# Backdoor draw turn map

Original GTO Gecko synthetic study, September 5, 2026. This study measures whether a five-card straight or flush exists among Hero's seven cards, not whether it is the best category or wins a showdown. A runout can belong to both events. Standard 52-card Hold'em, Hero 8h 7h, flop Kh 9c 2d. Codes: T=ten; c/d/h/s=clubs/diamonds/hearts/spades.

Only these five cards are conditioned on. All 47 remaining cards are uniformly distributed across unknown locations. Unknown opposing, folded and burn cards are integrated out, not assumed absent. There is no action-conditioned range or exposed opponent in the main enumeration. Each unordered two-card board completion is equally likely; both turn/river orders are possible. All 1,081 pairs are enumerated exactly; no Monte Carlo sampling or seed.

## Reproduce

Download all files in this folder into one directory. With Python 3 and Node.js 22 or later installed, run from that directory:

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

The generator overwrites only results.json, turns.csv and runouts.csv beside itself. It uses Python's standard library to inspect all five-card subsets. The independent Node verifier uses the fixture's rank-pair routes and suit counts, checks all published rows and totals, and checks the SHA-256 manifest when present. Run before changing files if checking the downloaded originals. A manifest mismatch after an intentional edit means the original package no longer matches; do not regenerate its manifest to hide an unexpected mismatch.

## Files and interpretation

- generate.py: complete generator and fixed input fixture.
- verify.mjs: independent fixture-specific verifier; not a general poker evaluator.
- results.json: exact counts and inputs.
- runouts.csv: all unordered pairs, with 0/1 flush, straight, intersection (both), and union (either) flags.
- turns.csv: all 47 turn cards; columns count qualifying river cards among 46 possibilities.
- turn-map.svg: six-group visual summary of turns.csv; equivalent values are in the article's HTML table.
- README.md: this method and limitations.
- MANIFEST.sha256: package integrity hashes for all other files.

## Hand check

Ten unknown hearts give C(10,2)=45 flush completions. The three distinct straight rank pairs are 5+6, 6+T and T+J, each with 4×4=16 combinations: 48 straight completions. Three all-heart pairs belong to both events, so union=45+48−3=90 of 1,081 (8.3256244%). Summing the turn ledger gives 180 successful ordered turn/river pairs out of 47×46=2,162, the same probability. Exactly 22 turn cards retain at least one route.

If two specific opponent cards are additionally exposed, the sample space changes to C(45,2)=990. For zero, one or two hearts among those cards, flush completion counts are C(10,2)=45, C(9,2)=36 or C(8,2)=28. These are separate conditional calculations, not rows of the main 47-card study. No claim is made about their straight counts or action-conditioned likelihood.

## Limits

One constructed flop and hole-card combination. No opponent range, betting, fold equity, position, stack, rake, payout, bounty, chance of seeing the river, policy recommendation or app output. Counts identify possible five-card formations, not clean winning outs; a higher opposing flush or straight may win. No completion of either target does not mean zero showdown equity: pairs or even unimproved cards could win against some opposing hands. Other flop structures need new enumeration. This is an off-table explanation, not live assistance.