Root-unit decoders: two lemmas, an erratum to #595, and open cases (partial; no κ claim)

[DreamingOfClouds]

No κ claim. This package is deliberately partial. It states two lemmas about exact decoders of root-line units under #446 Theorem C, an erratum to #595, one SAT closure, and the cases that remain open. Every result is labelled PROVEN, CHECKED, or ARGUED/OPEN.

python3 -m pip install -r research/root-unit-decoder-lemmas/requirements.txt
bash research/root-unit-decoder-lemmas/verify.sh /fresh/output/dir     # 35 PASS lines (2 in step 0), about 1 minute, single process; step 0 fails hard on any MANIFEST/pin mismatch

Results

id	result	status
A	Lemma A (T ≤ 2). For every admissible pairing B (not only B0), every exact decoder over Q (rational decoders) with at most two totals has Σ_j rank_F2 B[R_j, A_j] ≥ v, so its Theorem C cost is ≥ v. A3-A8 and D4 are closed for every pairing at windows h ≥ 5 (every window has h ≥ rank_F2 B ≥ ω and x*(h) is nonincreasing; D4 at windows 3-4 is not covered); D5-E7 stay open at T ≥ 3. Proof in the README (private lines peel; the shared part splits into two triangular pieces by the sign of α − β)	PROVEN; table arithmetic and 2,100 instance checks of the proof's inequalities CHECKED
B	Lemma B (lift identity). For every rational B′ on the support of B, v ≤ Σ_j rank_Q B′[R_j, A_j]. Corollary: a decoder cheaper than v needs blocks that no common rational re-weighting brings down to their F2 ranks (characteristic-2 cancellation). The Fano matrix (F2-rank 4, Q-rank 7) is included as an illustration only	PROVEN; identity checked exactly on 720 random pairs and on the dyadic E6/E7/E8 decoders (E8: F2 block-rank sum 72 < 120 ≤ every sampled Q-rank sum)
ERR1	Erratum to #595. Dyadic exact decoders use rank_Q G totals, below rank_F2 B0: E8 8 < 9 = rank_F2 B0, E6 6 < 7 = rank_F2 B0, so counting the totals through rank_F2 B0 (#595 README line 205: "the F2 decoder of the E8 pattern needs 9 totals (its F2 minrank is 9; c = 8 is UNSAT, checked in c6)") does not hold over Q. Exact 8-term decoders already appear in #717, #968 and #1077; the new point is the correction to that row. E8's cost 72 is attained by the 8-total decoder too, so the E8 cost is unchanged. (A second erratum to #595; #717's E1 is a different one.) The decoders are rebuilt in exact rationals	CHECKED
D5	No exact rank-3 decoder on D5 has all block ranks ≤ 4, so cost ≥ 13 and the D5 T = 3 windows h7-8 are closed. SAT, kissat UNSAT in 892 s, DRAT proof (1.71 GB, not shipped) verified by drat-trim (s VERIFIED). The generator, the CNF digest (f5e75d3b…7796) and the rerun and check commands are shipped; verify.sh regenerates the CNF and compares the hash, it does not solve	CNF regeneration CHECKED; solver runs RECORDED; encoding independently audited (SOUND-WITH-CAVEATS)
F′	Open problem. cert(M) ≥ ⌈2v/3⌉ for every exact rank-3 decoder would close every remaining T = 3 window (D6, D7, D8, E6, E7; window table in the README, max N(h)/v = 2/3). The obstruction is the isotropic lines with tied minimal 2-adic valuations	OPEN
R1	A condensed escape map: the hypotheses that are still live (H3 non-star decoders, about 20 copy-rank units on the 120 ports; H6 E8² Pauli, ×1.538, blocked by the I−K compensation circuit; H2; H4/H5; H10) with their thresholds, and three numeric targets	ARGUED (from our own analysis; not recomputed)

Scope

• The clique count result on the number of totals holds for pairings with B = 1 on the clique (B0-type); it is not proved here.
• For all pairings, Lemma A covers T ≤ 2.
• D5 T = 3 windows h7-8 are closed by a recorded, DRAT-verified SAT proof (CNF regeneration checked here); no transfer to D6 or E6 is claimed (their floors are w = 5).
• Everything else is open: D6-D8, E6 and E7 at T = 3, and T = 4, 5.

Frontier cited, not analysed. The best conditional κ among the open pull requests is 0.001676855229071321 (#1424, rohanarun; #1485, #1536, #1578 refine its endpoints), 3.08× under #595's bound of 5.1595e-3.

Follow-ups. The 2-adic certificate theorem, its sharp B3/C3 example, and further SAT closures.

Credits

Builds on #595, #446, #717 and #899 (format limits, Theorem C, package format), #968 (themilkman216), #1077 and #1180 (woklerx) and #1171 (3xcalibur-tech) for the E8 decoder floors that the H3 row builds on, and #1238 (verification format). Overlap check (decoder / lift / minrank / T ≤ 2 searches): #1177 (RossRichardson) is consistent with Lemma A in its F2 setting and treats a different object; #1471 is the same kind of F2/Q gap for the P112 graph; #669 and #1386 treat star decoders of the E8 unit, not root-unit decoders under Theorem C. We found no pull request that states Lemma A, Lemma B or the D5 closure.

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

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 08e134b4a0c7f04869dae703f2a2e688e53c44f0.

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

Reviewed scope: Written ordered-field two-total and Hadamard-lift lemmas, exact dyadic counterexamples to a totals-count premise, and CNF regeneration are accepted as partial research. The D5 UNSAT proof is not shipped or replayed here; its status remains author-recorded, and the later conjecture remains open.

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.