Certify the R439 K defect as the D-internal cross-term tally

[unpaidmoon]

Change and mathematical scope

Finite certificate, not a new kappa. On the three retained R439 helper-star words, the syntactic K_equals_D defect equals the D-internal cross-term tally on every target. Selected κ stays 0.001672660197076918. No paid-profile change and no weakened-contract adoption.

Let K be the helper-column mask left on each target by the frame-free replay of (A, scatter, B), N the register ids named by D writes to that target, and E the helper columns actually xored into the target when D runs alone from identity columns. The cross-term tally is C = E XOR (N restricted to helper columns). For an arbitrary D section, C equals the defect exactly when E = K.

Measured on all 111 targets of all three words: E = K, every helper register is restored at the end of D, and the defect/cross totals are 17,636 / 17,636 / 17,653 bits across 439 / 439 / 439 sectors. The owner checker agrees on the raw helper part and still reports syntactic K_equals_D false.

Task and overlap

Base: bb4bccbf42844d2801487450727c2f63e10d1d0f. Priority revision 6. Card bit-helper-contract, scope refined.

The card asks for one replacement H2/K implication or a counterexample. PR #4092 already showed that no sector-local or single-operation repair meets the syntactic flag, and its open question 1 asks whether that defect equals the restored-edge cross terms inside D. This package answers that comparison. It does not choose an admissible contract, which is why the scope is refined rather than a full interface replacement.

Nearest open work:

• Research: per-helper-sector census of the literal K_equals_D defect on retained R439 helper-star words (no new kappa) #4092 counts the defect by sector and leaves this comparison open. It does not execute D or identify the bits with delivered-versus-named columns.
• Show that one restored helper edge misses the literal K mask #3533 shows one abstract restored edge whose literal mask misses the delivered column. This tally is that mechanism on the three full words, including target-to-target reads that carry earlier helper columns.
• Research: price the first profitable helper-star interface word (no κ) #3891 prices a helper-star interface word. It is a different question.

Overlap refresh at 2026-10-20 01:38 UTC covered 803 open PR bodies and 2,630 discussion records. No open PR states this equality. Closed-PR body coverage was not refetched.

Validation

CPython 3.12.3, from the repository root:

python3 -B research/r439-k-cross-term-tally/verify.py
python3 -I -O -Werror -B research/r439-k-cross-term-tally/verify.py
python3 scripts/research_package.py --package research/r439-k-cross-term-tally

Expected line:

PASS_R439_K_CROSS_TERM_TALLY words=3 targets=111/111/111 equal_targets=111/111/111 defect_bits=17636/17636/17653 cross_bits=17636/17636/17653 sectors=439/439/439 helpers_restored=3/3 checker_K_equals_D=false/false/false mutation_breaks=3/3 synthetic_unequal=1 no_new_kappa

Both direct commands and the registered package replay passed here. Controls: SHA-256 pins; owner-checker agreement; one helper-edge deletion breaks E = K on each word; a synthetic h0 ^= h1, y ^= h0, h0 ^= h1 with an empty raw word separates the defect (h0) from the cross term (h1).

Not run: another helper-mixing lineage, native or all-size replay, a paid-profile recomputation, Lean, and hosted CI.

Credit

PR #4092 (Nick Petalas) for the sector census and the open comparison. PR #3533 for the one-edge mechanism. The words are the retained PR #967 / PR #1062 helper-star lineage, checked with research/e8-bit-p111-tuned. Prepared by unpaidmoon with substantial AI assistance (Cursor Grok). Apache-2.0.

Open next questions

1. Which extra hypothesis makes E = K true outside this triple? The synthetic edge shows an unconditional reading is false. Test one other pinned helper-mixing word before treating the equality as a legal-word identity. Admitting the hypothesis as the H2/K contract is a separate decision.
2. After the mask comparison is read as executed delivery, which H2 clauses still fail? The 612 target-to-target reads are a read-kind failure that this mask equality does not remove.

Workflow feedback (optional)

The handoff thread's explicit "smallest useful next task" lines are easier to order than prose DONE posts that leave a question in the last sentence. No separate session report.