Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864

[Dugongue]

This adds research/e8-signed-prefix, an offline, source-bound finite construction with

kappa = 125311832453733 / 125000000000000000 = 0.001002494659629864.

Against the frozen PR392 head ac55c57035c197f744390fe4bc9cf49df1b61816 (0.001000220224856110), this is an increase of 0.22739%. The snapshot includes PRs through 393. No claim is made against later work.

This is a conditional finite construction, not an unconditional integer-multiplication theorem or a Lean certificate. The inherited all-size supplier, dirty-bank, compiler, ordinary-call/borrowed-row restoration, precision and analytic interfaces remain conditional and are stated in the proof.

New contribution

We extend the positive target-prefix mechanism of PR390 to signed donor coefficients and apply it to PR376's latest R781 complex word, reconstructed from PR378. The selected witness has 25 disjoint pairs: 16 with q=+1 and 9 with q=-1. It does not claim a shared-donor/fanout gain or an unrelated new algorithm.

At a common initial zero frame, subtract q times the donor target; synthesize the exact difference of the two prefixes using source/write-version/frame tokens; then restore q times the donor at their common recovery frame. The initial donor value and its accumulated prefix cancel identically for arbitrary dirty inputs. All original helper/source frame visits remain, including visits whose scalar delivery cancels. Setup, recovery, endpoint adapters, work registers and phases are charged.

The local paid histogram delta is

rank:   1   2   3   4    6   7
calls: -8  -8  -7  -8  -14 +23

This preserves rank mass and strictly improves the moment for every exponent between zero and one. It removes 22 child calls per invocation, or 1,584 across the 72-invocation complex assembly. The actual dirty scalar operator, retained totals and helper cleanup are unchanged, so the original literal response K remains valid.

Composition and direct credits

• PR376: exact helper substitutions/frame repairs producing R781, and bit helper reuse.
• PR378, with PR365/374 ancestry: far-retimed complex source.
• PR377: compressed nine-stage complex assembly, phase/owner framework and finite invoice.
• PR381: nine-stage unsplit bit assembly and its paid checks.
• PR390: the preceding positive target-prefix restoration mechanism and signed dirty-column checks.
• PR387 / PR185: six fixed ordinary bootstrap levels and outer slack refinement.
• PR392: applies that refinement to the preceding prefix supplier and is the frozen comparison point.
• PR384 / PR315 / PR144 / PR23: retained exact pricing/outer engine lineage; additional compiler/geometry ancestry and original authors are credited in NOTICE.md and bundled notices.

We compose these ingredients through the actual paid profile and 47 strict outer constraints; we do not add reported kappa gains. The complex supplier binds. The bit coarse saving remains 255384852701747/250000000000000000.

Reproduction and validation

From the uploaded directory, with Python dependencies in requirements.txt, C++17 and Boost headers:

python -B verify.py /absolute/path/to/a-new-output-directory

Set CXX and BOOST_INCLUDE only if needed. The verification is offline and source dependencies are included. It checks this changed composition rather than running unchanged public PR pipelines.

The replay covers reconstruction/negative controls, all 1,021 independent dirty columns in both local orientations, private restoration, 950 exact Clifford frames and 3,140 directed edges, Gaussian endpoint phases, the full nine-stage global template, 56,232 bank assignments, all 14,709 live-family cover corrections, independent template decoding, the changed finite invoice, both exact rational moment engines, the bit component and all 47 outer inequalities. The manifest binds the package bytes.

Research for future workers

research-notes/ includes the sufficient signed-prefix, interval, source-dual and clean-quotient algebra; the C++ signed witness search; exact cut-readout search; source-dual scalar witness; and bounded negative results. The source-dual witness has no physical admission and contributes no kappa gain. The clean-quotient route explicitly constructs dirty-response compensation rather than assuming it exists.

Broader two-donor, interval, normalized clean-response and helper-readout screens produced no further admitted saving. Their exact scopes and counts are recorded; they are not global impossibility claims. Structural DAG retrieval is labelled source-only, with no new Lean certification.

The package is small, self-contained and contains no RAR/ZIP substitute for required source files. Original licenses and attribution are preserved.

[Mister-Meeseeks]

Thank you @Dugongue. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Signed target-prefix cancellation on R781.

Review and limits: Exact reconstruction, changed complex admission, bit replay and 47-constraint assembly; arbitrary helper dirt and signed coefficients matter.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: c3251c8f33fac9332fb3606c2dc0bd4f32de4615. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.