# Rationale and audit trail

## What the answer means

There are three different questions: a diffusion optimum, the winner of a specified numerical search, and an admissible purchase. They need not agree. The current decision explicitly makes positive wallet log growth at 80% honour a purchase condition. None of the tested candidates passes it. I therefore report no purchase as the operational decision, and disclose the better conditional chain candidate rather than claiming the old cell is mathematically optimal.

The search is bounded by the supplied ratio menu, not by a firm rule limiting RR to 3. It cannot prove a global optimum over all real-valued ratios, time-varying controls, stops or entry windows. A finite sample cannot provide that proof either. The documents do not specify a quantitative limit for “looks like gambling” or “shallow drawdown”; creating one would change the question. Manual fixed-size trading with ordinary stops is the conservative implementation reading, not a guarantee of acceptance by the firm.

The current decision supersedes notebook Section 23's cost and ranking. In particular, I did not silently preserve its 120-calendar-day first-cash limit while claiming to optimise the newer ranking. If that limit is still intended, it must be restored explicitly; the slow RR 3 candidate is not validated against it. Its funded median alone is approximately 29×3.51=102 calendar days using the historical ratio-level average pace, before assessment or retries. This multiplication is only a scale illustration, not the median of calendar first cash. The old notebook's 55% funded figure uses the wrong stop-cost convention for the present question.

## Alternatives rejected

**Read off a universal risk corner.** The derivative in Section 8 holds at fixed ratio, cost and diffusion assumptions for assessment pass probability. It does not optimise probability of a funded payout. The minimum request creates an increasing lower branch and decreasing upper branch. The daily cap is pathwise and cannot be guaranteed by a fixed 4% or 3% risk when overnight floating profit enters the denominator and can be given back.

**Declare the stop corner the deployed optimum.** An expectation of one day is not an almost-sure bound on holding time. The supplied history contradicts the intraday premise. Spread, swap, news exits and calendar pace interact with stop width. No permitted function reruns the actual stop/window policy through prices. Keeping the grounded deployed rule is better supported than claiming a new stop optimum. Taking the short side is inherited operationally; no swap credit finances the purchase conclusion.

**Funded risk tending to zero.** At positive cost and a fixed dollar minimum, this drives drift-to-variance against the trader and makes completion slow. Below the ridge, diffusion payout probability increases with risk. At zero cost it is flat there, not strictly decreasing everywhere.

**Use one root solver for the whole problem.** The minimum-request kink has one-sided derivatives; a clock introduces risk dependence beyond u; overshoots and rounded lattice transitions produce non-smooth changes. A root solver can miss the maximum entirely. Root finding is useful as a proposal mechanism on each smooth surrogate branch, followed by endpoints, kinks and chain evaluation.

**Search only the analytical ridge.** At RR 1, 1.65% funded risk gives 43.08% payout probability at the finer lattice, while the exact ridge, 1.6667%, gives about 42.89% on the default lattice. RR 3 also has a nearby peak below its ridge. Barrier arithmetic matters. Both-sided local enumeration is necessary. The strict current-decision restriction u_f≤.35m excludes the upper side from the admissible ranking, but evaluating that side remains useful if that operator restriction is relaxed: the firm itself allows a larger request.

**Treat the old half-point grid as exhaustive.** It misses the assessment cap r_a=.0315/q and the funded boundary r_f=(1/60)/q. These are natural added candidates. Conversely, quoting many decimal places from a local numerical maximum would imply false implementation precision: the lot step alone is material at this tier.

**Exploit floor overshoot or clamp equity without changing probabilities.** Tick enforcement shortens the loss leg and changes its hitting probability. Merely clipping the booked loss with the original win probability creates a different game. Funding begins from zero profit, and withdrawals restore the same static cushion.

**Optimise gross extraction or average dollars alone.** These ignore payout frequency, waiting, the fee wallet and the honour condition. A per-cycle payout probability is not payouts per month. At the chosen RR 3 candidate the finer chain gives 0.930 expected payouts and 107.57 expected funded trades, versus 0.719 and 45.18 for the current candidate. The higher chance of ever paying comes with substantially slower turnover. Both should be visible to an operator who says “frequent”.

**Treat the model winner as a deployable improvement.** Cost at RR 3 is provisionally held at the same 1.33 asides/trade measured for the deployed strategy. RR-dependent holding time may change this rate. Calendar timing and intraday equity are missing from the allowed chains. At the mean stop, one 0.01 lot corresponds to risk 10×34.8×.01/3000=.116%; nominal 0.5443% risk rounds down to .464% at that stop. The daily stop varies, so this is not a constant adjustment. Rounded execution requires re-evaluation, not a claim that .5443% can be placed exactly every day.

**Count seeds as independent market histories.** Twenty direction seeds reduce conditional simulation noise but reuse the same price history. Plain symmetric brackets have a special fair-coin property; clipped account brackets and calendar durations still do not supply independent histories. Account-length calendar blocks, pairing, and protocol matching are essential.

**Drop unfinished accounts and compare with eventual probability.** This preferentially loses long-lived paying funded accounts. Section 14.10 reports a worst bias of −23.5 percentage points. An unbiased estimate of the completed-account protocol still estimates the wrong eventual quantity unless corrected. No broker replays were run here; a prospective protocol-matched test is specified rather than fabricated.

**Announce agreement after changing the error bars.** The initial test failed. The block diagnosis explains some of that failure but is post hoc. Even 14% beyond two SE is not itself proof of an exact null model. Search-adjusted uncertainty belongs on the difference between candidates and on the set of plausible winners, not just on the winning cell's rate.

**Add accounts, buy a larger tier, or assume the add-on dominates.** $99 exceeds the $60 fee cap, so the next base tier is excluded without simulation. The $51 add-on passes that cap but changes both split and minimum request. The paper's per-fee split argument is not a general stochastic dominance theorem once the minimum binds. I therefore also checked its boundary candidates: it improves funded probability, but the highest expected honoured cash in that search is $41.01 against $51. No-purchase is the supported decision over evaluated designs, not a theorem that no untested product could work.

## Numerical work actually executed

Only the authorised Prime model functions were used for numerical work. Imports used `PYTHONDONTWRITEBYTECODE=1` from `/root/Projects/trading-framework`; the Tier constructor supplied wallet inputs. No broker, notebook, web, git mutation, price download or log was used. Only these two documents were written.

Inputs: stop 34.8 pips; commission $5/lot; spread .5 pip; A=1.33 extra round trips/trade; γ=`cost_share(34.8)*(1+A)`=.06695402298850575; B=3000; split=.7; target/floor=.09; `clip_up=True, clip_dn=True` for assessment; `clip_dn=True` funded; default 14-trade clock. These deliberately match the brief's operating facts, not the older paper's 60-pip example or the broker's commission-only accounting.

Executed search:

1. Ratios q in {.5,.75,1,1.25,1.5,2,2.5,3}. At each, assessment candidate r_a=min(.0315/q,.0455/(1+γ)). Funded candidates {.005,.01,.015,.02,.025,.03,.035,.04,(1/60)/q}, filtered by qr_f≤1/60 and r_f(1+γ)≤.0455.
2. At each ratio add r_f=((1/60)/q)j/20, j=10,…,20. This is a boundary-oriented search, not an exhaustive continuous search.
3. At RR 1 separately evaluate funded {.015,.016,.0165,1/60} and assessment {.03,.031,.0315}.
4. At RR 3 refine r_f=.0052+j(1/180−.0052)/19, j=0,…,19. The best evaluated default-lattice point is .00544327485380117. The endpoints and neighbouring sampled points do not beat it. This is the optimum of the evaluated set, not a certified optimum between all those points. The broader full-grid and adaptive search recommended in method.md remains the appropriate procedure before a new deployment decision.
5. Re-evaluate current, same-ratio funded refinement and the best funded candidate at units=40/c,80/c,160/c; re-evaluate both RR 1 assessment settings and RR 3 assessment at 160/c. Refined winner assessment pass probability is .40510568243297046.
6. For the $51/.9-split add-on, evaluate the same eight ratios and assessment cap candidates, with funded risks z×.35×100/(.9×3000)/q for z in {.9,.95,1}. Its best evaluated payout probability is .5204330328 at q=3, assessment .0105 and funded .0043209876543, on the default lattice. The maximum expected honoured cash across these 24 candidates is $41.00832557. Thus the standard-split winner is not the maximum-payout-probability cell across splits; both products fail the financial gate. Refine this add-on point at 160/c and run its wallet with `addon=True`, split=.9 and the same Monte Carlo settings.

Key convergence results:

| q, funded r | c/40 payout probability | c/80 | c/160 |
|---|---:|---:|---:|
| 1, .015 | .418073520 | .418197283 | .418197283 |
| 1, .0165 | .430975876 | .430824091 | .430839642 |
| 3, .005443274854 | .481702235 | .481863806 | .481863806 |

Numerical stability does not establish market-model validity. The selected RR 3 conditional values at 160/c are P(pass)=.4051056824, P(any funded payout)=.4818638065, E[gross lifetime withdrawal]=.05079802145, P(cash per purchase)=.1952057662 and expected cash at full honour=$43.21499101. At 80% honour expected cash is $34.57199281<$38. This supports rejection of that candidate even before the simulated wallet gate. The incumbent's expected full-honour cash is approximately $36.8, consistent with `current_decision.md`; its purchase cash probability is approximately 16.44%.

Reproduction of the main comparison (ordinary Python data construction, then the permitted numerical functions):

```python
from src.notebooks.reports.topics.prime_academy.modules import prime as P
g = P.cost_share(34.8) * 2.33
for q, ra, rf in [(1, .03, .015), (1, .0315, .0165),
                  (3, .0105, .00544327485380117)]:
    assessment = P.eval_chain(ra, g*ra, rr=q,
        clip_up=True, clip_dn=True, units=160/(g*ra))
    threshold = P.payout_threshold(rf, q, 3000, .7)
    funded = P.funded_chain(rf, g*rf, threshold, rr=q,
        clip_dn=True, units=160/(g*rf))
    wallet = P.wallet_mc(P.Tier(3000, 38, 51), ra, g*ra, .7,
        .8, 1000, n=20000, seed=13, rr=q,
        funded_risk=rf, funded_cost=g*rf,
        clip_up=True, clip_dn=True)
```

Wallet simulation, 500 model trades, 20,000 paths, seed 13, h=.8:

| Cell | E log growth | Ruin | Median wallet maximum drawdown |
|---|---:|---:|---:|
| Current 1 / 3% / 1.5% | −.455756 | 1.980% | $456 |
| 1 / 3.15% / 1.65% | −.479878 | 2.955% | $468.52 |
| 3 / 1.05% / .544327% | −.127157 | 0 observed | $228 |
| Add-on: 3 / 1.05% / .432099% | −.202224 | .010% | $255 |

The add-on's finer-lattice funded probability is .5203471487 and gross lifetime withdrawal .04692286131. Its assessment is the same as the standard RR 3 candidate. This higher conditional payout probability does not compensate for the increased fee under the purchase gate.

Zero observed ruin is not zero true probability. These are model-trade horizons, not 500 calendar days, and the wallet function does not implement stopping purchases permanently after a refusal. The negative results do not license deployment under the stricter policy; they fail the current declared gate already. Positive results would have required validating the actual stop-on-refusal policy and calendar before purchase. The current decision's −.477 figure and this run's −.456 are separate Monte Carlo outputs; I have not silently substituted them as identical runs.

## Grounding precedence

`rules.json` and paper Sections 2 and 16 specify the rule uncertainty. Paper Sections 3–8 supply the derivations; Sections 14–15, `prime.py`, `broker_map.json`, `broker_results.json` and notebook Sections 23–24 supply chain/broker definitions and checks. The vault records execution details, news policy, lot sizing and categorical prohibitions. `current_decision.md` supplies the latest objectives, cost stack and gate. Later platform/tick-floor facts in the brief take precedence over older unresolved or overshoot-based statements. None of these sources makes a conditional model ranking a statistical proof of an optimum or a promise of payment.