Decoder floors, case (b): excess 3 is paid (≥ 64 with ≤ 4 non-full corrections; E8 lifts ≥ 68; no κ claim)

[DreamingOfClouds]

Follow-up to #2816 and #2826 (both ours). Only research/one-sided-excess-three/ is added; nothing existing is changed. No κ claim, no supplier word, no SAT solver.

Result

Setting as in #2816: P112, case (b), characteristic 0, every number of terms. Δ is the ε-free excess over 64, n_nf the number of non-full corrections of the normalised decoder.

• All eight slack-3 multisets are dead. [7^5,6^5,2], [7^5,6^5,1,1], [7^4,6^6,3], [7^4,6^6,1^3], [7^9,4], [7^9,3,1] (d = 8) and [7^3,6^6,2], [7^3,6^6,1,1] (d = 7) do not occur as the minimum-weight basis of S_{≤7}. With 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 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, no d ≥ 1 multiset of slack ≤ 3 occurs. The kills are exact finite computations run in two independent implementations that agree on every rank histogram (two 6-flats plus ≤ 2 lines: span ≤ 4; two 6-flats plus a 3-flat: span ≤ 7 < 8, with exact span ≤ 4 on the 6 relaxed-rank-8 pairs (all 161,420 third flats); a 7-flat and a 4-flat: span ≤ 4; controls: two 7-flats reach rank 8 on 152 pairs, two 6-flats give span 3).
• Carrier Lemma for κ ≤ 3 (written proof; the finite inputs are re-run): at most Δ z-carrying members of a spanning family of total weight 64 + Δ, Δ ≤ 7, when the weight-≤ 7 part has cost ≤ 3. New ingredient: a Parity statement for three pairwise non-orthogonal lines (Q on E8/2E8).
• Δ ≤ 3 ⇒ n_nf ≤ Δ (decoder level) and Δ ≤ 3 ⇒ ρ_0 ≤ Δ (lift level).

Consequences:

• one-sided P112 decoders with ≤ 4 non-full corrections cost ≥ 64 (any number of full ones); in general ≥ 64 − max(0, n_nf − 4) ≥ 60 (was ≥ 59 in 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);
• lift level: cost_w(S) ≥ 64 + min(|I|, 4); every lift of the E8 root code has cost_w ≥ 68 (≥ 72 unless the P112 restriction has Δ ≥ 4); gain-law form |P120| − C(P120) ≤ 52.

The bound ≥ 64 at n_nf = 4 is tight: the root-swap family of #2816 §6 attains exactly 64 for n_nf = 1..4.

Proof and checks

Every claim in the README carries its proof (Lemma B for any d, the kill criterion, the Carrier Lemma with its case analysis, the peeling argument) and is labelled PROVEN / CHECKED / RECORDED / INHERITED. verify.sh re-runs from a fresh git archive: manifest, byte-identical regeneration of both data files, the slack-3 landscape (land3.py 3, exactly the eight survivors), the finite inputs of the Carrier and Parity Lemmas (all 112 roots and 8 frame directions: coloop structure and cocircuit sizes ≥ 5; Parity searches over all pairwise non-orthogonal triples and quadruples), and both implementations on the survivors' flat configurations and the controls, chunked (at most 3 processes, nice -n 10, timeout 1200 on each, no wait loops). Then check_logs.py prints one PASS/FAIL line per assertion.

Not re-audited by the second implementation, and covered by written proofs or citations instead: Lemma B, completeness of the direction list for |Off| ≤ 80 (#2826 §4), the bookkeeping of the landscape (land3.py, one implementation; its stages are the cited rules of #2816 and #2826), and the Parity and Carrier proofs.

Not claimed

Δ ≥ n_nf for n_nf ≥ 5; two-sided decoders; case (c) (see #2784); P104 and other punctured units; positive characteristic.

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