Research: two clean checkpoints reduce the 50-factor leaf bill 1728 to 1248 (no new kappa)

[Better-Call-Paul]

With 50 ordered factors and six scratch arrays, allowing two initially zero arrays instead of one lowers the exact minimum signed-leaf bill from 1,728 to 1,248 in the ordered binary, reserved-connector class. Every source and scratch array is restored. This extends #714/#788 by keeping a second clean checkpoint available while a clean ancestor remains reserved.

Status: recursive-interface research construction; no new κ. The full-capacity class theorem is V(2^s,s,c) = 4^(s-c) 3^c. The package supplies the proof, an exact planner and streaming instruction emitter, source pins, complete work/storage/conditional precision accounting, and negative controls. The extra zero array is an input requirement: this does not turn arbitrary dirty storage into clean storage for free.

The 50-factor witnesses save 480t + 332g_D - 258g_C before extra provisioning, admission, preparation and routing, where t is the common complete leaf price. Leaf-count optimality is not a practical speedup or a supplier improvement. All-size compilation, physical hosting, fixed-tape routing, actual clean-space supply and analytic transfer remain obligations.

Reproduce with Python 3.10+ and the standard library:

python3 -I -O -B research/two-clean-checkpoints/verify.py
python3 -B research/two-clean-checkpoints/test_package.py

Expected: PASS_TWO_CLEAN_CHECKPOINTS: 1619 trees; 192 rounded cases; n50/s6: 1728 -> 1248; no new kappa and PASS_TWO_CLEAN_PACKAGE.

Validation on macOS arm64 / CPython 3.14.7:

• Independent preorder enumeration of all 1,619 decorated trees through six factors; 723 exhaustive weighted minima and 32,039 resource admissions.
• Exact noncommutative symbolic products, nonlinear restoration, literal inverses, and larger witness execution.
• 137 comparisons with unchanged pinned Research: reusable clean checkpoint lowers weighted binary leaf bill 20 to 16 (no new kappa) #788 code.
• 192 actual pinned Uniform rounded prime-power cyclic inversion (no κ claim) #628 rounded-leaf executions against dense rational represented-factor solves, with exact source/scratch restoration, zero-source checks and counted phase accesses.
• Eighteen interface/instruction controls, a rounded dirty-as-clean counterexample, normal/optimized and relocated replay, three package-corruption controls, eleven exact Git source bindings, and selected-record integrity.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. OS-random seed 50, recursive interfaces. No pinned sprint issue existed; the local claim records three candidates, scope, stopping rule and overlap. Only research/two-clean-checkpoints/ is added. The selected frontier and upstream/ are unchanged.

Credit FanPUYen/#714, nightangelflowerwin-ops/#788, chafreaky/#628, 0xkilgore/#663, androolloyd/#673 and the concurrent all-dirty work #689/#730/#744/#758/#784. Exact pins and original notices are retained. Prepared for Paul Chan (@Better-Call-Paul) with substantial OpenAI Codex assistance, under Apache-2.0. No human peer review, Linux/remote CI, proof-assistant verification or full multiplication theorem is claimed.

The baseline run passed its first seven historical groups. After supplying
SymPy 1.14.0 in an isolated environment, gen4 passed all nine finite stages.
After setting the existing Homebrew Boost include path, the complete selected
frontier626-verify replay passed both suppliers, independent owner accounting,
helper interfaces and exact bindings. No supplier source was patched.
Final overlap refresh covered 638 open PRs through #825; #788's consumed
head is unchanged. Concurrent #805 changes the leaf contract and is credited
without importing its code or making a cross-contract runtime comparison.

The baseline also passed verify-certificates and verify-ternary. Remaining
unrelated historical searches were stopped during audit_early_sharing.py
to bound the sprint; later groups were not reached. No full make verify
pass is claimed. The scoped package and complete selected frontier passed.

The immediate pre-submission check also read #822
(8c5e1b557f59bca29139c4c2dcc65edfb808372b, @shhxera), which changes dirty
connector lifetimes. This contribution keeps ancestor connectors reserved
and changes the number of clean inputs; no cross-class optimum is claimed.