Lower PR189 endpoint frames plus finite leaf bootstrap: conditional κ = 0.0006679123

[sennemmi]

Status

Draft follow-up stacked on #194 head a8c87783785be6fe20c4a8bb2ac4b3eec295a179. The branch includes that base construction, so the PR diff against main contains both the base and this follow-up.

Under the retained conditional interfaces, the exact composed result is:

• κ = 6679123/10000000000 = 0.0006679123.
• Gain over Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194's 1668581/2500000000: 4799/10000000000.
• Gain over the pending Bit leaf bootstrap: queue standing + the deferred #200 composition at conditional κ = 6.773148e-4 #199 claim 267141/400000000 = 0.0006678525: 299/5000000000.

This is not claimed as the current public record. Open #197 claims the higher 135216063303877/200000000000000000 = 0.000676080316519385 on a different packed PR187 bit supplier. It currently has no reported checks; its description says the new workflow has not run and does not claim a fresh full make verify on current main. #199 is also open with no reported checks. I keep this PR in Draft and do not claim to beat #197.

Construction

A deterministic search selected 102 nonadjacent endpoint-frame lowerings from 2,534 structurally eligible candidates on PR189's physical bit word. Candidate costs use the complete physical chain order: donor operations, recipient gauge splice, then recipient operations. Each selected frame contains the node value span and actual predecessor frames, stays inside both physical successors, and preserves every terminal-sink write. The Boolean operation word is unchanged. The exact assignments, 70 added frame-basis records, histogram delta, and rational cost receipt are in research/paired-cube-twin-local-168/bit/frame-descent.json.

The descended bit supplier has exact coarse saving 167089706787281/250000000000000000. PR184's unchanged pricing gives selected coarse saving C = 1670897/2500000000 and initial effective saving a_0 = 6679380665463285737/10^22. The finite depth-2 ordinary-leaf wrapper uses the PR185 recurrence first applied to PR189 by PR199:

• a_(j+1) = (1-C)C + C a_j.
• a_1 = 16708962969977344607748096089/25000000000000000000000000000000.
• a_2 = 41772424988253556235173052470510821833/62500000000000000000000000000000000000000.
• The checks establish a_0 < a_1 < a_2 < C < 1-a_2; the assembly then rechecks all 47 strict constraints and seven margins.

The complex supplier remains 219037/312500000, so the bit supplier binds. The PR189 physical word is inherited from #189; the finite leaf wrapper is inherited from #185. This composes #199's already-public pricing recurrence with the new frame descent; the recurrence and wrapper are credited, not claimed as new work.

Reproduction and validation

Local lightweight checks passed: exact frame geometry and physical rows, 58 source pins, exact recurrence, source/certificate consistency, PR184 rational assembly, all 47 strict constraints and seven margins, and git diff --check. No full high-cost verifier was run locally.

Cloud verification is running on head 2d6aac706f274812f260792f456c1b9458ce3026:

• Full repository verification matrix: all 14 make verify-* groups on Python 3.11, 3.13 and 3.14, plus the three formal checks.
• Source-assisted certificate workflow: regenerates the complex word, exact lift, source-assisted certificate, PR189 bit certificate, and requires an unchanged tree.

Reproduce from the repository root:

python3 -m pip install -r research/source-assisted/requirements-round13.txt
python3 -B research/source-assisted-v4/verify.py
python3 -B research/paired-cube-twin-local-168/verify.py
make verify

Scope and attribution

The result remains conditional on PR184's analytic, recursion, precision, weighted-frame, fixed-tape routing, restored-row and all-size semantic interfaces, the PR189 physical word and contracts, and PR185's completed finite ordinary-leaf construction. A globally renumbered scalar transcript of the new complex word and a full Clifford/router replay are not included. This is a conditional finite witness, not an unconditional theorem or a global optimum.

PR184, PR189, PR193/194, PR185, PR199 and PR168-v4 notices and authorship are retained. The 102-frame search and certificate are credited to OpenAI GPT-6 (Codex). The finite wrapper follows PR185 (Rohan Arun, OpenAI Codex assistance); the recurrence/pricing pattern follows PR199 (Maxime Fleury, Codebuff assistance). Apache-2.0 applies; see the package NOTICE files for the full predecessor attribution.

[sennemmi]

Archiving this Draft as part of the research closeout. It explored lowering the PR #189 endpoint frames with a finite-leaf bootstrap on #194. At head a22b271a06c6e61b7dcd2ec4859e6ccd532b1f6a, full CI failed, including the joint-dual source pin assertion and recycled-bit replay error (complex_certificate missing phys); the terminal workflow also failed. Both inherited dependencies remain open. The branch, commits and certificates are retained; no further maintenance is planned. See the consolidated closeout in #261.