Stopped pair-star trees: conditional κ = 7.808981744031e-5

[tomdif]

Selected conditional result

κ = 7808981744031/100000000000000000 = 7.808981744031e-5.

This update adds a binary pair-star exclusion tree and deterministic integer-weighted legal carrier exchanges to icekylinx's PR104 stopped product-ring/rational-center framework, with Rohan Arun's PR107 comparison producer and PR100 balanced transfer. It improves pinned PR107 by 0.13553%. Subsequent PR108 and PR109 report stronger candidates; no current-record claim is made. Further combinations are being investigated separately.

Construction and complete accounting

The pair-star prefix/suffix chain is replaced by a binary exclusion tree. A fixed integer price table makes 604 legal cardinality-preserving matching exchanges and then reaches a fixed point. The full recursive profile is regenerated from every actual edge and frame operation.

• Complex h=24, m=576, 44,340 additions, 8,120 roots, 7,194 matched carriers.
• R=45,266, W=191,429,920, recursive rank=110,261,771,840, largest child=552.
• Exact complex saving 4880994746013/62500000000000000.
• All copied centers, both data connectors, inverse correction, signed wrappers and exact divisor 21 remain paid.
• Three nested row reserves have coefficient 1176, degree 4000 and exact positive gap 40024/25.

The larger role count compared with PR107 is offset by an improved complete child moment. This is an asymptotic exponent refinement, not a runtime benchmark.

Reproduce and evidence

python3 research/stopped-pairtree/verify.py
# equivalently:
make stopped-pairtree-verify

Requires Python 3.11+ and C++17; the new package uses the Python standard library. The original PR104/107 complex constructors and exact certificates are freshly reproduced in scratch, followed by the selected graph and byte-identical matching. The unchanged coarse bit producer and ordinary leaf are inherited, source-pinned inputs; their moments and row reserves are rechecked.

Independent audits check all 4,096,576 signed coefficients, actual binary-frame nondegeneracy and containment, all 7,194 carrier edges, all 45,266 role lifetimes, and the full 76,670-operation literal program. Constructive elementary-shear inversion proves the scalar arbitrary-dirty identity; four signed fixtures check its implementation. Finite full-factorization, atom-ring and ordinary-wrapper controls accompany the written transfer review. Independent rational arithmetic checks the full paid recurrence, adjacent-grid exclusions, all 47 positive constraint slacks and seven cost margins, with ten corruption controls.

The complete seven-stage selected gate passed in 14.778 seconds with zero frozen-file drift. Python 3.11/3.14 component reproductions agree. The inherited joint-dual integration gate passed in 160.792 seconds, preserving its mathematical payload and updating only the three root-file pins and their derived metadata hashes. All 331 imported files match original Git blob IDs.

• Source manifest SHA256: 7dc682264ac44b13c336643c17ffce06663e6fb936954ba94bf8575e3588b31d.
• Certificate SHA256: 28cb74eb564186321fee6adffb58c2160a192fe8c2334391ad7cb06f5e061833.
• Current head: b79fbc4b289cc4740e4fe92dd347c5f5f36becfb.

The expanded 60-job CI matrix is pending on this update. The preceding saturated-deferred head passed all 57 jobs. Default verification writes temporary outputs only; --record writes separate run receipts.

Credit and scope

Prepared for Thomas DiFiore with substantial OpenAI Codex assistance. icekylinx supplies PR104's opposite-bank factorization, stopped atom interface, rational centers and ordinary-leaf lineage. Rohan Arun supplies PR107 and exact refinement/balanced-transfer precedents. Zhihao Chen, RaD/hipotures, James Chang, Aurel Prosz/Paureel, eumemic, Swapnil Jain, Dominik Scholz, Douglas Colkitt, OpenAI, David Harvey, Joris van der Hoeven and all prior contributors remain credited in preserved notices. Original licenses and AI-assistance disclosures are archived unchanged.

The earlier saturated-deferred result, matching the concurrent PR103 value 6.398282083297e-5, remains intact alongside the merged-span/deferred-storage construction. See research/stopped-pairtree/PROOF.md, physical-proof.md and balanced-transfer.md for the new composition.

All PR104 newly proposed atom-streaming, stopped ordinary-interface and odd-grid contracts, plus inherited analytic and all-size fixed-tape hypotheses, remain explicit dependencies. The finite checks do not prove the entire multiplication theorem or constitute independent human review. Draft pending maintainer review.