# Existence and Construction of Reel Strips from Mathematical Specifications -- plain text, part 1 of 5

Pages 1-14 of 59.
Sections: Introduction; Definitions and the Rearrangement Invariant; Three-Layer Decomposition; Existence and Constructibility

Next part: https://gamemathemagics.com/papers/reel-strip-construction.part2.txt
Whole document in one file: https://gamemathemagics.com/papers/reel-strip-construction.txt (34,595 words)
Original PDF: https://gamemathemagics.com/papers/reel-strip-construction.pdf

---
Existence and Construction of Reel Strips from Mathematical Specifications
Christian Fenner Independent Researcher
Las Vegas, NV fenner.d.christian@gmail.com
August 28, 2026

Contents

1 Introduction

4

2 Definitions and the Rearrangement Invariant

6

2.1 Game Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Two Consequences of the Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Three-Layer Decomposition

9

3.1 Layer 1: Hit Rate via Window Coverage . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Layer 2: RTP via Conditional Count . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Layer 3: Volatility via Count Distribution . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 The Sequential Decoupling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Existence and Constructibility

12

4.1 Quasi-Contiguous Clusters and Symbol Independence . . . . . . . . . . . . . . . . . 12

4.2 Reel Length as a Derived Quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3 The Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Filler System and Total RTP

15

5.1 Filler Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.2 Filler RTP Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3 Multilinearity of Filler RTP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.4 RTP Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.5 Total RTP Targeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.6 Side Effects and Separation of Concerns . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 The Co-location Matrix

17

6.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.2 Windows as Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.3 Design Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.4 RTP Invisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.5 The Transposition Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1

6.6 Row-Sum-Preserving Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.7 How Q Enters the Hit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.8 Exact Order-2 Hit-Rate Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.9 How Q Enters Payout Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.10 The Attainable Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.11 Constructibility from Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.12 Reachability by Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.13 The E[c2] Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Hit-Rate Targeting via Co-location

26

7.1 Filler Presences: Fixed and Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.2 The Off-Diagonal as the Hit-Rate Handle . . . . . . . . . . . . . . . . . . . . . . . . 27

7.3 Achievable Hit-Rate Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.4 RTP Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.5 Order-3 Residual and Directed Correction . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Payout Volatility

28

8.1 Variance Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8.2 Exact CVwin from Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.3 Pre-Computable CVwin Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.4 Three Controls for CVwin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.5 Variance from Count Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.6 Concentration Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8.7 Invariant Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9 Full Construction Theorem

32

9.1 The Design Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9.2 The Target Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9.3 The Full Construction Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9.4 Pre-Computability of the Achievable Set . . . . . . . . . . . . . . . . . . . . . . . . . 35

9.5 Integrality and Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

9.6 Infeasibility Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10 Wilds

36

10.1 Linear Separability of Mean Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.2 Additive Correction to the Co-location Matrix . . . . . . . . . . . . . . . . . . . . . 37

10.3 RTP and the Moat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

10.4 Degenerate Window Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10.5 Wild as a Skeleton Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10.6 Framework Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

11 Value-Bearing Symbols

38

11.1 Definition and Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.2 Linear System for Per-Count Averages . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.3 Attainable Set and Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11.4 Distributional Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11.5 Expected Value of the Value-Bearing Component . . . . . . . . . . . . . . . . . . . . 41

11.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2

12 Scatters

42

12.1 Count Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.2 Scatter RTP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.3 Constrained Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12.4 Volatility Tunability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12.5 Full Payout Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12.6 Integration with the Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

13 Lines Correction

44

13.1 Per-Payline Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

13.2 The Correction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

14 Comparison to Prior Work

45

15 Worked Example: Construction of a Complete Game

46

15.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

15.2 H1 Premium Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

15.3 Wild Multiplier as Effective CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

15.4 Filler Allocation and Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

15.5 RTP Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

15.6 Three-Coordinate Targeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

15.7 Strip Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

15.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

16 Design Abstraction and Extensions

52

16.1 Multi-State Games as Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 52

16.2 Free Games with Locking Wilds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

16.3 What the Designer Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

17 Conclusion

55

17.1 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

17.2 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A Reel Strip Listing

57

3

Abstract
The reel strip design problem asks whether a slot machine’s symbol sequences can be constructed from mathematical specifications rather than found by search. We prove that they can. A rearrangement invariant decomposes the per-symbol design space into three sequentially decoupled layers (hit rate, RTP contribution, and payout volatility), each with a characterized attainable set. A co-location matrix captures the order-2 cross-symbol window statistics that control total hit rate, with a reachable by mixing achievable set established by exact splice linearity. The achievable set of (H, CVwin) pairs at fixed counts is reachable by mixing: any interior point is reached by mixing extreme-point strips, with precision linear in strip length. Payout volatility decomposes exactly into a hit-rate component and a conditional win-shape component CVwin, targetable through clustering distribution, directed filler arrangement, and label permutation at frozen RTP and hit rate. The main result is a full construction theorem: for any per-symbol targets and any global targets (RTP, hit rate, volatility) in a characterized achievable set, with constructibility decidable from the co-location matrix before any strip is built, a reel strip realizing the full specification exists and can be constructed deterministically. Wild symbols require only a linear correction. Scatter symbols have count probabilities characterized by the Poisson-binomial distribution with multilinear achievable ranges. Symbols carrying numerical values (cash scatters, multiplier wilds) are handled by a linear system that separately targets per-count expected values with full distributional control. The framework supports multi-state games as Markov chains, with each state’s reelset constructed independently and aggregate metrics computed from the stationary distribution. A worked example constructs a 5 × 3 ways-pay base game with compound wild multipliers, scatter triggers, and mixed paying depths, targeting RTP = 64%, hit rate = 1/4, and CVwin = 15 simultaneously through three sequentially decoupled controls in under 17 seconds on commodity hardware.
1 Introduction
A slot machine is a set of independently spinning reels, each displaying a window of symbols. The cyclic sequence of symbols on each reel, the reel strip, determines every mathematical property of the game: how often the player wins, how much the house retains, and how the wins are distributed between frequent small payouts and rare large ones. These properties emerge from the combinatorial interaction of the strips across reels over an infinite horizon, and casinos depend on their exactness. The game mathematician’s task is to design strips that hit precise regulatory and business targets while delivering an experience that makes losing money enjoyable, a tension between mathematical constraint and player psychology that the existing literature has addressed only partially.
The reel strip design problem asks: given a complete set of performance targets at the persymbol, per-event level, does a reel strip exist that realizes all targets simultaneously, and can it be constructed?
The existing literature formulates this as a global optimization problem. The available treatments of slot mathematics [16] are descriptive rather than constructive. Balabanov et al. [2] apply Genetic Algorithms to search for a strip achieving a target total RTP, using Monte Carlo simulation as the fitness function. The same group [3] extends this with Discrete Differential Evolution optimizing RTP, prizes equalization, and symbol diversity as a multi-criteria objective. Keremedchiev et al. [4] replace Monte Carlo with exact full-cycle computation. Kamanas et al. [5], from a separate research group, introduce Variable Neighborhood Search with two local search operators, achieving the current state of the art. As of 2025, the same group continues to present VNS-based RTP optimization as the frontier of the field [6], with separate control of hit rate and volatility listed as open future work. Separately, Groote et al. [7] model slot machines as probabilistic process specifications and compute exact RTP via quantitative model checking — rigorous evaluation of an existing machine, not construction from specifications. In all optimization approaches, the
4

target is a single number or a weighted multi-criteria objective, and the method is iterative search over symbol distributions. Kamanas et al. note that their algorithm’s behavior “in the case where special symbols appear in reels (e.g., wild, scatter) is also unknown.”
We present the first constructive theory of reel strip design. The framework decomposes the design space into functionally decoupled controls that target global RTP, total hit rate, and total volatility simultaneously. This solves the problem the existing literature addresses, while extending control to the per-symbol, per-event level: individual RTP contributions, individual hit-rate frequencies, near-miss probabilities, and per-event volatility profiles, all specified independently and all guaranteed constructible. Every feasibility bound is computable from the specification alone. When targets are feasible, a reel strip realizing the full specification exists and can be constructed deterministically. No simulation is required; targeted refinement replaces undirected search.
The key tool is a rearrangement invariant (Theorem 2.7) that decomposes the per-symbol design space into three sequentially decoupled layers, each with a characterized attainable set. At the global level, the total RTP is targeted via filler count allocation (with existence by the intermediate value theorem on a convex polytope), the total hit rate is targeted via the co-location matrix (an order-2 window statistic with provable bounds on the uncaptured higher-order residual), and the total volatility is targeted by selecting the operating point on the RTP iso-surface that achieves the desired variance, with the co-location matrix realizing the hit rate at that point. The following are the specific contributions.
5

Contributions.
1. A rearrangement invariant and two structural consequences that decouple hit rate from RTP (Section 2).
2. A three-layer independence theorem with full characterization of the attainable set per layer, at the per-symbol, per-event level (Section 3).
3. An existence and constructibility theorem for the reel strip skeleton from arbitrary feasible per-symbol specifications (Section 4).
4. Multilinearity of RTP across reels, with a convex-path existence proof for any target RTP within a pre-computable band (Section 5).
5. The co-location matrix as a complete order-2 characterization of cross-symbol window statistics, with reachability of the achievable set established by exact splice linearity (Section 6). The E[c2] lattice provides a closed-form residual for fine volatility control.
6. Hit-rate targeting via the co-location matrix at frozen RTP, with the order-3+ residual measured at ∼10−4 on the worked example and correctable by splice (Section 7).
7. Payout volatility decomposition into hit-rate and conditional win-shape (CVwin) components, with exact CVwin computation from design parameters and three per-reel controls at frozen RTP and hit rate (Section 8).
8. A full construction theorem: for any per-symbol targets (p, c, τ 2) within their attainable ranges and any global targets (RTP, hit rate, volatility) in a characterized achievable set, with constructibility decidable from the co-location matrix before any strip is built, a reel strip realizing the full specification exists and can be constructed deterministically (Section 9).
9. A linear system for value-bearing symbols (symbols carrying numerical values such as cash amounts or multipliers), independently targeting per-count expected values with n − K free parameters for distributional control (where K is the number of count levels) (Section 11).
10. A complete scatter theory with Poisson-binomial count probabilities, multilinear achievable ranges, rational-function constrained ranges, and additive RTP decomposition with the wayspay system (Section 12).
11. A worked example constructing a complete 5×3 ways-pay game with compound wild multipliers, targeting (RTP, hit rate, CVwin) = (64%, 1/4, 15) simultaneously through three sequentially decoupled controls, verified by exact enumeration on commodity hardware (Section 15).
2 Definitions and the Rearrangement Invariant
Let Ω denote a finite set of symbols and let R denote the number of reels in a slot game.
Definition 2.1 (Reel Strip). A reel strip for reel i is a cyclic sequence ri = (ri,1, ri,2, . . . , ri,Li) of symbols drawn from Ω, where Li is the reel length. Indices are taken modulo Li.
Definition 2.2 (Spin). A spin on reel i selects a position j ∈ {1, . . . , Li} uniformly at random. The reels are sampled independently. By convention, the selected position j corresponds to the top entry of the display window.
6

Definition 2.3 (Window). The window at position j on reel i with display height W is the multiset of W consecutive symbols {ri,j, ri,j+1, . . . , ri,j+W −1} (indices mod Li). There are exactly Li distinct windows on reel i.
For a symbol s ∈ Ω on reel i, let ni(s) denote the number of positions occupied by s on the strip (the stop count).

Definition 2.4 (Window Coverage). Let ai(s) denote the number of windows on reel i that contain at least one instance of symbol s. The window coverage is:

p(is)

=

a(is) . Li

(1)

This is the probability that a uniformly sampled window contains symbol s.

Definition 2.5 (Conditional Count). The conditional count of symbol s on reel i is the expected number of instances of s visible in a window, given that at least one is visible:

ci(s)

=

W · n(is) a(is)

.

(2)

The numerator W · n(is) is the total number of sightings of s across all Li windows: each of the ni(s) stops appears in exactly W windows (those whose top position falls within W positions above the
stop), regardless of how the stops are arranged on the strip.

Definition 2.6 (Symbol Percent and Mean Count). The symbol percent of s on reel i is the fraction

of reel positions occupied by s:

spi(s)

=

n(is) . Li

(3)

The mean count is the expected number of instances of s in a uniformly sampled window:

m(is)

=

W

·

sp(is)

=

W

· n(is) . Li

(4)

Since sp depends only on the stop count n and the reel length L, and W is a fixed display parameter, m is determined entirely by the composition of the strip, not the arrangement of symbols within it.

Theorem 2.7 (Rearrangement Invariant). The mean count m = p · c, and this product is invariant under any rearrangement of the stops of s on the strip. Rearrangement trades p against c along the rectangular hyperbola p · c = m.

Proof. From Definitions 2.4, 2.5, and 2.6:

a Wn Wn

p · c = · = = m.

(5)

La L

The quantity a (the number of non-empty windows) cancels. Since m = W n/L depends only on n, W , and L, none of which change when stops are repositioned, the product p · c is invariant.

Rearranging the stops of a symbol changes which windows contain it (a, and hence p) and how many copies appear in each non-empty window (c). Clustering stops increases c (more per window) while decreasing p (fewer distinct windows reached). Spreading stops decreases c toward 1 while increasing p toward W n/L. The product is always m.

7

2.1 Game Evaluation
A slot game evaluates the outcome of a spin by comparing the symbols visible in each reel’s window against a paytable. We consider the standard ways evaluation, in which every combination of symbol positions across reels constitutes a potential winning alignment.

Definition 2.8 (Paytable and OAK Ladder). A paytable assigns to each symbol s ∈ Ω a sequence of pay values v2(s), v3(s), . . . , vR(s), where vk(s) is the payout for a k-of-a-kind (kOAK) win. Wins are left-anchored : a kOAK of symbol s requires s to be present in the window on each of reels 1 through
k, and absent from the window on reel k + 1 (the blocker reel). An ROAK has no blocker.

Definition 2.9 (Ways Count). In a ways game, the number of winning alignments for a kOAK of

symbol s on a given spin is the product of the per-reel counts of s across the k paying reels. If reel

i shows xi instances of s in its window, the ways count is

k i=1

xi.

The

payout

is

vk(s) ·

k i=1

xi.

Definition 2.10 (Return-to-Player). The return-to-player (RTP) is the expected payout per unit

wagered over the infinite horizon. The total RTP is the sum of contributions from all symbols and

all OAK levels:

R

RTP =

RTP(ks),

(6)

s∈Ω k=2

where RTPk(s) is the expected per-spin payout from kOAK wins of symbol s, divided by the wager.

2.2 Two Consequences of the Invariant

Proposition 2.11 (Ways RTP is First-Order). In a ways game, the RTP contribution of a kOAK

win for symbol s is:

k

RTPk(s) = vk(s) · mi(s) · (1 − p(ks+)1), k < R,

(7)

i=1

and RTP(Rs) = vR(s) ·

R i=1

m(is).

RTP depends only on per-reel means mi (on the paying reels)

and presence pk+1 (on the blocker reel). It never depends on within-window co-occurrence, on the

conditional count c directly, or on any higher moment of the count distribution.

Proof. Let Xi denote the count of symbol s in the window on reel i. The payout of a kOAK is

vk ·

k i=1

Xi

when

s

is

present

on

reels

1

through

k

and

absent

on

reel

k + 1.

Since

the

reels

are

sampled independently (Definition 2.2):

k

k

k

E Xi = E[Xi] = mi.

(8)

i=1

i=1

i=1

The blocker reel k + 1 contributes the binary test P (Xk+1 = 0) = 1 − pk+1: it asks whether symbol s is present, not how many instances appear. Since each mi is rearrangement-invariant (Theorem 2.7), the RTP is independent of how symbols are arranged within any reel.

Corollary 2.12 (Reels 1–2 Are RTP-Free). No standard paytable awards a 1OAK. Therefore p1 and p2 never appear as blocker terms in (7), and reels 1 and 2 enter the RTP formula only through m1 and m2, which are rearrangement-invariant. The arrangement of symbols on reels 1 and 2 may be chosen freely without affecting any RTP quantity.

8

Corollary 2.13 (Blocker-Safe Clustering). A filler symbol f with minimum pay depth df uses reel k as a blocker for its (k−1)OAK event, for each k ∈ {df + 1, . . . , R}. Clustering f on reel i changes p(if) (at fixed m(if) = pi · ci), which changes the blocker term (1 − p(if)) if i is a blocker reel for f . Consequently, clustering f on reel i is RTP-preserving if and only if i is not a blocker reel for f :
ˆ Depth-2 fillers (df = 2) block on reels 3, 4, . . . , R. Clustering is RTP-safe on reels 1–2.
ˆ Depth-3 fillers (df = 3) block on reels 4, 5, . . . , R. Clustering is RTP-safe on reels 1–3.
In general, a filler with minimum pay depth df may be clustered freely on reels 1 through df without affecting any RTP quantity. This extends Corollary 2.12 from a per-reel statement to a per-symbol, per-reel statement: the relevant condition is not whether the reel is a blocker, but whether it is a blocker for the symbol being clustered.
Remark 2.14 (Constraint Chain on Blocker Reels). On reels 1 and 2, which never serve as blockers, the window coverage p does not appear in the RTP formula. Both the stop count and the arrangement are free from RTP’s perspective.
On reels k ≥ 3, the coverage pk enters (7) as the blocker term (1 − pk). The constraints cascade:
1. Fix p: preserves the blocker contribution to RTP and preserves hit rate.
2. Fix c (at that p): since m = p·c, fixing both p and c fixes m, which preserves the paying-reel contribution to RTP. RTP is now fully frozen.
3. Vary the count distribution: at fixed p and c, different physical arrangements produce different distributions over {1, . . . , W } with the same mean c. This changes per-event payout volatility while preserving both hit rate and RTP.
This is the mechanism by which all three layers remain functionally decoupled even on blocker reels.

3 Three-Layer Decomposition
The rearrangement invariant establishes that RTP is fixed under rearrangement. This section characterizes the full per-symbol design space: hit rate (Layer 1, controlled by coverage), RTP contribution (Layer 2, controlled by stop count at fixed coverage), and per-event volatility (Layer 3, controlled by count distribution at fixed coverage and stop count). These three layers are functionally decoupled at the per-symbol, per-event, per-reel level. Global game targets (total RTP, total hit rate, total volatility) are addressed in Sections 5–8.

3.1 Layer 1: Hit Rate via Window Coverage

For the standard left-anchored OAK ladder on R reels, the hit-rate frequencies are determined entirely by the window coverages p1, . . . , pR.

Proposition 3.1 (Hit-Rate Equations). The probability of an exact kOAK of symbol s is:

k
Hk(s) = p(is) · (1 − pk(s+)1),
i=1
R
HR(s) = pi(s).
i=1

k = 2, . . . , R − 1,

(9) (10)

Hit rates are functions of p alone.

9

Proposition 3.2 (Suffix-Sum Solution). Define Sk =

j≥k Hj =

k i=1

pi

(the

probability

that

the

chain reaches at least reel k). Then pk = Sk/Sk−1 for k ≥ 3, and p1 · p2 = S2, admitting one degree

of freedom. This degree of freedom does not affect any OAK hit rate (since Hk for k ≥ 2 depends

on p1p2 = S2, not on p1 and p2 individually). It controls the near-miss rate: the probability that

symbol s appears on reel 1 but not reel 2 is H1(s) = p1(1 − p2) = p1 − S2, which ranges over [0, 1 − S2]

as p1 ranges from S2 to 1. The split is therefore a tease-frequency control with no effect on any

paying outcome. Notably, the framework makes near-miss rates auditable: they are explicit design

parameters with known values, not emergent properties of a heuristic strip. This transparency is

relevant to responsible-gambling considerations, where near-miss frequency is a documented design

concern.

Remark 3.3 (Generalized Events). The OAK ladder is one event family. Any R-tuple of tri-state
constraints (symbol present / symbol absent / unconstrained) defines an event whose hit rate
factors as i∈SYM pi · i∈NOT(1 − pi). Near-miss patterns and non-contiguous events are handled identically.

3.2 Layer 2: RTP via Conditional Count

Two reels with identical p produce identical hit rates regardless of internal arrangement. They differ in RTP through c.
Theorem 3.4 (Independent RTP Control). For fixed window coverages p1, . . . , pR, the RTP (7) can be varied continuously by changing the stop counts n1, . . . , nR (and hence mi = W ni/Li) without altering any hit rate.
Proof. Increasing ni at fixed pi increases mi = pi · ci (since ci = W ni/ai increases with ni at fixed ai), which increases the RTP contribution of every paying rung involving reel i via (7). Since hit rates (9)–(10) depend only on p, they are unaffected.

Proposition 3.5 (Attainable Conditional Count). For a symbol with n stops on a reel of length L with display height W , the attainable conditional counts from contiguous-block placement are:

Wn

ck

=

n

+

k(W

−

, 1)

k = 1, 2, . . . , n,

(11)

where k is the number of contiguous blocks, each separated by at least W − 1 other symbols. Each block of nj stops covers nj + W − 1 windows (the block plus two endcaps of W − 1 windows, which overlap with the block itself ). Total coverage is a = n + k(W − 1). At k = 1 (one contiguous block), c is maximized at W n/(n + W − 1), approaching W as n → ∞. At k = n (all stops isolated), a = nW and c = 1. The attainable set is dense in [1, W ) as L → ∞, since for any rational target c∗ = p/q ∈ [1, W ), a choice of n making W n/c∗ an integer achieves c∗ exactly.

Corollary 3.6 (Attainable RTP Range at Fixed Hit Rate). At fixed window coverages p1, . . . , pR (fixing all hit rates), the RTP contribution of a kOAK of symbol s ranges over:

k

k

vk · (1 − pk+1) · pi ≤ RTPk(s) ≤ vk · (1 − pk+1) · pi · cmax,i,

i=1

i=1

(12)

where cmax,i = W ni/(ni + W − 1) is the maximum attainable conditional count on reel i. The lower bound corresponds to fully spread placement (c = 1, m = p). The upper bound corresponds to fully contiguous placement (c = cmax, m = p · cmax). This range is computable before construction and provides a feasibility check: a target per-symbol RTP at a given hit rate is achievable if and only if it falls within these bounds.

10

3.3 Layer 3: Volatility via Count Distribution
Definition 3.7 (Conditional Count Distribution). For symbol s on reel i, let Ci(s) denote the random variable giving the count of s in a randomly sampled window, conditioned on the window being non-empty. The distribution of C lives on {1, 2, . . . , W }.

Proposition 3.8 (Count Distribution at W = 3). At W = 3, the distribution of C on {1, 2, 3} is determined by (c, τ 2) via:

P (C

=

3)

=

1 2

(τ

2

+

c2

−

3c

+

2),

(13)

P (C = 2) = (c − 1) − 2P (C = 3),

(14)

P (C = 1) = 1 − P (C = 2) − P (C = 3).

(15)

The variance is bounded:

τm2 in = (c − ⌊c⌋)(⌈c⌉ − c),

(16)

τm2 ax = (c − 1)(W − c).

(17)

Proof. Three unknowns (P1, P2, P3) subject to three linear constraints: Pj = 1, jPj = c, j2Pj = τ 2 + c2. The system is full-rank; solving gives the stated formulas. The variance bounds
follow from the support constraint Pj ≥ 0: minimum variance concentrates mass at the integers
nearest c; maximum variance places mass at {1, W } only.

Proposition 3.9 (Attainable Payout Variance). For a kOAK event of symbol s with pay value vk,

the expected payout is:

k

E[Yk] = vk · (1 − pk+1) · mi,

(18)

i=1

which depends only on first moments (mi and pk+1). The second moment of the payout is:

k
E[Yk2] = vk2 · (1 − pk+1) · m2,i,
i=1

(19)

where m2,i = pi · (τi2 + c2i ) is the unconditional second moment of the count on reel i. Since reels are independent, the product of second moments factors across reels. The payout variance is:

Var(Yk) = E[Yk2] − E[Yk]2.

(20)

At fixed p and c (fixing hit rate and RTP), the per-reel τi2 takes values in a finite, computable set determined by the gap-pattern compositions of the cluster, contained within the moment-problem envelope [τm2 in, τm2 ax] (Proposition 3.8). Each realizable τi2 gives a distinct per-event payout variance. The designer selects the payout volatility within this interval as an independent third specification:
it does not affect the hit rate (a function of p alone) or the RTP (a function of m = p · c alone).

Remark 3.10 (Higher Display Heights). At W > 3, the support {1, . . . , W } has more values than constraints, introducing additional degrees of freedom. This provides further design flexibility without affecting the independence result, which relies only on the fact that τ 2 does not enter the RTP formula.

11

Remark 3.11 (Scatter Scope). The first-order sufficiency of Proposition 2.11 holds for pay functions multilinear in per-reel counts, which includes the entire OAK ladder. Scatter symbols, which pay based on how many reels show the symbol (a sum, not a product), have pay functions nonlinear in the total count. For scatters, τ 2 enters the RTP. However, scatters in practice are placed as isolated singles (c = 1), in which case the per-reel count conditioned on presence is degenerate at 1 (τ 2 = 0), and the scatter pays reduce to elementary symmetric functions of the per-reel presences pi. Under this convention, first-order sufficiency is restored.
3.4 The Sequential Decoupling Theorem
Theorem 3.12 (Three-Layer Sequential Decoupling). For pay functions multilinear in per-reel counts (the OAK ladder), the following three quantities can be specified independently per symbol per reel:
1. Window coverage p controls hit rate (Proposition 3.1). Coverage and conditional count are independently specifiable: the designer targets both p and c, and the construction derives n = pcL/W and selects the cluster geometry accordingly (Proposition 3.5). The invariant m = p · c = W n/L determines n; it does not couple p to c.
2. Conditional count c controls per-event RTP at fixed p (Theorem 3.4). Changing the number of stops n (and hence c = W n/a) changes m = p · c, which changes RTP, while p (and hence hit rate) remains unchanged.
3. The conditional count distribution controls per-event payout volatility at fixed p and c (Proposition 3.9). Different physical arrangements with the same p and c produce different distributions over {1, . . . , W }, and hence different per-reel variances τi2 and different payout variances. Since the RTP formula reads only mi = pi · ci (a first-order moment) and the hit-rate formula reads only pi, the count distribution is an independent third control.

4 Existence and Constructibility
Sections 2 and 3 characterized the per-symbol, per-event design space: what can be specified, what ranges are attainable, and why the three layers are functionally decoupled. This section proves that any specification within those ranges can be realized by an actual reel strip.

4.1 Quasi-Contiguous Clusters and Symbol Independence

Definition 4.1 (Quasi-Contiguous Cluster). A quasi-contiguous cluster of symbol s is a sequence of n stops on the reel in which consecutive stops are separated by at most W − 1 positions. At this spacing, the window neighborhoods of consecutive stops overlap, so the cluster produces a contiguous range of non-empty windows. The internal gaps between consecutive stops may be occupied by other symbols.

Proposition 4.2 (Cluster Coverage and Conditional Count). A quasi-contiguous cluster with span S (the distance from first stop to last) produces a contiguous range of a = S+W non-empty windows, extending W − 1 positions beyond each end. The window coverage is p = a/L = (S + W )/L, fixed by the span and the reel length. The conditional count is:

Wn

c=

,

(21)

S+W

12

which varies with the number of stops n packed into the span. More stops at fixed span increases c; fewer decreases it. The cluster contains S + 1 − n internal gap positions available for other symbols.
The characterization is uniform: p is determined by the span, c by the ratio of stops to coverage, and the number of internal gaps by the difference. The fully contiguous case (gaps = 1, span = n−1) gives a = n + W − 1 and zero gap positions. The maximally spread case (gaps = W − 1, span = (n − 1)(W − 1)) gives the lowest c and the most gap positions.
Definition 4.3 (Separation). Two clusters on the same reel are separated if at least W −1 positions not belonging to either cluster lie between them. Under separation, no window of height W contains stops from both clusters.
Proposition 4.4 (Symbol Independence). The count of symbol s in any window depends only on s’s own stop positions. The statistics ps, cs, and τs2 are therefore unconditionally decoupled across symbols: each is determined by its own stops alone, regardless of whether clusters are separated.
Proof. The count of s in window {j, j+1, . . . , j+W −1} is the number of those positions occupied by stops of s. This is a function of s’s positions alone. Other symbols’ stops do not affect s’s count in any window.
Remark 4.5 (Role of Separation). Separation is a design convenience, not a mathematical necessity for symbol independence. It ensures Q[s][t] = 0 for separated designed pairs, simplifying the colocation analysis. Dropping separation tightens the minimum reel length bound (22) to L ≥ ns and returns the designed-designed block of Q as an additional design handle. The framework supports both separated and non-separated designs.
Proposition 4.6 (Interleaving). Let symbol A be placed in a quasi-contiguous cluster with span SA, stop count nA, and G = SA +1−nA internal gap positions. A guest symbol B placed exclusively in A’s gap positions is characterized by the same mathematics:
1. B’s available positions are the G gap positions. B’s maximum stop count is nB ≤ G.
2. B’s coverage is a subset of A’s coverage range. No additional reel space is consumed; B lives entirely within A’s footprint.
3. B’s conditional count is determined by B’s own span within the gap positions: cB = W nB/(SB+ W ), where SB is B’s span within the gaps. The same span-determines-p, stops-determine-c characterization applies.
4. A’s statistics are unaffected by B’s presence: they occupy disjoint positions and A’s count in any window depends only on A’s own stops (Proposition 4.4).
The characterization is recursive: B’s internal gaps can host a third symbol C, subject to the same mathematics. The separation cost (W − 1 positions) is paid once for the outermost cluster, not per symbol.
4.2 Reel Length as a Derived Quantity
The reel length L is not a design input. It is derived from the specification.
Proposition 4.7 (Maximum Symbol Percent). A symbol with target (p, c) on a reel of length L occupies exactly n = pcL/W positions, giving a symbol percent of sp = pc/W = m/W . At the maximum attainable c (one contiguous block, cmax = W n/(n + W − 1) < W ), the symbol percent is strictly less than p. The footprint of a single contiguous block is n positions; with k blocks requiring k(W − 1) separation positions, the total footprint is n + k(W − 1).
13

Each symbol s with ks quasi-contiguous clusters, totaling ns stops, requires ns positions for its stops and at least ks(W − 1) positions of intervening symbols to ensure separation between clusters. Without interleaving, the minimum reel length is:

L ≥ ns + K · (W − 1),

(22)

s∈Ω

where K is the total number of clusters across all symbols. With interleaving (Proposition 4.6), the bound is tighter: guest symbols occupy positions that count toward the host’s gaps, reducing total footprint. Additionally, the total symbol percent of designed symbols must leave room for fillers:

sp(s) =

ns < 1.

L

s∈Ωdesigned

s∈Ωdesigned

(23)

Since virtual reel strips have no physical length constraint [1], L can be chosen as large as needed to satisfy both bounds.

Proposition 4.8 (Integrality via LCM). Realizing a target (p, c) for symbol s requires a = pL and n = pcL/W to be positive integers. For a set of symbols with rational targets, let D be the least common multiple of all required denominators. Then L = D (or any multiple of D) simultaneously satisfies all integrality constraints.

Proof. Each rational target p(s) = as/bs and c(s) = es/fs requires L to be a multiple of bs (so that a = p(s)L is an integer) and a multiple of bsfs/ gcd(W, esbs) (so that n = p(s)c(s)L/W is an integer).
Taking D = lcm of all such denominators across all symbols gives the result.

4.3 The Existence Theorem
Theorem 4.9 (Existence and Constructibility of the Skeleton). Let each symbol s ∈ Ω be assigned per-reel targets (p(is), c(is)) with c(is) ∈ [1, W ). Then for each reel i, a reel length Li and a placement of all symbols exist such that every target is realized exactly. The construction is deterministic.
Proof. Choose Li satisfying the integrality constraints (Proposition 4.8) and the footprint bound (22). For each symbol s, the target ci(s) determines the cluster geometry: by Definition 4.1, quasicontiguous clusters achieve any integer coverage a ∈ [ns + W − 1, nsW ], so c = W ns/a takes every value in a dense set. Place the clusters of each symbol on the reel, with at least W − 1 positions of other symbols between any two clusters. By Proposition 4.4, the placements do not interfere: each symbol’s (p, c) is realized by its own cluster structure. The remaining positions are filled by filler symbols (addressed in Section 5).
Remark 4.10 (Lattice Resolution). Targets are realized exactly when they fall on the lattice {k/L : k ∈ Z}. For targets off the lattice, the construction achieves the nearest lattice point, with granularity O(1/L). Since L is a derived quantity with no upper bound, the designer may choose L as large as needed: longer reels provide finer-grained target resolution at no cost beyond strip length.
Proposition 4.11 (Reel Concatenation). Concatenating a strip with itself K times produces a strip of length KL that preserves all game metrics exactly: p, c, τ 2, m, Q[s][t]/L, RTP, hit rate, and variance are all invariant. This holds because the strip is cyclic: at the join between consecutive copies, the last W − 1 symbols of one copy are identical to the first W − 1 symbols of the next (they are the same strip). Therefore the windows spanning the join are the same windows that appear at the cyclic wrap in the original strip. No new window types are created; the multiset of windows is

14