# River top pair: method and selected data

Research date: September 28, 2026. Article: https://gtogecko.com/blog/river-top-pair-value-study

## Question and setup
After a flop bet/call and turn check/check, is the smallest available river value bet a cheap default for weak top pair when BTN is checked to?

Six-handed NLHE cash; all starting stacks of 100bb; no ante. BTN opens to 2.5bb, SB folds, BB calls (pot 5.5bb); BB's flop check is imposed by the upstream library. BTN bets 1.8bb and BB calls (pot 9.1bb). Both check the turn. River effective stack 95.7bb. The original catalogue specifies 5% rake/4bb cap; original details and convergence are unknown. New conditional rivers apply no additional rake and neutral opponent profiles. They preserve the library's reached turn ranges, filter each player's turn check, remove river-blocked combos, and then solve the river afresh. This does not re-solve the upstream strategy.

The river BB can lead/check. Hero EV and response comparisons use BB's endogenous checked range. IP bets 3.6/6.1/11.4/22.8/95.7bb plus check; OOP and IP raises/reraises remain available under the current preset. Actual rounded sizes, not ideal percentages, define payoffs. Six non-spade boards (first two discovery, last four held out) plus six same-rank spade boundaries. These are not all blanks: some complete straights.

## Definitions and screen
Eligible: positive serialized BTN reach, exactly one pair at showdown matching the highest board rank (T or above), hole-card kicker 9 or lower, five distinct board ranks. No made straight/flush or two-pair hand qualifies. 207 primary exact-hand rows, 129 held out, 207 boundary rows. Each scenario is one case; hands are not independent replications. No missing scenarios or eligible outputs.

For each exact hand, gap(action)=max available returned actionEV minus that actionEV. Check EV includes the existing river pot as winnings and deducts only future investment; same-node differences cancel the common EV origin. Best action is a comparison benchmark, not certified equilibrium value. The predeclared screen passes a scenario if 95% or more of eligible marginal BTN reach has smallest-bet gap <= 0.1bb. Near-equal display context 0.02bb; sensitivity thresholds 0.05/0.2bb. Floating-point comparisons allow 1e-9bb to preserve equality at exact printed boundaries.

Means/quantiles within a scenario use exact-combo marginal range weights. They are not occurrence frequencies against all compatible opponent cards; no deal-frequency estimate or bb/100 result is claimed. Range cells separately report mass, legal physical-combo count, entry weight (mass/legal count), and conditional action frequencies. A zero mass is absence, not a fold or unavailable value. Response-table percentages for an exact hero hand remove conflicting BB combinations first. Worse/tied/better compare all seven cards, not nominal pair rank.

## Quantization and sensitivity
Inputs and action probabilities are serialized to two decimals. No absent hero hand is injected. Positive weights rounding to zero are omitted; omissions and absolute changes are retained in scenario summaries. Normalization is allowed only within the per-component rounding bound. Primary input weights remain on their authentic reach scale. Four explicitly post-analysis sensitivities proportionally scale each player's positive input range to max 1 before two-decimal rounding: A94-2-7club near-tie, J73-2-6club largest primary gap, A94-2-7spade boundary, A85-2-Jclub borderline cutoff. This changes approximation resolution, not convergence settings. All four original screen verdicts persist.

## Accuracy limits and checks
Original solver name/version/full-tree abstraction and achieved convergence unavailable. Conditional settings request accuracy 0.05 and maximum 500 iterations, 8 threads; achieved accuracy, stopping reason and solver version are not returned. Request values are not error bounds. Every public hand/action retains the returned EV even if its mix is not exactly indifferent.

Our screen flags support>=5% with gap>0.1bb. Primary IP root output has28 flags outside the eligible target set and zero within it; boundary output has119 flags, including one eligible K-heart/6-spade on K8327spade (small bet 18%, gap 0.104bb). Other nodes may have further inconsistencies; no full-tree equilibrium certificate is claimed. Unknown convergence, rather than perfect-looking target rows, limits strategic conclusions.

An independent implementation reconstructed native turn reach using exact rational arithmetic, recomputed all 16 scenario/sensitivity summaries, and evaluated 8,303 checks plus 1,410 selected full bet-response trees with all raises/reraises. Main Jd4d returned versus rounded-policy evaluations differ by under0.01bb for the displayed central bets; large wagers elsewhere amplify output quantization. Check discrepancies are not all proved to be rounding alone. All 676 displayed class cells, 48 selected/sensitivity EVs, 12 chart bars and nine response summaries independently matched. These checks establish arithmetic and stated output semantics, not convergence or real-player behavior.

## Files
- board-summary.json: twelve aggregate cases, counts, gaps, quantiles, sensitivity thresholds, flags and input-rounding summaries.
- sensitivity-summary.json: four selected input-quantization reruns.
- selected-hands.json: five displayed exact hands and available sensitivity rows, all six frequencies and EVs.
- range-views.json: four illustrative 169-class views, with true conditional mixtures and entry weights.
- range-composition.json: main and boundary both-player category masses/shares.
- response-composition.json: exact-hero-conditioned fold/call/raise shares and worse/tied/better masses at three sizes.
- local-ev-comparison.svg: primary scenario mean gaps, same 0–1.4bb scale.

These are selected authorized study findings. Full proprietary input trees and access details remain private. Header artwork is original AI-generated imagery, not a solver screenshot.

## Source context
- https://blog.gtowizard.com/river-refinements-when-your-opponent-wont-check-raise/ — different 40bb MTT comparison; illustrates response-model sensitivity, not replication of these cash outputs.
- https://piosolver.com/docs/viewer/numbers_in_piosolver/ — meanings of EV/frequency/matchups, not certification of this service.
- https://apps.apple.com/us/app/gto-gecko-poker-study/id6738748434 — current US educational study-product listing, version 1.0.142, checked September 28. Availability varies by plan; custom research nodes are not promised in the app.