# Existence and Construction of Reel Strips from Mathematical Specifications -- plain text, part 4 of 5

Pages 42-56 of 59.
Sections: Scatters; Lines Correction; Comparison to Prior Work; Worked Example: Construction of a Complete Game; Design Abstraction and Extensions; Conclusion

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The value-bearing EV does enter the total RTP: for cash scatters, it is an additive RTP component; for multiplier wilds, it scales the OAK payouts of the symbols the wild substitutes for. The designer specifies the per-count average targets as part of the overall specification, and the filler allocation (Section 5) accounts for the value-bearing RTP contribution when targeting the total RTP. The Full Construction Theorem (Theorem 9.4) applies with the value-bearing linear system as an additional post-skeleton step, and the value-bearing EV as a pre-computed component of the RTP target.

12 Scatters
A scatter symbol pays based on how many reels display it, regardless of position within each reel’s window. (When scatters are placed as singles with c = 1, the number of reels displaying the scatter equals the total scatter count on screen. For c > 1, these differ; the Poisson-binomial model below counts reels, not individual symbols.) It is excluded from the ways-pay evaluation and is never substituted by wilds. This section shows that the scatter’s count probabilities, RTP, and payout distribution are exactly computable, independently tunable, and fully integrated with the rest of the framework.

12.1 Count Distribution

Proposition 12.1 (Poisson-Binomial Characterization). A scatter symbol on R independent reels

with per-reel presences p1, . . . , pR has count distribution given by the Poisson-binomial probability

generating function:

R

P (exactly k) = [xk] (1 − pi + pix).

(64)

i=1

The PMF is computable in O(R2) by sequential convolution. The probability of any threshold event

P (≥ n) =

R k=n

P (k)

is

a

sum

of

multilinear

terms.

Proposition 12.2 (Multilinearity of Count Probabilities). P (exactly k) is multilinear in p1, . . . , pR: each pi appears to at most the first power. Its extremes over any box i[li, ui] are therefore attained at vertices. The exact achievable range of any count probability requires at most 2R evaluations.

12.2 Scatter RTP
Proposition 12.3 (Scatter RTP). The scatter RTP is:

R

RTPscatter = P (exactly k) · pay[k],

(65)

k=0

which is linear in the PMF and hence multilinear in the pi. The total game RTP decomposes

additively:

RTPtotal = RTPways + RTPscatter.

(66)

The two components are computed from disjoint symbol sets and are independently targetable.

42

12.3 Constrained Ranges
Proposition 12.4 (Constrained Ranges). Fixing P (k∗) = t and solving for one presence pj = (t − qk)/(qk−1 − qk), where qm is the PMF at count m with pj = 0, any other count probability P (m) becomes a rational function of each remaining pi with quadratic numerator and affine denominator. The extrema of P (m) over each pi ∈ [li, ui] are attained at the two box endpoints or at the (at most 2) roots of the derivative’s numerator. The constrained range of P (m) at fixed P (k∗) is computed by a coordinate-wise sweep: for each variable pi, evaluate at the two box endpoints and the (at most 2) derivative roots, giving 4 candidates per variable. Sweeping all R − 1 remaining variables yields at most R · 4R−1 evaluations. For R = 5 this is 1,280 evaluations, trivially pre-computable. The result is a coordinate-wise sweep, not a global bound; the joint extremum may lie at a non-axisaligned point. For exact global bounds, Lagrange stationarity on the multilinear constraint yields a polynomial system solvable in fixed dimension.

12.4 Volatility Tunability
Proposition 12.5 (Scatter Variance at Fixed RTP). The scatter payout variance is:

Var(Yscatter) = P (k) · pay[k]2 − RTP2scatter.
k

(67)

At fixed scatter RTP (one multilinear constraint on R presences), asymmetric per-reel presences produce different count distribution shapes with different variances. The achievable variance range at fixed scatter RTP is a pre-computable interval, tunable by the choice of per-reel presences on the RTP iso-surface.

12.5 Full Payout Distribution
Proposition 12.6 (Scatter Distribution is Targetable). The scatter payout distribution has at most R + 1 atoms (one per count level k = 0, . . . , R). Each atom’s probability P (k) is controlled by the per-reel presences via the Poisson-binomial. Each atom’s value is controlled by the paytable or, for value-bearing scatters carrying cash values, by the linear system of Section 11. Both are independently targetable, so the full scatter payout distribution is exactly computable and exactly controllable.

12.6 Integration with the Framework
Proposition 12.7 (Decomposition of Game Metrics). The complete game’s metrics decompose into three sequentially decoupled systems and one exactly computable coupling term:
1. Scatter metrics (controlled by per-reel presences pi alone):
ˆ Scatter hit rate: P (≥ n) = k≥n P (k), from the Poisson-binomial. ˆ Scatter RTP: k P (k) · pay[k], multilinear in the pi. ˆ Scatter variance: k P (k) · pay[k]2 − RTP2scatter, tunable by asymmetric pi on the RTP
iso-surface.
No co-location matrix is involved. The scatter fires on a cross-reel count, not a within-window pattern.
2. Ways metrics (controlled by filler counts and co-location matrix Q):

43

ˆ Ways RTP: multilinear in filler counts (Section 5). ˆ Ways hit rate: targeted via Q off-diagonal (Section 7). ˆ Ways CVwin: targeted via clustering distribution, directed filler arrangement, and label
permutation at frozen RTP and HR (Section 8). σ2 derived.
3. Total RTP: RTPtotal = RTPways+RTPscatter. Additive, since the two systems evaluate disjoint symbol sets.
4. Coupling terms (controlled by the skeleton-filler block Q[scat][f ]):
ˆ P (ways win AND scatter win): on reels showing scatter, one cell is occupied, reducing the effective window for paying symbols. The joint probability is exactly computable from the per-reel window content distributions, captured by Q[scat][f ].
ˆ Cov(Yways, Yscatter): nonzero and negative (scatter presence reduces paying symbol counts). Exactly computable from the skeleton-filler co-location block.
Total hit rate: P (any win) = P (ways) + P (scatter) − P (both), all terms computable in O(L) from the constructed strip, without simulation. Total variance: Var(Ytotal) = Var(Yways) + Var(Yscatter) + 2 Cov, all terms computable in O(L) from the constructed strip, without simulation.
The Full Construction Theorem (Theorem 9.4) applies with the scatter as an additional skeleton symbol, its RTP as a pre-computed additive component, and its count probabilities as independently targetable parameters. The packing constraint (54) absorbs the scatter’s stop counts into the skeleton footprint.

13 Lines Correction
The framework as developed uses ways-pay evaluation: a kOAK win counts the product of per-reel symbol counts in the window. In lines-pay games, wins are evaluated along P fixed paylines, each selecting one cell per reel. This section provides the correction factor that converts between the two models. The framework applies to any display height W and any number of reels R.

13.1 Per-Payline Evaluation

For a game with P paylines on an R-reel, height-W grid, each payline selects one row per reel. Under a uniform spin, the probability of symbol s appearing at a specific cell on reel i is ns(i)/Li.
The expected payout for a kOAK of s on a single payline is:

RTPl(isn)e,k

=

vk(s)

·

k i=1

ns(i) Li

·

1 − n(sk+1) Lk+1

,

k < R,

(68)

and the total lines RTP is P times this sum over all symbols and OAK levels. The blocker term uses reel k + 1 only (left-anchored evaluation), matching the ways-pay convention (Definition 2.10).

44

13.2 The Correction Factor

Proposition 13.1 (Lines Correction). The per-symbol, per-OAK correction factor from ways to

lines is:

λs,k

=

P Wk

·

1

− n(sk+1)/Lk+1 1 − p(sk+1)

,

(69)

where the numerator uses the per-cell absence probability (lines blocker: one cell on reel k + 1) and the denominator uses the per-window absence probability (ways blocker: any cell in the window on reel k + 1). The correction is per-symbol and per-OAK-level: no single scalar corrects all symbols at all depths simultaneously.

Proof. The ratio RTPlines/RTPways for symbol s at depth k is:

P

· (n/L)k · (1 − n/L) (W n/L)k · (1 − p)

=

P Wk

·

1

− ns(k+1)/Lk+1 . 1 − ps(k+1)

(70)

The paying-reel terms simplify because (n/L)k/(W n/L)k = 1/W k. The blocker terms differ because the lines blocker checks one specific cell (n/L) while the ways blocker checks the entire window (p = W n/L under singles).

Corollary 13.2 (All-Ways vs All-Lines). When P = W R, the correction factor is λs,k = W R−k · (1 − p/W )/(1 − p) for k < R. At k = R, there is no blocker reel and λ = 1: all-ways and all-lines agree. For k < R, λ > 1 because the per-cell blocker is weaker than the per-window blocker. The divergence is approximately W R−k, reflecting that a ways game counts only the paying reels while a lines game counts the full R-cell path.

Remark 13.3 (Scope of the Lines Correction). The correction factor λs,k fully resolves the RTP computation for lines games: the total RTP under lines evaluation is a known, pre-computable function of the symbol counts and the payline geometry. The co-location matrix, the hit-rate targeting, and the volatility control operate on the ways-pay model. For lines games, the RTP targeting (Section 5) applies with λ as a multiplicative correction.
However, hit rate and volatility under lines evaluation are not resolved by λ alone. The hit rate in a lines game is the probability that at least one payline achieves a kOAK, which depends on which symbols occupy which specific rows in the window (not just which symbols are present). Paylines sharing cells create correlations that the per-window co-location matrix does not capture. The payout variance under lines evaluation similarly depends on per-payline covariances arising from shared cells. A complete lines-game theory for hit rate and volatility would require extending the co-location framework to per-cell, per-row statistics. This remains open.

14 Comparison to Prior Work
The existing literature on reel strip design consists of five optimization papers from two research groups (Bulgarian Academy of Sciences and University of Macedonia), plus one formal-verification approach (Eindhoven), all treating the strip as given or searching for one via metaheuristic optimization of a scalar or multi-criteria objective.
Balabanov et al. [2] apply a Genetic Algorithm to search for a symbol distribution achieving a target RTP, using Monte Carlo simulation as the fitness function. The search space is the set of symbol count vectors. Each candidate is evaluated by simulating 106 spins. The method finds

45

approximate RTP matches but provides no feasibility guarantee, no hit-rate control, no volatility control, and no characterization of the attainable set.
Keremedchiev et al. [4] replace Monte Carlo evaluation with exact full-cycle computation, eliminating simulation variance. The search remains a Genetic Algorithm over symbol distributions targeting RTP alone. The improvement is in evaluation speed and accuracy, not in the scope of what is targeted or guaranteed.
Kamanas et al. [5] introduce Variable Neighborhood Search (VNS) with two local search operators (swap and shift), achieving the current state of the art in RTP convergence speed. The 2025 survey by the same group [6] presents VNS as the frontier of the field, with independent hit-rate and volatility control listed as open future work.

GA [2] GA+Exact [4] VNS [5] This paper

RTP targeting Hit-rate targeting Volatility targeting Per-symbol control Existence guarantee Feasibility bounds Wild symbols Method Evaluation

approx no no no no no
unknown search
Monte Carlo

exact no no no no no
unknown search
full cycle

exact no no no no no
unknown search
full cycle

exact yes yes yes yes pre-computed linear correction construction closed form

The difference is not incremental. The prior work searches for a strip matching one target, with no guarantee of success and no control over the remaining degrees of freedom. The present framework characterizes the full achievable set, targets all three global metrics simultaneously alongside per-symbol specifications, guarantees existence when targets are feasible, detects infeasibility when they are not, and constructs a realization deterministically.
Remark 14.1. Hit-rate and volatility targeting are standard commercial deliverables — game mathematicians routinely adjust reel strips to meet these targets via iterative manual tuning or proprietary tools. What the prior academic literature lacks is not the practice but the theory: no published work establishes constructibility with characterized attainable sets, pre-computable feasibility bounds, or existence guarantees. The contribution of this paper is the mathematical foundation, not the design goal.
The co-location matrix, the three-layer decomposition, the RTP iso-surface, the reachability of the achievable set via splice linearity, and the E[c2] lattice have no precedent in the slot-design literature.

15 Worked Example: Construction of a Complete Game
This section constructs a complete game from specifications, exercising every section of the framework. The game is a 5 × 3 ways-pay machine with 35 = 243 ways, bet = 1 credit, and total RTP = 94% (64% base game +30% free games). Three design coordinates are targeted simultaneously: RTP = 64.00%, hit rate = 1 in 4.0, and CVwin = 15.0. The free-game RTP is determined by the free-game reelset and trigger frequency; the specific EV of the free-game mode depends on those

46

strips, which are constructed independently. Changing the free-game contribution would modify the free-game strips, not the base-game strips built here.

15.1 Specification
Paytable. Ten paying symbols in two depth classes (Table 1).

Symbol Depth 2OAK 3OAK 4OAK 5OAK

H1

2

0.40 1.00 4.00 10.00

M1

2

0.15 0.50 2.00 6.00

M2

2

0.10 0.30 1.00 4.00

M3

2

0.05 0.20 0.60 2.50

A

3

—

0.10 0.40 1.50

K

3

—

0.10 0.30 1.00

Q

3

—

0.05 0.20 0.80

J

3

—

0.05 0.15 0.60

10

3

—

0.05 0.10 0.40

9

3

—

0.05 0.10 0.30

Table 1: Paytable (pays per way per credit wagered). Depth classes: {H1, M 1, M 2, M 3} at d = 2 (pay from 2OAK), {A, K, Q, J, 10, 9} at d = 3 (pay from 3OAK). Within each depth class, label permutations are HR-neutral: 4! × 6! = 17,280 assignments.

Wild. Multiplier wild on reels 2–4 only. Back-loaded: P (any wild on screen) ≈ 1/31. Per-reel presence pwild = 0.004, 0.007, 0.022 on reels 2, 3, 4 respectively. Clustering: c = 1.0 (reel 2, singles), c = 1.2 (reels 3–4). A moat of W − 1 = 2 positions separates the wild cluster from H1.
Scatter. p = 0.10 on all five reels, placed as singles. Cash pays: 3 = $1, 4 = $2.50, 5 = $10. Trigger probability P (3+) = 1/117. Cash RTP: 0.93%.

15.2 H1 Premium Symbol
Five per-OAK hit rate targets: P (2OAK) = 1/15, P (3OAK) = 1/30, P (4OAK) = 1/250, P (5OAK) = 1/600, and a near-miss target P (miss on reel 2 only) = 1/150 (symbol present on reels 1, 3, 4, 5 but absent from reel 2). Five equations in five unknowns (per-reel effective presence p1, . . . , p5), solved by the ratio chain (Proposition 3.1). Per-reel clustering targets ci set independently. The resulting skeleton specification (Table 2):

15.3 Wild Multiplier as Effective CC

The wild carries a compound multiplier: when k wild stops appear in a window, the multi-

plier is

k j=1

ωj

.

By Definition 11.2, the per-count average for multiplicative evaluation is ω¯k =

E[ ωvisible | count = k]. The compound multiplier folds into the wild’s effective conditional count:

ceff = P (1) · 1 · ω¯1 + P (2) · 2 · ω¯2,

(71)

where P (k) is the probability of count level k given presence. This ceff replaces cwild in the additive formula meff = mnative + pwild · ceff (Proposition 10.1), which sets the floor for every paying symbol’s

47

R1 R2 R3 R4 R5

L

511 1500 720 2067 510

peff 0.5284 0.2000 0.3681 0.1451 0.2941

cH1

1.2 54/49 1.8

1.8

1.8

nH1 108 108 156 153

90

pwild — 0.004 0.007 0.022 —

cwild

—

1.0

1.2

1.2

—

nwild

—

2

2

18

—

nscat

17

50

24

69

17

nfiller 386 1340 538 1827 403

Table 2: Skeleton specification. Per-reel L is derived from integrality constraints: the largest L under a design cap such that n = pcL/W and a = pL are integer for every designed symbol on that reel. Reel 2 is concatenated K = 2, reel 3 K = 2, reel 4 K = 3 (Proposition 4.11): reel 2 for a minimum of 2 wild stops, reel 4 for finer distributional control over the multiplier values from {2, 3, 5}; peff = pnative + pwild.

effective m on that reel. The compound across reels is handled by the standard product meff in the RTP formula.
At uniform ω per reel (all stops carry the same value), the per-count averages are ω¯1 = ω and ω¯2 = ω2. The per-reel values are solved to hit RTP = 64% at the given R4 distribution; CVwin emerges from the resulting compound structure. The solution uses non-uniform stop values on reel 4 from {2, 3, 5}: two stops at ω = 2, five at ω = 3, and eleven at ω = 5. Count-2 windows on reel 4 see compounds ranging from 2 × 3 = 6 to 5 × 5 = 25 depending on which stops are visible — this dispersion drives CVwin above the uniform baseline. Reel 4 is concatenated K = 3 (Proposition 4.11) to provide 18 stops for finer distributional control; reels 2 and 4 are both concatenated, reel 2 for a minimum of 2 wild stops. Reels 2 and 3 carry uniform values ω2 = 1.93 and ω3 = 2.90, solved for RTP = 64%. The per-stop values are assigned from {2, 3, 5}; the compound multiplier in each window is computed by exact enumeration over all Li windows per reel, not by the linear system of Section 11 (which targets per-count arithmetic means, not the arithmetic mean of products arising from multiplicative compounding). The non-uniform stop value distribution is the primary CVwin lever.
15.4 Filler Allocation and Scatter
Nine fillers are allocated on the RTP iso-surface via the intermediate value theorem (Proposition 5.8), with blend parameter t = 0.19 between uniform and pay-weighted distributions. Total filler RTP: 15.6%. Combined with H1 at 47.5% and scatter at 0.93%: 64.00% base game RTP.
Scatter constrained range. The Poisson-binomial structure (Proposition 12.1) constrains the achievable scatter-count distribution. At per-reel p = 0.10: P (3+) = 1/117, P (3) = 1/123, P (4) = 1/2,222. A target of P (4) = 1/500 is infeasible at this P (3): the ratio P (4)/P (3) is constrained by the Poisson-binomial coupling.

48

15.5 RTP Verification
Full enumeration over all Li windows per reel yields exact meff and peff per symbol per reel, including the compound wild multiplier folded into the CC (Table 3).

Symbol

RTP 5OAK freq Any win freq

H1 M1 M2 M3 A K Q J 10 9 Scatter

47.5% 6.9% 3.6% 2.0% 1.0% 0.7% 0.5% 0.4% 0.3% 0.2% 0.93%

1/600 1/800 1/810 1/1050 1/1130 1/1150 1/1080 1/1220 1/1200 1/1200
—

1/9 1/14 1/14 1/16 1/71 1/70 1/70 1/72 1/75 1/73 1/117

Total 64.000%

Table 3: RTP per symbol, verified by exact enumeration.

15.6 Three-Coordinate Targeting

The variance decomposition identity (Proposition 8.1) gives σ2 = (RTP2/p)(1 − p + CVwin2), where

p is the hit rate and CVwin is the conditional coefficient of variation of the win amount given a win.

The three coordinates (RTP, p, CVwin) are targeted by three sequentially decoupled controls.

Hit rate structure: H = S1/Ω. The hit rate decomposes into S1 =

s

ds i=1

p(si)

,

the

expected number of winning symbols per spin (pure order 1, reads presences only), divided by

Ω = S1/H = E[#winning symbols | win], the overlap factor (one scalar, all co-location). On the

present design at the singles operating point: S1 = 0.497, H = 0.305, Ω = 1.63, overlap cost 38.7%

of the na¨ıve disjoint sum. RTP reads mi on paying reels and p on the blocker reel only; S1 reads pi

on paying reels. These are disjoint: presences on non-blocker reels move S1 (and hence H) without

touching RTP.

Hit rate targeting. The achievable hit-rate range at RTP = 64% is ∼1/3 (all fillers as

singles) to ∼1/10 (all fillers maximally clustered on non-blocker reels). Filler clustering trades p

against c at fixed m = p · c (Theorem 2.7), so RTP is exactly preserved. Clustering is restricted to

non-blocker reels (Corollary 2.13): depth-2 fillers may cluster on reels 1–2; depth-3 fillers on reels

1–3. Target: 1/4.0. Each filler’s per-reel cf is a separate design parameter (Definition 5.1); the per-symbol cf values are chosen so that the resulting per-symbol presences pf = mf /cf yield the

target HR through the inclusion–exclusion formula. The splice construction (Proposition 6.32) fills

gaps between lattice points at O(1/n) resolution.

HR-preserving moves and the volatility kernel. On each reel with the others fixed, H is

exactly linear in the per-reel window-type counts (the full vector of W -gram frequencies, not just

the pairwise co-location Q). The order-2 contribution H(2) is a function of Q alone (Corollary 6.19);

49

the order-3 terms depend on triple co-occurrence, a strictly finer statistic. At the H(2) level, every gradient ∂H(2)/∂Qr[s, t] is negative: more co-location always lowers H (shared windows make wins coincide). The HR-preserving moves at fixed presences form a linear subspace (the kernel of the gradient), spanned by leverage-weighted trades: increase Qr[s, t] by ε/gr[s, t] and decrease Qr′[s′, t′] by ε/gr′[s′, t′], where g = ∂H(2)/∂Q. Net ∆H(2): exactly zero; the residual H − H(2) (measured at ∼10−4 on this design) is correctable by the splice construction. Both H(2) and E[Y 2] are linear in the co-location entries on each reel, so each pair (s, t) carries an exact efficiency ratio: volatility gained per unit of hit rate spent. The HR-preserving volatility trade is explicit: raise a highefficiency pair (e.g., H1 adjacent to M1), lower a low-efficiency one (e.g., H1 adjacent to 9), net ∆H(2) = 0, net ∆E[Y 2] > 0.
CVwin targeting. The per-reel multiplier value distribution is the coarse CVwin control. At uniform ω per reel: CVwin ≈ 11 (the baseline). Introducing non-uniform stop values on reel 4 from {2, 3, 5} creates compound variance across count-2 windows. The distribution is the degree of freedom: [0×1, 2×2, 2×3, 14×5] gives CVwin ≈ 14.5; shifting stops toward ω = 3 ([0×1, 2×2, 5×3, 11×5]) gives CVwin = 15.0. RTP is preserved exactly across distributions because it reads only per-count means; CVwin reads per-count variances, which the distribution controls at frozen RTP. Reel 4 is concatenated K = 3 (Proposition 4.11) to provide 18 stops for this distributional control. The compound multiplier is computed by exact window enumeration.
The residual at fixed (RTP, HR). With counts fixed and the wild moat frozen (standard practice, ∼1.6% of filler cells), the diagonal s E[Ws2] is exactly invariant. The entire residual freedom is in the cross terms s≠ t E[WsWt], which are linear in the per-reel co-location entries (on each reel with the others fixed). Under the wild-moat convention, the cross terms account for 100% of the residual variance. The H(2)-preserving kernel moves change these cross terms at ∆H(2) = 0, giving CVwin control at frozen RTP and H(2), with the order-3 residual correctable by splice.
Decomposition check. At the final operating point: fHR = 0.33%. The game is deep in the premium regime: 99.7% of variance comes from win shape (CVwin), driven by the compound wild multiplier.
Remark 15.1 (Multipliers as a Volatility Lever). Value-bearing symbols such as wild multipliers provide a clean way to increase volatility without touching the hit rate. The maximum singleevent exposure is bounded by the product of the maximum per-reel multiplier, the maximum per-reel count, and the top pay value. In principle a co-occurrence with the premium symbol could further increase this exposure; the co-location matrix and the 3-gram targets (Section 6) make this computable exactly.
Increasing the frequency of the maximum-exposure event raises CVwin directly: the multiplier inflates the right tail of the conditional win distribution without changing symbol presence. This gives the designer a spectrum from a minimum wild exposure win (a single low-valued wild on the rarest reel) to the maximum compound exposure (all wild reels active, all at maximum value), with the frequency of each controlled by the per-reel presence pwild and the Section 11 value assignment.
However, the more RTP that is concentrated in rare compound events, the less realized RTP the player experiences per session. A player encountering only the common portion of the pay distribution will ruin faster, even though the long-run RTP is unchanged. Past a threshold, further concentrating RTP into the extreme tail has diminishing returns: a 1,000× win and a 10,000× win produce similar subjective impact, but the latter locks ten times the RTP into an event the player is unlikely to witness.
The quantities that capture this tradeoff are not volatility metrics but experiential ones: expected session length at a given bankroll, probability of triggering a feature within a budget, probability of reaching a target multiple of the initial bankroll before ruin. These questions are
50

computable from the payout distribution that the framework constructs. The CVwin coordinate tells the designer where the game sits on the spectrum between frequent-modest-wins and raremassive-wins; the experiential analysis tells the designer where it should sit for the target audience.
A practical illustration: in a ways game, ensuring that each reel can produce a full stack of the premium symbol (count = W ) means the player will occasionally see a full reel of the top symbol land on one or two reels. This teases the maximum-pay event—all reels stacking simultaneously— and gives the player a visible goal. But if the per-reel probability of a full stack is q, the probability of all R reels stacking is qR, which can be made astronomically small while keeping single-reel stacks common. The maximum-pay event contributes negligible RTP (it is too rare to matter), so nearly all RTP is concentrated in achievable wins. The game remains volatile—the CVwin is still high from the compound multiplier and the moderate wins—but the teased jackpot does not starve the player’s session.
15.7 Strip Construction
Each reel comprises quasi-contiguous clusters (Definition 4.1) of designed symbols (H1, wild, scatter) separated by moats of filler positions. The moat ensures that no window contains stops from two different designed symbols, so the additive formula meff = mnative + pwild · ceff holds exactly for designed symbols. Filler stops in the moat may share windows with the wild, so their effective contribution includes a multiplicative correction E[Xf · M ] that the additive formula understates. This is handled correctly by the exact window enumeration used for the compound multiplier (Section 11), not by the closed-form formula.
The filler arrangement is constructed by the splice method (Proposition 6.32). On each reel, two extreme-point filler arrangements are built sharing the same skeleton and counts: strip A (fillers spread as singles, maximizing H) and strip B (fillers grouped into runs, minimizing H). The target hit rate H∗ = 1/4.0 falls between the two extremes. The splice AnABnB at ratio nB/(nA + nB) hits the target at resolution O(1/n). For the present game, 2 copies of strip A and 1 copy of strip B (n = 3) achieve H = 1/4.0 with RTP drift 0.000pp.
The clustering distribution τ 2 of the H1 cluster may be chosen independently at each reel: minimizing τ 2 (uniform gaps) minimizes the marginal variance of the H1 paying event, while maximizing τ 2 (concentrated gaps) maximizes it. This is an additional volatility lever at frozen RTP and hit rate.
The full strips (per-reel L as in Table 2) are given in Appendix A. Remark 15.2 (Pipeline Speed). The entire pipeline — from an arbitrary game specification to verified strips — runs in under 17 seconds:
1. Count allocation (RTP targeting): bisection on the filler scale factor, ∼60 ms.
2. Strip construction: skeleton placement plus filler assignment, < 100 ms per reel.
3. Hit-rate targeting: build strips A and B, measure HR by exact inclusion–exclusion (∼1 ms), compute splice ratio. Total < 1 s.
4. CV targeting: distribution search over multiplier value assignments, each requiring an omega solve (∼1 s per candidate in Q-space). ∼10 s for the full enumeration.
5. Verification: exact enumeration of all windows per reel, < 1 s per reel.
Total wall time from specification to verified strips: under 17 seconds on commodity hardware. This compares to hours or days of Monte Carlo simulation in the current industry workflow, with no guarantee of convergence to the target.
51

15.8 Summary

Target Value Control

Section

RTP 64.00% Filler allocation + multiplier solve

5, 11

Hit rate 1 in 4.0 Filler clustering + Q flow construction 2, 6, 4

CVwin 15.0

Stop value distribution + concatenation 8, 11

σ

19.3 Derived from decomposition

8

fHR

0.33% Premium regime

8

Table 4: Three-coordinate targeting: three sequentially decoupled controls.

The entire construction—from specification to verified strips—runs in under 17 seconds on commodity hardware. Every step is closed-form or finite enumeration. No Monte Carlo simulation is required; no iterative search is performed. The framework targets all three design coordinates simultaneously through sequentially decoupled controls that compose in sequence, each preserving what the previous established, on a game with compound wild multipliers, scatter triggers, and mixed paying depths.
16 Design Abstraction and Extensions
A natural concern is that a constructive framework, by imposing mathematical structure, restricts the designer’s creative freedom. The opposite is true. When strips are constructed to order from specifications, the designer works in experience space rather than symbol space. The reel strip becomes a compiled artifact, not a hand-tuned one. This section illustrates the level of abstraction the framework supports.
16.1 Multi-State Games as Markov Chains
Modern slot games are multi-state: a base game, one or more free-game modes, pick bonuses, and progressive features, connected by trigger events. The game’s long-run behavior is a Markov chain over states, where each state has its own reelset and the transitions are determined by trigger probabilities. Remark 16.1 (Per-State Specification). In the present framework, each game state is an independent design problem. The designer specifies, for each state:
ˆ Per-symbol targets: (p, c, τ 2) per reel, controlling the event-level experience (how often each symbol appears, in what clusters, with what visual density).
ˆ Global targets: (RTP, hit rate, volatility) for that state, controlling the payout profile (how much of the house edge is returned, how frequently, with what variance).
The framework constructs each state’s reelset independently by Theorem 9.4. Let E[payouti] denote the expected payout per spin in state i, and let wi denote the wager in state i (wi = 1 for the base game, wi = 0 for free-spin states). The Markov chain’s stationary distribution π then determines

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every aggregate metric:

RTPgame =

i πi · E[payouti] , i πi · wi

(72)

Hgame = πi · Hi.
i

(73)

The denominator sums only the wagered states; free-spin states contribute expected payout to the numerator but no wager to the denominator, correctly capturing their effect on the overall return. The overall volatility is computable from the per-state variances and the transition structure: Vargame accounts for both within-state variance (per-spin variance at each state’s reelset) and between-state variance (the RTP differences between states, which produce session-level variance as the game transitions between high- and low-paying modes). Both components are functions of the per-state targets and the transition matrix, all specified by the designer.
The same aggregation applies within a feature. A free-game sequence with locking wilds passes through a sequence of conditional reelsets (one per locked-wild configuration). Each conditional reelset has its own per-symbol targets and global metrics. The feature’s aggregate RTP, hit rate, and volatility are computed from the sub-chain over conditional states. The designer specifies the experience at every level of the hierarchy: per-symbol within each state, per-state within each feature, per-feature within the game.

16.2 Free Games with Locking Wilds
Consider a free-games feature with locking wilds and retriggers, a common modern game structure:
1. Trigger : scatter symbols on the base game trigger N free spins. 2. Locking wilds: each wild that lands during free games remains locked in place for the remaining
spins, progressively improving the reelset. 3. Retrigger : scatter symbols during free games add additional spins. 4. Conditional reelsets: as wilds lock, the effective reelset changes. Each configuration of locked
wilds defines a distinct game state.
The designer specifies this feature entirely in experience space:
ˆ Base game RTP, hit rate, volatility (the everyday experience). ˆ Free-game trigger frequency (e.g., 1 in 200 spins). ˆ Free-game per-spin RTP (typically much higher than the base game). ˆ Retrigger probability per free-game set. ˆ Expected number of wilds locked by end of session. ˆ Overall game RTP (the regulatory requirement, typically 88%–96%).
The framework compiles these into reelsets:

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1. The overall RTP constraint determines the relationship between base-game RTP, free-game RTP, trigger frequency, average session length, and retrigger probability. This is a linear system in the per-state RTPs weighted by the Markov chain’s stationary distribution [13].
2. Each per-state RTP, hit rate, and volatility target is realized by a reelset constructed via Theorem 9.4.
3. The locking-wild progression is a sequence of conditional reelsets, each differing from the previous by the addition of locked wilds at specific positions. Since wild positions are skeleton positions, each conditional reelset is a known modification of the base free-game skeleton, and the framework’s per-state constructibility applies to each.
4. The scatter trigger frequency is a per-symbol coverage target (pscatter), achievable by the skeleton constructibility theorem (Theorem 4.9).
The designer never touches a reel strip. The experience specification determines the mathematical targets, and the framework constructs the strips that realize them.
16.3 What the Designer Controls
The framework supports the following design axes, all independently specifiable:
ˆ Per-symbol hit rate: how often each symbol appears in a window (Layer 1, coverage p).
ˆ Per-symbol visual density: how many stops of each symbol appear when it hits (Layer 2, conditional count c).
ˆ Per-event volatility: the spread of per-symbol counts across windows (Layer 3, count variance τ 2).
ˆ Near-miss frequency: the probability of a symbol appearing on k − 1 of k required reels, controlled by per-reel coverage targets.
ˆ Total RTP : the house edge, targeted to arbitrary precision via filler allocation.
ˆ Total hit rate: the win frequency, targeted via the co-location matrix.
ˆ Total volatility: the payout variance, targeted via count allocation on the RTP iso-surface and fine-tuned by clustering and co-location rotations.
ˆ Feature triggers: scatter frequencies, bonus trigger rates, retrigger probabilities, all specified as per-symbol coverage targets.
ˆ Multi-state payout profiles: per-state RTP, hit rate, and volatility, with the overall game RTP constrained by the Markov chain.
Each axis is independently adjustable within its pre-computable attainable range. The “feel” of the game is not lost by the mathematical framework; it is parameterized by it. Every design intuition (“I want the free games to feel generous but volatile,” “I want near-misses on the premium symbol,” “I want the base game to be tight with frequent small wins”) translates into a numerical target that the framework can realize.
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17 Conclusion
This paper presented the first constructive theory of reel strip design from mathematical specifications. The rearrangement invariant decomposes the per-symbol design space into three sequentially decoupled layers (coverage, conditional count, count distribution), each with a characterized attainable set. The existence and constructibility theorem guarantees that any specification within these ranges can be realized by an actual reel strip. The filler system targets total RTP via the intermediate value theorem on a convex allocation polytope. The co-location matrix captures the order-2 window statistics that control hit rate, and the achievable set in (H, E[Y 2]) is reachable by exact splice linearity, so any interior target is reached by mixing. The full construction theorem assembles these results: for any per-symbol targets and any global targets (RTP, hit rate, volatility) in the characterized achievable set, a reel strip realizing the full specification exists and can be constructed deterministically. No simulation is required; targeted refinement replaces undirected search.
Wild symbols are handled by a linear correction to each paying symbol’s effective count, with a closed-form adjustment for all-wild ways and a pre-computable degenerate window adjustment to the co-location margin laws. Symbols carrying numerical values (multiplier wilds, cash scatters) are handled by a linear system targeting per-count expected values, with concatenation providing arbitrary distributional resolution. The worked example (Section 15) constructs a complete game targeting (RTP, hit rate, CVwin) = (64%, 1/4, 15) with compound wild multipliers from {2, 3, 5}, demonstrating all three coordinate controls operating simultaneously. The framework supports multi-state games as Markov chains, with each state’s reelset constructed independently and the aggregate metrics computed from the stationary distribution.
17.1 Open Problems
Several directions remain for future work. Lines-game hit rate and volatility. The lines correction factor λs,k (Section 13) resolves RTP
for lines games. Hit rate and volatility under lines evaluation depend on per-cell, per-row symbol placement and on correlations between paylines sharing cells. Extending the co-location framework to capture this row-level structure is the natural next step for lines-game support.
Hold-and-spin features. Hold-and-spin mechanics (“coin” features, lightning links) involve a sequential process: symbols land, lock in place, and respins continue until no new symbols appear. The locked-symbol accumulation is a spatial point process on the reel grid, and the feature’s termination condition (no new landings) creates a geometric-like distribution over session lengths. Modeling the feature’s RTP, hit rate, and volatility requires extending the present framework to handle correlated, state-dependent reelsets evolving within a single feature activation.
Generalized evolving-state features. The locking-wild free-game example (Section 16) illustrates one pattern of state evolution. A general theory would treat each feature as a controlled Markov chain over reelset configurations, with the designer specifying the transition structure and per-state experience targets. The framework’s per-state constructibility (Theorem 9.4) provides the foundation, but the joint optimization of transition probabilities and per-state targets to achieve aggregate feature metrics (total feature RTP, expected session volatility, retrigger dynamics) remains an open design problem.
Pair moments under a skeleton. The attainable set of pair moments E[csct] under a skeleton is exactly decided for free reels but open for the skeletoned case at production strip lengths. Closing this certifies the hit-rate band and bounds the cross-term share of E[Y 2].
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17.2 Closing Remarks
The existing literature treats reel strip design as a search problem: propose a strip, evaluate it, iterate. This paper reframes it as a construction problem: specify the targets, verify feasibility, build deterministically. The shift from search to construction is made possible by the structural decomposition of the design space into functionally decoupled, characterized layers, each with existence guarantees and pre-computable bounds.
The practical implication is that the game designer’s task changes from trial-and-error strip adjustment to specification of the player experience. The mathematical framework does not replace design intuition; it provides a compiler that translates intuition into guaranteed-correct implementations.
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