# Prospect 51: the order-run bonus can change the best hold

**Research addendum, 30 September 2026.** This uses the staged v3.5 Prospect 51 rules and paytable. It changes no game rule or engine. The payoff below is expected **gross jbits** conditional on a known opening hand, before the already-paid one-jbit entry. The archive-lock award, when present, is fixed by the opening order and therefore cancels when holds are compared.

## Exact comparison

Consider the ordered opening **8♥–9♥–10♥–J♣–4♥**. The table enumerates every unordered base-poker redraw from the 46 identities outside the opening hand. Order-run completions are counted as ordered rotor draws, then multiplied by the expected free-spin award from the prior exact note: \(5+802746/2349060=298763/55930\) jbits per trigger.

| Hold | Base poker gross | Order-run chance | Expected order-run bonus | Combined gross |
|---|---:|---:|---:|---:|
| Four hearts: positions 1, 2, 3, 5 | \(27/23=1.1739130435\) | 0 | 0 | \(27/23=1.1739130435\) |
| Four ascending ranks: positions 1, 2, 3, 4 | \(35/46=0.7608695652\) | \(4/46=2/23\) | \(298763/643195=0.4644983248\) | \(1576301/1286390=1.2253678900\) |

The **base-only optimum** among all 32 masks is the four-heart hold. The **base-plus-bonus optimum** among all 32 is the 8–9–10–J hold, ahead by exactly \(66191/1286390\approx0.0514548465\) expected gross jbits. The base disadvantage is \(19/46\); four of the 46 one-card replacements complete an order run. Thus a triggered bonus exceeding \(19/4=4.75\) jbits changes the preference. The game's fixed 5-jbit trigger component alone exceeds that threshold; the independent fresh-spin poker award increases the margin.

This is a conditional strategy comparison, not a general return or profitability claim. The browser engine rejects out-of-range 32-bit words before reducing modulo each inventory size, then applies an affine rotor map with a coprime multiplier. Under uniform browser random words, that map is a permutation at each step, even when its offset depends on earlier draws; every ordered replacement sequence is therefore uniform. The bonus spin uses a fresh 51-card inventory. This does not claim that the displayed auto-hold setting is optimal or certify the browser's random source empirically.

The convenience **auto-hold is not an optimal-play rule**. In fact, standing
pat is strictly dominated in expected gross for every paying opening that is
exactly one high pair, two pair, or three of a kind: keep the equal-rank
group(s) and redraw the other cards. The retained cards preserve at least
the current base payout, and a higher category has positive probability.
For one high pair, one of its two undealt rank-mates makes at least trips;
for two pair, any undealt mate makes a full house; for trips, two undealt
cards of another common rank make a full house. Each event has positive
probability in the 46-card redraw pool.
Duplicate ranks prevent an order-run bonus under either action, while the
archive award is fixed by the opening order. On J♠–J♥–3♦–7♣–9♠, standing
pat pays exactly 1 gross jbit; holding only the two jacks has exact combined
gross expectation 1172/759 ≈ 1.544137 jbits, the best of all 32 masks for
this opening. The player can release the automatic holds before drawing;
the checkbox does not lock the decision.

## Method and verification

For every mask, the verifier enumerates all \(\binom{46}{r}\) unordered base-award completions, where \(r\) is the number of unheld positions, and scores them with an evaluator written separately from the v3.5 engine. It counts favorable **ordered** run completions over \((46)_r\) possibilities using the 20 allowed ascending/descending rank ladders and the available rank multiplicities. Exact integer fractions are compared by cross multiplication; decimal values are display aids only.

The 32 masks form a useful exact check: each final unordered five-card subset \(S\) corresponds to precisely one mask, namely the opening identities in \(S\). Therefore their completion counts sum to \(\binom{51}{5}=2,349,060\), their category counts reproduce the full-deck census, and their base gross sums total 802,746 jbits. The verifier checks these identities for each of seven frozen openings. For three openings, including the strategy reversal above, it also compares the independent category scorer against the v3.5 engine on **all 2,349,060 mask/completion outcomes**. It checks the engine's `drawOdds` total and held identities for every mask, and directly compares run counts with the engine for all masks drawing at most two cards.

Two deterministic rotor traces additionally check full settlement in the frozen engine. From the example hand, holding 8–9–10–J and drawing Q♥ makes a straight and order run; a prescribed two-pair free spin gives 4 base + 7 bonus = 11 gross jbits. Holding the four hearts and drawing A♥ makes a flush without an order-run spin and gives 6 gross jbits. These are integration fixtures, not the expectation calculation.

Run `node research/project-51/prospect-51/verify_prospect_holds.cjs` from an extracted complete-site directory. The script rewrites `prospect_hold_receipt.json`, which includes exact numerators and denominators for all **32 masks × seven hands**, and `prospect_switch_32_masks.csv`, a compact view of all masks for the example above. Inputs are the frozen site files `area-51/prospect-51/data.js` and `engine-v2-7.js`; the independent deck, evaluator, and run counter do not import the engine's scoring logic.

## Site-ready wording

> The hold-odds panel reports the ordinary poker award for the selected hold. An order run can also trigger a free spin, so its expected bonus can change which hold has the highest combined gross award. For example, from 8♥–9♥–10♥–J♣–4♥, holding four hearts has the higher base award expectation, while holding 8–9–10–J has the higher expectation after the order-run bonus. These are exact odds under independent uniform draws, not measured player returns.

The panel's existing “Expected base gross” label accurately describes its current calculation. Any strategy comparison presented as **total** expected gross must include the order-run contribution. The engine and auto-hold behavior remain unchanged in this round.