# A5s four-bets: selected study evidence

Retrieved September 21, 2026. Twelve six-handed NLHE cash configurations use equal starting stacks of 50bb or 100bb, no ante and a 2.5bb open. BTN three-bets to 7.5bb; the blinds use 12bb. All uninvolved players fold. Catalogued rake: 5%, capped at 4bb. Original solver version, postflop abstraction and achieved convergence are unavailable.

## Planned comparison

Discovery: BTN versus BB and CO versus BTN, at both depths. Held out: HJ versus BTN, UTG versus BTN, CO versus BB and BTN versus SB, at both depths. Primary: 48 hand/node rows for A2s–A5s. Rival: twelve A9s rows. Reference: 24 AA/AKs rows. All primary and rival rows have positive opening reach. These are twelve related nodes from two stack models, not 84 independent experiments.

The hypothesis asked whether A5s has no greater local small-four-bet regret than A9s in at least 75% of held-out nodes. Nominally, it does in seven of eight held out and eleven of twelve overall. One difference is only 0.00076bb, inside the predeclared 0.02bb near-tie band. Overall, ten differences favor A5s by more than 0.02bb, one is nearly tied, and one favors A9s.

A separate test asked whether automatically making the small A5s four-bet stays within 0.1bb of the best stored action in every node. It fails in five, including one borderline 0.10162bb case. Four remain above 0.2bb.

## Arithmetic and denominators

Local regret = maximum stored action EV minus small-four-bet EV. Action amounts are total preflop contributions. Fold EV is zero at the current decision; the opening raise is committed. At 100bb BTN versus BB, A5s call EV is 0.41413bb and raise-to-24bb EV is 0.08084bb: a 0.33329bb difference. Jam and fold remain in the comparison.

This compares one action against stored continuations. It is not a complete policy evaluation, a win rate, or a result against humans. No missing or zero-reach row is replaced by a convenience floor.

Weights are marginal hand-class reach. Multiplicity is six for pairs, four for suited hands and twelve for offsuit hands. The grids show one selected 100bb BTN–BB example. Hero’s conditional action mix fills each cell; entry weight separately indicates reach. Villain’s single color indicates membership, not its next action. Both ranges were checked against upstream reach products within 0.0001 export weight resolution.

Blocker counts enumerate compatible exact combinations with As5s or As9s, applying the supplied class weight to each suit combination. They omit folded-player card bunching. Response counts describe the exported policy; they are not pure-bluff EVs or proven optimal responses.

## Numerical limits

The current action graph, placeholder frequency encoding and EV units were checked against the application decoder and an independent implementation. No positive-reach hero action played at least 10% has an EV gap greater than 0.1bb. Six such discrepancies occur in later defender responses across five configurations, with a maximum of 0.2024bb. Those limit response-policy interpretation. At a 5% frequency threshold, the borderline 50bb UTG–BTN A5s four-bet is also flagged: 5.95% support with a 0.10162bb gap. This screen does not certify convergence.

Export resolution is 0.00001bb for EV and 0.0001 for frequency. Printed precision is not accuracy. The 0.02bb near-tie band and 0.05/0.1/0.2bb sensitivity screens are editorial conventions.

Price, position, ranges, menus and continuations change together across configurations. This is not a causal depth, position or blocker experiment. No ICM, bounty or multiway inference is made.

## Files

- selected-hands.json: 84 selected/reference rows, with action frequencies, EVs, weights and losses.
- node-summary.json: twelve comparisons and wheel-ace sensitivity screens.
- range-views.json: two selected explanatory range grids.
- range-composition.json: disjoint hand-class composition groups.
- blocker-counts.json: response masses before and after exact hero card removal.
- fourbet-loss.svg: twelve A5s losses on a shared 0–0.4bb scale.

The selected values allow the displayed arithmetic to be checked; they do not reproduce a full-tree solve. Full proprietary exports, catalogs, access paths and private ledgers are not published. The header is AI-generated editorial artwork. Numerical figures are deterministic.