# Folded-stack ICM calling study

Selected public evidence for **The Folded Stack That Changes Your All-In Call**, GTO Gecko, September 16, 2026.

## Setup and coverage

Three-player NLHE tournament final table, button folds, small blind chooses its largest effective wager, big blind folds or calls. Seven matched families, fourteen native library scenarios; all 2,366 BB hand-class rows have positive weight. The five held-out families contribute ten scenarios / 1,690 rows. The main discovery families have SB/BB stacks 40/20 and 10/30; held-out families have 10/20, 30/10, 20/30, 5/40 and 40/5.

Stacks in the tables are **after each 0.12 bb ante, before blinds**. Stored pre-ante stack vectors add 0.12 bb to every listed stack. Blinds are 0.5/1 bb. No bounty is modeled. The source catalogue describes a 1,000-entry tournament paying 150 places. Three players remain, with payouts 15.19%, 10.96%, 7.94% of the original pool, totaling 34.09%.

The main SB 40/BB 20 effective wager is a total 20 bb, putting BB all-in although SB retains 20 bb. BB's call adds 19 bb. Pot before calling is 21.36 bb and after calling 40.36 bb. Encoded calls are increments; raises are total wagers. Main smaller SB raise menus differ: to 3.5 bb with BTN 5 versus to 2.5 bb with BTN 10. Both permit fold, limp and effective shove. These are different native solutions, not an isolated causal test of a single feature. Changing BTN also changes total chips and earlier ranges.

## Units and weighting

Stored action EVs are gains relative to folding, measured in **percentage points of the remaining prize total**, not bb. Fold EV is 0. A stored 0.270 pp is 0.270% of the remaining prizes: $27 if those prizes total $10,000. Multiplying by 0.3409 expresses the value in percentage points of the original prize pool.

All BB hand-class weights in the selected nodes equal 1. Pair/suited/offsuit classes have 6/4/12 combinations, totaling 1,326. Calling share is sum(combinations × weight × call frequency) / sum(combinations × weight). It is a marginal combination summary, not an occurrence frequency with mutual card removal or folded-button card effects reconstructed.

SB reached-shove weights equal SB entering weight × shove frequency. The four displayed range views publish only the selected illustration. In SB membership grids,100% means membership; it is not the original probability of shoving. The separate entry weight retains that information. No equity advantage or blocker mechanism is asserted from these marginal grids.

## Local regret calculation

For a hand h in the shorter-BTN destination, V*(h)=max(fold EV,call EV). If q(h) is the larger-BTN call probability, imported-mix regret is V*(h) − [(1−q(h))×fold EV + q(h)×call EV]. Means and quantiles use BB marginal combination weights. This benchmark is the best stored action, not the destination's exported mixed policy. The native mixed policy can itself have positive local regret.

For the main SB 40/BB 20 pair, mean imported-mix regret is 0.0294828356 pp, versus approximately 0.00001021 pp for the destination's exported mix. The study does not evaluate a complete altered strategy, an adapting opponent, or tournament results. No bb/100 or real-money performance estimate follows.

The predeclared diagnostic screen asked whether at least 95% of the *larger-BTN calling mass* could call at a local action loss ≤0.1 pp in the shorter-BTN node. Five pairs fail; the two 5 bb confrontations pass. `pair-summary.json` preserves 0.05/0.1/0.2 pp sensitivity shares. This screen uses source-call weighting, unlike the all-hand imported-mix mean. Thresholds are study choices, not universal error definitions. The direction hypothesis is supported in all 7 pairs; individual hand frequencies need not move in the same direction.

## Independent static-ICM illustration

Top-three prizes are normalized to 100. With BTN post-ante stack B, SB 40 and BB 20, BB's terminal chip vectors after folding/winning/losing are respectively `[B,41.36,19]`, `[B,20,40.36]`, `[B,60.36,0]` in BTN/SB/BB order.

Enumerate finishing orders using each remaining player's chips divided by total remaining chips at each finish. Weight the corresponding prize by each order probability. A zero-chip eliminated player receives third prize. With no ties, break-even win probability is `(Ufold−Ulose)/(Uwin−Ulose)`.

| BTN | BB fold value (%) | BB win value (%) | BB lose value (%) | No-tie break-even |
| --- | ---: | ---: | ---: | ---: |
| 5 |34.12498409|39.28901025|23.29128777|67.72024161%|
|10 |32.84247110|38.35065088|23.29128777|63.42355422%|

These derived absolute outcome shares are distinct from the stored relative action EVs. They omit ties and future play; they do not establish any hand's equity against the native shove range. A 0.125 bb ante sensitivity gives 67.70866% and 63.41096%, preserving one-decimal interpretation. Catalogue 0.12 bb stacks independently reproduce the stored prehand equities within 0.000006 normalized percentage points.

## Files and limitations

- `node-summary.json`: 14 scenario summaries and native menus.
- `pair-summary.json`: 7 matched comparisons and local-regret sensitivity results.
- `selected-hands.json`: 22 illustrative hand/node rows, including the 65s frequency reversal.
- `range-views.json`: four illustrated BB/SB range views; entry weights, class multiplicity and conditional frequencies.
- `range-composition.json`: hand-group weighted combination totals.
- `calling-range-comparison.svg`: the aggregate chart, generated from the pair summary.

Frequencies are quantized in 0.4 percentage-point steps; EVs in 0.002 pp steps. Ten selected BB hand/action entries have positive support but regret above 0.02 pp, maximum 0.082 pp. Unknown original solver version, abstraction, achieved convergence and future-game settings limit interpretation. Full original HRC archives were unavailable in this run. The current packed library was independently decoded and all 2,366 BB rows, 676 range cells, chart values and primary calculations audited. Full proprietary trees and private reproduction remain outside this public bundle.

Original AI-generated header artwork is illustrative. The numerical chart, ranges and tables are deterministic representations of the selected study data. Public narrative sources discuss ICM concepts and limitations; they are not sources for this run's measured values.