# Method

## 1. Formulation

Use `current_decision.md`’s latest ordering, not an invented weighted utility: maximise funded probability of any payout; choose assessment pass probability, allowing a one-percentage-point tie in favour of fewer trades; choose the largest wallet-admissible tier. Report payout timing, count and wallet drawdown separately. “Any payout” is not payouts per month. Section 23’s earlier 120-calendar-day requirement is a sensitivity, not a constraint retained in the current decision.

Decisions are common reward/risk (q>0), stage risks (r_a,r_f>0), stop ℓ, entry window H, tier bankroll/fee (B,F), split α, and integer concurrent accounts n. Equity profit e, target T=.09, floor depth D=.09, cost c=γr, and gross win u=qr are fractions of initial B. Dollar equity is E=B(1+e). Wallet starts W₀=$1,000.

**Firm constraints, pathwise:**

- While alive, e>−D; pass requires e≥T. Every tick satisfies Eₜ≥.95Eprevious-day. The latter includes unrealised gains surrendered overnight, commissions and aside costs.
- Completed trading days d≥3 in each stage; idle interval<30 calendar days; payout elapsed time≥14 calendar days; gross request M≥100/(αB); largest winning day bmax≤.35M. Conservatively require uₐ≤.35T during assessment. These gross-win tests are stricter than net-profit tests.
- Lots L satisfy L≥.01 and L/.01 integer, with 10ℓL/B≤r; position dollar notional≤30E; nB≤$300,000. Tier/split choices belong to `rules.json`’s finite menu.
- Each opening/closing time t satisfies |t−release|≥10 minutes; use the all-currency calendar and step aside before every touched hourly bar. Prohibited-act indicators≤0: EA/HFT, reverse arbitrage, swap abuse and gambling-focused conduct. Use manual execution absent permission for automation. The last two are discretionary classifications, not probabilities the chain certifies.

**Operator/wallet constraints:** ordinary entries per day≤1, n≤1 until an honoured payout, nF≤W, F/W≤.06, simulated ruin≤.05, and purchase only if G(.8)>0, where G(h)=E[log(W₅₀₀/W₀)] at honour probability h. After refusal, further purchases≤0. Fixed stage sizing forbids loss-driven escalation. Shallow drawdown and acceptable appearance have no supplied numerical bounds; do not manufacture them.

Assume funding resets e=0; tick floor; both assessment clips; no recurring fee; payout clock starts at funding. If the 35% rule is funded-only, remove the assessment cap. Net-day accounting relaxes it to (q−γ)rₐ≤.0315. An earlier clock helps; recurring fees hurt. Treat lot minimum as confirmed by the paper’s later platform observation, despite the older JSON’s “unverified”. Calendar, actual spread, swap and discretionary acceptance remain unresolved.

## 2. What calculus settles

A fair bracket wins with probability p: pu−(1−p)r=0 gives p=1/(1+q). Subtracting cost gives mean μ=−γr and variance v=qr²=ur. Thus the diffusion exponent λ=−2μ/v=2γ/u. With T=D,

P=1/(1+exp(2γD/u)),

Pᵤ=P(1−P)2γD/u²>0, and Pᵧ=−P(1−P)2D/u<0.

At fixed q and γ the assessment risk is therefore a corner:

rₐ≤min[.0455/(1+γ), .0315/q].

Here .0455=.05(1−D) is a sufficient daily-loss allowance **without overnight floating profit**. The true pathwise constraint can be tighter. This is no theorem that funded risk, or wallet growth, should be maximised.

Let k=$5/lot commission, s spread in pips, and σ daily price standard deviation in pips. Then γ=(k/10+s)/ℓ and dγ/dℓ=−(k/10+s)/ℓ²<0. Fair price hitting time has expectation qℓ²/σ². Under an *expected-duration* limit≤1, ℓ*=σ/√q. This neither guarantees intraday closure nor optimises calendar growth. Overnight swap w per night and mean nights ν(ℓ) replace the numerator by k/10+s−wν(ℓ); variable aside frequency also destroys the simple monotonicity. Retain the grounded stop max(.75σyesterday,30 pips), H=07:00–10:00 UTC, and short side without assuming forecast skill or swap income.

For funded probability A, put m=100/(αB), h=.35 and M=max(m,u/h):

A=(exp(λD)−1)/(exp(λ(D+M))−1).

Below u=hm, M=m. Differentiating log A with respect to λ gives D/(1−exp(−λD))−(D+m)/(1−exp(−λ(D+m)))<0: x/(1−exp(−λx)) increases with x. Since dλ/du<0, A increases with u. Above the ridge, set z=2γD/u and C=exp(2γ/h). Then A=(exp(z)−1)/(Cexp(z)−1), dA/dz=exp(z)(C−1)/(Cexp(z)−1)²>0 and dz/du<0. Thus A decreases. The diffusion optimum is a **kink**, not a smooth stationary point. At zero cost the lower branch is flat. This corrects Section 8’s unqualified decreasing claim.

For a shared stage up-step, cash probability PA has an interior stationary point only where P'/P+A'/A=0; compare endpoints and kinks too. Separate stage risks remove that forced tradeoff. For the paper’s growth surrogate, define fee share f=F/W₀, cash probability π, conditional cash/wallet y, expected cycle time τ, and J=(1−π)log(1−f)+πlog(1−f+y). Growth J/τ has condition

τ[π'log((1−f+y)/(1−f))+πy'/(1−f+y)]−Jτ'=0.

Primes mean derivatives along the chosen smooth parameter branch. Fee share affects this optimum, not PA itself. No interior maximum or uniqueness is guaranteed.

## 3. Exact computation

At state e use legs a=min(qr,T−e+c) for assessment and a=qr funded; b=min(r,e+D−c). Positive-leg clipping gives win probability b/(a+b), not 1/(1+q). Transitions are e+a−c and e−b−c. With transient transition matrix Q, upper-exit vector v and pass vector x, x=v+Qx implies (I−Q)x=v. Occupancy propagation gives exit times. `eval_chain` and `funded_chain` implement this; funded propagation includes a clock, then payout resets. If a cycle pays with probability A, P(K=k)=(1−A)Aᵏ and E[K]=A/(1−A).

Search the original finite ratio/risk grid, discard infeasible cells, add assessment cap intersections and funded ridge points, then refine both sides and competing local peaks. Enumeration is cheap relative to broker replays and catches lattice traps; a single boundary search misses clock/overshoot peaks. Root finding belongs only to smooth diffusion branches. The unclipped assessment’s 3% trap (40.2% versus 42.0% at 2.5%, current decision’s cost convention) illustrates why calculus is insufficient.

Use lattice units c/40, c/80 and c/160 and check convergence; “exact” means exact for that rounded chain. Also inspect lot rounding. The supplied chains omit calendar-state daily-equity checks and use 14 **trades**, not calendar days. Strict implementation needs equity/day/clock/maximum-day-profit state; these existing functions cannot certify it. Do not call a chain optimum globally executable.

## 4. Inference

Declare one comparison before new outcomes: H₀: Δ=P(A)−P(B)≤0. Pair cells on identical calendar blocks, each at least twice the longer account span, clustering **all seeds together**. With block successes S and completed counts N, estimate each pooled rate ΣS/ΣN. For b blocks, sΔ is the sample standard deviation of paired differences in influence contributions (S−P̂N)/mean(N), divided by √b. Reject only if Δ̂>2sΔ and at least ten blocks exist. This is an approximate one-sided 2.3% test; use cluster-t calibration for few blocks, which is more conservative.

Normal planning power at true advantage δ is Φ(δ/sΔ−2), where Φ is the standard-normal CDF. At the grounded assessment planning SE of 1.7 percentage points: power is 50% at 3.4 points, 83% at 5, 94% at 6; 80% requires about 4.8 points. This is **not** a measured paired SE; estimate covariance before assigning those powers to A−B.

Match model and broker costs, clipping, replay length and exclusion of unfinished accounts. Completed-only rates are not eventual probabilities: funded truncation reached −23.5 points. An eventual-rate claim needs protocol-bias correction with its uncertainty, or prospective complete follow-up. Compare model residuals under the same protocol before transporting a ranking. Freeze one comparison or use simultaneous block-bootstrap intervals for all searched contrasts; report the set of cells not distinguishable from best. Resampling seeds alone is invalid. RR 1’s plain-bracket coin symmetry does not remove account/calendar dependence.

The original test failed: 39% outside two seed SEs. The later 73 resolvable cells and 14% failures are a diagnosis, not a new successful preregistered test. Four funded blocks cannot identify a funded optimum. Extra seeds cannot fix that, nor measure firm honour, spread or intrabar breaches.

## 5. Worked application

At B=$3,000, α=.7: m=4.7619%, hm=1.6667%. Use γ=`cost_share(34.8)*2.33`=6.6954%, including 1.33 asides and 0.5-pip spread; no swap credit. The search and reproducible calls are in `rationale.md`.

| Cell: q / assessment / funded | Pass | Any funded payout |
|---|---:|---:|
| Current: 1 / 3% / 1.5% | 39.32% | 41.82% |
| Same-ratio refinement: 1 / 3.15% / 1.65% | 39.65% | 43.08% |
| Best evaluated, standard split: 3 / 1.05% / 0.5443% | 40.51% | 48.19% |

The last improves funded probability by 6.37 points; it is a finite-search standard-split optimum, **not proof of a continuous or empirical optimum**. Assessment distribution cap binds; funded minimum binds while qr=1.6330%<1.6667%; daily nominal caps, $38/$1,000 fee cap and aggregate exposure are slack. Calendar/daily-equity/discretionary compliance remains uncertified. Higher RR reduces loss size for similar up-step but slows accounts: median funded payout is 29 trades versus 19; historical ratio-specific pace is 3.51 versus 2.03 calendar days/trade. The $51 add-on's best sampled boundary candidate gives about 52.0% funded probability but only $41.01 expected honoured cash against its $51 fee; the add-on does not open the gate either.

Under the current gate and evaluated designs, the operational optimum is **no purchase, n=0**, not an asserted new live setting. The best candidate has G(.8)=−.127 and expected honoured lifetime cash $34.57<$38, before unresolved execution costs. Staying out gives G=0. If the earlier 120-day constraint is reinstated, this slow candidate needs separate calendar validation and cannot be admitted from these chains. Current RR 1 / 3% / 1.5% also remains gated off.