# Multiway fold-equity response ledger

Created 2026-09-06 (Europe/Oslo) for GTO Gecko. Original synthetic teaching model.

## Question and assumptions

Can identical overall fold rates identify the chance that two defenders both fold? No. A acts before B; B's overall count pools its responses after A folds and after A calls. All tables contain 100 units of probability mass, **not 100 observed hands**. There is no sampling error, random seed, empirical fit or solver run.

Hero bets the remaining 50 chips on the river into a 100-chip pot. Both defenders cover Hero and may only fold or call. Hero receives zero at showdown on every called branch, including ties being excluded. No rake, fees, raises, subsequent betting, tournament payouts or bounty model. No cards, ranges or hand-selection model. It is an abstract response/payoff construction, not a recommendation or estimate of player behavior. Checking EV is not modeled.

## Files and reproduction

Place these seven files in one folder: `generate.py`, `verify.mjs`, `results.json`, `response-tables.csv`, `response-tree.svg`, `README.md`, `MANIFEST.sha256`.

Python 3.11+ and Node 22+; standard libraries only. From that folder:

```
python generate.py
node verify.mjs
```

The generator rewrites only JSON and CSV beside itself, so it also works when called using an absolute path from another folder. The verifier checks the numeric outputs independently and, when present, checks the file manifest. Generation should preserve the supplied bytes; the manifest is intentionally not regenerated by the public generator. If editing the example, a mismatch against the published manifest is expected.

## Construction

Fix A and B overall folds at 60/100. Let x be the count where both fold. In FF, FC, CF, CC order the table is `(x, 60-x, 60-x, x-20)`. Requiring nonnegative cells yields 20 ≤ x ≤ 60. The sensitivity data include all 41 integer x values; arbitrary fractional probability mass permits every real value in this interval. Featured x values 20, 36 and 60 illustrate the low endpoint, independence and high endpoint; they are illustrative choices, not measured typical values.

`P(B folds | A folds) = x/60`. Multiplying by 60/100 yields x/100. The chip payoff is +100 if FF (uncalled Hero bet returned), −50 otherwise; expected value is `(100*x - 50*(100-x))/100`. Featured results are −20, +4, +40 chips. Break-even is q=1/3; with A fold=3/5, B's conditional break-even rate is 5/9. These calculations are exact rational numbers. The diagram depicts only x=36 and has the same values as the article's complete table.

## Evidence and boundaries

- MIT 18.05 Spring 2022, Reading 3, conditional multiplication and independence: https://ocw.mit.edu/courses/18-05-introduction-to-probability-and-statistics-spring-2022/mit18_05_s22_class03-prep.pdf
- GTO Wizard, The Math of Multi-Street Bluffs, single-street pure-bluff accounting: https://blog.gtowizard.com/the-math-of-multistreet-bluffs/

Accessed 2026-09-06 Europe/Oslo. The sources support the mathematical method; they do not supply these ledgers. No competitor performance or product comparison is made.

The independent verifier reconstructs/validates the finite tables, identities, literal featured results and CSV parity. This verifies arithmetic under assumptions, not empirical accuracy, strategic optimality or the behavior of any GTO Solutions product. Sampling uncertainty in real branch counts remains a separate issue. A general HUD percentage cannot be transplanted to a specific hand or betting node without a matching definition, scenario and information set.

Original code, data and diagram created for GTO Gecko. External sources retain their own rights.