[petalas]

Summary

Adds research/r439-star-kdefect-sectors/: a complete per-helper-sector census of the syntactic K_equals_D failure carried by the three retained R439 helper-star words (seed19 candidate from the PR967 lineage; PR1062 composition and candidate), under the pinned owner checker's own definitions (research/e8-bit-p111-tuned/code/xcheck_p.py).

Established (finite certificate on three named words at base bb4bccbf42844d2801487450727c2f63e10d1d0f):

word	D gates	defect bits (K ⊕ N)	defective helper sectors	sector-local KS(h) pass	single-gate repairs	single-bit repair
seed19 candidate	1,165	17,636	439 / 439	0 / 439	0	no
PR1062 composition	1,165	17,636	439 / 439	0 / 439	0	no
PR1062 candidate	1,195	17,653	439 / 439	0 / 439	0	no

• Global, not local: all 439 helper sectors are defective (sector sizes 4–68 targets); a per-helper sector-local replacement of H2/K cannot lift to the global flag — any replacement must carry cross-sector hypotheses, consistent with the retained global PR1194 route.
• No single-operation repair: exhaustive enumeration of all single-D-gate removals and all single-bit toggles yields zero repairs on every word.
• Localization: the raw helper part K is bit-identical across all three words (0 differing bits) although phase-B replacement rewrites B (712→578) and C (2,574→2,440); identical D leaves the defect exactly at 17,636 bits; the star rerun (+30 D gates, N differing in 173 bits) moves the defect by 17 bits. The failure lives entirely in D versus the fixed raw helper component.
• H2 clause split: frame clauses are clean on all words (0 nonzero-frame gates, 0 start violations); H2 fails purely via destination/read clauses (442/442/472 non-target-writing and 612 non-helper-reading D gates).

An independent GF(2) all-column replay agrees with the pinned owner checker on K_equals_D, J_identity and U_block_form for all three words, and full exact-legality replay passes as a positive control (the words are legal while carrying the quantified failure).

Related open work

• Show that one restored helper edge misses the literal K mask #3533 — abstract three-gate restored-edge K-mask mechanism; this PR counts that mechanism's consequence across full retained words (no numerical conflict; different object scale).
• research: pin dirty R439 helper K mismatch #3904 — pins one D-inclusive per-target example (target 112, helper column 260). Checked numerically against this census: under the checker's official raw definition W0 = (A, scatter, B) (D excluded), used here, that bit pair is not in the defect set (column 260's sector has 57 defective targets, not including 112). Different raw-word definitions; no contradiction, and the distinction is documented in the package README.
• Research: price the first profitable helper-star interface word (no κ) #3891 — prices a helper-star interface word; addresses the paid-profile side, not the syntactic K failure decomposition; no overlapping number.

Validation

• python3 -I -B research/r439-star-kdefect-sectors/verify.py and the -O variant: PASS_R439_STAR_KDEFECT_SECTORS words=3 defect_bits=17636/17636/17653 defective_sectors=439/439/439 KS_pass=0/439/0/439/0/439 single_op_repairs=0 controls=5 no_new_kappa (~2 s each, standard library, CPython 3.12.15).
• python3 -B scripts/research_package.py --package research/r439-star-kdefect-sectors: PASS_REGISTERED_RESEARCH_PACKAGE outcome=completed.
• python3 -B scripts/check_research_commit.py --package research/r439-star-kdefect-sectors --check-source-history: PASS_RESEARCH_COMMIT_BYTES.

Scope and limits

Finite certificate on three named words; not a proof that every weakened per-operation contract fails (a changed contract with named cross-sector hypotheses escapes it); no supplier admission; no κ, paid-profile or selected-frontier change; no native, hosted, all-size or cold-replay execution (not requested by this package). Inputs bound by SHA-256 in SOURCES.json.

Prepared by Nick Petalas with AI assistance.