# Method: choosing and testing the Prime Academy settings

**Symbols.** T = D = 9%: target and static floor, as fractions of the bankroll B. r_a, r_f: risk per trade in the assessment and in the funded stage. u = RR·r: the up-step. γ: cost as a share of the risk, so c = γr (base 2.87%: commission plus 0.5 pip over the 34.8-pip mean stop). θ = 2γ/u. α = 0.7: the split. m1 = 100/(αB): the minimum request, 4.76% at $3,000. M: the payout threshold. t_w: the payout clock, in trades. P: pass rate. a: P(any payout), the chance a funded cycle pays. K: payouts per account. Numbers come from `prime.py` unless a paper section is cited.

## 1. Formulation

**Variables.** RR_a, r_a, RR_f, r_f, tier B, accounts n. The stop (0.75σ, floored at 30 pips) and the 07:00–10:00 UTC window come from the walk-forward and enter only through γ and pace.

**Objectives.**
- **O1.** Maximise P(cash) = P(RR_a, r_a) · a(RR_f, r_f). E[K] = a/(1−a) rises with a, so the same a maximises payouts per purchase and minimises fees per payout.
- **O2.** Among near-ties, prefer fewer trades to a pass and to the first payout.

**Hard constraints.**

| # | inequality | source |
|---|---|---|
| R1 | r(1+γ) ≤ 0.05(1−D), so r ≤ 4.42% | rule: daily loss, review reading |
| R1s | (1+g+γ)r ≤ 0.05·E_prev, g the floating gain in stops at the reset: safe if r ≤ 2.24% (RR 1), else P(breach per account) ≤ 1%, a breach needing g ≥ g* = (0.05E_prev − (1+γ)r)/r: 0.49–0.64 at 3%, 0.11–0.22 at 4% | rule: strict reading, assumed |
| R2 | RR·r_a ≤ 0.35T = 3.15% | rule: 35% in the assessment, assumed to apply |
| R3 | request ≥ M = max(m1, RR·r_f/0.35) | rule |
| R4 | one position at a time; 3 trading days; t_w = 14; news asides in γ | rule, operator |
| R5 | floor on the tick: the stop leg is clipped | operating fact |
| W1–W3 | fee ≤ 6% of the wallet; ruin ≤ 5%; log growth > 0 at 80% of payouts honoured | wallet |

R2 counts the gross win: commission is charged at the open and a trade lasts 2.03 calendar days, so the closing day books all of RR·r. Lots and margin never bind (r ≥ 0.12%; about 7% allowed at 1:30). A 14-day clock is about 7 trades at the pair's pace. Zero edge: legs a and b win with probability b/(a+b).

## 2. What calculus settles

**Step 1: the stages separate.** The funded account opens at the nominated balance, so P(cash) is a product of two positive factors sharing no variable. Each is maximised alone; no joint grid.

**Step 2: the assessment is a corner.** In the diffusion limit (Proposition 14), P = 1/(1+e^{θD}), and

  dP/du = θD·e^{θD} / (u(1+e^{θD})²) > 0 for every γ > 0.

RR and r enter only through u. So u_a goes to the tightest cap:

  u_a* = min(0.35T, RR·4.42%) = **3.15%** at RR 1 (4.42% if R2 does not apply).

The cap's shadow price, dP/du at 3.15%, is 1.29: **0.13 points of pass rate per 0.1% of risk**. The chain agrees: 45.38% at 3.00%, 45.57% at 3.15%.

On the clipped chain, RR 0.75, 1, 1.5 and 3 along the cap pass at 45.68, 45.73, 45.78 and 45.86%. So RR is settled by what the diffusion omits:
- Off RR 1 the broker's brackets fall 1.1 points short of fair at RR 0.5 and 1.4 at 1.5 (Table 28). That is an adverse drift Δp(1+RR) of 1.6% and 3.5% of the risk, about the size of γ. At RR 1 the coin side is fair by construction.
- RR < 1 needs a larger r, which tightens R1. RR > 1 makes trades slower. **So RR = 1.**

It breaks only without the target clip: 3% becomes a lattice trap at 40.2%, because three wins net only 8.74%. Clipped, it rises at every 0.1% step from 2.5% to 4.5%.

**Step 3: the funded optimum is a kink.** In the diffusion limit, a = (e^{θD}−1)/(e^{θ(M+D)}−1), with M = max(m1, u/0.35).
- **Below u = 0.35m1:** M is fixed. a falls in θ, and θ falls in u, so a rises with u.
- **Above it:** θM = 2γ/0.35 = κ is constant. With A = e^{θD}, a = (A−1)/(Ae^κ−1), and da/dA = (e^κ−1)/(Ae^κ−1)² > 0. A falls in u, so a falls with u.

Hence **u_f* = 0.35·m1 = 1.667% at $3,000, for any γ > 0**. No first-order condition holds there. `funded_model` gives slopes of +0.4 points per 0.1% on the left and −1.2 on the right, so erring low is cheaper.

Where it breaks:
- **(a) The clock.** For t_w trades only the floor absorbs, with variance t_w·u·r, so at fixed u a smaller r survives better. At u = 1.5%, RR 1, 1.5 and 3 give a = 52.4, 53.7 and 55.7%, with median first payouts near 37, 53 and 130 calendar days. The clock also pulls the RR 1 optimum below the kink.
- **(b) The lattice.** After n trades with w wins, equity is (2w−n)r − nγr. A payout is allowed when 2w−n ≥ m1/r + nγ, so the required excess j (parity of n) drops at

  **r_edge(j, n) = m1/(j − nγ).**

  At n = 14, j = 4 that is 1.323% at base cost (a = 51.5% at 1.30%, 52.7% at 1.35%). It predicts every jump the line searches found (1.253, 1.349, 1.413, 1.555% at γ = 1.44, 3.35, 4.5, 6.7%), and a [1.642, 1.667]% window at t_w = 7.
- **(c) The level:** diffusion 59.3% at 1.5%, chain 52.4%.

**Step 4: the rest are corners.**
P and a are tier-free (Proposition 1) except through m1; W1 and W2 admit only $3,000 at a $1,000 wallet (fee 3.8%; $10,000: 9.9%). The window enters only through γ, so the tightest spread wins. n = 1 by the operator's rule.

## 3. What must be computed exactly

| where | computation | cost | what it misses |
|---|---|---|---|
| assessment, clipped | `eval_chain` at the cap, and at the cap less a margin | 0.1 s | nothing: monotone |
| funded ridge | r_f over [0.6, 1.1]·0.35m1 at 0.05%, plus a point above each r_edge, per input scenario: γ ∈ {1.44, 2.87, 3.35, 4.5}%, t_w ∈ {7, 14} | 8 s per scenario | risk that varies per trade |

A root find fails: with a kink and a sawtooth no first-order condition holds. The half-point grid missed the tooth tops (1.10%, 1.35–1.45%) and the 1.642–1.667% window.

**Line search at base cost, t_w = 14:** a = 52.87, 51.54, 52.72, 52.77, 52.63, 52.36, 51.99, 51.42 and 50.67% at r_f = 1.10, 1.30, 1.35, 1.40, 1.45, 1.50, 1.55, 1.60 and 1.667%. Across the eight scenarios the argmax wanders from 1.05% to 1.50%, yet 1.5% stays within 2.2 points of each best.

**The lot grid.** At 1.5% the lots run 0.10–0.15, so flooring to 0.01 leaves 10/11 to 1 of the nominal risk. A nominal risk just above an edge trades below it on some days, which a fixed-r chain cannot see.

**Rule.** Admit r_f ∈ [r_edge(4,14)/0.909, 0.35m1], which is [1.456, 1.667]% at base cost, and take its highest a: **1.5%**. This holds while γ ≤ 3.63%; above that the pick is 1.6%.

## 4. Inference

**The optimum has no sampling error.** It is an exact function of the inputs: γ, t_w, the rule readings, and a drift fixed at zero. Its uncertainty is the input uncertainty pushed through the chains (scenarios, regret of the chosen cell), not a resampled history. The 2.8 years resolve inputs (aside rate, pace, cost) and model checks with large predicted effects.

**What it cannot resolve.**
- **The drift.** 467 deployed closes give the win rate a standard error of 2.31 points. Overturning the assessment corner needs Δp ≥ γ/2 = 1.44 points. Power at 2 SE: 9%; 80% needs about 9,450 trades (53 years). Zero edge is imposed, not measured.
- **Plateau and corner differences:** 0.4 points (funded 1.4% against 1.5%), 0.86 (assessment 3% against 4%).

**Testing "cell A beats cell B".**
1. Pair the cells with common random numbers: same path, seeds and account cycling.
2. Take differences d_k over K calendar blocks, each twice the longer account span.
3. The null is Δ_model, the chains under the broker's protocol. Not zero, and not the exact difference: truncation biases funded rates by up to 23 points.
4. z = (d̄ − Δ_model)/SE_d, SE_d = sd(d_k)/√K, against t on K−1 degrees of freedom.
5. Declare a claim only if |Δ_model| ≥ MDD = (t_{K−1} + 0.84)·SE_d. Otherwise the chain decides.

Before a run, bracket SE_d² = s_A² + s_B² − 2ρs_As_B over ρ in [0, 0.9]. Assessment 3% against 4% (s = 1.72, 1.22): SE_d 2.11 to 0.82, MDD 5.9 to 2.3, against Δ = 0.86, so not resolvable (power 18% even at ρ = 0.9). Funded 1.5% against 4%: Δ ≈ 21 against SE_d ≤ 3.14, so resolvable.

**Declared tests, news-on replay** (the current decision's cohort and seeds):
- **T1, assessment model check,** as the current decision declares it (block SE 1.7; power 50, 83, 94% against 3.4, 5, 6-point shortfalls).
- **T2, the funded ridge.**
  - d = P̂_f(1.5%) − P̂_f(4%) at RR 1, paired, in 226-day blocks (K = 4).
  - Null: the exact chains at γ = 1.44% × 2.33 = 3.35% differ by 51.22 − 28.92 = 22.30 points. Less the 1.66-point protocol bias (Section 14.10: −2.72 and −1.06), Δ_model = **20.6**.
  - Reject the ridge if d < 20.6 − 2.35 × 3.14 = **13.2** (one-sided 5%, t on 3 degrees of freedom).
  - Power: 82% against a half-size ridge, about 100% against none. News off it read 19.7 against 23.0, z = −1.06.
- **T3, drift alarm.** Reject zero edge if the short-side |p̂ − 0.5| > 2 SE. At 467 closes, 80% power only against 6.6 points: an alarm, not an estimate.

## 5. Worked application

| variable | method | current pick | status |
|---|---|---|---|
| RR | 1 | 1 | Step 2 |
| r_a | **3.0%** | 3% | R2 corner at 3.15%, held 5% below (1.74 pips); R1 slack; R1s unmeasured (g* 0.49–0.64) |
| r_f | **1.5%** | 1.5% | minimum request binding (M = 4.76%); funded 35% slack (1.5 < 1.667); daily slack (g* 2.30 > 1) |
| tier, n | $3,000, 1 | same | W1 and W2 slack (3.8%; ruin 0.11%) |

**At base cost:** P = 45.38% (9.65 trades, median 8 to pass). a = 52.36%, E[K] = 1.10, E[W] = 6.66%, first payout at a median of 18 trades. P(cash) = 23.8%, cash 1.67 times the fee. Wallet growth (`wallet_mc`, seed 13, 20,000 paths) +0.179 with every payout honoured, +0.018 at 80%.

**The optimum equals the current pick.** r_a is 0.15% under the corner (0.19 points), r_f 0.17% under the kink (clock and lattice).

**Corrections to the current reading.**
1. The corner is 3.15% gross, not the 3.24% that the net arithmetic ("a 3% win is 2.91%") allows.
2. The 1.74-pip margin does not cover weekend gaps (median 9 pips, paper Section 16). Confirm the venue fills the target at its own price, or go flat before weekends in the assessment.
3. Above γ = 3.63% the funded pick is 1.6%. At 6.7% the best is 1.65% (42.86% against 41.81%), but the gate is closed there.

**Under the other readings:**
- 35% rule funded-only: r_a = 4% (+0.86 points), once R1s is measured (g* 0.11–0.22).
- Equity-based 35%: r_a ≤ 1.575%, P = 41.68%.
- 7-trade clock: a = 54.48% at 1.5%, against a best of 55.01% at 1.35%, which no test can resolve.