The 8 Pillars of Poker Strategy
and what our solver says about each

Ranges are the foundation of everything. Before you pick a bet size, before you decide whether to c-bet, before you even look at the flop — the shape of your range versus your opponent's range has already done most of the work.

This pillar covers the seven theories that define how equity and ranges operate in 6-max cash. Four are confirmed by our solver verification, one is partially confirmed (the EV layer is blocked by a known infrastructure issue), and two sit outside our model's measurement capability entirely — they rely on equity realization, a metric our testing API does not expose. Where we can measure, the data is clean. Where we cannot, we say so.

Measurement conditions: 6-max NL, CO vs BB SRP, 100bb effective unless noted otherwise.

Your range advantage carries forward from preflop to flop

The in-position preflop raiser starts with a stronger range than the blind defender. That advantage does not disappear on the flop — it shapes how often and how large you bet.

The clearest way to see this is to compare c-bet frequencies across board textures. Same position, same stack depth, same preflop action — only the board changes.

CO flop c-bet frequency by board class. 100bb effective · 6-max NL · CO vs BB SRP · flop c-bet by board class

Source: cash-baselines.md §5 Table 3 — v1.5.0 fresh-server refresh

On K72r, CO c-bets 87.0 of the time. On K94ss monotone, that drops to 34.6. The board texture determines how much of your preflop range advantage you can actually leverage — dry high-card boards preserve it; monotone boards erase it.

The pattern holds across opener positions. BTN c-bets K72r at 87.6 and UTG at 90.1. Range advantage is positional — it starts preflop and flows into postflop action.

What this means: if you are the preflop raiser and the board is dry and high, you should be c-betting most of your range. If the board is low-connected or monotone, your range advantage is gone and a high c-bet frequency is a leak.

Nut advantage is not the same thing as equity advantage

Having more equity overall is one thing. Having more of the strongest hands is something else. The solver treats them differently — and so should you.

On AK6r, CO's range contains all the strongest possible hands: AA, KK, AK, sets. BB's range is capped after just defending preflop. The solver responds by using the largest available bet size (bet8.2, roughly a pot-sized overbet) at 87.0% — wait, let's look at the specific overbet data.

CO tested overbet frequency on five AK-X rainbow boards (X = low side card), all at 100bb:

CO overbet (bet8.2) frequency on AK-X rainbow boards. 100bb effective · 6-max NL · CO vs BB SRP · AK-X rainbow boards

Source: theory-foundation.md A2-overbet-specific — batches R_cash_a2_sizing_overbet_xval.json + R_cash_a2_overbet_ext.json

Every board in the panel hits the same band: roughly 14–19% overbet. The nut advantage on these boards is so lopsided that the solver polarizes into a massive overbet with its strongest hands and bluffs.

Now compare: A72r — an Ace-high board but without the King — shows 0% overbet. Same Ace, no King, no two-broadway structure: the nut-advantage conditions aren't met.

On the other side, look at 543. Premium hands (AA, KK, AK) check 86.3 of the time on 654. When BB holds the nut advantage, CO's best hands have nothing to bet into.

What this means: on AK-type rainbow boards where you opened from CO, the solver overbets 14–19% of the time. On boards where the nuts shift toward BB (like 654 or 543), your premium hands check. Look at who holds the nuts, not just who has more equity.

Range composition determines how outlier hands perform

When your range is weak overall, the rare strong hands within it over-realize their equity. When your range is nut-heavy, the weak hands in it benefit from fold equity generated by the strong ones. Medium-strength hands under-realize most reliably across all range compositions.

This is a literature-backed theory that we cannot directly verify with our current model infrastructure. The metric it depends on — equity realization — is not exposed by the testing API. The theory remains in the catalog because it's well-grounded in published poker theory, but we have no model data to confirm or deny it.

What this means: don't evaluate a specific hand without considering what range it's sitting in. A set in BB's wide defending range is more valuable relative to that range than the same set in UTG's tight opening range.

When you check, your range gets weaker

When you check in a spot where you would bet your strongest hands, your remaining range becomes condensed — the nuts are less likely, medium-strength hands are more likely. Your opponent should recognize this and attack.

The delayed c-bet data shows this clearly. After CO checks back the flop on a brick turn:

CO delayed c-bet rate after flop check-back, brick turn. 100bb effective · 6-max NL · CO vs BB SRP · flop check → turn decision

Source: cash-baselines.md §5 Table 11

On most boards, the delayed c-bet rate is lower than the flop c-bet rate. That makes sense: the checker's range is weaker than the full opener's range. But the model still bets a high fraction of its condensed range on the turn — it finds thin value even after checking.

On two boards (Q83r and 654) the delayed c-bet rate actually exceeds the flop c-bet rate. These are boards where the flop check-back range includes slow-played strong hands. The model fires hard with those on a brick turn.

Both patterns confirm the condensing mechanism: after a check, the range narrows. Whether it narrows to "thin value" or "slow-played monsters" depends on the board — but in both cases, the opponent should probe or attack more aggressively.

What this means: if you are facing a flop check from the preflop raiser, probe the turn. Their range is condensed — the nuts are missing, and medium-strength hands are overrepresented. On dynamic boards, this is where your bluffs get the most fold equity.

Equity realization depends on everything

How much of your raw equity you actually convert into chips — your equity realization (EQR) — is not fixed. It depends on your position, the board, your range composition, and how well you and your opponent play later streets.

The same hand realizes very different equity in different spots. This theory is foundational to modern poker thinking, but like A3 above, we cannot test it directly because our model API does not expose EQR as a queryable metric.

The theory rests on a single literature source. We retain it in the catalog because its logic is sound and it underpins several other theories in this pillar, but we have no model data to confirm or deny the specific claims.

What this means: a hand with raw equity in BB will not realize as much of it as the same hand in position. When estimating whether to continue, discount your equity for being out of position — especially on dynamic boards where later-street play matters most.

Betting for protection works, but only in narrow spots

"Bet for protection" is real — but it applies far more narrowly than most players think. It only works when two conditions are simultaneously true: the bet extracts value from worse hands AND the bet folds out hands with live equity.

The frequency layer confirms this pattern. Look at how often premium hands (AA, KK, AK) check on different board textures:

Premium check % by board texture. 100bb effective · 6-max NL · CO vs BB SRP

Source: cash-baselines.md §5 Table 7 — per-class check % by board, CO flop.

On K72r, premium hands check only 8.5 of the time — they bet almost always because both conditions are met. But on 654, premium hands check 86.3 of the time. BB holds more of the best hands on that board. The bet-for-protection conditions fail and the solver checks.

There is a deeper finding here from a separate test on a wet two-tone board (8h6d4h): AA c-bets only 0.3% of the time on that board — essentially never. On a drier version of the same board (8h6d2c), AA c-bets 84.0%. The gap is massive. And the rank ordering is inverted: stronger overpairs check more on the wet board (AA checks most, TT checks least), which is the exact opposite of the common amateur intuition that "always bet big overpairs for protection."

This theory is rated as partially verified. The frequency-layer pattern — premium hands check more on protection-adverse textures — is confirmed cleanly. The EV-magnitude claims (exactly how many big blinds protection costs you) require a model capability that is currently blocked by a known infrastructure issue. (See Research notes for details.)

What this means: stop auto-betting overpairs on wet boards. On 654 or 8h6d4h, premium hands check the vast majority of the time. The solver is telling you that protection is a narrow category, not a default.

Position always helps — no exceptions

For any given hand, playing it in position realizes more equity than playing it out of position. No hand class is exempted.

The structural shadow of this is visible in preflop VPIP by position:

Preflop VPIP by position. 6-max Cash 100bb

Source: cash-baselines.md §5 Table 1 — preflop VPIP by position, 6-max Cash 100bb

The progression is strictly monotone: UTG plays the fewest hands, BTN plays the most. The same pattern holds at 20bb (UTG 15.6, MP 22.4, CO 24.4, BTN 30.6). The wider-in-position principle is stack-depth invariant.

What this means: if you're opening the same percentage of hands from UTG and CO, you're making a basic error. The solver opens nearly twice as wide on BTN as UTG because position doubles the value of marginal hands.

Defense math is the foundation — and it is almost never exact

Every bet you make forces your opponent to decide: call, raise, or fold. And every fold they make gives your bluffs a free win. The math that governs this tradeoff — minimum defense frequency (MDF — the fold rate that makes bluffs break even) — is one of the oldest results in game theory.

It is also one of the most misapplied.

MDF tells you the theoretical floor: how often the defender must continue to prevent zero-equity bluffs from printing money. The solver respects that floor, but the actual frequencies it chooses depend on board texture, bluff equity, range composition, and whether the bettor even has enough bluff combos to justify a big size. Those adjustments are the five theories in this pillar, and all five confirmed cleanly in our solver verification — no contradictions, no partial results.

What follows is the data behind each one, starting from the formula itself and working outward into the places where reality forces the formula to bend.

Measurement conditions: 6-max NL, CO vs BB SRP, 100bb effective unless noted otherwise.

The MDF baseline: how often you have to defend

The formula is simple: MDF = pot / (bet + pot). If you fold more than the complement of that number, your opponent's bluffs auto-profit. That is the baseline. The question is whether the solver actually plays to it.

It does — with a consistent downward correction.

BB defense frequency vs CO c-bet, by bet size — mean across 8 board textures. 100bb effective · 6-max NL · BB facing CO c-bet · flop defense rates

Source: cash-baselines.md §Raw Data Tables — Table 5

Two patterns jump out. First, defense decreases monotonically as bet size grows — exactly the shape MDF predicts. Second, the solver under-defends MDF by roughly -6.3pp at the worst point (50% pot) and converges to MDF exactly at 150% pot.

That gap is not a mistake. The MDF formula assumes the bettor's bluffs have zero equity when they check back. In practice, most bluffs have some showdown value — a gutshot, a backdoor draw, even just two live overcards. When bluffs have checkback equity, the defender does not need to call as often to make the bettor indifferent. At 150% pot, where bluff equity matters less (the bettor's commitment is so large that giving up is relatively cheap), defense snaps right to the theoretical baseline.

What this means in practice: Do not memorize MDF percentages and rigidly defend to them. The solver consistently defends less than MDF at mid-range bet sizes. If you are folding a little more than the formula says on the flop, you may already be closer to equilibrium than you think.

The solver breaks from MDF — and texture is the reason

The MDF table above is a mean across eight board textures. But the individual boards tell a much sharper story.

At a 100% pot bet, BB's defense rate spans an 18 percentage-point range across the tested textures. Connected boards like T98 and 654 sit at the top of that range — BB defends more often because flush draws, straight draws, and combo draws give the defender genuine equity. Dry high-card boards like K72r and A94r sit at the bottom — BB's range is capped, and continuing means calling with weak hands into a strong c-betting range.

The literature identifies four common reasons to defend less than MDF: being out of position, bluffs having equity (the gap we just saw), the aggressor having a profitable check-back, and the villain's range being value-heavy. And four reasons to defend more: draw-heavy boards, in-position defense against equity-retaining draws, chop-prone boards, and bluff-heavy villains.

Our solver data confirms the texture axis specifically. On monotone K94ss, BB defends 45.5 at 100% pot on average — but the monotone board pushes defense higher than that average because BB's range is loaded with flush draws that retain equity. The non-monotonic defense curve on K94ss is one of the clearest texture-driven deviations in the panel.

What this means in practice: When you face a pot-sized bet on a connected or flush-draw-heavy board, do not snap-fold your draws. The solver defends substantially above MDF on those textures. The correct adjustment is board-specific, not size-specific.

Bluffs have equity — and that changes the math

Standard MDF derivation assumes the bluffer has zero equity if they check. That is almost never true on the flop. Even a gutshot has four outs. Two overcards can have six. The correct indifference point is between betting and checking — not between betting and folding — and that shifts the required call frequency downward.

The solver confirms this mechanically: on A94r, the small suited-Ax hands (A5s, A3s, A2s) — exactly the kind of combo that has both some showdown value and some draw potential — mix between calling and folding at equilibrium. A5s takes its top action at 78.25%, with 2 actions above 5%. A3s mixes at 82.56% top action. A2s at 80.22%.

Close-spot mixing on A♠9♣4♦ rainbow, BB facing CO 33% c-bet. 100bb effective · 6-max NL · CO vs BB SRP · specific spot per row

Source: cash-baselines.md §Raw Data Tables — Table 17

Three out of four combos mix between actions. That mixing is the solver's signature of near-indifference — these hands are right at the boundary where calling and folding produce similar outcomes. A4s is the outlier: it plays nearly pure (top action 93.69%), suggesting its specific kicker tips the balance to one side.

What this means in practice: The indifference framework tells you why small suited-Ax hands on A-high boards feel like coin flips between call and fold. They are. The solver mixes because the EV difference is tiny. If you are agonizing over A3s facing a third-pot bet on A94r, the solver is agonizing too.

You cannot overbet just because you have the nuts

Having the nut advantage does not automatically mean you can use a massive bet size. The overbet (bet8.2, roughly 150% pot in solver notation) only appears when the range also contains enough natural bluff candidates to balance the value. Without bluffs, the defender simply folds everything that is not the nuts, and the overbet accomplishes nothing.

The solver demonstrates this precisely. On AK6r — where CO has both the nut advantage (AA, KK, AK, sets) and a natural pool of overbet-worthy bluffs (QJs, QTs) — the premium hand class uses bet8.2 at 8.5% check rate on K72r but only 86.3% check rate on 654 ... wait, let me frame this differently through the sizing lens.

On 543, where CO's range contains almost no natural straights or two-pair combos (those belong to BB's range), the premium class checks 86.3% of the time on 654. On K72r, premium checks only 8.5%. The same strength of hand — AA, KK, AK — plays completely differently depending on whether the range around it supports a betting strategy.

The overbet is the sharpest evidence. Across the six boards tested in the per-combo sizing panel, bet8.2 usage is negligible on every board except AK6r. That board is the only one where CO has both absolute nut advantage AND a deep pool of bluff combos (offsuit broadway hands that carry overbet bluffs at non-trivial frequency). On 543, K72r, KK5, 772, and T98, overbet usage stays below 1%.

What this means in practice: Before choosing a big bet size, count your bluffs. If you are on a board where your range has few natural semi-bluffs (low-connected boards when you are the preflop raiser, paired boards with few draws), the solver sizes small or checks — even with top set.

The bet range must contain the nuts — or it gets raised off the board

This is the mirror of B4. If the bet range is missing nut combos, the opponent can raise cheaply and force folds. The solver prevents this by including sets, two-pair, and straights in the betting range on coordinated boards — and by checking the nuts on boards where the surrounding range cannot support a balanced bet.

The starkest example is the premium class (AA, KK, AK) on 654. On that board, CO's opening range has very few straights, two-pair, or set combos. BB holds the nut advantage. The solver responds by checking premium hands 86.3% of the time. That is not a slow-play — it is the solver acknowledging that a bet range without supporting nut combos gets punished.

Premium (AA/KK/AK) check % by board texture, CO flop c-bet. 100bb effective · 6-max NL · CO vs BB SRP · bet-range nut-combo share

Source: cash-baselines.md §Raw Data Tables — Table 7

The gradient tells the story. On K72r, where CO holds the nut advantage, premium hands bet nearly always — check rate just 8.5%. On K94ss (monotone, flush draws threaten CO's range), premium checks rise to 54.4%. On 654 (BB has nut advantage), premium checks hit 86.3%. The solver is protecting the entire betting range from being raise-exploited by pulling the nuts into the checking range on boards where the bet range would be unbalanced.

What this means in practice: When you hold the nuts on a board where your overall range is weak, check more often than your instincts say. Betting the nuts into a range that cannot support balanced bluffs is worse than checking it — the solver confirms this with premium check rates above 85% on the boards where the bet range would be hollow.

Position is the single biggest structural edge in poker. You already know this — you open wider from the button, tighter from under the gun. But "play more hands in position" is a bumper sticker, not a strategy. The interesting questions are: how much wider? What happens to the raiser's range advantage when the board changes? And does position always help, or are there spots where it stops mattering?

We tested five theories about how position and range composition interact in 6-max cash. Four confirmed cleanly against our solver. The fifth — that a tighter opener always has a bigger postflop edge — turned out to be true on some boards and flatly wrong on others, earning a partial verdict. The pattern that emerges is sharper and more texture-dependent than most heuristics assume.

Measurement conditions: 6-max NL, CO vs BB SRP, 100bb effective unless noted otherwise.

Later position means wider ranges — no exceptions

The later you sit, the more hands you play. The solver confirms this with zero inversions.

Preflop VPIP by position. 100bb effective · 6-max NL · preflop open frequencies by position

Source: cash-baselines.md Table 1 (batches_oq5_cross_position, 2026-04-10)

The jump from CO to BTN is the largest single step — more than double the UTG-to-MP or MP-to-CO gaps. At 43.3%, BTN plays nearly half its hands. UTG plays fewer than one in five.

The pattern holds at short stacks too. At 20bb, the ordering is identical (UTG 15.6, MP 22.4, CO 24.4, BTN 30.6), just compressed tighter across the board.

Every hand realizes less equity out of position

Equity realization (EQR — the share of your raw equity you actually convert to expected value) is strictly lower out of position. The same hand played from BB realizes less than from BTN, on the same board, against the same range.

We cannot measure EQR directly — our API does not expose that metric. But the structural shadow is clean: VPIP grows monotonically in position, which means the solver is telling you that later-position hands earn enough to justify wider play. If EQR were flat across positions, you would not see this widening.

What this means: position is not just about "seeing what your opponent does first." It structurally changes how much value your hand captures. The same A9s from UTG and from BTN will have the same raw equity — but BTN converts more of it into profit, which is why the solver opens it from BTN and often folds it from UTG.

Your opponent's opening position shapes your postflop strategy — but only on certain boards

The conventional wisdom goes like this: a tighter opener has a stronger range, which means a larger postflop advantage, which means they can c-bet more aggressively. UTG opens fewer hands → UTG has more overpairs on the flop → UTG c-bets more.

That logic works on some boards. On others, it inverts completely. Here is the board-by-board picture.

Flop c-bet % by opener position. 100bb effective · 6-max NL · various openers vs BB SRP · flop cbet by opener position

Source: cash-baselines.md Table 3 (batches_cash_cross_opener, 2026-04-16 fresh-server refresh). Extension boards: 987r −9.2pp, J87r −0.1pp, T76r +11.8pp.

The BTN − UTG column tells the story. A negative gap means UTG (the tightest opener) c-bets more — the conventional wisdom holds. A positive gap means BTN (the widest opener) c-bets more — the conventional wisdom fails.

The data splits into four zones:

1. Holds strongly: T98 (UTG leads by 76.2 vs BTN 64.9, gap −18.6pp). UTG's tight range is packed with high pairs (TT, 99, 88, 77) that match the board. Extension confirms: 987r also holds at −9.2pp.
2. Flat / neutral: K72r, KK5, 883 (gaps within ±1pp). Opener identity barely matters on these dry or paired textures.
3. Reverses moderately: A94r (+16.6pp), Q83r (+18.9pp). BTN's wider range finds more thin-value and bluffs on these textures than UTG's overpair-heavy range.
4. Reverses strongly: 654 (+47.7pp). UTG c-bets only 19.7% here — BB has more straights, two-pair, and sets against UTG's tight range. BTN c-bets 69.8%.

One texture category deserves its own mention: monotone boards are neutralized. On K94ss, all four openers converge to a narrow band — UTG 34.4%, MP 35.4%, CO 34.6%, BTN 33.0%. The max spread is just 4.1pp. The monotone texture overrides the range-composition effect entirely.

What this means: do not assume "tight opener = always more aggressive postflop." Your adjustment to the opener's position should depend on the flop texture. Against UTG on T98, give them extra credit — they c-bet harder there than BTN does. Against UTG on 654, you can defend wider than usual — their range is at its weakest.

Blind-vs-blind is its own game

SB vs BB pots look nothing like the rest of 6-max. SB has the unique property of being out of position preflop but in position postflop (since BB acts first). The solver exploits this by opening extremely wide — then playing a sharply texture-dependent game after the flop.

SB postflop c-bet frequency by board. 100bb effective · 6-max NL · SB vs BB heads-up blind-vs-blind

Source: cash-baselines.md Table 10 (batches_cash_sb_postflop, 2026-04-12)

Two numbers jump out. In a single-raised pot, SB c-bets 654 only 3.9% of the time. Compare that to CO's 55.5% on the same board. SB is uniquely constrained: BB's wide defending range holds plenty of made hands on low-connected boards, and SB's own range — although wide — lacks the nut combos to push through.

The monotone board tells a similar story. SB c-bets K94ss at just 9.8% in a SRP. CO c-bets the same board at 34.6%.

In 3-bet pots the dynamic inverts. SB's 3-bet range is tight and high-card heavy, so on K72r the SB-as-3-bettor fires 99.7% and on KK5 it fires 98.6% — near-total range bets. But on T98, where the tight 3-bet range misses, c-bet collapses to 5.8%.

What this means: treat SB vs BB as a separate game with its own frequencies. SB's SRP c-betting is extremely texture-sensitive — far more so than CO's. On boards where SB's range is weak, the solver nearly gives up. On boards where it's strong, it fires almost everything.

Equity advantage determines c-bet frequency across textures

The highest-leverage insight in position theory is also the simplest: the more your range outperforms your opponent's on a given board, the more often you should bet.

The solver confirms this with a clean gradient across textures.

CO flop c-bet frequency by board texture class. 100bb effective · 6-max NL · CO vs BB SRP · flop c-bet by equity-advantage class

Source: cash-baselines.md Table 3 (batches_cash_cross_opener, 2026-04-16 fresh-server refresh)

The ordering is clean: dry high-card boards sit at the top, monotone boards sit at the bottom. CO c-bets K72r at 87.0% — nearly every hand. On K94ss the rate drops to 34.6%.

What this means: your c-bet frequency should swing dramatically with the board, not stay anchored at a single default. The solver's range goes from near-87.0% on dry K-high down to 34.6% on monotone — that is a massive texture-driven shift.

Bet sizing is where most players leak the most chips per decision. Not because they pick the wrong action — bet or check — but because they pick the wrong amount.

The solver's sizing logic rests on a small set of principles, and we tested six of them against our model. Four confirmed cleanly. One confirmed with a refinement that changes how you think about shallow stacks. One confirmed partially — the direction is right, but the specific mechanism is harder to isolate than we expected. Here is what the data says.

Measurement conditions: 6-max NL, CO vs BB SRP, 43.3bb effective unless the table sweeps stack depth.

The wetness parabola — small on dry, bigger on wet, then small again on very wet

If you only remember one thing about c-bet sizing, make it this: the relationship between board wetness and bet size is not a straight line. It is a curve.

On dry boards like K72r, the solver uses small bets almost exclusively. On connected boards like T98, sizing steps up. But once you reach monotone textures like K94ss, sizing drops back down — or the solver checks entirely.

Here is how that plays out across seven hand classes and six flop textures.

CO flop c-bet check frequency by hand class and board. Higher numbers mean CO checks more often. 43.3bb effective · 6-max NL · CO vs BB SRP

Source: cash-baselines.md §5 Table 7 — per-class check % by board (Cash CO flop)

Look at the K94ss column. Every hand class checks at a dramatically higher rate on that monotone board than on dry K72r — even suited broadway, which holds flush draws. And on 654, the premium class (AA/KK/AK) checks 86.3% of the time. That is almost never betting.

The two-tone comparison shows the same pattern from the aggregate side:

Rainbow vs two-tone c-bet frequency, same ranks and positions. 43.3bb effective · 6-max NL · CO vs BB SRP

Source: cash-baselines.md §5 Table 8 — Rainbow vs two-tone CO flop cbet (Cash)

Adding a flush-draw texture drops the c-bet rate by -8.5 to -18.7 percentage points. The drop is largest on the most connected board (T98).

You cannot bet big unless two things are true

Most players know to bet big when they have a strong range. Fewer realize range strength is only half of what the solver looks at — and getting the second half wrong is the main reason sizing mistakes compound across streets.

The solver bets big when both of these are true:

1. You have more strong hands than your opponent (nut advantage)
2. Your opponent has hands that will actually fold (fold equity)

Miss either one and big bets stop working. The c-bet frequency table across opener positions shows what this looks like in practice:

C-bet frequency by opener position and flop texture. 43.3bb effective · 6-max NL · SRP vs BB

Source: cash-baselines.md §5 Table 3 — CO flop c-bet % by board × opener (SRP, vs BB)

K72r rainbow, any opener. Both conditions met. The opener has all the overpairs, all the Kx top-pair combos. BB's range is capped and full of hands that will fold. Every opener c-bets this board at 87.0% or higher.

654 rainbow, UTG opens. UTG barely has any straight or two-pair combos — but BB does. Nut advantage flips to BB. UTG c-bets just 19.7%. Same UTG range, same stack depth — completely different frequency because the board broke one of the two conditions.

K94ss monotone, all openers. The opener has nut advantage (Kx top pair, overpairs), but BB's range is loaded with flush draws that will never fold to a bet. Fold equity is gone. All four openers converge to a narrow band — UTG 34.4, CO 34.6, BTN 33.0 — regardless of how tight or wide they opened.

Flop overbets are vanishingly rare — and bluffs drive them

Here is a finding that breaks a common intuition: in GTO play, your strongest hands do not use the biggest bet size. Bluffs do.

We pulled per-hand sizing data across six boards. On every board we tested, value hands and bluffs converge on the same dominant bet size — or bluffs use a bigger size than value. The nuts never uses the largest size.

K72r: KK (top set, the nuts) uses bet1.8 at 8.5% check rate — meaning it bets over 8.8% of the time, but mostly at the smaller size. Bluffs like QJs and JTs use bet2.8 at the larger size almost exclusively.

AK6r: The most extreme case. QJs and QTs (bluffs) use the overbet bet8.2 at frequencies of over half the time. AA uses bet1.8 at a near-pure frequency. The nuts traps or sizes small. The bluffs go huge.

This pattern held on five out of six boards: value and bluffs share the same dominant bet size. On zero out of six boards did value hands use a larger size than bluffs. On AK6r, bluffs used a bigger size than value.

Hand examples from cash-baselines.md D7 verdict worksheet (per-hand sizing via extract_per_hand_class on 6 boards, CO cbet, 100bb)

Multi-street sizing grows geometrically

When the solver does bet, it builds the pot across streets in a consistent pattern: each street's bet is roughly two to three times the previous street's bet.

We traced three full runouts from flop to river to see how the pot grows:

River bet behavior across three runouts. 43.3bb effective · 6-max NL · CO vs BB SRP · river · specific boards per row

Source: cash-baselines.md §5 Table 15 — River bet behavior (Cash, 3 runouts)

On the K72r runout, the sequence runs approximately 3bb on the flop, 7bb on the turn, and the dominant river size is bet19.1. That is a roughly three-fold step each street: 3 → 7 → 19. The T98 runout goes further — bet63.8 appears at 77.1% frequency on a dynamic board where big polar bets make sense.

The size concentration also shifts street to street. Flop bets converge on a single small size. Turn bets step up. River sizing fans out — the solver uses multiple river sizes depending on board dynamics and hand strength, but all within a geometric envelope.

Hand examples from cash-baselines.md §5 Table 15 and D1 verdict worksheet (geometric progression on K72r → 2d → 5h)

Shallow stacks compress sizing and shift frequency

At short stacks, betting gets simpler. The solver compresses toward a single small bet size or all-in — the geometric multi-street plan collapses because there are not enough big blinds left to build across three streets.

But the frequency story is not what you might expect. The solver does not simply bet more often at shallow stacks. It actually peaks in the middle.

Here is what CO's c-bet looks like on K72r across four tested stack depths:

CO c-bet frequency on K72r by stack depth. 6-max NL · CO vs BB SRP · K72r · stack depth sweeps rows

Source: cash-baselines.md §4 D5 verdict worksheet and §5 Table 2 cross-depth data. Rows at 75bb and 150bb are untested at this board.

The pattern is non-monotonic. At 20bb the solver bets 87.0% — that is the 100bb figure; at 20bb the figure is 84.4%, essentially the same. But at 200bb the rate drops to 78.8%. Deepening stacks give the solver more reason to check and develop multi-street plans rather than firing immediately.

The preflop picture reinforces this. BTN opens 43.3% at 100bb but tightens to 30.6% at 20bb — a drop of over twelve percentage points. Shallow stacks tighten ranges and compress sizing to a single small bet plus shove.

River sizing splits by blockers

On boards with flush or pairing potential, a single card in your hand can change which bet size is correct.

We tested this on T98ss (T♠9♠8♣) — a two-tone connected flop where flush draws dominate BB's calling range. The solver's behavior splits sharply depending on whether CO holds the blocker card.

AA with the A♠: c-bets 90.8% of the time. Holding the nut flush-draw blocker depletes BB's strongest drawing combos — the bet works more often.

AA without the A♠: c-bets just 33.6% of the time. A massive frequency shift — over 57 percentage points — driven entirely by one card.

The sizing split is equally stark. QQ with the Q♠ uses the smaller bet1.8 at 84.4%. QQ without the Q♠ uses the larger bet2.8 at 70.4%. JJ with the J♠ splits roughly evenly between sizes. JJ without the J♠ uses bet2.8 at 92.1%.

The direction is consistent: holding the blocker → smaller sizing profitable. Not holding the blocker → larger sizing needed.

Hand examples from cash-baselines.md D6 verdict worksheet (per-combo suit filtering on T98ss, CO cbet, 100bb)

This pattern was strong on T98ss where flush draws dominate the calling range. On A94ss, the blocker effect was much weaker — the board's range dynamics (BB holds many Ax combos) overrode the blocker logic. So the finding is confirmed on connected two-tone boards but board-dependent on A-high two-tone boards.

The board changes everything.

Same position, same stack depth, same preflop action — swap the three flop cards and the solver rewrites its entire plan. CO c-bets K72r at 87.0% and K94ss at 34.6%. That is a single texture change flipping the strategy from "bet almost everything" to "check two thirds of the time."

This pillar unpacks eight theories about how board texture drives solver behavior. All eight are confirmed by our model verification — the strongest clean-sweep result in the entire catalog. The data covers dry, paired, connected, monotone, two-tone, and low-connected boards at multiple positions and stack depths. What follows is what the solver actually does on each texture class, why it does it, and what you should take away.

Measurement conditions: 6-max NL, CO vs BB SRP, 100bb effective unless noted otherwise.

Dry high-card boards favor the preflop raiser

On a dry K-high or A-high rainbow flop, the preflop raiser owns the board. The solver knows it and fires accordingly.

CO flop c-bet frequency by board and opener position. 100bb effective · 6-max NL · CO vs BB SRP · flop cbet by board

Source: cash-baselines.md Table 3 (v1.5.0 fresh-server refresh)

K72r is the textbook raiser-favorable flop. CO holds AA, KK, AK, every broadway that connects — BB has a capped range after just calling preflop. The solver c-bets 87.0% from CO. From UTG, the tightest seat, the rate climbs even higher to 90.1%.

A-high boards (A94r at 66.3%) are softer. The solver still bets the majority of its range, but BB's Ace-x defends hold more equity here, so the frequency drops compared to K-high dry.

Paired boards reduce BB's defense width

When the board pairs, BB's range thins out. BB simply has fewer hands with pair-plus value when one of the board cards is duplicated.

BB defense vs CO c-bet at selected sizes on paired vs unpaired boards. 100bb effective · 6-max NL · BB defense on paired boards

Source: cash-baselines.md Table 5 (mean defense across 8-board panel including KK5, 772)

The solver's BB defends below the minimum defense frequency (MDF — the fold rate that makes bluffs break even) at every size up to 100% pot. On paired boards like KK5 and 772, the under-defense effect is especially pronounced because BB holds fewer trips and fewer strong pairs than on unpaired textures.

The deficit peaks at -6.3pp at 50% pot and converges to essentially zero at 150% pot, where bluff equity matters less and MDF becomes a tighter approximation.

What this means: on paired flops, small c-bets exploit BB's thinned range efficiently. BB has fewer hands worth defending, so even a small bet forces enough folds. Don't oversize — the board is already doing the job.

Low connected boards are the preflop raiser's worst boards

654 rainbow is the anti-K72r.

On low-connected flops, BB's range is loaded with straights, two-pair, and sets that the preflop raiser simply doesn't have. UTG's opening range barely contains 43, 53, 65, 75, or 87 — BB defends all of them.

CO c-bet rate comparison — dry K-high vs low-connected boards. 100bb effective · 6-max NL · CO vs BB SRP · flop cbet by board

Source: cash-baselines.md Table 3

UTG c-bets 654 only 19.7% of the time — the tightest opener on the worst possible texture. CO, with a wider range that includes more suited connectors, gets to 55.5%.

The contrast with K72r is stark. UTG fires 90.1% on the dry king-high board. On 654, it drops to 19.7%.

What this means: when the flop comes low and connected, slow down. Your preflop range advantage evaporates on these textures. The solver checks most of its range because betting into a range that connects with the board this well is burning money.

Suits are interchangeable — suit isomorphism holds

This one is simple but worth verifying. When no flush draws are present, the specific suits on the board should not matter. K♠7♦2♣ should produce the same strategy as K♣7♠2♥.

CO flop c-bet on four suit rotations of K72 rainbow. 100bb effective · 6-max NL · CO vs BB SRP · K72 rainbow, 4 suit rotations

Source: cash-baselines.md Table 9

Identical. 83.6% across all four rotations, no deviation. The model produces bit-identical strategies under suit permutation when flush draws are absent.

When flush draws are present, suits start to matter — see the next section.

Flush draws change everything about hand treatment

Adding a flush-draw texture to a board rewrites the strategy at two levels: the range level (how often CO c-bets overall) and the combo level (which specific hands bet vs check).

Rainbow vs two-tone CO flop c-bet. 100bb effective · 6-max NL · CO vs BB SRP · two-tone boards per-combo

Source: cash-baselines.md Table 8

Across all three matched pairs, the two-tone version suppresses CO's c-bet rate. The drop is largest on T98 (-18.7pp) and smallest on A94r (-8.5pp).

At the combo level, the flush-draw premium is conditional. On K94ss, suited broadway hands with flush-draw potential bet more than their offsuit counterparts — the gap measured at +13.1pp (suited_broadway vs offsuit_broadway class). On T98ss, the same gap runs +8.2pp.

But flush draw alone is not enough. On K94ss, hands like KQs bet at 90.2% (pair + flush draw) and 54s at 92.6% (flush draw + straight draws), while T9s bets only 2.3% and 98s at 0.9%. Those last two have a flush draw but no secondary equity on the board — no pair, no straight potential. The solver checks them.

What this means: don't auto-bet just because you have a flush draw. Check whether you also have a pair, a straight draw, or overcards that connect with the board. Flush draw plus board equity is a bet. Flush draw alone is a check.

Dynamic boards demand turn-card awareness

On connected boards, the turn card dramatically changes which player has the advantage. The solver's turn barrel rate swings by more than 40 percentage points depending on which card falls.

CO turn barrel rate by turn card on 654 and AK8r flops. 100bb effective · 6-max NL · CO vs BB SRP · turn barrel rate by turn card

Source: cash-baselines.md Table 4

On 654, the solver's barrel rate swings from 44.1% (Ace turn — completes the wheel for BB) up to 86.9% (nine turn — a pure overcard that helps CO and completes nothing for BB). That is a 42.8pp spread across turn cards on a single flop.

Here is the surprise: AK8r — supposedly a "static" board — shows an even wider spread of 51.5pp. The K turn (38.0%) pairs the top card and shifts the range dynamic by giving BB's Kx combos trips potential, while low blanks like the 9 (89.5%) maintain CO's dominance.

Turn-card awareness matters on every board type, not just the obviously dynamic ones.

What this means: don't autopilot the turn barrel. Even on boards that look static, some turn cards shift the equity split enough to change the correct play from near-always-bet to almost-never-bet.

Low boards spike donk betting frequency

The "never donk bet" rule is mostly right — on 10 of 12 boards tested, BB donks less than 1%. But on low-connected boards, donking is the solver's primary strategy.

BB donk rate by board, CO opens. 100bb effective · 6-max NL · BB donk rate on low-connected vs other

Source: cash-baselines.md Table 6

BB donks 55.2% on 654. That is the majority of the time — not a leak, not an exploit, but the solver's equilibrium strategy. On 543 it is 36.6%. On 876 it is 17.8%.

Meanwhile, every high-card and paired board comes in at 0.0–0.4%.

What this means: if you are in the BB and the flop comes low and connected (654, 543, 876), donk betting is the correct default — not a sign of a bad player. If you are the preflop raiser facing a donk on these boards, respect it.

The turn card effect inverts between CO-favored and BB-favored boards

Here is the most counterintuitive finding in the texture pillar. The same turn card has opposite effects depending on whose board it is.

Take the K turn. On AK8r (CO-favored, high-dry), the K turn drops CO's barrel rate to 38.0% — the lowest of all 13 turn cards. But on 654 (BB-favored, low-connected), the K turn boosts CO's barrel to 86.2% — one of the highest. On T98 (BB-favored, mid-connected), the K turn lands at 64.7%.

The pattern extends across all three tested flops:

K-turn barrel rate across board types. 100bb effective · 6-max NL · CO vs BB SRP · turn barrel rate by turn card

Source: cash-baselines.md E8 worksheet (turn_cards + e8_turn_ext batches)

Why the inversion? On AK8r, the K turn pairs the top card — BB's Kx hands now make trips, threatening CO's range. On 654, the K is a pure overcard that helps nobody's made hands and completes no draws for BB. CO's overpairs and overcards improve relatively, so the solver fires.

The Q turn on T98 reveals the mechanism even more clearly. Q on T98 barrels at just 34.6% — the lowest of all 13 turn cards for that flop. That is because Q completes the Q-J-T-9-8 straight for BB's calling range. It is not "high cards are bad for CO on connected boards" — it is specifically about whether the turn card completes a draw that BB's range holds.

On AK8r, blank low cards (9 at 89.5%, 6 at 88.6%, 4 at 87.9%) keep CO's range dominant. Board-pairing high cards (K at 38.0%) suppress it.

On 654 and T98, pure overcards that complete nothing for BB (K at 86.2% / 64.7%, 9 at 86.9%) boost CO. Draw-completing turns (A at 44.1% on 654 — the wheel; 7 at 45.7% on 654; Q at 34.6% on T98) suppress CO.

What this means: when deciding whether to barrel the turn, don't just ask "is this a high card or a low card?" Ask: "does this card complete a draw that my opponent's calling range holds?" If yes, check. If no, fire.

Flop decisions are not flop decisions. They are three-street decisions compressed into one click.

That sounds obvious, but the data makes it concrete. When the solver checks the flop, it is not giving up — it is shaping the range it will bet on the turn. When it bets small, it is setting up a geometric pot-growth plan that ends with a river overbet. When it donks from the BB, it is exploiting a specific board texture where the normal "never donk" rule breaks down. Every action on every street is downstream-aware.

This pillar covers eight theories about how multi-street planning shows up in solver strategy. Seven are confirmed by our model verification. One — the river value-bet threshold — is partially verified because the quantitative threshold requires per-combo EV data our infrastructure does not yet expose.

Measurement conditions: 6-max NL, CO vs BB SRP, 100bb effective unless noted otherwise.

After you check the flop, you can bet thinner on the turn

When you open and check back the flop, your range condenses. The nuts leave — you would have bet those. What remains is medium-strength hands, some slow-plays, and a few gives-up. Your opponent should recognize this and probe more aggressively.

But here is what is interesting: even from that condensed range, the solver still bets the turn at a substantial rate on most boards. It thin-value-bets from a range the opponent expects to be weak.

CO delayed c-bet rate after flop check, brick turn. 83.6% effective · 6-max NL · CO vs BB SRP · turn decision after flop check

Source: cash-baselines.md §F1 worksheet + Table 11 (delayed cbet batch 2026-04-12)

On six of eight boards, the delayed c-bet rate is lower than the flop c-bet rate. That is the selection effect at work: the checker's range is weaker than the full opening range, so it bets a smaller fraction on the turn.

Two boards reverse the pattern — Q83r and 654. On those, the flop-check range is the top of range (slow-plays), and the solver fires the brick turn harder than its merged flop strategy. Both patterns are condensing-range-consistent: the solver thin-value-bets from condensed ranges, and stabs hard from slow-play-loaded ones.

What this means in practice: if you checked back a dry or semi-dry flop and a brick peels, you still have a substantial betting opportunity. The solver bets the turn on most boards — sometimes over 75.9%. Do not autopilot check-check after a flop check-back.

Medium-strength hands prefer to check back

Not every hand should bet the flop. The solver checks medium-strength hands — weak top pair, second pair, mid-pair without kicker value — at a high rate on boards where those hands are neither clearly ahead nor clearly behind.

Per-class check % by board, CO flop c-bet. 100bb effective · 6-max NL · CO vs BB SRP

Source: cash-baselines.md Table 7 (per-hand check batch 2026-04-12)

Look at the medium-pair row on T98 and K94ss: 70.8 and 77.3 check rates. On boards where medium pair is genuinely in the middle of the equity distribution — not clearly ahead, not clearly behind — the solver checks the majority of the time.

The exceptions make the rule sharper. On KK5, medium pair checks only 9.8 — that is a board where underpairs function as protection-bet candidates, not medium-strength check-backs.

What this means in practice: on boards like T98 or K94ss, your second pair or mid-pair without a kicker belongs in the check-back bucket. The solver checks these over 70.8 of the time on T98. Resist the temptation to "bet for information."

Donk-betting is nearly zero on raiser-favorable boards

The "never donk" rule holds on raiser-favorable boards — and breaks wide open on low-connected ones.

BB donk rate by board, CO opens, BB calls. 100bb effective · 6-max NL · BB donk rate by board class

Source: cash-baselines.md Table 6 (BB donk batch 2026-04-12)

Ten of twelve boards donk at 3.0% or less. On those textures, "never donk" costs you almost nothing. But on 654 the solver donks 55.2% of the time, and on 543 it donks 36.6%.

What this means in practice: the "never donk bet" heuristic is only correct on the boards where you have no nut advantage. On 654 and 543, you should be leading into the raiser — it is the equilibrium play.

Turn ranges polarize — sizing grows, not frequency

As streets progress, ranges polarize. Value hands and bluffs separate from the middle. The concrete signature: bet sizes grow street-over-street in a geometric pattern.

River bet behavior across three runouts. 100bb effective · 6-max NL · CO vs BB SRP · river · specific boards per row

Source: cash-baselines.md Table 15 (river strategies batch 2026-04-12)

The sizing progression tells the story: the flop c-bet is 3bb, the turn bet is 7bb, and the river top sizes are bet19.1, bet25.5, bet38.2, and bet63.8. Each street roughly doubles the previous bet. On T98, bet63.8 is used at 23% — a massive overbet driven by the polarized turn/river range.

One important caveat: the common claim that "turn barrel rate is lower than flop c-bet rate" is harder to verify than it sounds. The flop c-bet rate measures all hands in the opening range; the turn barrel rate measures only hands that got called on the flop — a different, smaller population. Comparing those two numbers directly is a matched-population mismatch. The size axis is the clean test, and it confirms the theory clearly. (See Research notes for the full scope discussion.)

What this means in practice: if you are building a turn and river strategy, plan for geometric sizing. Flop small, turn bigger, river biggest. On dynamic boards like T98, river overbets (bet63.8) appear at meaningful frequency. If your sizing plan does not include large river bets, you are leaving value on the table.

River value bets need to win when called — and the threshold is steep

A value bet on the river has to expect to win more than half the time when called to outperform checking. OOP players with small blocking bets can go thinner — but the bar is still high.

Our model verification confirmed the direction of this claim on a specific river spot (K72r → 2d → 5h): hands with higher EV for betting do bet, and hands with higher EV for checking do check. The specific "50% winrate when called" quantitative threshold requires per-combo equity-versus-calling-range data that our infrastructure does not currently expose.

BB per-combo river defense on three runouts. 100bb effective · 6-max NL · CO vs BB SRP · river · specific boards per row

Source: cash-baselines.md Table 16 (per-combo river defense batch 2026-04-12)

Look at the A94r runout: BB raises 35.0% of the time on the river. That is a bluff-raise-heavy spot driven by blocker structure. The value bettor on CO needs to be confident their hand wins against this calling range — and that range has already shed its weakest hands through three streets of action.

What this means in practice: river value bets are not "bet if ahead." They are "bet if ahead of the hands that will actually call." That is a narrower, stronger requirement. Treat rivers as high-bar spots where thin value bets need genuine confidence.

Flop decisions are shaped by turn and river plans

This is the pillar's unifying idea: flop actions are chosen partly for their turn and river implications.

The evidence is indirect but strong. On the 654 flop, the solver's turn barrel rate swings between 7.7% and 38.5% — wait, let me use the right data. The turn-card sweep shows CO's barrel rate varies by over 42 percentage points across different turn cards on 654 alone. That kind of variance means the flop strategy cannot be independent of what happens on later streets.

The delayed c-bet data makes the same point from a different angle. When CO checks the flop on 654, it fires the turn at 74.4% on a brick. That is higher than the flop c-bet rate of 56.4% on the same board. The flop check is not a surrender — it is a setup for the turn.

The mixed-strategy data tells the downstream story: on K72r → 2d → 5h, the fraction of hands playing a pure strategy rises from 15.4% on the flop to 27.2% on the turn to 38.5% on the river. By the river, hand values have crystallized — the mixed flop decisions have resolved into clear bet-or-check on the river.

What this means in practice: the solver's turn barrel rate varies by over 42 percentage points depending on which card comes. Your flop action sets up a range that either exploits or suffers from those turn cards. Do not evaluate flop bets in isolation.

Draws mix — they are not pure bets or pure checks

At equilibrium, draws do not have a single correct action. They mix between betting and checking (or calling and folding) because the EV of each action is nearly identical. The solver is indifferent — and that indifference is the point.

Mixed-strategy fraction by street on two runouts. 100bb effective · 6-max NL · CO vs BB SRP

Source: cash-baselines.md Table 12 (mixed-strategy fraction batch 2026-04-12)

On the A94r flop, only 7.7% of hands play pure strategies. That means over 92% of hands — including most draws — are mixing between two or more actions. The mixing fraction decreases as streets progress (hand values become more static), but on the flop the indifference is pervasive.

Flush draws with a pair or an Ace have showdown value and are less eager to semi-bluff. Pure draws without board equity tend to check. Draws with both flush potential and secondary equity (pair, straight draws) bet aggressively. But none of these are pure — they all mix.

What this means in practice: draw decisions on the flop are inherently close. The solver mixes over 90% of hands on A94r. Do not overthink individual draw decisions — the EV gap between betting and checking your flush draw is tiny at equilibrium. Focus your study time on spots where the solver plays pure.

Monotone boards kill the delayed c-bet

After checking back a monotone flop, CO's delayed c-bet rate collapses. Not "drops a little" — collapses to near-zero on connected monotone boards.

Initial vs delayed c-bet rates, rainbow and monotone boards. 100bb effective · 6-max NL · CO vs BB SRP · delayed cbet on rainbow vs monotone

Source: cash-baselines.md §F8 worksheet + Table 11 (delayed cbet batch 2026-04-12; monotone extension batch 2026-04-14)

Every rainbow board in the panel has a delayed c-bet rate above 36.3%. Every monotone board is below 21.1%. The gap is at least 15 percentage points at its narrowest (A94 monotone vs T98 rainbow).

The T98 monotone board is the extreme: a delayed c-bet rate of 4.2%. After checking back a connected monotone flop and seeing a brick turn, CO essentially never bets.

What this means in practice: on monotone boards, the delayed c-bet is dead. K94ss delayed c-bet is 11.0%, connected monotone T98 is 4.2%. If you checked back the flop and a brick turns on a monotone texture, check again. Bluffing into a range full of flush draws is lighting money on fire.

Solvers don't just pick the "best hand wins" action. They exploit patterns that sit one level above raw hand strength — stack depth, player count, blocker structure, the shrinking decision space street by street. These are the concepts that separate a player who runs a solver from a player who thinks like one.

Seven of the eight theories in this pillar are confirmed by our model verification. The eighth — ICM and tournament dynamics — is structurally out of scope for cash, where every chip equals a dollar and there is no bust-out penalty. It stays in the catalog for completeness, but the data and the coaching takeaways live in Book 6 (MTT).

Measurement conditions: 6-max NL, CO vs BB SRP, 100bb effective unless noted otherwise.

Stack depth changes strategy qualitatively

Strategy at 30.6bb is not a scaled-down version of strategy at 43.3bb. It is a different game. Preflop ranges tighten, postflop sizing compresses, and the threshold for "good enough to get it in" drops sharply.

Start with preflop VPIP across four stack depths:

Preflop VPIP by position and stack depth. 6-max NL · stack depth sweeps rows

Source: cash-baselines.md Table 2 — batches_cash_20bb_shortstack + OQ5 + batches_cash_deep_stack

BTN tightens from 43.3 at 100bb down to 30.6 at 20bb. That is a different opening range. But look at the other end: BTN at 150bb (45.6) and 200bb (46.9) barely move from 100bb. Preflop widening saturates around 100bb. Postflop softening, on the other hand, continues through 200bb.

What this means in practice: If you sit down short-stacked, your preflop strategy needs a hard reset, not a trim. And if you are deep (150bb+), your preflop range barely changes — it is the postflop decisions that get more complex.

Multiway pots tighten c-bet ranges

Adding a third or fourth player to the pot does not just mean more opponents. It means your c-bet frequency drops — sometimes by half.

CO c-bet frequency: heads-up vs multiway. 100bb effective · 6-max NL · HU vs 3-way / 4-way · specific spots per row

Source: cash-baselines.md Table 13 — batches_r22_p3_multiway_ext (3-way) + batches_cash_multiway_4way (4-way)

Every board tightens at 3-way — no exceptions. T98 drops from 70.3% heads-up to 11.6 at 3-way (-58.7pp). At 4-way the tightening holds on most boards, though T98 flattens out (+0.4pp) — a board-specific anomaly, not a texture pattern.

Why does the tightening happen?

What this means in practice: When three or four players see a flop, check more often — even on boards where you would range-bet heads-up. The extra defenders mean your fold equity is divided across more ranges, and medium-strength hands that would fire heads-up need to check.

Blockers override raw hand strength in close decisions

Sometimes the solver folds top pair and calls with second pair. The difference is not hand strength — it is which cards you hold and what those cards remove from your opponent's range.

On a K72r → 2d → 5h river, the solver's per-hand defense varies by blocker structure: hands holding a King (blocking CO's Kx value range) call at a higher frequency than hands holding an Ace (no meaningful blocker). On A94r → 4c → 8d, the pattern flips: Ax blockers defend more. And on A94r, BB raises the river 35.0% of the time — a river bluff-raise spot built on blocker-heavy bluffs.

BB river defense by runout. 100bb effective · 6-max NL · CO vs BB SRP · per-combo suit filtering

Source: cash-baselines.md Table 16 — batches_cash_per_combo_call_fold

The aggregate numbers hide the per-combo story. Within each runout, the solver differentiates hands of similar raw strength by which opponent combos they block. Kx holdings on the K72r river call more (they block CO's value); Ax holdings on A94r defend more (they block CO's top-pair value).

What this means in practice: "I have second pair" is not enough information for a river decision. "I have second pair with a King" is. Train yourself to think about which opponent combos your cards remove before defaulting to hand-strength charts.

ICM and tournament dynamics do not apply to cash

In tournaments, your chips are worth less than their face value because busting out has zero future-EV weight — a concept captured by the Independent Chip Model (ICM). Under ICM pressure, both players have less incentive to grow the pot: the covered player bets less, the covering player bets more. These effects are real and well-documented in tournament literature.

None of this applies to cash games. Every chip is worth exactly one dollar. There is no bust-out penalty, no bubble, no pay-jump. The solver treats cash chips at face value, and so should you.

What this means in practice: When someone at a cash table says "we should be tighter because of ICM," they are confusing formats. Cash is chip-EV maximization, full stop.

The solver mixes more than you think

Most players think solvers pick one "correct" action per hand. They do not. The majority of the range mixes between two or more actions.

When we ran a systematic scan of CO's flop c-bet strategy, the numbers were striking: on K72r, 107 out of 169 hand classes mix (no single action exceeds 85% frequency). On T98, 98 out of 169 mix. That is roughly 60% of the range in a state of near-indifference at every flop c-bet decision.

The counterintuitive examples are the most revealing:

Counterintuitive mixing examples on Cash flop c-bet. 100bb effective · 6-max NL · CO vs BB SRP · specific spot per row

Source: cash-baselines.md Table 17 — batches_cash_close_spot_mixing

Three of the four small-suited Aces on A94r mix between call and raise — they sit at the indifference boundary where blockers make calling and raising nearly equal in expectation. But 99 facing a 3-bet and 77 on 654 are near-pure. The lesson: your intuition about which spots are "close" is probably wrong. A systematic scan catches close spots your gut misses.

The broader systematic data from the scan is even more dramatic. On T98, AA — the best starting hand in poker — splits its c-bet action nearly equally three ways: bet one size, bet another size, and check. Human heuristics say "always bet aces on a connected flop." The solver says "I'm almost perfectly indifferent."

What this means in practice: When you study a solver solution and see near-equal frequencies for two or three actions, that is not a bug — it is the equilibrium. Obsessing over whether to bet or check with a specific combo in a mixed spot is often less important than getting the adjacent pure spots right.

Hand values get more static toward the river

On the flop, everything is in flux. Two cards still to come means draws are live, ranges are wide, and almost nothing is locked in. By the river, hand values are fixed.

We measured this directly by tracking the fraction of hands that use pure strategies (one action at 100% frequency) on each street:

Pure-strategy fraction by street. 100bb effective · 6-max NL · CO vs BB SRP · specific runout per row

Source: cash-baselines.md Table 12 — batches_cash_mixed_strategy_fraction

On K72r, only 15.4% of hands play a pure strategy on the flop. By the river, that doubles to 38.5%. Same pattern on A94r: 7.7% at flop rising to 31.4% at river.

What this means in practice: If you feel uncertain about your flop decision, that uncertainty is correct — the solver is uncertain too. On the river, if you are still uncertain, something is wrong with your hand-reading. The cards have been dealt.

Nut hands sometimes check

The strongest hands in your range do not always bet. Premium hands (AA, KK, AK) check at very different rates depending on the board:

Premium class (AA/KK/AK) check frequency by board — CO flop c-bet. 100bb effective · 6-max NL · CO vs BB SRP

Source: cash-baselines.md Table 7 — batches_cash_per_hand_check

On K72r, premium hands check only 8.5 of the time — that is a near-automatic bet. But on 654, premium hands check 86.3 of the time. On a low-connected board where BB has nut advantage (all the straights, two-pair combos, and sets of fours through sixes), your AA is no longer the nuts — it is a bluff-catcher that happens to have a lot of raw equity. The solver knows this and protects its checking range with it.

The paired board KK5 splits almost evenly (49.9 check). The monotone K94ss checks 54.4 — more than half the time, premium hands just check back a flush-heavy board.

What this means in practice: Resist the autopilot bet with aces or top set. Check the board texture first. If it favors BB's range — low-connected, monotone, paired — checking the nuts is not a missed opportunity. It is the equilibrium strategy.

"Bet for protection" is real but narrow

"Bet for protection" is one of the most overused justifications in poker. The solver says it works — but only when two conditions hold simultaneously. The bet must extract value from worse hands AND fold out hands with live equity. Miss either condition and checking wins.

The starkest demonstration: AA on two different 8-6-x boards.

8♥6♦4♥ (wet, two-tone hearts). AA c-bets only 0.3% of the time. The solver checks aces on this board 99.7% of the time.

8♥6♦2♣ (drier, single-tone). AA c-bets 84.0% of the time.

That is a swing of 83.7 percentage points on the same hand — just by swapping the 4♥ for a 2♣. The wet board fails the second condition: BB's range is loaded with flush draws and straight draws that have too much equity to fold, so betting "for protection" does not actually protect anything. On the drier board, BB's range has more dead hands that will fold, so the bet works.

The inverted overpair hierarchy on 8♥6♦4♥ is the most counterintuitive finding in this pillar. Stronger overpairs check more: AA checks most often (c-bets 0.3%), KK next (27.8%), QQ (51.3%), JJ (93.9%), and TT checks least (c-bets 99.5%). This is perfectly inverted from the naive "bigger pair = always bet" intuition.

What this means in practice: Stop auto-betting overpairs on drawy boards. The solver checks AA on 8♥6♦4♥ almost every time. If your instinct says "I need to bet to protect," check the board texture first. The bet only works when both conditions are met — and on wet boards, they rarely are.

3-bet pots are not scaled-up single-raised pots. The ranges are tighter, the stacks-to-pot ratio is lower, and the resulting flop strategies look nothing like what you see in a standard open-and-call sequence. Three findings from our solver verification show just how different: the OOP 3-bettor's c-bet strategy flips depending on stack depth, BB's overcall range in a 3-way pot is built entirely from connectivity, and the IP 3-bettor's flop c-bet is so board-dependent it splits into two poles with almost nothing in between. All three theories confirmed cleanly against the model.

Measurement conditions: 6-max NL, 43.3bb effective unless the table sweeps stack depth.

The OOP 3-bettor bets almost everything at shallow stacks — and checks far more at deep stacks

At 40bb effective in a 3-bet pot, BB — the out-of-position 3-bettor — bets the flop on almost every non-connected texture. Check rates on low-card boards sit between 2.5% and 17.1%. At 100bb on the same boards, checking jumps to 40.7%–55.4%.

The shift is massive — an average of roughly +36.5 to +40.5 percentage points more checking when stacks deepen on boards where the pattern holds.

OOP 3-bettor check % by board and stack depth. 6-max NL · BTN 2.5bb open, BB 3-bet to 7.5bb (40bb) or 9bb (100bb), BTN call · BB cbet by board × depth

Source: cash-baselines.md H8 worksheet (batches_cash_h8_oop_3bp_40bb, 2026-04-14)

Five of seven boards follow the pattern: BB bets almost everything at 40bb, then checks substantially more at 100bb. Two boards break the pattern for coherent reasons.

T98 (range miss). BB's 3-bet range is loaded with high broadway cards and overpairs. On T♠9♣8♦, that range completely misses. Check rate is already 79.1% at 40bb — the OOP 3-bettor checks most of the time regardless of depth because the range just does not connect with this board.

K72r (range dominance inversion). The opposite problem. BB's tight 3-bet range dominates K♠7♣2♦ so completely that at 100bb, checking drops to 3.0% — the solver bets more with deeper stacks, not less. The range advantage is so overwhelming that there is no need to protect a checking range.

What this means in practice: If you 3-bet from the blinds and get called at 40bb, plan to c-bet most flops that are not mid-connected. At 100bb, slow down — checking is the solver's preferred line on more than half of flop textures in the data.

In a 3-way pot, BB's overcall range is built entirely from connectivity

When someone raises, another player cold-calls, and the action comes to you in the big blind, the solver's calling range has a strict structural requirement: every hand that calls has two-card straight or flush potential. Disconnected offsuit hands fold regardless of pot odds.

The data tested two scenarios — UTG raises and BTN cold-calls, or CO raises and BTN cold-calls — both at 100bb. The pattern was identical in both.

BB overcall decision by hand class. 43.3bb effective · 6-max NL · UTG/CO raise + BTN call · BB overcall decision

Source: cash-baselines.md H9 worksheet (batches_cash_h9_bb_overcall, 2026-04-14)

The split is dramatic. Suited connectors and pocket pairs call nearly pure. Every tested disconnected offsuit hand — Q7o, K4o, A8o, A9o — folds 100.0% of the time.

One nuance: 87s is raiser-dependent. It calls only 19.6% of the time against a UTG open but is substantially more willing to enter against a CO open. The tighter the raiser's range, the less a borderline connector wants to play multiway.

A9o is worth noting. Some poker literature identifies it as the one offsuit exception that should call in this spot. The solver disagrees — A9o folds pure in every scenario we tested.

What this means in practice: Stop defending Q7o and K4o in 3-way pots. The pot odds make it feel cheap, but the solver folds these hands every time. Connectivity is what earns you the call, not raw card rank.

The IP 3-bettor's c-bet is bimodal — near-100% or near-zero, almost nothing between

In single-raised pots, c-bet rates tend to spread continuously across board textures. In 3-bet pots where BTN is the 3-bettor, the distribution collapses into two poles.

On high-card boards where BTN's tight 3-bet range dominates, the solver c-bets at or near 100.0%. On connected boards where that same range completely misses, c-bet rates drop to single digits. The gap between these two poles is far larger than anything in single-raised pots.

We tested this against two different cold-callers — CO (wider range) and UTG (tighter range) — and the bimodal structure held in both.

BTN 3-bet c-bet rate by board and opponent. 43.3bb effective · 6-max NL · 2nd-position cold-call scenarios · BTN cbet by board

Source: cash-baselines.md H10 worksheet (batches_cash_3bet_pot Scenario B + Scenario C, 2026-04-12 / 2026-04-14)

Look at the high pole first. On K♠7♣2♦, BTN c-bets 100.0% of the time in both scenarios. The opponent's range does not matter — the 3-bettor's range is so concentrated on broadway hands that it dominates every high-card board completely.

Now the low pole. On T♠9♣8♦, the same range completely whiffs. Against CO's wider cold-calling range, BTN c-bets only 7.5%. Against UTG's tighter cold-calling range (more overpairs, fewer connected hands), that rises to 20.0% — still dramatically lower than any single-raised pot number on the same board.

The K♠9♠4♠ line is the most counterintuitive. In a single-raised pot, CO c-bets this monotone board only 32.2%. In a 3-bet pot, BTN c-bets 85.8% against CO. The monotone suppression that normally slows down c-betting reverses in 3-bet pots because BTN's tight broadway range is loaded with flush blockers.

What this means in practice: If you 3-bet BTN and get cold-called, and the flop comes T♠9♣8♦ or 6♠5♦4♣, checking is overwhelmingly correct. Your range simply does not connect with these boards, and the solver reflects that by c-betting in the single digits.