Conditional κ ≈ 0.001721588647: nine additional paired helper extractions

[mason-hale]

Change and mathematical scope

Composes nine additional paired input/output extractions on Jeremy Diviney’s PR2908, preserving every existing restored endpoint. Three use the existing frames; six become available after 13 explicit inherited-frame substitutions. The complete per-call change is 9H1 + 18H3 − 18H4, with nine helpers changing width9→8 and all nine additional private buffers charged.

The final literal T74 allocation gives the conditional bound

κ = 0.001721588647197801880605 − 10⁻⁴⁶ − 10⁻⁷⁰,

an increase of 0.000001190309843563994674 (about 0.06919%) over PR2908 at 51309592971e3b5be935599eed2a95404dc7781e. The complete stock is 66304 + 408 + 8 + 82 + 2 = 66804. All 271728 helper roles and actual owner residuals are included in the paid histogram.

This is a conditional finite supplier/outer certificate. The inherited all-size, compiler and theorem obligations remain. No selected-frontier change, practical timing claim, packing-optimality claim or unconditional multiplication theorem is made.

Task and overlap

Explicit donor assignment: improve the latest conditional κ. Task card complex-new-helper-inventory, refined to the paired endpoint/inventory mechanism; canonical investigation Discussion2612. Base 0706e744cbda1e4d0da5a6c569ad9b47876c6960.

Immediate predecessor: PR2908, which extracts helper408 and asks whether another independent direction can be preserved. This package supplies nine more, with literal gates, exact frame edits, a realized owner fill and full 82-buffer admission. The frozen word includes helpers 424,428,412,331,299,320,511,304,499 in that order.

The source chain also pins PR2894 at 18d03c447886895ef9abf888398178e0428533ca and PR2833 at e9307b48d7d1b690939935ecef0f8e524e6d92f4. All frozen predecessor sources remain unchanged. The separate rank-seven helper408 investigation concerns a different endpoint reduction; it is not claimed here. Final freshness coverage is recorded in evidence/FRESHNESS.json, with the usual limitation that metadata cannot establish universal novelty.

Validation

• Exact rational free-operator identities at all nine stages; final 632-input forward and dual audits for both providers; 36 semantic rejection controls.
• Fresh full five-stage native replay for both final providers, including phases, wrappers, local replay and cleanup.
• Exact 271728-role owner check, actual 82-buffer namespace and66804 physical stock; five owner-corruption controls; independent directed rational moment with explicit tails and outward rounding.
• Complete source-bound admission: rebind, discovery, block norm, actual prefix, stock, independent outer checks, rejection controls and final release. The public replay uses an explicitly authenticated unchanged retained parent/native cache from the earlier fresh replay; it does not describe that retained work as fresh in this invocation.
• The registered bounded replay reruns local/owner/moment checks and binds stored full-release evidence. The complete public command separately reruns native and integrated admission.

python3.12 -B scripts/research_package.py --package research/paired-helper-extension
python3.12 -B research/paired-helper-extension/reproduce.py --repo . --work /tmp/paired-complete

See the package README for exact Git fetches, dependencies, explicit retained-cache reuse and evidence. Tested on macOS arm64/Python 3.12.14. No independent human review or proof-kernel verification is claimed; hosted CI is separate from the local results.

Credit

Jeremy Diviney’s paired extraction, exact structural checker and integration are the immediate foundation. Retains Jeremy Kothe’s PR2894/2833 supplier and cleanup, Yiyang Lu’s shared-kernel/native/source tools, Dugongue’s packing, Marius-Paul Dumitrean’s owner/retired-helper lineage, Philippe Van Velzen’s cost work, Tekalign’s owner/admission checkers, icekylinx’s free-operator auditor and the credited native/Clifford authors. Exact source pins and full attribution are in SOURCES.json and CREDITS.md.

Mason Hale, with substantial OpenAI Codex assistance, including parallel search and cross-checks. Apache-2.0; retained notices are preserved.

Open next questions

The package’s Open next questions identify profitable remaining paired directions and better literal allocations. Recompute on the final 82-endpoint word and charge the complete cost before pursuing another candidate; neither a modular nominee nor a fractional packing root is an admitted bound.