# Rationale: the alternatives considered and rejected

Every number here comes from `prime.py` at base cost γ = 2.87% unless it cites the paper, the notebook or `broker_map.json`.

## Optimisation

**1. A full grid over (RR_a, r_a, RR_f, r_f), ranked by a declared rule (the method in use today).**
- **Kept:** the exact chains it is built on.
- **Rejected as the search**, for three reasons.
  - **It spends evaluations on dimensions calculus has already removed.** P(cash) factors into the two stages (Step 1). Along the assessment cap, the clipped chain gives 45.68–45.86% from RR 0.75 to RR 3, so RR does nothing to P there.
  - **Its 0.5% lattice is too coarse for the funded ridge.** The funded ridge is a sawtooth, with teeth at r_edge(j, n) = m1/(j − nγ). The half-point grid saw 1.0%, 1.5% and 2.0% and missed the tooth tops at 1.10% (52.87%) and 1.40% (52.77%). It also missed the [1.642, 1.667]% window at a 7-trade clock.
  - **A grid reports an argmax that looks precise but is not stable.** Across plausible inputs, the funded argmax moves from 1.05% to 1.50% while the value moves by under 2.2 points.

**2. The diffusion closed forms alone (`p_pass_diffusion`, `funded_model`, `growth_model`).**
- **Kept:** for structure. They give the corner, the kink and the fact that only u matters.
- **Rejected:** for levels and locations.
  - **The level is wrong.** The funded diffusion reads 59.3% at 1.5% against the chain's 52.4%, because it has no 14-trade clock.
  - **The location is wrong.** The diffusion puts the optimum at the kink, 1.667%. The chain, with the clock, has it lower: a is 50.67% at 1.667% against 52.36% at 1.5%.
  - **It cannot see the lattice teeth.**

**3. A root find on a first-order condition.**
- **Rejected.** The assessment has none: dP/du > 0 everywhere, so the optimum is a corner and the useful number is the constraint's shadow price, 0.13 points per 0.1% of risk.
- The funded optimum is a kink in the diffusion and a sawtooth on the chain, so no derivative is zero at the maximum. A root finder would stop on a tooth edge.

**4. A continuous optimiser (Nelder–Mead or Bayesian optimisation) on the chains.**
- **Rejected** for the same non-smoothness.
- It would also give the argmax a spurious precision, when the argmax moves by 0.45 points of risk with the cost and the clock reading.
- A one-dimensional line search with the edges placed from the closed form costs about 8 seconds a scenario and misses nothing the chain can represent.

**5. Maximise the wallet's expected log growth (`wallet_mc`) directly.**
- **Rejected as the objective.**
  - It assumes a share of payouts honoured, which nothing measures.
  - It carries Monte Carlo noise.
  - It rewards a high funded risk, which contradicts the operator's objective of frequent payouts. Notebook section 23: +0.577 at 4%/4% against +0.311 at 4%/1.5%.
- **Kept as a constraint check:** ruin 0.11%, and growth +0.179 or +0.018 at 100% or 80% of payouts honoured.

**6. Payouts per unit of calendar time (renewal-reward) as the objective.**
- **Rejected.** It rewards short funded lives, that is, buying accounts faster, which spends fees.
- The objective kept is expected payouts per purchase, E[K] = a/(1−a). It rises with a, so maximising a is the same thing, and it also minimises fees per payout.

**7. Moving RR away from 1, either the funded RR 3 at 0.5% (55.7%, the map's best) or a higher RR at fixed u.**
- **Rejected.** At fixed u = 1.5%, RR 3 gains 3.4 points of a through the clock, but its median first payout moves from about 37 calendar days to 130.
- The broker measures a per-trade shortfall off RR 1: −1.4 points of win rate at RR 1.5, an adverse drift of 3.5% of the risk, larger than γ. The modelled gain would not survive it.

**8. The assessment at the exact corner, 3.15%, or at 3.24% under the net reading.**
- **Rejected.** The gain is 0.19 points (45.38% to 45.57%), below anything a test can resolve.
- The rule it approaches eliminates the account or forfeits payouts when broken.
- 3.24% breaks the cap under the gross reading, which is the right one when commission is booked on the day the trade opens.

**9. A breach-proof assessment risk under the strict daily-loss reading, r ≤ 2.24%.**
- **Rejected as the default.** It costs about 1.4 points (44.0% against 45.38%) to remove a risk that has not been measured.
- **Held as the fallback** if the measured breach rate at 3% exceeds 1% an account.
- The other fix is to close a trade before the daily reset when its floating gain exceeds g*·r. That is a stopping rule on a martingale, so it keeps the trade fair.

**10. The minimax-regret funded risk, 1.35%**, whose maximum regret across the eight input scenarios is 1.12 points against 2.15 at 1.5%.
- **Rejected.** 1.35% sits just above the first edge, 1.323%. At the $3,000 tier the 0.01-lot grid floors a trade's risk to as little as 10/11 of nominal, so 1.35% trades below the edge on some days. The fixed-risk chain cannot see that.
- The lot-robust window is [1.456, 1.667]%, and 1.5% is its best point. The regret gap between the two, under 1.1 points, is itself below any resolvable difference.

**11. A 7-trade payout clock**, reading the rule's 14 calendar days at 2.03 days a trade.
- **Rejected as the base reading.** Fourteen trades is conservative.
- Under 7 trades, a rises by about 2 points and the best moves to 1.35%, a 0.5-point gain that no test can see.

## Inference

**12. Choosing the optimum from the broker's cells directly (the empirical argmax).**
- **Rejected**, for four reasons:
  - Block standard errors are 1.2–3.8 points.
  - Protocol truncation biases the funded rates by up to 23 points.
  - The 1.5% funded cell has 4 blocks.
  - Taking the maximum of 128 noisy cells selects noise.
- The broker tests the model. The model chooses the cell.

**13. Standard errors between seeds, and a bootstrap over seeds.**
- **Rejected.** The seeds share one price path, so both understate the error. This is how the declared test in Section 14.10 failed 39% of cells.

**14. A block bootstrap of the argmax over the history.**
- **Rejected.** With 4 to 33 blocks the resampled argmax only reflects the flatness of the plateau.
- The chains are exact, so the only uncertainty worth propagating is in the inputs (γ, the clock reading, the rule readings). The method propagates it by scenario and reports the regret.

**15. Testing cell against cell with independent standard errors.**
- **Rejected.** It throws away common random numbers. The paired, block-clustered difference, tested against the model's own difference under the protocol, is the correct and more powerful statistic.
- Even paired, 3% against 4% in the assessment (0.86 points) is unresolvable: the minimum detectable difference is 2.3–5.9 points. The method declares that in advance rather than running an underpowered test.

**16. Estimating the edge from the history and optimising under it.**
- **Rejected.** 467 deployed closes give 9% power against the smallest drift that would move the corner (1.44 points of win rate). 80% power needs about 9,450 trades.
- Zero edge stays imposed. The drift test is kept only as an alarm for gross failure.

**17. A hierarchical model that pools the broker's cells into one fit.**
- **Considered and deferred.** Pooling cannot remove the single shared price path, which is the binding limit, and it adds priors the operator would have to accept.
- It would help only once a second pair or a longer history exists.