≥ 64 (CONDITIONAL on Decoder floors, case (b) for every number of terms: one-sided P112 decoders cost ≥ 64 without corrections (unique optimum) and with ≤ 2 non-full corrections (no κ claim) #2816).
• K2. One-sided excess pays for corrections: Δ ≥ n_nf for n_nf ≥ 3 (Decoder floors, case (b) for every number of terms: one-sided P112 decoders cost ≥ 64 without corrections (unique optimum) and with ≤ 2 non-full corrections (no κ claim) #2816, Decoder floors, case (b): the six remaining d = 8 multisets of #2816 are dead (≥ 64 with ≤ 3 non-full corrections; E8 lifts ≥ 67; no κ claim) #2826).
• K3. (1,1), m = 10, non-disjoint |J_P ∩ J_Q| ≥ 2: cost ≥ 64 (the disjoint case is closed here).
• K4. P120 versions of the one-sided results, and P112 certification of Rank-9 realisations of the E8 pattern are B-form or the A2 family (classification; no κ claim) #2819.
• K5 (batching, H×). A single-call decoder containing a cancelled residual costs ≥ 64 + 7.
• K6. A per-term floor for label windows h ≥ 12 (break-even 51 / 46 / 42 at h = 12 / 13 / 14 is recorded pricing, not re-run). The full master over all tables also needs h = 10, 11 (Decoder floors, closures: punctured E8 units (Model C) to p = 46 (all m) and p ≤ 56 (merged m ≥ 8), windows h = 10–11 (no κ claim) #2639) and K6; Decoder floors, closures: punctured E8 units (Model C) to p = 46 (all m) and p ≤ 56 (merged m ≥ 8), windows h = 10–11 (no κ claim) #2639's per-term floors are not by themselves shown to give 72 on P120.

Relation to philippeVV's R1–R4 (#1587 follow-up 8).

• R1 and R2 are not needed for the master: universality replaces every realisation by N₀. Universality does not prove R1 or R2. R2 is Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212.
• R4's m ≥ 10 part is K-P120; its P112 shadow is K1 ∧ K2, and its first open configuration is K3 (non-disjoint). R4's h ≥ 10 part is Decoder floors, closures: punctured E8 units (Model C) to p = 46 (all m) and p ≤ 56 (merged m ≥ 8), windows h = 10–11 (no κ claim) #2639 plus K6.
• R3 is not addressed here.

Credit: philippeVV (gain law; lift cases; R1–R4), #1212, #2338, #2724, #2816, #1633, #1224, #968, #1714. Lean Lemma OC is #2694.

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