# Double-backdoor c-bet study Published by GTO Gecko / GTO Solutions AS. Study date: September 7, 2026. Article: https://gtogecko.com/blog/double-backdoor-cbet-study These are selected and aggregated views of the owned precomputed library. They are not a new solve, a complete database export, or an equilibrium certificate. Source solver version and achieved convergence were not present in the export. ## Model and selection Six-max no-limit Hold'em cash, BTN vs BB single-raised pot, 100bb before posting, 5.5bb flop pot, 97.5bb behind, 5% rake capped at 4bb, no ante or ICM. The pot and current app configuration correspond to a 2.5bb BTN open, BB call and folded 0.5bb SB. BB's root allows only check. BTN actions are bet 4.1bb, bet 1.8bb, check. Facing 1.8bb, BB can raise to 6.4bb, call or fold. Later raise options remain in the retrieved flop tree. Sizes outside the menu were not tested. The current report contained 1,755 board classes. Fifteen unpaired rainbow classes were chosen in advance, with no missing boards: - Discovery: Ks9h2d, Ks7h4d, KsJh3d, As7h4d, As9h2d, AsTh3d. - Validation: Qs8h3d, Qs6h2d, QsTh4d, Js7h2d, Js9h4d, Js8h3d, Ts6h3d, Ts8h2d, Ts7h4d. This is a purposive board matrix, not a random or representative sample. The held-out boards belong to different high-card families from discovery. No rule was changed after exploration to fit validation. For each board, eligible hands are suited with both hole-card ranks absent from the flop. Exclude made straights and any hand that can complete a straight with just one extra card. Include a hand only if two new ranks can complete a straight using both hole cards. All four suits must have positive reach and aligned action EVs. On a rainbow board three suited versions have a backdoor flush draw, while the missing-suit version does not. This yields 237 matching rank groups and 948 exact-combo cases. Cases from one board are not independent board observations. No eligible group had unequal reach across its four suits. ## Arithmetic and weights Raw amounts and EVs are converted using the current app's divide-by-10 rule. Probability vectors are stored to two decimal places. Normalize a valid vector by its sum when rounding produces 0.99 or 1.01; do not assign missing EVs zero. Displayed EV precision (0.001bb) is not achieved numerical accuracy. For an exact combo h: bet frequency = p(bet 4.1) + p(bet 1.8) bet edge = max(EV(bet 4.1), EV(bet 1.8)) - EV(check) action loss = max(all available action EVs) - EV(selected action) Average the three backdoor-flush versions of each rank group by their combo reach weights and compare with the missing-suit version. Average those matched differences within a board by group reach. Separately reported frequency means use eligible combo reach weights. All across-board means give each board equal weight. These are range-combo composition measures, not deal frequencies or hero-specific matchup probabilities. They omit folded-player joint card bunching and are never presented as bb/100 hands. Examples from selected-hands.json: - Ks9h2d, 8s7s: 1.570 - 1.553 = +0.017bb for best bet minus check. - Ks9h2d, 8c7c: 1.228 - 1.234 = -0.006bb for best bet minus check. - As7h4d, KdTd: 1.532 - 1.423 = 0.109bb small-bet loss; 1.532 - 1.188 = 0.344bb large-bet loss. - Ts7h4d, Ah2h: small-bet loss 1.261 - 1.146 = 0.115bb; large-bet loss 1.261 - 1.248 = 0.013bb. Local loss uses the retained continuations. Changing an entire policy changes reach and may invite opponent adaptation; this study does not evaluate that. The preregistered screening tolerances were 0.05bb for near-equal actions and 0.10bb for a material local gap, with 0.02/0.20bb sensitivity. They are practical study cutoffs, not solver error bounds or universal blunder thresholds. board-summary.csv includes per-board means, reach-weighted 95th percentile, maximum loss and fractions strictly above each threshold. Across nine held-out boards, mean small-bet loss is 0.022085bb; mean share over 0.05bb is 16.0038%, and over 0.10bb is 0.9259%. Quantiles describe variation among combos; they are not confidence intervals. No selected eligible small-bet loss exceeds 0.20bb. ## Range views The four HTML/SVG matrices show selected current BTN and BB ranges on two example boards. Hand-class action frequencies use legal exact-combo reach weights. Class entry weight is total reach divided by legal combinations after board removal, not divided by a fixed six/four/twelve when cards are blocked. Full cells preserve action mixes; hatched cells are outside the range. Grid averages intentionally hide individual suit differences, so consult the selected exact-combo table for the focal hands. BB's root check has probability one in this restricted tree. Its range at both response nodes equals the root range. Response mass is sum(reach × normalized action probability), not a fold probability conditional on a specific hero hand. Pair categories are mutually exclusive made-hand classifications; they do not measure equity, nut advantage, or the strength of draws. ## Verification and limits Independent code checked all 948 cases, 237 groups, 15 board means, the restricted root checks and response ranges, all four range matrices (676 cells), and the key EV arithmetic. Legal cards, blocker removal, array order, probability totals, pot accounting and units were also checked. Floating-point ties use a 1e-9 tolerance in the positive-edge diagnostic. The raw retrievals, hashes, frozen manifest, transformations and independent verification scripts remain in the owner's private reproducibility bundle. These public selections permit auditing the displayed arithmetic; they do not let a reader independently rerun the entire proprietary retrieval. No other positions, stacks, rake, two-tone/paired boards, multiway or tournament models were analyzed. No turn policy or fresh convergence study was performed. The observed betting-frequency mean increases on 15/15 boards. The change in bet-minus-check EV is positive on 13/15 boards, only +0.004871bb across all 15. Without achieved convergence, small EV differences should not be treated as precise evidence that a particular suit must take one action. Figures and range renderers were built from owned components and the selected study results; no third-party charts were copied.