# Paired-flop check-raise defense: selected study data

GTO Gecko, September 7, 2026. [Read the study](https://gtogecko.com/blog/paired-flop-check-raise-defense).

## Model and coverage

Six-max no-limit Hold’em cash, heads-up flop, 100bb before posting. The opener raises to 2.5bb; the small blind folds 0.5bb and the big blind calls. The flop pot is 5.5bb. Configured rake is 5%, capped at 4bb, with no ante.

The big blind is forced to check at the root. The opener can check, bet 1.8bb or bet 4.1bb. We follow the 1.8bb bet and a big-blind raise to 6.4bb total. The opener faces a 13.7bb pot and can fold, call 4.6bb or re-raise to 15.6bb total. Raise amounts are total street commitments, not increments.

The main matrix has 18 BTN boards: nine rank patterns, each in rainbow and two-tone form. Six board classes from three rank patterns are discovery; twelve classes from six other rank patterns are held out. Four CO boards provide a position check with the same available sizes. The sample is purposive, not representative or chance-weighted. This cash configuration is available only at 100bb in the inspected catalog.

Eligible hands have different hole ranks, neither matching a board rank, and no direct straight or flush draw. Backdoors remain. All hands share the board pair. The sample contains 6,905 eligible BTN hand-board cases and 1,148 CO cases. Suit variants from one board are not independent evidence.

## Calculations

For each exact hand, local fold loss is the highest legal action EV minus fold EV. Source amounts divide by 10 into big blinds. Source increments of 0.01 become 0.001bb; this is numerical resolution, not achieved accuracy. Fold EV is zero from the current decision; prior contributions are already spent.

Within a board, frequencies and losses use the acting player’s exported combo reach. Across boards, each class receives equal weight. Percentiles use a mixture in which each board has equal total mass and its hands have normalized within-board reach. They are not averages of individual board percentiles. These are marginal-range descriptions, not card-compatible occurrence frequencies or bb/100 hands.

The frozen shortcut screen required loss of at most 0.1bb for at least 95% of eligible mass. The qualifying share was 76.7624% across all 18 BTN boards and 77.4942% across the twelve held-out boards. No tested BTN or CO board passed the 95% screen. BTN mean loss was 0.089295bb; the mixture median was zero, the 90th percentile was 0.339bb, and the 95th percentile was 0.525bb. This screen is an editorial tolerance, not a universal mistake definition.

Probabilities and weights are exported to two decimals. Action vectors totaling 0.99 or 1.01 are normalized. BB raised reach is reconstructed as entering reach times raw raise probability. We compare it with BB’s next own decision range after an opponent re-raise; the opponent’s action probability is not multiplied into BB’s marginal reach.

Both inputs and the descendant weight are rounded. Validation therefore tests whether product and target rounding intervals overlap, rather than applying a target-only 0.005 tolerance. All comparisons are consistent within quantization. Exact latent ranges are unavailable.

## Files and meanings

- `board-summary.csv`: all 22 selected decision summaries, with units in the headers.
- `selected-hands.json`: illustrative exact combos, normalized frequencies, action EVs and fold losses. A hand can appear in more than one article comparison.
- `range-views.json`: two selected 169-class views on K♠ K♥ 2♦. BTN cells show actual defense mixtures. BB cells show retained raised reach with 100% membership, not a new decision. Entry weight equals weighted mass divided by legal combinations after board removal.
- `range-composition.json`: marginal composition before and after each player’s relevant action. These whole-range summaries are not conditioned on a selected hero hand.
- `current-strength.json`: current five-card rank against BB raising combinations compatible with each specified hero hand. Ahead, tied and behind are not showdown equity; no turn or river runouts are evaluated.
- `fold-cost.svg`: board shares losing more than 0.1bb by folding. Direct draws are excluded from the denominator. Different boards change ranges and eligibility, so differences are not causal texture effects.

## Limits

Solver name, version, achieved convergence and tree construction details are not provided. No frequency or EV has a verified equilibrium error bound. Tiny EV gaps do not establish precise indifference.

This study does not re-solve a restricted policy, evaluate an entire changed strategy, measure human errors, establish a win rate or compute causal blocker or flush-draw value. The article separates stored outputs, derived arithmetic and interpretation.

The figures are selected findings from the owner’s authorized library. Header artwork is original Gemini-generated editorial imagery and contains no numerical evidence. Complete proprietary trees, retrieval endpoints and internal working ledgers are excluded from publication.