# Existence and Construction of Reel Strips from Mathematical Specifications -- plain text, part 3 of 5

Pages 28-41 of 59.
Sections: Payout Volatility; Full Construction Theorem; Wilds; Value-Bearing Symbols

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Remark 7.4 (Wild Caveat). In the presence of wilds, the effective blocker presence pesff = ps + pwild−Q[s][wild]/L reads the skeleton-filler entry Q[s][wild]. Filler-only rotations (four filler symbols a, b, c, e ∈/ {wild}) change only filler-filler entries and leave every Q[f ][wild] untouched, preserving pfeff and hence RTP. This requires F ≥ 4 fillers.
7.5 Order-3 Residual and Directed Correction
The order-2 hit rate H(2) is a function of Q alone (Corollary 6.19). The full hit rate H includes order-3+ terms not captured by Q. After rearranging fillers to achieve a target Q (and hence a target H(2)), the actual hit rate H may differ from H(2) by the order-3 residual.
Remark 7.5 (Residual Magnitude and Resolution). The inclusion–exclusion for the total hit rate terminates exactly at order W . At W = 3, the order-3 residual between H(2) (from Q) and the true H depends on triple co-occurrence across reels. On the worked example design (Section 15), the measured residual is |H − H(2)| ≈ 10−4; the magnitude is small because the order-3 terms are products of three per-reel presences with alternating signs and blocker attenuation. This residual is not a fundamental limitation: the splice construction (Proposition 6.32) hits the true H directly, bypassing Q as an intermediary. At n blocks, the HR resolution is O(1/n). At n = 5, the error is ∼0.02%. At n = 50, ∼0.002%. Precision costs strip length, linearly. The target is hit exactly when it is a rational multiple of 1/n. No iteration, no cell swaps, no convergence argument — more blocks is the closed-form answer.
Proposition 7.6 (Exact Hit-Rate Construction). For any target H∗ in the achievable hit-rate band, there exists a finite splice AnABnB with |H(AnABnB ) − H∗| < ϵ for any ϵ > 0, where n = nA + nB = O(1/ϵ). RTP is preserved exactly (all blocks share the same counts). The construction is deterministic: build the two extreme-point strips, compute the ratio, concatenate. Evaluation cost scales as O(nLbase) per reel, which is linear in precision — at n = 50 (ϵ ≈ 0.002%) with Lbase ≈ 2000, evaluation remains sub-second.
Remark 7.7 (Two Structural Invariants). Throughout the hit-rate targeting pipeline, two invariants hold:
1. RTP is preserved exactly: every operation preserves counts, and RTP reads only counts and the (frozen) skeleton.
2. The skeleton is untouched : only filler positions are modified. Every per-symbol target from Section 4 remains exact.
The hit-rate targeting step adds a third achieved target (total hit rate) without disturbing the first two (per-symbol specifications and total RTP).

8 Payout Volatility

8.1 Variance Decomposition

The industry-standard volatility index is defined as the unconditional per-spin standard deviation

σ of the payout (scaled by wager), sometimes multiplied by a jurisdiction-specific constant. This

metric conflates two distinct player experiences: how often wins occur, and how dramatic they are.

The identity

σ2 = RTP2 p

1 − p + CVwin2

(46)

28

decomposes σ2 into a zero-inflation term (1 − p)/p driven by hit rate, and a win-shape term CVwin2 driven by the conditional payout distribution, where CVwin = σY |Y >0/E[Y |Y > 0] is the coefficient of variation of the payout given a win.

Proposition 8.1 (Hit-Rate Share of Volatility). The fraction of σ2 attributable to hit rate alone

is:

1−p

fHR = 1 − p + CVwin2 .

(47)

On flat paytables (CVwin ≈ 0.5), fHR ≈ 75%: hit rate is the volatility. On premium paytables (CVwin ≈ 8), fHR ≈ 1.5%: the win shape dominates. The paytable determines which regime the game is in.

Remark 8.2 (Same VI, Different Experience). Three games with identical σ, RTP, and VI can deliver completely different player experiences: frequent small wins with occasional jackpots (high CVwin, high p), moderate wins at moderate frequency, or rare uniform payoffs (low CVwin, low p). CVwin isolates the win drama; σ does not.
Remark 8.3 (On the Industry Volatility Metric). The volatility index, proportional to σ, is the industry’s standard measure of how “swingy” a game feels. But as Proposition 8.1 shows, the majority of σ2 on typical paytables comes from the hit rate — from the zeros in the payout distribution, not from the wins. A game with a perfectly flat paytable (every win pays identically) can register as “high volatility” simply by having a low hit rate. Yet the player’s experience of that game is not swingy: when they win, they always win the same amount. The swings are in the waiting, not in the winning.
A player does not experience a zero-pay spin as a volatile outcome — it is simply a loss, identical to every other loss. What the player perceives as volatility is the variation among wins: sometimes a small return, sometimes a large one. This is precisely what CVwin measures — the dispersion of win sizes relative to the average win.
We propose that CVwin is the natural replacement for the volatility index as a design metric. It is targetable at frozen RTP and hit rate (Section 8.4), pre-computable from the specification (Proposition 8.7), exactly computable from the constructed strip (Proposition 8.5), and it measures the quantity the player actually cares about: how dramatic the wins feel when they hit. The unconditional σ2 remains derivable from (RTP, p, CVwin) via (46) for any regulatory or analytical purpose that requires it.

Definition 8.4 (Three Design Coordinates). The natural targeting coordinates are (RTP, p, CVwin):

1. RTP is exact from counts (Theorem 2.7).

2. Hit rate p is exact via the inclusion–exclusion formula, which terminates at order W (Section 7).

3. CVwin is the conditional shape parameter, controlled by clustering distribution, filler arrangement, and label assignment (Subsection 8.4).

σ2 (and hence VI) is determined by their combination via (46). It is a derived quantity, not a design target.

29

8.2 Exact CVwin from Design Parameters
Proposition 8.5 (Exact Conditional Volatility). CVwin is an exact function of the design parameters, computable in O(|S|2 · R · L) without simulation:

CVwin2

=

E[Y 2] p · M2

−

1,

RTP M= ,
p

(48)

where E[Y 2] is assembled from per-reel factors computed in a single O(L) pass per reel:

k

E[Y 2] = vs2,k mˆ 2,s,i · Bs,k+1 + 2

vs,k vt,l Ji(s, t, k, l),

s,k

i=1

(s,k)<(t,l)

i

(49)

diagonal

cross terms

where

mˆ 2,s,i

=

1 Li

pos Cs,i(pos)2 is the unconditional second moment of the count (summed over

all positions in one pass), Bs,k+1 is the blocker factor, and Ji is the per-reel joint factor:

ˆ

Paying

reel

(both

present):

Ji

=

1 Li

pos Cs,i(pos) · Ct,i(pos).

ˆ

Blocker

reel

(s

absent,

t

present):

Ji

=

1 Li

pos: Cs=0 Ct,i(pos).

ˆ Both blocked (k = l, reel k+1): Ji = P (Cs,i = 0, Ct,i = 0).

ˆ One ladder ended, other paying (k < l, reels k+2 . . . l): s unconstrained, t paying, Ji =

1 Li

pos Ct,i(pos) (the unconditional first moment of t’s count, since only t’s ways contribute

to the cross term).

ˆ One ladder ended, other blocked (k < l, reel l+1): t blocked, Ji = 1 − pt,i.

ˆ Both ladders ended (reels past max(k, l)+1): Ji = 1.
Proof. E[Y 2] = E ( s,k Ys,kPays,k)2 . Expanding the square and using reel independence, each term factors into per-reel contributions. Each per-reel factor is computed from a single pass over L positions. The same-symbol cross-depth terms vanish (mutually exclusive: a symbol achieves exactly one OAK depth per spin). All per-reel factors are exact expectations over L positions; no approximation is introduced.

Remark 8.6 (Skeleton–Filler Decomposition). E[Y 2] decomposes as E[Ys2kel]+2 E[Yskel ·Yfill]+E[Yfi2ll]. The first two terms are fixed by the skeleton. The third is the design freedom controlled by the
filler arrangement.

8.3 Pre-Computable CVwin Bounds

Proposition 8.7 (Theoretical CVwin Range). At fixed (RTP, p, paytable), the achievable CVwin is bounded by the moment problem on the pay-value support. With M = RTP/p and single-event

payouts in [vmin, vmax] (where vmax is the maximum achievable payout including ways, i.e. vmax =

maxs,k vs,k · W k):

CVwin,

2 max

=

(M

−

vmin)(vmax M2

−

M) .

(50)

This outer bound applies to the single-event payout distribution and requires no strip. Cross-event

contributions (multiple symbols winning simultaneously) are captured by the cross-term sum in (49).

The inner bound is obtained by sweeping the three controls below through their attainable ranges

and evaluating (49).

30

8.4 Three Controls for CVwin
All three operate at frozen RTP.

Proposition 8.8 (Control 1: Clustering Distribution (per reel)). At fixed (ps, cs) for designed symbol s on reel i, the gap pattern within the cluster determines τs2,i. The per-reel second moment m2 = τ 2 + c2 ranges over a computable interval. At W = 3 with count levels {1, 2, 3}, the fraction
α3 of count-3 windows ranges from max(0, c − 2) to (c − 1)/2, giving:

m2 ∈ 3c − 2 + 2 max(0, c−2), 4c − 3 .

(51)

The compound effect across d paying reels is i m2,i: multiplicative, not additive. RTP and hit rate are unchanged (m = p · c and p are fixed).

Proposition 8.9 (Control 2: Directed Filler Arrangement (per reel)). The total skeleton–filler co-occurrence f Q[designed][f ] per reel is fixed by the skeleton geometry. The distribution across fillers is free: placing expensive fillers (M1, M2) in windows adjacent to the premium designed symbol (H1) increases the cross terms in (49); placing cheap fillers (9, 10) there decreases them. Intuitively, co-locating high-paying symbols creates windows where multiple large pays fire simultaneously, amplifying win-shape dispersion; spreading them apart makes large co-occurrences rarer and reduces volatility. This compounds across paying reels through the per-reel joint factor Ji. RTP is unchanged (counts are invariant); hit rate is preserved by restricting to filler-only rearrangements (Remark 7.4).

Proposition 8.10 (Control 3: Label Permutation (global)). Among fillers with identical per-reel count profiles, permuting pay-value assignments preserves RTP, Q, and hit rate, changing only the payout variance. The change operates through third-party cross terms: for each other symbol g, the contribution shifts by vg(vf2 − vf1)(Q[f1][g] − Q[f2][g]). This is a coarse, one-time, global control.
Remark 8.11 (Combined Leverage). All three controls compound. Controls 1 and 2 are per-reel and multiply across paying reels. Control 3 is global. The combined CVwin range at fixed RTP, HR, and skeleton is determined by the product of per-reel m2 ranges (Proposition 8.8), the filler arrangement freedom within the skeleton co-occurrence budget (Proposition 8.9), and the label permutation set (Proposition 8.10). The range scales with the paytable’s pay dispersion: higher vmax/vmin ratios amplify all three controls.

8.5 Variance from Count Allocation
Proposition 8.12 (Count Allocation as the Primary σ2 Control). From the decomposition (46), σ2 depends on p through RTP2(1 − p)/p, which varies by orders of magnitude across the achievable hit-rate range. The filler count allocation (Section 5) moves p on the RTP iso-surface; this is the dominant σ2 control. The three CVwin controls provide the shape adjustment at each chosen hit rate.
Proposition 8.13 (Pre-Computable Variance Interval). The achievable σ2 interval at fixed RTP is computable from the paytable and filler budget:
1. The filler allocation set is a product of R box-constrained simplices (Corollary 5.4). Along any edge, the RTP is affine (Lemma 5.2).
2. Find crossing points where RTP(t) = R∗.

31

3. At each crossing point, evaluate p and the CVwin range from Proposition 8.7.

4. Compute σ2 from (46) at the extremes of both p and CVwin.

This gives an inner approximation. The number of edge evaluations is R ·

F 2

· V R−1, independent

of L, where V is the number of vertices of each box-constrained simplex factor.

8.6 Concentration Bounds
Proposition 8.14 (Concentration of the Empirical Distribution). With σ2 exactly known and the per-spin payout bounded by b, Bernstein’s inequality [14] gives:

1n

nt2

P

n Yi − RTP ≥ t
i=1

≤ 2 exp

− 2σ2

+

2 3

bt

.

(52)

8.7 Invariant Summary
At the conclusion of targeting, four quantities are simultaneously achieved: 1. Per-symbol specifications (p, c, τ 2): fixed by the skeleton (Section 4).

2. Total RTP : fixed by count allocation (Section 5), preserved by all subsequent operations.

3. Total hit rate: realized by co-location construction (Section 7), preserved by hit-rate-invariant operations.

4. Conditional payout shape CVwin: targeted by clustering distribution, directed filler arrangement, and label permutation, all at frozen RTP and hit rate.
σ2 and VI are derived from (RTP, p, CVwin) via (46).

9 Full Construction Theorem
This section assembles the results of Sections 4–8 into a single existence and constructibility theorem for complete games.

9.1 The Design Parameter Space
Definition 9.1 (Design Parameter Space). The design parameter space D is the set of all tuples (n, τ 2, Q1, . . . , QR) where:
1. n = (n(si)) is a matrix of per-symbol, per-reel stop counts satisfying s n(si) = Li for each reel i.
2. τ 2 = (τs(i)) is a matrix of per-symbol, per-reel conditional count variances. The attainable τ 2 values at fixed (p, c) form a finite, computable set determined by the gap-pattern compositions of the cluster. The moment-problem bounds [τm2 in, τm2 ax] (Proposition 3.8) provide an outer envelope; the realizable values are a subset.
3. Qi is a co-location matrix for reel i, consistent with the counts n(i) and the skeleton structure, achievable by a physical strip (verified constructively).

32

9.2 The Target Map

Definition 9.2 (Target Map). The target map Φ : D → R3 sends each design parameter tuple to:

Φ(n, τ 2, Q1, . . . , QR) = (RTP, H(2), CVwin),

(53)

where RTP is computed from n alone (Proposition 2.11), H(2) is the order-2 hit rate computed from {Qi} (Corollary 6.19), and CVwin is computed from n, τ 2, and {Qi} (Proposition 8.5). RTP and CVwin are exact functions of the design parameters. H(2) is an exact function of Q; the true hit rate H includes higher-order terms not captured by Q, with the residual H − H(2) measured at ∼10−4 on the worked example (the magnitude depends on the design’s per-reel presences and
pay depths). The splice construction (Proposition 6.32) targets the true H directly at resolution
O(1/n).

Definition 9.3 (Achievable Set). The achievable set A is Φ(D) ⊆ R3, parameterized by the skele-

ton and the resulting filler budget Ni = Li −

n(i)
s∈Ωdesigned s

on each

reel.

A

target

specification,

consisting of per-symbol targets (p(si), c(si), τs2(i)) satisfying the packing constraint (54) and global

targets (R∗, H∗, CVw∗ in), is feasible if the per-symbol targets lie within their attainable sets (Proposition 3.5 for (p, c); gap-pattern enumeration for τ 2) and (R∗, H∗, CVw∗ in) ∈ A.

9.3 The Full Construction Theorem
Theorem 9.4 (Full Construction). Let each designed symbol s be assigned per-reel targets (ps(i), cs(i), τs2(i)) within their attainable ranges (Sections 3–4), subject to the packing constraint that the total designedsymbol footprint leaves positive filler budget on each reel:

sps(i) < 1 for all reels i,
s∈Ωdesigned

(54)

where sps(i) = p(si)c(si)/W is the symbol percent of s on reel i (Proposition 4.7). Let (R∗, H∗, CVw∗ in) be global targets. The construction pipeline proceeds as follows:

1. RTP band (pre-computable): R∗ must lie in [RTPmin, RTPmax] (Definition 5.5). If not, the target is infeasible at these skeleton counts; the binding constraint is reported.

2. HR outer bound (pre-computable): H∗ must lie in the hit-rate band at the filler counts determined by R∗ (Definition 7.1). This bound is computed from the transportation polytope of pairwise co-locations. At finite L, integrality and connectivity may exclude some interior points; concatenation K times (Proposition 4.11) multiplies all Q entries by K, making the integer lattice K-fold finer without changing any game metric. At sufficient K, the integerachievable set fills the continuous bound to O(1/KL) resolution. Targets outside the band are certified infeasible at any K.

3. CVwin targeting: CVw∗ in is achieved at the chosen (R∗, H∗) via clustering distribution, directed filler arrangement, co-location along the HR-preserving kernel, and multiplier value distribution (Section 8.4). All controls operate on integer lattices with resolution O(1/L); at strip length L = O(1/ε), any target in the achievable range is hit to precision ε. Splicing provides additional O(1/n) refinement between lattice points.

4. Constructive certificate: extreme-point strips are built at the HR band endpoints, and the target (H∗, CVw∗ in) is reached by splicing at the appropriate ratio (Proposition 6.32). Precision is O(1/n) where n is the number of blocks.

33

RTP is achieved exactly. Hit rate and CVwin are achieved to O(1/L) precision from the integer lattice of co-location entries, clustering values, and multiplier assignments. At L = O(1/ε), all three coordinates are within ε of their targets. σ2 is derived from (RTP, p, CVwin) via (46).
The pipeline is deterministic: it either produces a strip or reports which bound is violated. Constructibility of any candidate Q is decidable before a strip is built (∼60 ms to reject, ∼2.6 s to accept and build) (Proposition 6.27, Section 6.11). Any target between two constructively verified operating points is achievable at L = O(1/ε) for precision ε, by lattice density and splice linearity. The full pipeline from specification to verified strips runs in ∼17 s.
Proof. The proof follows the pipeline steps, showing that each preserves the targets established by previous steps.
Step 1: Skeleton. For each designed symbol s on each reel i, the target (ps(i), c(si), τs2(i)) determines a quasi-contiguous cluster with a specific span (fixing p and c) and a specific gap ordering (fixing τ 2). By Theorem 4.9, this placement exists and is constructible. Each symbol’s statistics depend only on its own stops (Proposition 4.4), so clusters may be separated or interleaved as the design requires.
Step 2: Filler counts. Since R∗ lies in the RTP band, the intermediate value theorem on the convex allocation polytope (Proposition 5.8) guarantees a filler count vector n∗ with RTP(n∗) = R∗ at lattice resolution O(1/L).
Reachability. By Corollary 6.33, the achievable set of (H, E[Y 2]) pairs at fixed counts is reachable by mixing (Corollary 6.33). Any target in the interior is reached by splicing extreme-point strips (Proposition 6.32). The derived coordinate CVwin is a smooth function of these, so any interior (H, CVwin) target is reachable. Resolution is O(1/n) where n is the number of splice blocks, made arbitrarily fine by increasing n.
Step 3: Filler arrangement. The target hit rate H∗ is achieved in two stages: (a) Direct construction. For each filler f on each non-blocker reel, the clustering cf is chosen so that pf = mf /cf yields the target per-reel presences. Since mf = W nf /L is fixed by counts, the designer solves for the cf values that produce H∗ via the inclusion–exclusion formula. The filler stops are then placed as quasi-contiguous clusters at the target cf , exactly as for designed symbols. This produces a single strip achieving the target HR at any lattice point where the required cf values are realizable (integer-compatible gap patterns). (b) Continuous interpolation via splicing. When H∗ falls between two lattice points, two strips A and B at nearby achievable HR values are spliced: AnABnB achieves any rational convex combination of H(A) and H(B) at resolution O(1/n) (Proposition 6.32). Both strips share the same skeleton and counts, preserving RTP exactly. The splice fills the gaps in the lattice, making the achievable set dense rather than discrete. In both cases, RTP is preserved exactly (counts are invariant) and Proposition 6.32 guarantees the splice mixture is exact when the skeleton pins the boundary cells. Step 4: Verification. The constructed strips realize:
ˆ RTP = R∗: determined by counts alone, fixed in Step 2, preserved by Step 3.
ˆ H = H∗: the splice ratio targets the true hit rate directly at resolution O(1/n) (Proposition 7.6). Precision costs strip length, linearly.
ˆ CVwin = CVw∗ in + O(1/L): all CV controls (clustering, co-location, multiplier distribution, k1) operate on integer lattices with O(1/L) step size. Any target between two constructively verified CV values is achievable at L = O(1/ε).
All four steps are deterministic and finite.
34

9.4 Pre-Computability of the Achievable Set
Proposition 9.5 (The Achievable Set Is Pre-Computable). The achievable set A can be computed from the paytable, the skeleton specification, and the reel lengths, before any strip is constructed:
1. The RTP range is the interval [RTPmin, RTPmax] from the filler allocation polytope (Definition 5.5).
2. At each RTP value, the hit-rate range is determined by the co-location freedom at those counts (Definition 7.1).
3. At each (RTP, hit-rate) pair, the volatility range is determined by the count allocation on the RTP iso-surface (Section 8.5), the attainable τ 2 range (Proposition 3.8), and the hit-ratepreserving rotation freedom (Theorem 6.20).
Under the continuous relaxation D˜ (real-valued counts on the allocation polytope, τ 2 on the moment interval, Q on the LP relaxation of F), the image A˜ = Φ(D˜) is compact and connected. The integerachievable subset AZ is an O(1/L) net in A˜ (Proposition 9.6). The LP relaxation is a necessary condition for the Q coordinate ; the complete test requires integrality and connectivity .
9.5 Integrality and Resolution
Proposition 9.6 (Lattice Resolution). All design parameters are ultimately integers (stop counts, Q entries, gap positions). The continuous achievable set A is approximated by the integer-achievable subset AZ to resolution O(1/L) in each coordinate. As L → ∞, AZ becomes dense in A, and every target in the interior of A is achievable exactly.
Proof. The RTP resolution is O(W/L) per count change (Proposition 5.8). The τ 2 resolution is O(1/a) per gap reordering, where a = O(L). The Q entry resolution is integer-valued with Q[s][t] ∈ {0, 1, . . . , L}, giving per-entry resolution O(1/L) in the normalized quantity Q[s][t]/L. Since L is a derived quantity with no upper bound (Proposition 4.8), the resolution can be made arbitrarily fine.
Proposition 9.7 (Joint HR/Volatility Targeting at Fixed RTP). Given a strip achieving target RTP and H(2), the conditional payout shape CVwin (and hence σ2) can be adjusted to arbitrary precision without changing either RTP or H(2), by the following procedure:
1. Concatenate the strip K times (Proposition 4.11). This preserves RTP, H(2), and σ2 exactly (all normalized quantities Q[s][t]/L are unchanged), while multiplying the number of integer Q values by K.
2. Apply hit-rate-preserving rotations satisfying condition (37). At pay depth d = 3, solutions to condition (37) are abundant at every concatenation factor K (Remark 9.8). Each rotation is realized by a singles-preserving transposition on the strip (no two stops of the same filler land within W − 1 positions after the swap). Each such rotation changes σ2 by O(1/L), and L = KL0 is now K times larger, providing K-fold finer variance resolution. For games with mixed pay depths, the available rotations are depth-specific (d = 3-preserving but not d = 2-preserving); a single non-consecutive δ = 1 seed move breaks the mod-K lattice on the off-diagonal entries, restoring the availability of consecutive-entry (depth-universal) rotations for all subsequent moves. The seed move introduces a one-time O(1/L) perturbation to H(2) at non-target depths, correctable by a subsequent compensating move.
35

Since RTP depends only on counts (Theorem 2.7) and counts are preserved by both concatenation and rotations, RTP is exact throughout. Since H(2) depends on (Q[s][t]/L)d and rotations satisfying condition (37) preserve this sum at the relevant pay depth, H(2) is preserved throughout. The variance resolution is O(1/KL0), arbitrarily fine for sufficiently large K.
Remark 9.8 (Physical Operations vs. Algebraic Scaling). A natural objection is that concatenation scales every Q entry by K, so any rotation must use δ = K to preserve divisibility, yielding a variance step of K/(KL0) = 1/L0 — no improvement. This reasoning is incorrect. Concatenation is a strip-level operation; rotations are physical operations on the strip. A single transposition of two filler stops in one copy of the concatenated strip changes exactly those windows containing the swapped positions, modifying the relevant Q entries by ±1 regardless of L. The variance step is (va − vc)(vb − ve)/L = (va − vc)(vb − ve)/(KL0), which genuinely shrinks with K. After such a move, the Q entries are no longer multiples of K (e.g., 5K → 5K ± 1), and condition (37) admits δ = 1 solutions on non-K-divisible entries. Existence: at K = 2, the quadruple (A, C, E, B) = (0, 6, 4, 6) → (1, 7, 3, 5) satisfies (A3 + C3) − (E3 + B3) = 0. After such a move, entries are no longer K-divisible, and consecutive-entry solutions become available.
9.6 Infeasibility Detection
Remark 9.9 (When Targets Are Infeasible). If the target specification lies outside the feasible set, the construction pipeline detects this at the step where the remaining freedom cannot accommodate the remaining targets:
ˆ Per-symbol footprints violate the packing constraint (54): detected before construction begins.
ˆ R∗ outside the RTP band at the resulting filler budget: detected in Step 2.
ˆ H∗ outside the hit-rate band at the required counts: detected in Step 3.
ˆ CVw∗ in outside the achievable CVwin range at the required (counts, Q): detected in Step 1 (clustering) or Step 3 (rotations).
In each case, the pipeline reports which constraint binds and the achievable range at that constraint, enabling the designer to adjust the infeasible target.

10 Wilds
A wild symbol substitutes for every paying symbol. This section shows that the wild enters the framework through two additive corrections: one to each paying symbol’s mean count m, and one to the effective co-location matrix Q. Both are closed-form. The moat around the wild cluster handles the RTP interaction.

10.1 Linear Separability of Mean Count

Proposition 10.1 (Effective Mean Count). For a paying symbol s, the effective mean count per

window is:

mesff = mnsative + mwild,

(55)

where mnsative = W ns/L and mwild = W nwild/L. The wild’s contribution is additive and arrangementindependent.

Proof. By linearity of expectation: E[Cseff ] = E[Csnative] + E[Cwild] = mnsative + mwild.

36

10.2 Additive Correction to the Co-location Matrix
Proposition 10.2 (Effective Co-location). Every window containing the wild effectively contains every paying symbol (via substitution). The effective co-location matrix for paying symbol pairs is:

Qeff [s][t] = Qnative[s][t] + Q[wild][wild] − Qoverlap[s][t] for all paying pairs s ̸= t, (56)
where Q[wild][wild] is the number of windows containing the wild (fixed by the skeleton), and Qoverlap[s][t] counts windows containing the wild AND both s and t natively. The overlap term prevents double-counting of windows at the edges of the wild cluster where native fillers and the wild coexist. The overlap count is pre-computable from the skeleton geometry and is O(1) per wild cluster (at most 2(W − 1) windows per cluster, corresponding to the entry and exit edges).
Proof. A window either contains the wild or it does not. In windows without the wild, only native co-occurrences contribute. In windows with the wild, every paying symbol is effectively present, so every pair co-occurs. Summing Qnative[s][t] + Q[wild][wild] double-counts windows where both s and t are natively present AND the wild is present. Subtracting Qoverlap[s][t] corrects this.
Remark 10.3. For highly clustered wilds (cwild ≥ 2), most wild-containing windows have ≥ 2 wild stops and at most one native symbol, so Qoverlap[s][t] = 0 for most pairs. The correction is nonzero only for filler pairs that appear together in the 2(W − 1) edge windows per cluster. The correction is O(K/L) per pair (where K is the number of wild clusters), exactly computable, and must be included for correctness.
Corollary 10.4 (Effective Diagonal). The effective diagonal (per-symbol presence count) is:

Qeff [s][s] = Qnative[s][s] + Q[wild][wild] − Q[s][wild],

(57)

where Q[s][wild] counts windows containing both native s and wild (by inclusion-exclusion). Under the moat (Q[s][wild] = 0 for non-adjacent symbols), this simplifies to Qnative[s][s] + Q[wild][wild].

10.3 RTP and the Moat
Corollary 10.5 (RTP Under Wild Presence). RTP is computed by substituting mseff for ms in the RTP formula (7), with one correction: all-wild ways (no native symbols on any paying reel) are counted once per paying symbol but awarded once at the highest pay s∗. The correction subtracts the overcounted all-wild contributions:

k

∆RTP = −

vk(s) · mw(ii)ld · (1 − pseff,(k+1)).

s̸=s∗ k

i=1

(58)

When the wild has no presence on reel 1, mw(ii)ld = 0 and the correction vanishes. The effective blocker presence pesff = ps + pwild − Q[s][wild]/L is determined by the skeleton geometry. Under the moat, fillers adjacent to the wild cluster have Q[f ][wild] > 0 (they share edge windows with the wild), producing the degenerate window correction of Proposition 10.6. This is pre-computable from the skeleton before any filler is placed.

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10.4 Degenerate Window Correction

Proposition 10.6 (Degenerate Windows from Wild Clustering). When the wild has cwild > 1, some windows contain multiple wild stops and fewer than W distinct symbols. For a filler symbol

s adjacent to the wild cluster (in the moat), let ds denote the co-occurrence deficit: the count of

degenerate windows containing s, weighted by the shortfall in distinct neighbors. The margin law

becomes:

Q[s][t] = (W − 1) · Q[s][s] − ds.

(59)

t̸=s

The deficit ds is an integer computable from the wild’s cluster structure. At W = 3, each degenerate window contributes deficit 1.

10.5 Wild as a Skeleton Symbol
The wild is placed in the skeleton with its own per-reel targets (p(wii)ld, c(wii)ld, τw2(ilid) ), subject to the same constructibility results as any designed symbol (Theorem 4.9). Its footprint enters the packing constraint (54). The moat (Definition 4.3) separates the wild cluster from other skeleton symbols, ensuring Q[wild][s] = 0 for other designed symbols. Fillers fill the remaining positions, including moat positions adjacent to the wild.

10.6 Framework Compatibility
Proposition 10.7 (Framework Under Wilds). The framework of Sections 2–9 extends to games with wild symbols via two additive corrections:
1. mseff = ms + mwild replaces ms in the RTP formula, with the all-wild overcounting correction (Corollary 10.5).
2. Qeff [s][t] = Qnative[s][t] + Q[wild][wild] − Qoverlap[s][t] replaces Q[s][t] in the hit-rate formula (Proposition 10.2), with the degenerate window correction for moat fillers (Proposition 10.6).
Both corrections are pre-computable from the skeleton. The filler-filler block of Qnative retains its full design freedom. The Full Construction Theorem (Theorem 9.4) holds with the wild included in Ωdesigned.
Remark 10.8. Kamanas et al. [5] noted that their algorithm’s behavior “in the case where special symbols appear in reels is also unknown.” In the present framework, wilds are two additive corrections to m and Q, both closed-form, both pre-computable from the skeleton geometry.

11 Value-Bearing Symbols
A value-bearing symbol is a symbol whose stops carry a numerical value in addition to their identity. Common instances include cash amounts (scatter pays) and multipliers (wild multipliers), but the framework applies to any symbol whose stops carry a designable numerical parameter. The symbol’s identity determines where it participates in OAK evaluation. The numerical value determines how much it contributes when it does. This section shows that the numerical values are independently targetable via a linear system over the stop values, and that the attainable set of per-count averages is pre-computable.

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11.1 Definition and Evaluation
Definition 11.1 (Value-Bearing Symbol). A value-bearing symbol s has, on each reel i, a set of n(si) stops, each carrying a numerical value ωj ∈ R≥0. The symbol’s identity (s) determines its coverage p, conditional count c, and count distribution τ 2 via the framework of Sections 2–4. The numerical values {ωj} determine the expected payout conditional on the symbol’s count in a window.

Definition 11.2 (Per-Count Average). For a value-bearing symbol with count k in a window (i.e.,

k of the W cells contain the symbol), the per-count average ω¯k is the expected sum (or product,

depending on the evaluation rule) of the numerical values of the k visible stops. For additive

evaluation (cash scatters):





k

ω¯k = E ωvisible,j count = k .

(60)

j=1

For multiplicative evaluation (wild multipliers):


k
ω¯k = E ωvisible,j
j=1

 count = k .

(61)

11.2 Linear System for Per-Count Averages
The stops of a value-bearing symbol in a quasi-contiguous cluster occupy specific positions. At W = 3, windows containing the symbol have counts 1, 2, or 3. Each stop participates in windows of potentially multiple count levels (a stop near the cluster edge appears in count-1 and count-2 windows simultaneously).
Proposition 11.3 (Per-Count Targeting via Linear System). The per-count averages (ω¯1, . . . , ω¯K), where K ≤ W is the maximum count in any window, are linear functions of the stop values (ω1, . . . , ωn). The coefficient matrix A ∈ RK×n has entry Ak,j equal to the fraction of count-k windows containing stop j. When A has full row rank, the system Aω = ω¯ is solvable for any target in the attainable polytope {Aω : ω ≥ 0}, with n − K free parameters available for higher-moment targeting or satisfying additional constraints. The attainable polytope is pre-computable from the cluster geometry (Proposition 11.4). For multiplicative evaluation (wild multipliers), substituting uj = log ωj and targeting the per-count geometric mean log ω¯kGM = j Ak,juj reduces to the same linear system with the same rank conditions. The geometric mean is the natural target for multiplicative evaluation; if the arithmetic mean of products is required, the system is polynomial of degree k in the stop values.
Proof. The per-count average ω¯k is, by definition, the mean over all count-k windows of the sum of visible stop values. This is linear in the stop values: ω¯k = j Ak,jωj. The coefficient matrix A is determined by the cluster geometry, which is fixed before numerical values are assigned.
Full row rank holds when the count levels are sufficiently differentiated by the cluster geometry. For quasi-contiguous clusters with mixed gaps, A has full row rank at n ≥ W . For contiguous clusters, full rank requires n ≥ 2W − 1. For singles (c = 1), the system is trivially solvable (one count level, K = 1). Rank deficiency occurs only for very small clusters or perfectly symmetric repeating patterns.

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11.3 Attainable Set and Resolution
Proposition 11.4 (Attainable Per-Count Averages). The attainable set of per-count averages (ω¯1, ω¯2, ω¯3) is a polytope determined by the cluster geometry and the bounds on individual stop values (ωj ≥ 0, and any upper bound imposed by the game design). The polytope is pre-computable from the skeleton specification.
Proposition 11.5 (Resolution via Reel Length). Each stop carries one numerical value, so the granularity of the per-count average is O(1/nk) where nk is the number of stops participating in count-k windows. Since nk scales with L (longer reels have more stops), the resolution of the percount averages is O(1/L). Doubling the reel length (Proposition 4.11) doubles the number of distinct values available and halves the targeting resolution.
Remark 11.6 (Weight Tables Are Subsumed). Industry practice assigns values to symbol stops via a weight table: a single probability distribution from which every stop’s value is drawn independently. When k stops are visible (count = k), the expected sum is k · E[w] and the expected product is E[w]k. The per-count averages are locked in a ratio determined by k; they cannot be targeted independently.
The linear system of Proposition 11.3 is strictly more powerful. Each physical stop receives its own fixed value, and different stops contribute with different weights to different count levels. The per-count averages are independently targetable because the coefficient matrix generically has full row rank: the count levels see different weighted combinations of stop values. A weight table is the special case where all stops are assigned values from the same distribution; the framework does not require this restriction.

11.4 Distributional Control

Proposition 11.7 (Distributional Control). At a single count level k, the nk stops participating in count-k windows contribute to the empirical distribution of visible values with weights Ak,j. Within one count level, targeting J moment constraints requires J linearly independent columns of row k of A, which is generically available for J < nk. Across count levels simultaneously, the same stop values ωj appear in all rows of A, so moment targets at different count levels are coupled through the shared unknowns. The number of simultaneously targetable constraints (moments across all levels combined) is bounded by the rank of the stacked system, which equals K (the number of count levels) when A has full row rank, and grows with n as additional moment equations are appended.

Proof. Each moment target at count level k adds one row to the constraint matrix. The j-th moment at level k is i A(kj,i)ωij for the additive case (or i Ak,iuji under the log transform). The rank of the stacked system determines how many targets are simultaneously achievable. With n
stop values and K · J moment-count constraints, the system is solvable when the stacked matrix
has full row rank, which holds generically for K · J < n.

Remark 11.8 (Per-Reel and Global Volatility of the Value-Bearing Component). The per-reel variance of the value-bearing component at count k is Var(ω|count = k), targetable by the distributional control above. The per-reel total value-bearing variance combines the per-count variances weighted by the count distribution:

W

Var(Ωs) = P (count = k) · Var(ω|k) + ω¯k2 − E[Ωs]2.

(62)

k=1

40

The count probabilities P (count = k) are controlled by Layer 3 (τ 2). The per-count variances Var(ω|k) and means ω¯k are controlled by the linear system (Proposition 11.3). All factors are independently adjustable, so the per-reel value-bearing volatility is fully targetable. The global value-bearing volatility across reels follows from the independence of reel spins.

11.5 Expected Value of the Value-Bearing Component

Proposition 11.9 (Value-Bearing EV). For additive evaluation (cash scatters), the expected contribution of the value-bearing component on a single reel is:

W

E[Ωs] = P (count = k) · ω¯k,

(63)

k=1

where P (count = k) is determined by the count distribution (Layer 3, Proposition 3.8) and ω¯k (the per-count arithmetic mean) is determined by the linear system (Proposition 11.3). For multiplicative evaluation (wild multipliers), the RTP contribution requires the per-count arithmetic mean of the product, which is degree k in the stop values and not controlled by the log-transform linear system. The log-transform targets the geometric mean; if the arithmetic mean is required, the system is polynomial and must be solved directly, though it remains solvable with n − K free parameters.

11.6 Applications
Remark 11.10 (Cash Scatters). A cash scatter is a value-bearing symbol with additive evaluation. Each scatter stop carries a cash value. The scatter’s total payout on a spin is the sum of visible values across all reels. The per-reel expected contribution is controlled by the linear system (Proposition 11.3). The cross-reel total is the sum of independent per-reel contributions (since reels spin independently), so the scatter’s overall EV and variance are computable from the per-reel value-bearing EVs.
Remark 11.11 (Scatter Count Probabilities). Unlike OAK symbols (evaluated per reel per window), scatters are typically evaluated by total count across all reels. The probability of exactly j scatters across R reels is a convolution of the per-reel count distributions. Since each reel’s scatter count distribution is determined by its coverage p(sic)atter and its count distribution, the cross-reel scattercount probabilities are pre-computable. The probability of triggering a feature (e.g., 3+ scatters) is a known function of the per-reel coverages.
Remark 11.12 (Multiplier Wilds). A multiplier wild is a value-bearing symbol with multiplicative evaluation. Each wild stop carries a multiplier value (2×, 3×, 5×, etc.). When k multiplier wilds appear in a window, the total multiplier is the product of their values. Under the log transform uj = log ωj, the per-count geometric mean is targeted by the same linear system (Proposition 11.3). The per-reel expected multiplier given count k is exp(u¯k), independently targetable at each count level.
Remark 11.13 (Integration with the Framework). Value-bearing symbols are placed in the skeleton with per-reel targets (p, c, τ 2), exactly as non-value-bearing designed symbols. The numerical values are assigned after the skeleton is constructed, by the linear system (Proposition 11.3). The placement and the numerical values are independent in the following sense: the skeleton determines where the symbol appears and how it clusters, and the linear system (Proposition 11.3) determines what values the stops carry. The placement does not constrain the values, and the values do not affect the placement, the co-location matrix, or the hit rate.

41