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Where characteristic 2 starts to pay: a T² threshold for root-unit decoders (E6 ties, E7 first gain found; no κ claim) - #2125
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Mister-Meeseeks merged 1 commit intoOct 16, 2026
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…coders (E6 ties, E7 first gain found; no κ claim) Package research/char2-threshold. Exact F2 costs of the full-orthogonal-pattern pairings of root units against T^2 (D4-D6 and E6 tie, E7 49 < 63 first gain found, E8 72 < 120), exact minimum cost of a Fano Levi-graph realisation over Q (9 at T = 4, 13 at T = 5 with rank(PQ^T) = 5, against v = 7), numerical realisability of Sp4/Sp6 recorded. Law is an explanation plus evidence; no claim that T <= 6 cannot beat v. No kappa claim. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
This was referenced Oct 14, 2026
This was referenced Oct 15, 2026
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Where characteristic 2 starts to pay: a T² threshold for root-unit decoders (E6 ties, E7 first gain found; no κ claim)
Package
research/char2-threshold. Model, cost formula and lemmas are those of the merged #1590 (exact decoders, Lemma B, ERR1) and of #2111 (three-total decoders); this PR adds the explanation of where an F2 gain over v first appears.The law (argued). For root units the dense F2 cost of the full orthogonal pattern is about T × (block rank), the block rank is about T, so a gain over v needs more than about T² lines.
Table (CHECKED, exact; cost of an admissible pairing for the exact decoder M = Gram): D4/D5/D6 tie (cost = v = 12/20/30); E6 ties exactly at 36 = 6² (stored basis and B; B0 costs 37 in the first-independent basis); A4–A6 above v (B0 upper bounds 16/25/36; annealing found 16/21/34); E7 is the first gain found (49 < 63, T = 7; also checked in #1590, G3.E7-dyadic); E8 (72 < 120) is known (#1590 ERR1). The E6 B0 value 37 is checked in #1590 (G4.E6). F4 ties at 24 (recorded).
Fano Levi graph (CHECKED, exact): it realises over Q at T = 4; the exact minimum F2 cost of the stored realisations (rank(PQᵀ) = T, all T blocks nonempty) is 9 (T = 4) and 13 (T = 5) against v = 7: no gain at that size.
Realisability (RECORDED, numerical): the Sp4 doily (15 points) realises at T = 4 and not at T = 3; the 63-point Sp6 realises at T = 7, and Levenberg–Marquardt found no realisation at T = 6 or 5. No exact rational realisation is shipped, so these are not CHECKED.
Scope. The searches (annealing, cliques, lattice pools) are not exhaustive and are recorded, not re-run; the table costs are costs of specific B (upper bounds on the F2 minimum). There is no claim that T ≤ 6 can never beat v; the law is an explanation plus evidence. No κ claim, no window closed. Context: #606, #1698, #1830.
Verify.
bash research/char2-threshold/verify.sh /fresh/dir(Python standard library andcc; seconds; step 0 checksMANIFEST.sha256andSOURCES.jsonand stops hard on mismatch; 10 PASS lines including step 0).Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.
🤖 Generated with Claude Code