One-sided padding lemma: P112 to P120 at cost exactly +8 (L120 suffices for Delta >= n_nf)

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research/one-sided-padding-lemma/MANIFEST.sha256

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+# sha256 of package files
+b056475ca54f685a26ff1de6ad2986fd31c4d448e02bac428afd85b9cd2443d1  README.md
+e86d8e5a057cfe111013d9ff499ba4898f88c4e40a775afbf3249fb3dedc9edd  PROOF_MAP.md
+1ea58e398f40239c0945d495a807b33bcadeb6d5dcd147a12bf71ff06a9ed7ec  NOTICE.md
+c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4  LICENSE
+a9029fbebed80d5530d9a770340843cfd032aac6af297d237cc24b758537ea81  requirements.txt
+daade0beeb1e35574b216b8c7ee60a51c633e46bad63503fb3232f2ab4b27778  checks/pad_audit.py
+402a7ebf3913ef62da52a7b20e8f2f794a4e202b24fac10ebd8a29c5f6d45039  checks/pad.py
+b84b5d55ae302ffdf40ee1323c15f3b57180b9e9624a857a5556e10a73b553b8  expected/pad_audit.out

research/one-sided-padding-lemma/NOTICE.md

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+# Notice
+
+Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0 (`LICENSE`, copied from the repository root).

research/one-sided-padding-lemma/PROOF_MAP.md

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+# Proof map
+
+| Claim | Status | Proof / check |
+|---|---|---|
+| (1) Padding: one-sided P112 decoder of cost c gives one-sided P120 decoder of cost exactly c+8 | proved; checked | Exactness: rows of P112 unchanged, row f_i = (2/mu)(mu e_i) = 2e_i for i in I, row f_j = 2e_j for j not in I, so P_E Y_E^T = C_E and q_k = C_E y_k (one-sided). Cost (rank B[R_k,A_k] = dim U_k - dim(U_k cap l(A_k)^perp)): non-corrected term unchanged (A_k grows, perp shrinks, U_k cap perp = 0 already); corrected term +1 (A_k gains f_i, l(A_k) = W_i + z_i = F_2^9, beta(z_i,z_i)=1); unit term exactly 1. Total c + |I| + (8-|I|) = c+8. Merging corrected terms with the same i is required first (an unmerged second term would also lose its epsilon, +1) and never raises cost (#2816 section 3.1). Check: `checks/pad.py` (5 decoders), `checks/pad_audit.py` (12 decoders incl. a split-correction decoder that exercises merging); both assert exactness over Q and equality of two cost computations. |
+| (2) Lift form cost_w(S+) <= cost_w(S) + 8 - |I| | proved | Minimal basis of S extended by 0 keeps supports and weights; p_h^{E8} - ext(p_h^{X_I}) lies in span of delta_{f_j} (j not in I), each of weight 1. |
+| (3) L120 suffices for the one-sided P112 floor 64+|I| for all I, i.e. Delta >= n_nf for all Delta | proved (sufficiency only) | (2) with the #2816 section 7 identity; no P120 decoder cost model is used on this route. Reading the corollary as "one-sided P120 decoders cost >= 72" additionally uses the dense-beta P120 cost model (imported, not covered by the P112-only theorem). Chain: Delta >= n_nf <= L120 <= [Delta + D <= 7 implies rho_0 <= Delta]; converse not claimed. |
+| (4) delta delta^T padding is two-sided | proved | delta_{f_i} not in C(E8): codewords have |Off| >= 53. q = C_E 2e_j stays in col C_E. |
+| L120 | OPEN | Residue: 38 OPEN rows of the #3049 ledger; 7 prior-only rows; 7 E6-slack rows; d=0 sliver Z = ker rho cap S nonzero with D >= 1, kappa <= 6. |
+
+Checks cover the 12+ decoders: coordinate; root swaps k=1..4; basis changes; overcomplete m=11; shift decoders; split swap.
+Related: #2816, #2826, #2862, #2985, #3049, #3029 (conditional only on L120 via this lemma).

research/one-sided-padding-lemma/README.md

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+# One-sided padding lemma: P112 to P120 (no kappa claim)
+
+Statement and a two-implementation check; nothing here claims a kappa improvement.
+
+**Lemma (padding).** Every one-sided P112 decoder (case (b)) of cost c becomes a one-sided P120 (E8) decoder of cost exactly c + 8: after merging corrected terms with proportional y, for each frame line f_j with j not in the corrected set I add the term (delta_{f_j}, y = 2e_j); for i in I the corrected term gets p_k(f_i) = 2/mu. The new q = C_E * 2e_j is the E8 frame codeword, so the decoder stays one-sided.
+
+**Lift form.** S containing the code of X_I gives S+ (extend by 0 on f_j, j not in I, plus the unit vectors delta_{f_j}) containing C(E8), with cost_w(S+) <= cost_w(S) + 8 - |I|.
+
+**Consequence (sufficiency only).** L120 (every lift S containing C(E8) has cost_w >= 72) suffices for the one-sided P112 floor 64 + |I| for every I, i.e. Delta >= n_nf for all Delta, using only the #2816 section 7 identity (corrected cost = cost(X_I) - |I|). The converse is not claimed. This makes the conditional in #3029 conditional on L120 alone.
+
+**Remark.** Padding with delta delta^T is two-sided: delta is not a codeword (codewords have |Off| >= 53), so it enlarges col Q. The earlier remark that the reduction "does not transfer" referred to that two-sided padding.
+
+**OPEN (L120).** Not proved. Residue: the 38 OPEN rows of the #3049 ledger (one d=4 [7^5,4]; one d=6 [7^7,6]; four d=7; 32 d=8), the 7 prior-only rows (killed only by prior arguments), the 7 E6-slack rows (E6-cover rows, conditional), and the d=0 sliver Z = ker(rho) intersect S nonzero with D >= 1, kappa <= 6.
+
+Cites: #2816, #2826, #2862, #2985, #3049, #3029.
+
+Run `bash verify.sh` (needs `sympy`; `PYTHON=/path/to/python`). Runtime about 2 s. `PROOF_MAP.md` maps each claim to its proof and check.