[aliromman]

This enabling proof removes the grid-only single-crossing assumption S1 for every positive-integer fragmentation invoice in the fixed p16 controller class. The complete moment is strictly increasing on t in [4/5,1] and has exactly one zero with C=1-t in (0,1/20). This makes the class's raw-moment sufficiency test rigorous. It produces no new native word or κ improvement.

Proof, certificate receipt, source pins and checker. Base: canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority 7.

Fix p=16, a=7 or 8, v=112, d=8, 18 native calls, the retained 408 helper endpoints and eight rank-8 center jobs per call. Each main/helper net increment may be split into any positive integer partition of that increment. Keep all other targets and paid jobs fixed. Partitions may differ across fields and calls; no realizable native word is asserted for each partition. Descents, nonnested excess cost, extra jobs, changed endpoints and coded interfaces are outside the theorem.

The key certificate uses weighted tails B_k=sum_(r>=k) r(T_r-P_r). Splitting a paid rank cannot decrease any tail. The unsplit comparison has B_k=2432 for k=2,...,8 and B_k>=32480 for k=9,...,145. Summation by parts proves strict interval monotonicity. Exact integer-power enclosures give:

f_unsplit(19/20) <= -1994616351/524288 < -3804
f_partition(19/20) <= f_unsplit(19/20)
f_partition(1) = 752.

Thus every invoice in the class has one crossing. The proof replaces sampling; it does not add more digits to the existing root.

Use CPython 3.12 and the standard library:

python3.12 -B scripts/research_package.py --package research/fragmentation-single-crossing
python3.12 -B scripts/check_research_commit.py --package research/fragmentation-single-crossing

Both registered modes passed under Python 3.12.13 with warnings as errors and -O. Expected verifier marker: PASS_FRAGMENTATION_SINGLE_CROSSING. Checks bind the complete source invoice, all 144 tails, 96 positive integer partitions, 22 exact root intervals and a rational exponential bound. Changed high-rank cost, missing helper credit and a wrong partition total are controls. Committed-byte/export checks, both direct exported modes and git diff --check passed. Full-history validation was not run in the shallow checkout. No native, Lean, compiler, outer-ledger or selected replay ran.

PR #5226 supplies the fragmentation framework and S1 statement. Its source attribution names Claude Opus 5.5 via Claude Code; no human authorship is inferred from the submission account. Cao Yifan (@intqwq) supplies PR #4610 and the consumed invoice. Retained controller/native credit is preserved. New proof and checks: Ali Romman (@aliromman) with substantial OpenAI Codex assistance. Apache-2.0; author-checked, not independent acceptance. Scoped claim.

[jwr404]

Independent review (James Reed, with Claude Code). Head b0edd31a04fe4705846df7673b4a0ec2442260d0.

Verdict: the proof is correct, and the certificate numbers reproduce independently.

Proof check

• Summation by parts. Σ_r (r c_r) h_t(r) = B_1 h_t(1) + Σ_{k=2}^{145} B_k [h_t(k) − h_t(k−1)] − B_146 h_t(145), and the boundary terms vanish (h_t(1) = 0, B_146 = 0). ✓
• h is increasing. h_t' = x^(t−2)[1 − (1−t) ln x] > 0 on [1, 145] for t ≥ 4/5, given ln 145 < 5. I checked the rational partial sum Σ_{j≤12} 5^j/j! = 2837864809/19160064 > 145. ✓
• Partition step. Only paid jobs are split. Parts at least k sum to at most R·[R ≥ k], so paid tail mass can only drop and every B_k can only rise. ✓ At t = 19/20, subadditivity of x^t makes paid moments only grow, so f_partition(19/20) ≤ f_unsplit(19/20). ✓ Together with f(1) = 752 > 0, this gives exactly one zero in (19/20, 1) and none elsewhere in [4/5, 1].

Independent numbers

My own code builds the unsplit invoice from the PR's controller formula:

• 224 at rank 145;
• 1,680 paid at rank 1;
• 15 controller pairs;
• 18·232 paid at rank 8.

	this check	PR
B_k, k = 2..8	2432 (constant)	2432
min B_k, k ≥ 9	32480	≥ 32480
B_1 (mass)	752	752
f_unsplit(19/20) (50-digit Decimal)	−3804.42952805	inside [−997308605/262144, −1994616351/524288] ✓
pinned source ARITHMETIC.json	mass 752, min B_k = 32480 (at k = 74), f(19/20) ≈ −10501.5	consistent with "current min tail 32480"

The PR verifier passes locally under python3.12 -I -O -W error (PASS_FRAGMENTATION_SINGLE_CROSSING).

A useful corollary outside this class

The lemma (nonnegative weighted tails, one positive, rank bound keeping h_t increasing, so r_max < e^5 for t ≥ 4/5) also applies directly to:

• the Ordinary BIT saving 0.003475458 via three finite leaf backends #4814 BIT raw invoice: max rank 127, min B_k = 26160, mass 753;
• the summed complex + BIT invoice of Joint bit+complex supplier cannot beat the average of the two p16 invoices (negative result) #5298's model A: min B_k = 32480, f(19/20) ≈ −19587.9.

So the uniqueness of those roots, which #5298 only grid-checks, also follows from this lemma. I'll note that on #5298.

Minor: the scope sentence "helper paid jobs cancel their same-rank credits" could say explicitly that helper targets stay fixed and only helper paid jobs are partitioned. The proof uses exactly that.