Finite bit construction: exact R439 four-helper fusion below the pair floor

[thakrarsagar]

Change and mathematical scope

One connected four-helper preparation reduces the frozen PR941 R439 word from 11,949 to 11,946 complete ADDs, including preparation and restoration. It crosses PR941's 11,948-ADD disjoint-pair floor, whose exact integer dual is replayed here, by leaving the disjoint-pair class. No new kappa; the selected frontier is unchanged.

The contribution is a small finite construction plus an exact bounded search: all 20,160 GL(4,2) bases for each of 64 prescribed unions of two PR941 helper pairs, 1,290,240 fully priced combinations. The exact minimum in that declared class is 11,946. This is not the cheapest known R439 circuit: PR965 and PR967 already report smaller words using different triple and star searches. Their savings are not combined here.

The winner acts on helpers [269,88,288,153] with three XORs, replacing two independent XORs. The injection saves five ADDs and the additional preparation/restoration costs two. Whole-word forward growth increases 70,083 -> 70,355, reverse growth stays 10,548,832, and the 104-bit local / 192-bit nine-stage caps still pass. All 439 helpers, nine COPY/ERASE pairs, frame paths, paid histogram and rank mass 5,129 remain unchanged.

Validation

Reproduce from the repository root with Python 3.11+ and the standard library:

python3.11 -I -B research/r439-helper-quartets-seed07/verify.py

Expected JSON fields: status=PASS_R439_HELPER_QUARTETS, combinations=1290240, columns=661, native_ADDs=11946, saved_ADDs=3, no_new_kappa=true. The README includes the independent C++17/Boost native command and exact expected output.

Completed verification:

• Direct Gaussian enumeration of GL(4,2), all realizing words and shortest-distance edge inequalities; independently inverted matrices and set-valued prices reproduce the entire census.
• PR941's 96,141-edge integer pair-bound certificate; all 661 arbitrary-input D columns in both directions; unchanged owning E8/frame checks; exact complete costs and strict precision inequalities.
• Native 5/9/11-stage checks on both words: 883/1,327/1,549 formal columns per word, 96 rejected bridge controls, identical paid ranks and copy lifetimes. Apple clang 21 / Boost 1.85.0, with 186 non-system dependency hashes recorded.
• Normal and optimized isolated Python replay; 11 built-in negative controls, one charged neutral control, four external package corruptions rejected; 43 source pins; 25 existing selected-record tests and nine finite-bit-contract tests.

Only research/r439-helper-quartets-seed07/ is added. Base 2921874, OS assignment seed 7, bit-supplier track. No pinned sprint issue exists; the claim is local. Current PRs and all-state overlap searches were refreshed through #1043. Adjacent #1023 concerns bit hosting widths, #1027 concerns R480 six-wire integer circuits, and #1043 changes the helper stock and source boundary. The newly opened #1040 retunes one target after PR967's stars, reaching a lower 11,780 ADDs; this contribution freezes the target basis and fuses two PR941 helper pairs. Its different object and lower count are credited in the README. No repository-wide artifacts were regenerated and upstream/ is unchanged.

As in PR915/926/941, the original syntactic H2 and literal K extractor deliberately reject this semantic D replacement; the predicates were not weakened. Supplier adoption still needs a whole-hosted emission/audit, complete finite/precision invoices, semantic-interface acceptance and outer assembly. Inherited all-size/compiler, common-label/mutable-source realization, hosting, tape and analytic assumptions remain. No full multiplication proof, supplier cold replay, full repository suite, Linux CI success or independent human peer review is claimed.

Credit Gareth Davidson for PR915/926/941, Scott Woolsey Biggart for the two-sided mechanism, Ross Richardson for the GL(3,2) search pattern, eumemic and DreamingOfClouds for the selected R439 lineage, and the source contributors retained in the pinned manifests. Prepared by Sagar Thakrar / @thakrarsagar with substantial OpenAI Codex assistance in mathematics, implementation, verification and writing. New material is Apache-2.0. No priority or best-known-circuit claim is made.