Research: all-puncture E8 star decoder exclusion and p=56 classification (seed 49, no new kappa)

[3xcalibur-tech]

Summary & Status

• Status: Research note / scoped negative result (no new $\kappa$; certificates/selected-result.json and upstream/ are unchanged).
• Track & Seed: Track 40–49 (Structural decoder research), OS random seed 49.
• Base Commit: ce7ad1b6ab09452c3478723a2e96ed5ff212dc08
• Predecessors Extended: PR Prove E8 star-decoder puncture thresholds through 35 deleted lines #631 (Chi Zhao, 0 <= p <= 35), PR Scoped negative result: classify the E8 rank-five boundary at 36 punctures #875 (Philipp / @mightyOil, p = 36), and PR Scoped negative result: E8 star-copy ratio cannot improve through 55 punctures (no new kappa) #939 (Ross Richardson, 0 <= p <= 55).

What Changed / Was Learned

PR #939 excluded the $E_8$ root-only star label-decoder from improving on the 9-puncture ratio $63/111$ up to $p \le 55$ and stopped at $p = 55$ because at $p = 56$ a minimum-weight codeword $c_0 \in C$ ($\operatorname{wt}(c_0)=56$) can vanish on the 64 live ports $L = {0,\dots,119}\setminus P$, dropping the live binary label rank $\dim_{\mathbb{F}_2}(C|_L)$ from $9$ to $8$. Moreover, the pairwise ($k \le 2$) capacity inequality $M_a + M_b \ge 84 - p$ used in PR #939 fails at $p = 72$ ($27/48 = 9/16 < 63/111$) and across $78 \le p \le 119$ (collapsing to $0$ for $p \ge 84$).

This package (research/e8-star-all-punctures-seed49/) resolves the remaining range $56 \le p \le 119$ and classifies the $p = 56$ dimension-loss boundary:

1. Proposition 1 (Live-Quotient Star Cost Equivalence under Dimension Loss): For any puncture set $P$ with $0 \le p \le 119$ ($1 \le v = 120 - p \le 120$) and restriction kernel $K_P = {c \in C : \operatorname{supp}(c) \subseteq P}$ of dimension $k_0 = 9 - \dim_{\mathbb{F}_2}(C|_L) \in {0,\dots,8}$, surviving rational star rank $s_P(c)$ is constant on cosets of $C / K_P \cong C|_L$ and vanishes iff $c \in K_P$. Thus the minimum star cost over $(9-k_0)$-element bases of the live quotient $C|L$ equals the minimum star cost $C{\text{star}}(P)$ over 9-element bases of $C$.
2. Proposition 2 (Complete Two-Orbit Classification at the $p=56$ Boundary): The $255$ puncture sets of size $p = 56$ with $\dim_{\mathbb{F}_2}(C|L) < 9$ are the supports of the $255$ weight-$56$ codewords of $C$ (all with $k_0 = 1$, $\dim{\mathbb{F}_2}(C|_L) = 8$), which partition under $W(E_8)$ into two orbits:
   • Orbit 1 ($135$ even-frame codewords): 511-reader rank histogram ${0:1,; 7:112,; 8:398}$, cumulative $\mathbb{F}2$-dimensions $(1,1,1,1,1,1,1,8,9)$, exact optimal basis $(0,7,7,7,7,7,7,7,8)$ of star cost $57$ on $64$ ports ($57/64 = 0.890625 > 63/111$). All $56$ single-port restorations ($p=55$) restore $\dim{\mathbb{F}_2}(C|_L)=9$ with cost $58$.
   • Orbit 2 ($120$ root-nonorthogonality supports $S_a = {i : |\langle L_i, L_a \rangle| = 4}$): live ports ${a} \cup a^\perp$ ($1+63=64$ lines), 511-reader rank histogram ${0:1,; 6:56,; 7:198,; 8:256}$, cumulative $\mathbb{F}2$-dimensions $(1,1,1,1,1,1,8,8,9)$, exact optimal basis $(0,6,6,6,6,6,6,6,8)$ of star cost $50$ on $64$ ports ($50/64 = 25/32 = 0.78125 > 63/111$). All $56$ single-port restorations ($p=55$) restore $\dim{\mathbb{F}_2}(C|_L)=9$ with cost $51$.
3. Proposition 3 (Sharp Subspace Support-Union Hierarchy, $k=1,\dots,9$): The minimum support union of a $k$-dimensional subspace $W_k \subseteq C$ is sharply
   $$(U_1, U_2, U_3, U_4, U_5, U_6, U_7, U_8, U_9) = (56, 84, 100, 108, 112, 116, 118, 119, 120),$$
   attained simultaneously by a single nested full flag $0 \subset W_1 \subset \dots \subset W_9 = C$.
4. Theorem 4 (All-Puncture $0 \le p \le 119$ Ratio Exclusion): Imposing $\delta_{d_i} \le p$ and $\sum_{i=1}^k M_{d_i} \ge U_k - p$ ($1 \le k \le 9$) across all $24{,}310$ sorted rank profiles (with PR Prove E8 star-decoder puncture thresholds through 35 deleted lines #631's $72-p$ for $0 \le p \le 8$) proves $C_{\text{star}}^{\text{live}}(P)/(120-p) = C_{\text{star}}(P)/(120-p) \ge 63/111$ for all $0 \le p \le 119$, with equality possible only at $p = 9$.

Reproducible Verification Commands

python3 -I -O -B research/e8-star-all-punctures-seed49/verify.py
python3 -I -O -B research/e8-star-all-punctures-seed49/independent_check.py
python3 -I -O -B research/e8-star-all-punctures-seed49/test_package.py

Scope, Attribution & AI Disclosure

• Scope: Applies strictly to the real $E_8$ root-only star label-decoder cost model $C_{\text{star}}(P)$ defined in PR Prove E8 star-decoder puncture thresholds through 35 deleted lines #631 and PR Scoped negative result: E8 star-copy ratio cannot improve through 55 punctures (no new kappa) #939; does not claim a formal proof of the asymptotic multiplication theorem or exclude non-star/multi-stage decoders.
• Attribution: Credits Chi Zhao (PR Prove E8 star-decoder puncture thresholds through 35 deleted lines #631), Philipp / @mightyOil (PR Scoped negative result: classify the E8 rank-five boundary at 36 punctures #875), and Ross Richardson (PR Scoped negative result: E8 star-copy ratio cannot improve through 55 punctures (no new kappa) #939), with retained sources pinned in SOURCES.json.
• AI Disclosure: Prepared by @3xcalibur-tech with substantial AI assistance from Antigravity (Google DeepMind / Gemini). Licensed under Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at ab74c73b5b512f459b9be97680f41390d45a3bf3.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Live-quotient equivalence, support-union hierarchy and all-puncture E8 star cost exclusion, conditional on the explicitly inherited parabolic classification.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.