# PP292 addendum A1 — rarity ordering and the bug's category declines

30 September 2026 · Mathematical addendum to [PP292](PP292_the_declared_card_as_a_wild_card_2026-09-29.md) · Poker 51 v0.4

This note adds a direct proof and a category comparison. It does not replace the deposited PP292 edition or change the playable rules. The known wild-card paradox is credited in PP292 to S. Gadbois; no claim of priority is made here. The calculations concern a finite card model, not the Riemann hypothesis.

## Rules and comparison

The physical 52-card deck has 51 natural cards and marker Ⅰ in place of 2♣. A marker hand has Ⅰ and four distinct natural cards. A virtual substitution must be distinct from those natural cards, except for the expressly allowed five-of-a-kind cases. Under any proposed total order of the exclusive poker categories, a marker hand takes its highest reachable category. Ordinary hands keep their exclusive categories. A strictly rarity-consistent order puts a category above another only when its resulting count is smaller; the proof below also rules out weak orders that allow equal counts.

For the bug comparison, the deployed standard category hierarchy and rank tie-breakers apply. The baseline is the **same physical five-card hand** with Ⅰ read as ordinary 2♣. “Up,” “same,” and “down” here refer to category, not rank or kicker strength within a category.

## Wild Ⅰ: no rarity-consistent category order

Write P, D, T, F, and Q for one pair, two pair, three of a kind, full house, and four of a kind. Among five-card hands drawn only from the 51 natural cards, their counts are:

| Category | P | D | T | F | Q |
|---|---:|---:|---:|---:|---:|
| Natural count B | 992,640 | 111,672 | 49,632 | 3,384 | 564 |

Each of these rank-multiplicity categories is invariant under a transitive relabeling of ordinary cards. Removing one card therefore keeps exactly 47/52 of its 52-card hands. In P, D, T, F, Q order, the natural counts above are 47/52 times the ordinary 52-card formulas

```text
13 C(4,2) C(12,3) 4³,
C(13,2) C(4,2)² 11·4,
13 C(4,3) C(12,2) 4²,
13 C(4,3) 12 C(4,2),
13·12·4.
```

The relevant four-natural-card partitions and counts are:

| Rank partition | Count | Categories reachable with Wild Ⅰ |
|---|---:|---|
| 2+1+1 | 76,032 | P, D, T |
| 2+2 | 2,592 | D, F |
| 3+1 | 2,304 | T, F, Q |
| 4 | 12 | Q, five of a kind |

These counts use the four-card deletion factor 48/52 = 12/13:

```text
M_211 = (12/13) 13 C(4,2) C(12,2) 4² = 76,032
M_22  = (12/13) C(13,2) C(4,2)²      = 2,592
M_31  = (12/13) 13 C(4,3) 12·4        = 2,304
M_4   = (12/13) 13                     = 12
```

For any category order, resulting counts C satisfy

```text
C_P >= 992,640
C_D <= 111,672 + 76,032 + 2,592 = 190,296
C_T <= 49,632 + 76,032 + 2,304 = 127,968
C_F <= 3,384 + 2,592 + 2,304 = 8,280
C_Q <= 564 + 2,304 + 12 = 2,880
```

If higher categories must be no more common, P is forced below D and T, F is forced above D, and Q is forced above T. Thus the 2+2 marker hands go to F rather than D; the 3+1 hands cannot go to T; and all 76,032 hands of type 2+1+1 go to whichever of D and T is higher.

If T is higher, it has 49,632 + 76,032 = 125,664 hands against D's 111,672. If D is higher, it has 111,672 + 76,032 = 187,704 hands against T's 49,632. In either case the higher category is **strictly more common**. This contradiction rules out all orders, including weak rarity consistency. The [all-order search receipt](RANKING_SEARCH_v1.json) remains a separate finite regression check.

This theorem concerns rarity ordering, not the validity of a game. Poker 51 declares a fixed hierarchy and tie-breakers. Every physical hand then has a maximum among its legal readings, so every showdown has a well-defined, transitive score.

## The bug: 41,895 category declines

The [published census](WILD_CENSUS_v1.json) gives 69,277 category increases among the 249,900 hands containing Ⅰ. A direct count gives 41,895 decreases. Let j be the number of natural twos among the other four cards:

| j | Decreases | Reason |
|---:|---:|---|
| 1 | 38,481 | 3 × C(44,3) non-ace candidates, minus straight or flush rescues |
| 2 | 3,366 | C(3,2) × [C(48,2) − C(4,2)] |
| 3 | 48 | Any fourth natural card |
| **Total** | **41,895** | |

For j = 1, the three remaining natural cards have no ace and come from the 44 cards of ranks 3 through K. A straight rescues 3 × 4 × 4³ = 768 of these candidates: the rank triples are 345, 346, 356, and 456. A flush rescues 3 × C(11,3) = 495; 3 × 4 = 12 hands are counted in both rescues. The decline count is therefore 3 × C(44,3) − (768 + 495 − 12) = 38,481. For j = 2, the sole non-declining choice is two natural aces: the bug changes twos full of aces into aces full of twos. For j = 3, all 48 choices decline from four twos.

Consequently, against the ordinary-2♣ reading, the bug moves **69,277 up**, **138,728 stay in the same category**, and **41,895 down**. “Same category” does not mean equal rank or kickers. For example, Ⅰ, 2♦, 6♠, 9♥, Q♣ is a pair of twos with ordinary 2♣, but ace-high under the bug: no straight or flush completion is available, so the bug reads as an ace. This is a consequence of the declared rule, not an evaluator error.

These hand comparisons do not establish game returns, fair odds, player advantage, or optimal dealer play. The current game rule is unchanged.