[unpaidmoon]

Change and mathematical scope

Proved, for the net splitting tax at exponent t = 1/2 and every integer m ≥ 1: there is a unique τ_m > 1 with g(m) = g(m+1), the continuous peak at that tie is z = m + θ_m with 0 < θ_m < 1/2, and on (m, m+1) the best integer quotient is m when z < m + θ_m and m + 1 when z > m + θ_m.

The argument is a written proof in research/half-exponent-quotient-tie/PROOF.md. It uses the envelope facts (I1)–(I3) from merged CrocSwap#6769 and the explicit critical curve

q(u) = (2σ - 1) u^2 (2σ + 1 - 4 σ u) / (2 σ u - 1)^2,

where u is the square root of the optimizing remainder. That curve is strictly decreasing and strictly convex, so the remainder square root is strictly convex in the quotient. The envelope is then strictly larger at z + 1/2 than at z - 1/2. The ratio g(m)/g(m+1) is strictly increasing in σ, which places the tie before the midpoint.

This is not a κ change, not a paid-profile improvement, and not a supplier admission. The selected frontier stays 363005817/100000000000.

Not proved: θ_m < 1/2 for an exponent other than 1/2; the rate 1/2 - θ_m ∼ c/m; any inventory application.

Task and overlap

Base commit e54523517d0f03e58352e647c403a5af3e525a57. Quoted source: research/sharp-net-splitting-tax/PROOF.md at that commit (blob d49a79b19c791eafbc009fd9fdab17bd3e0d8e2e). No dependency on unmerged CrocSwap#7135.

Session r276, item 9. Ordered eligible reading, after dropping topics already claimed, delivered, or owned by unpaidmoon PRs and fork branches: (1) bit-new-register-model, (2) complex-rank-shape-decoder, (3) selected-stock-envelope, (4) uniform-transfer-audit, (5) census-pruning-audit, (6) replay-runtime, (7) portable-replay, (8) overlap-receipt, (9) scope-navigation, (10) scalar-to-moment. Open CrocSwap#1166 handoffs inspected through 2026-11-11T13:19Z were already claimed or were unpaidmoon families. Item 9 is scope-navigation.

Scope is changed. The current heading in docs/ci-verification.md still calls the CrocSwap#6030 replay current, while certificates/selected-result.json names CrocSwap#7000. That sentence was not edited: research/historical-selected-replay-gap pins live digest c7d3027e9ad8bcbff68f07aa9576b30fb2f3c76c44d876d40c7fdbf4a3e2228a and requires authenticated_input to reject an unrecognized update of docs/ci-verification.md. The card trigger forbids rewriting a frozen package to change historical wording.

The delivered piece is the t = 1/2 transition left open by merged CrocSwap#6769 (“compare the two optimized candidates for a fixed rational t”). Open CrocSwap#7135 proves the general-exponent staircase and states that θ_m < 1/2 is not proved for all t and m. Its finite t = 1/2, m ≤ 30 certificate is not this all-m proof. No fork cursor/* branch on this topic was present at push.

Validation

CPython 3.12, standard library only.

python3 -I -W error -B research/half-exponent-quotient-tie/verify.py
python3 -I -O -W error -B research/half-exponent-quotient-tie/verify.py
python3 -B scripts/research_package.py --package research/half-exponent-quotient-tie
python3 -B scripts/check_research_commit.py --package research/half-exponent-quotient-tie

Observed: PASS_HALF_EXPONENT_QUOTIENT_TIE (twice), PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES at aca0cc65310d5053a5de2aedd15bcdfa3670f3e7.

The verifier checks cleared polynomial identities for v^2 - u^2 = q(u) and H = 0, the factored formulas for q' and q'', the discriminant identity 4σ^2(5-8σ), the exact point σ = 3/2, u = 1/2, q = 2, and a sign-rejection control. It does not enumerate m and does not prove the general-exponent case. No selected cold replay, Lean, or Docker was run.

Credit

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Envelope facts: CubeJerry, CrocSwap#6769. The half-window question and the general-exponent staircase without the midpoint side: Nick Petalas, open CrocSwap#7135. This proof does not use that pull request as an input. Apache-2.0. No independent human or proof-kernel review.

Open next questions

• For 0 < t < 1 and every integer m ≥ 1, is θ_m < 1/2? The t = 1/2 proof uses an explicit radical curve. The same comparison would follow from convexity of μ(q) = (s_q^{1-t} - t) / (1 - s_q^{1-t}) at σ = ζ(m+1/2). A finite sample is not a proof. Blocker: no closed form for s_q when t ≠ 1/2.
• At t = 1/2, what is c in 1/2 - θ_m ∼ c/m? Start from the third derivative of log g on q(u). Blocker: that expansion is not in this package.

Workflow feedback (optional)

The historical replay gap’s live pin on docs/ci-verification.md makes the obvious current-status sentence repair collide with a package that must keep rejecting that path. A one-line note in the CI doc’s header, pointing at the selected record without editing the pinned body, would separate navigation from that frozen rejection.