# Arena synthesis note: the optimization and inference method

Date 2026-09-29. Candidates: candidate-1 Opus, candidate-2 Fable, candidate-3 GPT (Codex).
Judge: GPT (Codex), read-only. No dropouts.

## Convergence

All three: the stages separate; the assessment is a corner at the 35% cap with the pass rate
monotone in the up-step; the funded payout chance has a kink at u = 0.35 m_1 in the diffusion
limit with no first-order condition; below the kink the exact chain is a sawtooth, so the funded
risk needs a fine line search and the assessment one exact evaluation; the optimum has no
sampling error given the inputs, so inference is about inputs and model faults; "A beats B" is a
paired difference in calendar blocks. All three qualify the paper's Proposition "the
distribution cap sets the cash frequency" (a decreasing in u) to the branch above the kink.

Divergence: candidate-1 and candidate-2 keep RR 1 in the funded stage (time, verifiability);
candidate-3 finds RR 3 at 0.54% funded (48.2% against 41.8% at the 6.7% cost) and leaves the time
constraint out of its formulation. candidate-2 moves the assessment to 3.24% on a net reading of
the cap; candidate-1 reads it gross at 3.15%.

## Pick

**Base: candidate-3**, as the judge recommended. The picker's first lean was candidate-1, for its
lattice-edge formula, its declared funded test and its weekend-gap catch. Two things reversed
it. The judge checked and found errors in candidates 1 and 2 (candidate-1's cap row off by about
0.15 points and its claim that 1.5% is the best of its admissible interval, when 1.456% reads
higher; candidate-2's net-win expression wrong for RR other than 1, a zero-drift duration
formula used where the drift is not zero, a misquoted z-score, and a slack constraint called
binding), and none in candidate-3; the picker reproduced the cap row and the sawtooth and agrees.
And candidate-1's declared test T2 checks whether the broker reproduces the model's ridge,
which is a model check, not a test that one cell beats another; candidate-3 declares a
superiority test with a paired block estimator.

## Grafts

From candidate-1: the closed-form tooth positions r_edge = m_1 / (j - n gamma), the cap's
shadow price (0.13 points per 0.1% of risk), the gross reading of the cap with its reason
(commission at the open, a two-day trade), the weekend-gap caveat on the margin under the cap,
and the plateau-not-ridge reading of the funded pick. Rejected: its 1% breach allowance on the
strict daily rule, its two accounts, and its power claim on the strict-daily test.

From candidate-2: the edge-and-cost sensitivity table (one point of win rate equals two points
of cost share), the dominance-region reading of "A beats B" (sign constant over the input
region: settled by the chain; sign flips: settled by the measurement that pins the input, e.g.
the spread at 0.33 pip for 1.5% against 1%), payout timing reported beside the payout chance,
and the live stop rule as an alarm. Rejected: the 3.24% assessment (net reading), between-seed
errors for account-level tests, the net-win algebra.

From the operator, added to the formulation: the time constraint T1 (median time to first cash
within tolerance), which candidate-3 treated as a sensitivity. It is what returns RR_f to 1.

## Verification

Recomputed by the picker from `modules/prime.py`: the cap row (45.50 / 45.57 / 45.69 / 45.83% at
RR 0.75 / 1 / 1.5 / 3, u = 3.15%); the sawtooth (jump between 1.30% and 1.35%, edge 1.3236%);
the plateau 1.10% to 1.50% within 0.7 points, best 52.87% at 1.10%, 1.5% at 52.36% with the
first payout at 18 trades against 23; the fixed-up-step RR effect (37% to 48% from RR 0.5 to 3
at u = 1.6%); the lower branch rising (29% at 0.6% risk to 43% at 1.66%); candidate-3's RR 3
cell (pass 40.5%, any payout 48.2% at gamma 6.7%). The 35% rule's wording and the commission
timing were read from rules.json and the paper. Not verified, and next: the strict daily
breach rate, the aside rate on a narrower calendar, the live spread.