# Method: what calculus settles, what the chain must compute, what the data can say

Notation (paper Table 4): T = D = 9%, r risk per trade, RR reward-to-risk, u = RR·r the
up-step, γ cost as a share of the risk, m₁ = 100/(αB) = 4.762% the minimum request on the
$3,000 tier, τ(RR) calendar days a trade (broker_map.json: 2.03 at RR 1), P the pass rate,
a = P(any payout). Every value is `prime.py` at the 34.8-pip mean stop, floor on the tick,
target clipped, γ = 2.87% (0.5-pip spread, no asides) unless named: 1.44% commission only,
6.70% with 1.33 asides. Subscripts a, f: assessment, funded.

## 1. Formulation

Variables: (RR_a, r_a), (RR_f, r_f), tier B, accounts n. The entry window stays 07:00–10:00 UTC,
where τ and the spread are measured.

Objectives, the operator's, in order: (O1) funded, maximise a, ties to payouts per calendar day;
(O2) assessment, maximise P_a, ties within a point to fewer days; (O3) gate: wallet log growth
g > 0 at 80% of payouts honoured, at the measured cost. O1 and O2 separate: P(cash) = P_a·a_f and
the cells share no variable once RR may differ by stage. Only O3 couples them, through days to
first cash.

| # | source | inequality | at γ 2.87% |
|---|---|---|---|
| C1 | rulebook, daily 5% | r(1+γ) ≤ 0.05(1−D) | r ≤ 4.42% (4.26% at γ 6.7%) |
| C1s | rulebook, strict | r(1+γ) + G ≤ 0.05·E_prev, G an overnight gain given back | G ≤ 0.64 of the stop at 3%, 0.51 at 3.24%, 0.22 at 4% |
| C2 | rulebook, 35% cap in the assessment (assumed; unverified) | RR_a·r_a·(1−γ) ≤ 0.35T | u_a ≤ 3.243% |
| C3 | rulebook, 35% cap funded | M ≥ max(m₁, RR_f·r_f/0.35); shapes a, bounds nothing | kink at u_f = 1.667% |
| C4 | venue | rB/(10ℓ) ≥ 0.01 lot; margin ≤ B(1−D) | r ≥ 0.116%; slack |
| C6 | operator | median days to first cash ≤ 120 | |
| C7 | operator | n = 1 until a payout is honoured; more trades at equal rate | |
| C8 | wallet | F ≤ 0.06W₀; ruin ≤ 5% | $3,000 only ($99 is 9.9%) |

C2 takes the day net of commission, as current_decision.md does; gross gives 3.15%.

## 2. What calculus settles

**Step 1.** A trade has mean −γr and variance RR·r² (paper §3), so the barrier exponent is
λ = 2γr/(RR·r²) = 2γ/u (Prop. 14). r cancels: only u and γ reach any barrier probability.

**Step 2, the assessment is a corner.** With T = D, P = 1/(1 + e^{λD}) and
∂P/∂u = P(1−P)·2γD/u² > 0 everywhere: no interior optimum; P rises until a constraint stops it.
C2 binds before C1 whenever RR_a ≥ 0.73, so u_a* = 0.35T/(1−γ) = 3.243%. Chain: 43.37% at 2%,
45.38% at 3%, 45.73% at 3.243%, 46.24% at 4% (C2 broken).

**Step 3, splitting u_a.** P barely sees the split (chain at u = 3%: 44.87% at RR 0.5, 45.38%
at RR 1, 45.63% at RR 3). Time does: E[N] = TD/(u·r) = TD·RR/u² and τ(RR) both rise in RR, so
days = TD·RR·τ(RR)/u² is increasing in RR, and the corner is the smallest RR_a keeping
r_a = u_a/RR_a inside C1: RR_a ≥ 0.73. At u = 3.24%: 11.7 days at RR 0.75, 16.7 at RR 1, 82 at
RR 3. Pulling the other way, outside the calculus: C7 and part 4 (only at RR 1 is a random-side
trade an exact coin; at RR 0.5 the broker's win rate sits 3.5 SE under fair). RR_a = 1 is a
choice of verifiability costing 5 days.

**Step 4, the funded stage has a kink, not a first-order condition.** a ≈
(e^{λD}−1)/(e^{λ(M+D)}−1), M = max(m₁, u/0.35) (Prop. 15). Left of u = 0.35m₁, M is fixed and λ
falls as u rises, so a rises; right of it M grows linearly and dominates, so a falls (cost-free:
65.4% for all u ≤ 1.667%, 44.0% at u = 4%). The maximum is the kink u_f* = 1.667%, where the
one-sided derivatives differ in sign. Nothing is set to zero.

**Step 5, splitting u_f.** At fixed u_f, RR_f enters only through the clock: the 14-trade wait is
14·u·r/(M·D) of a cycle, so a smaller r (larger RR) raises a while trades per payout and τ(RR)
both grow. C6 pins RR_f at 1 (part 5).

**Step 6.** The stop is the walk-forward stop of Corollary 10, not Prop. 16's corner, so γ is a
measured parameter. Tier: the C8 corner. n = 1 by C7.

**Where the limit breaks.** (i) Lattice: the request is cleared by an integer number of net
wins, k·r(1−γ) minus the losses' cost ≥ M, so a(r) is a sawtooth whose teeth move with γ.
(ii) The clock is in trades: with the wait set to zero the chain is nearly RR-invariant at fixed
u (55.4–56.4% at u = 1.5%); with it, 46.3% at RR 0.5, 52.4% at RR 1, 55.7% at RR 3. (iii) The
minimum request is the kink itself: `funded_model` there reads 59.9% against the chain's 50.7%,
and the exact optimum is elsewhere. (iv) Clipped legs near the floor cut the win probability
b/(a+b): at 3.243% the limit reads 46.02% against 45.73%. (v) One 0.01-lot step is 0.116% of
bankroll; finer risk is the same trade.

## 3. What must be computed exactly

The chain costs 0.03 s an assessment cell, 0.3 s a funded cell.

**Assessment: one evaluation at the corner** u_a*, its lattice neighbours, at each γ, plus the
u-invariance row as a check on the split. A grid wastes every cell above C2; a root find has
nothing to find.

**Funded: a line search in r_f at RR_f = 1** over [0.6%, 2.5%] in 0.01% steps (190 points, a
minute), at each γ. Not the grid: the half-point lattice straddles the teeth (the argmax at
γ 2.87% is 1.13%, which no half-point sees). Not a root find or golden section: both assume
unimodality, and a(r) has eight to twenty local maxima here. The interval ends at 2.5% because
Step 4 proves decline above the kink and the chain confirms it from 1.7%. The result is read
against C4.

**The ridge across RR_f** is the same scan per RR, reported in days. **The wallet** is
`wallet_mc` at the chosen cell only, as the O3 gate. None sees the spread, swap or P(paid):
inputs.

## 4. Inference

**Where the uncertainty lives.** Given the per-trade win probability p, the cost share γ and the
rule readings, every rate is exact. The optimum's uncertainty is input uncertainty propagated
through the chain, not sampling error of accounts. An edge ε = p − ½ enters the drift as
−γr + 2εr: one point of win rate is two points of γ. At the grid cell:

| ε | γ_eff | P_a(3%) | a_f(1.5%) | cash/fee |
|---|---|---|---|---|
| −2 pt | 6.87% | 39.1% | 41.2% | 0.94 |
| −1 pt | 4.87% | 42.2% | 47.4% | 1.26 |
| 0 | 2.87% | 45.4% | 52.4% | 1.67 |
| +1 pt | 0.87% | 48.6% | 56.4% | 2.20 |

**What 2.8 years resolve, after §14.10.** (a) The engine: at RR 1 on the coin side each trade
is an exact coin whatever the price did, so the 20 seeds are independent except through clipped
trades and the between-seed SE is close to valid: 4% cell 47.2 ± 1.35 against 48.1 (z −0.65),
funded 1.5% cell 52.5 ± 3.3 against 52.7 under the protocol (z −1.0). (b) τ = 2.03 ± 0.006 days.
(c) Asides A = 1.33 ± 0.05 (Poisson, 467 trades): γ at the measured spread is 6.7 ± 0.3%.
(d) The market: the deployed side is always short, so fairness is a claim about EURUSD drift
over two days, tested on one path of 467 trades, SE 2.3 points; the seeds share the path and add
nothing, at RR ≠ 1 even on the coin side (Table 28). (e) The low-risk funded cells (4 blocks,
open account dropped) are not evidence; the chain does not need them.

**What cannot be resolved.** ε at the scale that matters. Power of the 467-trade test at |z| > 2:
7% against 1 point, 14% against 2, 51% against 4.6, 80% against 6.5; 80% power at 1 point needs
19,600 trades, 109 years. The spread and swap are outside the data. So the decision must be
robust over ε ∈ [−2, +2] points and γ ∈ [1.44%, 6.7%], and cell orderings are tested by
dominance over that region.

**Testing "cell A beats cell B".** Δ(p, γ) = f_A − f_B is a deterministic chain output; H₀:
Δ ≤ 0; evaluate Δ over the region. Constant sign: settled by the chain, no replay adds to it.
Sign flip: unresolved, and the measurement pinning the flipping input settles it, not more
history. Results: 4% beats 3% on P_a at every γ_eff ≥ 0.87% (Δ −0.2 to −2.3 points), flipping
only above ε = +1.4, so the 3% choice rests on C2 alone. 1.5% against 1.0% funded flips at
γ ≈ 2.4%, a live spread of 0.33 pip: 1.0% wins commission-only (57.0 against 55.4), 1.5% wins
at 2.87% and above. 1.63% beats 1.5% only at γ 6.7%, by 1.2 points.

**Declared tests.** T1, engine: the two chosen cells through the broker, coin side,
|broker − protocol expectation| < 2 between-seed SE; power 84% against a 4-point fault at the
assessment cell, 78% against 9 points at the funded cell. T2, market: short-side win rate on 467
trades, reject at |z| > 2 (outside 45.4–54.6%); power as above, declared weak. T3, live: stop
buying if the running win rate falls below 0.5 − 1/√n after n ≥ 100 trades (a 10-point shortfall
at 100, 5 at 400). The funded stage's only test is the payout record.

## 5. Worked application

Against RR 1, 3.0% assessment, 1.5% funded, $3,000 tier, one account.

**Assessment.** The optimum is the C2 corner r_a = 3.243% at RR 1 (3.376% at γ 6.7%): P_a
45.73% against 45.38% (+0.35 points), 8.2 trades against 9.65, 16.7 days against 19.5. C2
binds; C1 is slack; C1s tightens, a breach needing an overnight gain over 0.51 of the stop
against 0.64, which is measurement 3 of current_decision.md. On the lot grid: 0.27 lots (3.13%
effective) against 0.25 (2.90%); floored lots always sit under the cap. If C2 is funded-only, C1
binds and the pick is 4.0% (46.24%, 12 days).

**Funded.** The kink 1.667% is not the exact optimum (50.7% there). The line search puts the
argmax of a on a plateau whose peak moves with γ: 0.87% (57.8%) at 1.44%, 1.13% (52.9%) at 2.87%,
1.43% (51.6%) at 3.35%, 1.64% (43.2%) at 6.7%. Over r_f ∈ [1.1%, 1.5%], a is within 0.5 points
of its maximum at 2.87% and within 2.4 in every scenario, less than one point of ε moves it
(4.5) or the best funded measurement resolves (3.3). The plateau is a tie on O1,
broken upward by time: 1.5% against 1.13% gives a first payout at 36 against 47 days, life 110
against 174 days, 1.00 against 0.65 payouts per 100 days. Above 1.5% a leaves the plateau
(50.7% at 1.667%, 46.3% at 2%) as C3 binds. So r_f = 1.5% at RR 1: the grid pick, as the upper
edge of a plateau rather than the top of a ridge. Across ratios the per-RR argmax rises from
50.3% (RR 0.5) to 55.8% (RR 3) while the first payout moves from 29 to 116 days; RR 3 breaks
C6, so RR_f = 1.

**Tier, accounts, gate.** $3,000 (C8 binding at 3.8% of the wallet), one account (C7).
`wallet_mc`, 20,000 paths, 500 trades, at (3.24%, 1.5%): ruin 0.2%, first cash median 120 days
(grid 132), g +0.22 at full honour, +0.04 at 80% (grid +0.18, +0.02). At γ 6.7% both cells read
g −0.25 to −0.48: the gate stays closed, as current_decision.md found.

**Net.** The method moves the assessment one lot step, 3.0% → 3.24%, worth +0.35 points and
3 days, and leaves the funded cell, the ratio, the tier and the gate where the grid put them.
What the grid lacked: its funded cell sits on a plateau whose position depends on an unmeasured
spread, and no history of this pair can order cells closer than 2 points of win rate.