# Existence and Construction of Reel Strips from Mathematical Specifications -- plain text, part 2 of 5 Pages 15-27 of 59. Sections: Filler System and Total RTP; The Co-location Matrix; Hit-Rate Targeting via Co-location Next part: https://gamemathemagics.com/papers/reel-strip-construction.part3.txt Previous part: https://gamemathemagics.com/papers/reel-strip-construction.part1.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 --- exactly K copies of the original multiset. The operation multiplies the number of stop positions by K, providing K-fold finer granularity for any quantity that depends on individual stop assignments (filler counts, value-bearing symbol values, gap orderings). Since concatenation is always available and introduces no side effects, the designer may assume arbitrarily fine resolution at every stage of the construction pipeline. Proof. Let r = (r0, . . . , rL−1) be the original cyclic strip and rK the K-fold concatenation of length KL. For any position j in rK , the window (rjK , rjK+1, . . . , rjK+W −1) equals (rj mod L, r(j+1) mod L, . . . , r(j+W −1) mod L) because the strip repeats with period L. Therefore the multiset of windows in rK is exactly K copies of the multiset in r. Every quantity computed as a ratio (count/KL, Q entry/KL, etc.) equals the corresponding ratio in the original (count/L, Q entry/L), since both numerator and denominator scale by K. Remark 4.12 (What This Theorem Does Not Address). Theorem 4.9 constructs a skeleton: each designed symbol placed with its target (p, c). It does not yet address total game RTP (which depends on the filler composition, Section 5), total game hit rate (which depends on cross-symbol co-occurrence, Sections 6–7), or total game volatility (Section 8). These are handled by subsequent sections, each building on the skeleton without modifying it. 5 Filler System and Total RTP The skeleton from Section 4 places every designed symbol with its target (p, c). Every remaining position on the reel is occupied by a filler symbol : a paying symbol whose count is determined by the RTP allocation and whose clustering is a design parameter controlling hit rate. 5.1 Filler Placement Definition 5.1 (Filler Symbol). A filler symbol f ∈ Ω is a paying symbol whose placement is not fixed by the skeleton. On blocker reels (reel df + 1 for a depth-df symbol), cf = 1 (singles): no two stops within W − 1 positions, so pf = mf = W nf /L is determined by counts alone. On non-blocker reels (1, . . . , df ), cf ≥ 1 is a design parameter: clustering reduces pf at fixed mf (Theorem 2.7), which is the primary hit-rate control (Corollary 2.13). Every position not occupied by a designed symbol or a wild is assigned to a filler. No reel position is left blank. The total number of filler positions on reel i is Li − s∈Ωdesigned ns(i). These positions are distributed among F filler symbols, each with its own pay values in the paytable. 5.2 Filler RTP Formula The RTP formula reads mi = W n(fi)/Li on paying reels (arrangement-invariant) and pi on blocker reels. On blocker reels, fillers are singles (cf = 1), so pf = mf = W nf /L — determined by counts alone. On non-blocker reels, pf does not enter the RTP formula (there is no blocker term). Therefore the filler RTP is a pure function of counts regardless of clustering: RTPk(f ) = vk(f ) · k i=1 W nf(i) Li ·  1 − W nf(k+1) Lk+1   , k < R. (24) 15 No arrangement parameters appear. The total filler RTP is: RTPfiller = R RTP(kf ) . f k=2 (25) The total game RTP is the sum of the skeleton contribution (fixed by Section 4) and the filler contribution: RTPtotal = RTPskeleton + RTPfiller. (26) 5.3 Multilinearity of Filler RTP Lemma 5.2 (Multilinearity). The total filler RTP is multilinear in the per-reel filler count vectors: with all other reels held fixed, RTP is an affine function of any single reel’s filler counts. Proof. In (24), each OAK rung involves each reel at most once: reel i contributes either W n(fi)/Li (linear in n(fi)) as a paying reel, or 1 − W n(fi)/Li (affine in nf(i)) as a blocker. No rung contains a product of two quantities from the same reel. The total filler RTP, a sum of such rungs, is therefore affine in each reel’s filler counts with all others fixed. Remark 5.3. Multilinearity has a practical consequence beyond finding extrema. It means the marginal RTP contribution of each filler on each reel is a constant (at fixed counts on other reels). Moving one stop of filler f from reel i to reel j changes the RTP by a predictable, pre-computable amount. Each candidate reallocation can be scored in O(1) time from a pre-computed marginal table, rather than requiring a full RTP recomputation. 5.4 RTP Band Corollary 5.4 (Extrema at Vertices). Filler RTP is multilinear across reels (Lemma 5.2): affine in each reel’s count vector with the others held fixed [12]. Over the box-constrained simplex f nf(i) = Ni, 0 ≤ n(fi) ≤ ⌊Li/W ⌋, an affine function’s extrema are attained at vertices [15]. Since reel-by-reel optimization preserves this property (each reel’s optimum is at a vertex of its own box-constrained simplex, regardless of the others’ positions), the global extrema are attained at vertices of the product polytope. Definition 5.5 (RTP Band). The RTP band is the interval [RTPmin, RTPmax] of total game RTP values achievable by varying the filler composition within the box-constrained simplex, with all designed-symbol placements fixed. Proposition 5.6 (RTP Band Width). The RTP band has positive width whenever at least two filler symbols have distinct pay values. The width depends on the spread of filler pay values and the proportion of the reel occupied by fillers. A larger filler region (lower skeleton footprint) and a wider spread of filler pay values produce a wider band. Proof. If two fillers f1, f2 have distinct pay values, then moving one stop from f1 to f2 on some reel changes the filler RTP (since the marginal RTP contribution of f1 and f2 differ on at least one rung). The floor and ceiling vertices therefore differ, and the band has positive width. Remark 5.7 (Feasibility Check). The RTP band provides a pre-construction feasibility check. Before any filler is placed, the designer computes the band from the skeleton and the paytable. If the target total RTP falls outside the band, the skeleton must be adjusted (by modifying designed symbol counts) before construction can proceed. If it falls inside, the filler system is guaranteed to reach it. 16 5.5 Total RTP Targeting Proposition 5.8 (Total RTP Is Achievable). Any total RTP within the RTP band is achievable to within the lattice resolution δ(L) = O(1/L). Proof. The filler allocation has R(F − 1) continuous degrees of freedom (each of R reels distributes its filler budget among F filler symbols, subject to one sum constraint per reel). The feasible set of allocations forms a convex polytope P (the Cartesian product of R simplices). Let xmin, xmax ∈ P be the vertex allocations achieving RTPmin and RTPmax respectively (Corollary 5.4). Since P is convex, the path x(t) = (1 − t) xmin + t xmax, t ∈ [0, 1], (27) lies entirely within P. The filler RTP along this path is a continuous function RTP(t) of one real variable, with RTP(0) = RTPmin and RTP(1) = RTPmax. By the intermediate value theorem, for every target RTP∗ ∈ [RTPmin, RTPmax], there exists t∗ ∈ [0, 1] such that RTP(t∗) = RTP∗. The continuous allocation x(t∗) may have non-integer counts. Rounding via the largest-remainder method (floor all counts, then increment those with the largest fractional parts until nf = Ni) preserves the budget constraint exactly and changes each count by at most 1. Each unit change shifts the total RTP by vk · k · mk−1 · (W/L) · (1 − pk+1) per symbol per OAK level. The aggregate resolution δ(L) depends on the paytable; since L is a derived quantity with no upper bound (Proposition 4.11), the resolution can be made arbitrarily fine. 5.6 Side Effects and Separation of Concerns Remark 5.9 (Filler Allocation and Hit Rate). Changing filler counts changes filler window coverages (pf = W nf /L), which changes the total game hit rate. The filler allocation is not chosen for RTP alone: it is selected as the operating point on the RTP iso-surface where the target hit rate falls within the co-location-achievable range and the target volatility falls within the clusteringachievable range. The joint selection is formalized in the Full Construction Theorem (Section 9). Remark 5.10 (Skeleton Preservation). The filler allocation modifies only filler counts. The designed symbols’ stop counts, placements, coverages, conditional counts, and count distributions are unchanged. Every per-symbol target established by the skeleton (Section 4) is preserved exactly. Remark 5.11 (Pre-Computability). The RTP band and the per-reallocation RTP change are computable from the specification and the skeleton before any filler is placed. The designer knows, before construction begins, whether a target total RTP is achievable alongside the per-symbol targets. 6 The Co-location Matrix Sections 4 and 5 construct a skeleton and target total RTP. The remaining global targets are total hit rate and total volatility, both of which depend on which symbols share windows. This section introduces the mathematical object that captures that structure. 17 6.1 Definition and Basic Properties Definition 6.1 (Co-location Matrix). For a reel strip r of length L with display height W , the co-location matrix Q is the |Ω| × |Ω| symmetric matrix defined by: Q[s][s] = #{j : s ∈ window(j)}, (28) Q[s][t] = #{j : s ∈ window(j) and t ∈ window(j)}, s ̸= t. (29) The diagonal records per-symbol presence counts. The off-diagonal records pairwise co-occurrence counts. Q is the complete order-2 window statistic of the strip. Proposition 6.2 (Margin Laws). Under the singles constraint (no symbol appears twice in any window), the following identities hold: Q[s][s] = W · ns, (30) Q[s][t] = (W − 1) · Q[s][s], (31) t̸=s W Q[s][t] = · L. (32) 2 s 3). At display height W , each window is a clique of size W , contributing +1 to W 2 off-diagonal entries. The triangle decomposition generalizes to a W - clique decomposition. The results of this section hold at any W ; the triangle language is used for concreteness at W = 3. 6.3 Design Freedom Proposition 6.6 (Q Decomposition Under a Skeleton). When a skeleton is present (designed symbols at fixed positions), the co-location matrix decomposes into three blocks: 18 1. Skeleton-skeleton: entries Q[s][t] where both s and t are designed symbols. These are fully determined by the skeleton geometry and cannot be changed. 2. Skeleton-filler: entries Q[s][f ] where s is a designed symbol and f is a filler. The skeleton determines which windows contain s; the filler arrangement determines which filler occupies the remaining cells in those windows. These entries are partially constrained by the skeleton but admit design freedom in the filler assignment. 3. Filler-filler: entries Q[f1][f2] where both are fillers. These are the main design space. Under singles (which holds for fillers on blocker reels), the margin laws apply exactly in this block. The skeleton contribution to the total Q is pre-computable in O(L) from the skeleton geometry alone. The filler contribution is the design freedom. The total is additive: Qtotal = Qskeleton + Qfiller. Remark 6.7 (Degenerate Windows). When a designed symbol has c > 1, some windows contain multiple instances of that symbol, violating the singles constraint locally. These degenerate windows affect the margin laws for both skeleton and filler entries: a filler symbol appearing in a degenerate window has fewer than W − 1 distinct neighbors, reducing its row sum (see Proposition 10.6 for the explicit correction). The degenerate window count and the per-filler deficit ds are computable in advance from the skeleton geometry. By Corollary 2.13, filler clustering is restricted to non-blocker reels, so degenerate filler windows occur only on reels where p does not enter the RTP formula. On blocker reels, fillers remain as singles. Proposition 6.8 (Off-Diagonal Freedom). The number of free parameters in Q is n 2 − n, where n = |Ω|. The diagonal is fixed by counts (30), and the row-sum law (31) imposes n linear constraints on the n 2 off-diagonal entries. Proof. There are n 2 off-diagonal entries (upper triangle). Each of the n row-sum constraints fixes one linear combination of the off-diagonal entries in that row. The total mass constraint (32) is implied by summing the row-sum constraints, so it is not independent. The remaining freedom is n 2 − n. Remark 6.9. For n = 8 filler symbols, the freedom is 28 − 8 = 20 dimensions. For n = 10, it is 45 − 10 = 35. This is the space within which hit rate and volatility can be moved at frozen RTP. 6.4 RTP Invisibility Proposition 6.10 (Q Cannot Move RTP). Any change to the off-diagonal block of Q (at fixed diagonal) preserves RTP exactly. Proof. The RTP formula (7) reads each reel through mi = W ni/L (on paying reels, invariant by Theorem 2.7) and pi = ai/L (on blocker reels). For filler symbols on blocker reels (singles), pi = W ni/L = mi, determined by counts alone. For designed symbols, pi is fixed by the skeleton (which is not modified). In both cases, the quantities RTP reads are determined by either the counts or the skeleton, neither of which changes when the filler arrangement changes. The off-diagonal entries of Q, which record pairwise co-occurrence, do not appear in the RTP formula. Therefore any rearrangement of fillers that preserves counts preserves RTP exactly. Remark 6.11. This is the structural foundation of the co-location approach: the off-diagonal block of Q is the design space that moves hit rate and payout variance while being invisible to RTP. The asymmetry arises because RTP is multilinear in per-reel counts (Proposition 2.11), and pairwise co-occurrence is an order-2 quantity that does not enter a multilinear formula. 19 6.5 The Transposition Lattice Definition 6.12 (Transposition). A transposition exchanges the symbols at two positions on the strip. It preserves all symbol counts and hence all diagonals and all row sums. Proposition 6.13 (Lattice Rank). The lattice of achievable Q-differences generated by transposi- tions has rank exactly n 2 − n, equal to the off-diagonal freedom (Proposition 6.8). No hidden linear invariant obstructs movement: any achievable difference is an integer combination of transposition deltas. Proof. A rotation on quadruple (a, b, c, e) changes four entries: Q[a][b] and Q[c][e] by +δ, Q[a][e] and Q[c][b] by −δ. This difference vector lies in the integer kernel of the unsigned vertex-edge incidence matrix B of Kn (the row-sum constraints t≠ s Q[s][t] = rs are exactly Bq = r, so any difference in kerZ B preserves all row sums). For non-bipartite Kn (n ≥ 3), the rank of B over Q is n (Proposition 6.8), so dim(kerQ B) = n 2 − n. The rotation generators (alternating 4-cycles in Kn) span kerZ B: any kernel vector can be reduced to zero by subtracting generators, using completeness of Kn to route any edge-pair through a 4-cycle. At n = 4: kerZ B has rank 4 2 − 4 = 2; the three alternating 4-cycles satisfy one linear dependence, spanning the full kernel. For general n ≥ 3: any kernel vector is reduced to zero by iteratively subtracting 4-cycle generators, using completeness of Kn to route each nonzero edge-pair through a common vertex. Remark 6.14. The lattice rank establishes that transpositions span the full design freedom. It does not establish which specific Q matrices are achievable, only that the achievable set has the expected dimension. 6.6 Row-Sum-Preserving Rotations Definition 6.15 (Rotation). A row-sum-preserving rotation on Q selects four distinct symbols a, b, c, e and an integer δ, and applies: Q[a][b] ← Q[a][b] + δ, Q[c][e] ← Q[c][e] + δ, (33) Q[a][e] ← Q[a][e] − δ, Q[c][b] ← Q[c][b] − δ. (34) The rotation preserves all row sums, all diagonals, all symbol counts, and hence all RTP quantities. It is valid if all four resulting entries remain non-negative. Every transposition induces a rotation (or a sum of rotations) on Q. Remark 6.16. Rotations are the atomic operations in the off-diagonal design space. Any two Q matrices with identical margins are connected by a sequence of rotations. 6.7 How Q Enters the Hit Rate The total game hit rate depends on per-reel moments through the inclusion-exclusion principle. Proposition 6.17 (Hit Rate via Inclusion-Exclusion). For a game with R identical reels and pay depth d (minimum OAK level), the probability of at least one win is: W H = (−1)k+1 d Ui[S] , k=1 S⊆Ω i=1 Li |S|=k (35) 20 where Ui[S] = #{windows on reel i containing every symbol in S}. The order-1 terms U [{s}] = Q[s][s] are fixed by counts. The order-2 terms U [{s, t}] = Q[s][t] are the off-diagonal entries of Q. Terms of order 3 and above are not determined by Q. Proof. The inclusion-exclusion over symbol subsets gives the probability that at least one symbol is present on all d paying reels. For a subset S, the probability that every symbol in S is present on reel i is Ui[S]/Li. Independence of the reels gives the product across reels. Under singles, a window has at most W distinct symbols, so U [S] = 0 for |S| > W and the sum terminates at k = W . Remark 6.18. The formula extends to per-reel Qi (non-identical reels) by replacing (Q[s][t]/L)d with d i=1 Qi[s][t]/Li. For symbol-dependent pay depths ds, the pairwise intersection term for symbols s (depth ds) and t (depth dt, dt > ds) uses Q entries on reels 1, . . . , ds and the deeper symbol’s marginal presence on reels ds+1, . . . , dt: P (As ∩ At) = ds i=1 Qi[s][t] Li · dt i=ds+1 Qi[t][t] . Li The worked example has depths d ∈ {2, 3}, so this form is exercised directly. Corollary 6.19 (Order-2 Hit Rate). The order-2 hit rate is the truncation of (35) at k = 2: H(2) = Q[s][s] d − Q[s][t] d . (36) L L s∈Ω s 2Ad for d ≥ 2. The hit-rate-preserving rotations are those between asymmetric entries where the convex increase from the smaller entries is exactly compensated by the concave decrease from the larger entries. The consecutive-entry condition provides a constructive family of such rotations. With n 2 − n off-diagonal degrees of freedom and hit rate being one scalar, the null space of hit-rate-preserving moves has dimension at least n 2 − n − 1. This family of moves is available for volatility fine-tuning (Section 8), though count allocation and clustering provide the dominant variance controls. 6.9 How Q Enters Payout Variance Proposition 6.23 (Per-Reel Payout Second Moment and Q). For analysis of cross-symbol co- occurrence effects, consider the per-reel linear surrogate X = s vs · Cs, where Cs is the count of symbol s in the window. This is not the ways-pay payout (which involves products across reels) but captures the per-reel contribution to variance. The second moment is: E[X2] = vs2 · Q[s][s] L · (τs2 + c2s ) + 2 Q[s][t] vsvt · , L (39) s s 1, E[CsCt] depends on the joint count distribution within shared windows and is not simply Q[s][t]/L; the designed-designed cross terms are handled as fixed skeleton contributions (Proposition 6.6). Corollary 6.24 (Volatility Control via Off-Diagonal). At fixed counts (fixed diagonal) and fixed per-symbol distributions (fixed τs2, c2s), the payout variance is an affine function of the off-diagonal entries of Q. Increasing Q[H1][H2] (two high-pay symbols share windows more often) increases variance. The off-diagonal block is the handle for total payout volatility at frozen RTP and frozen per-event volatility. 6.10 The Attainable Set Not every matrix satisfying the margin laws is achievable. Realizability is constrained by integrality and the window-content structure. Definition 6.25 (Feasibility Conditions). A candidate matrix Q satisfying the margin laws is subject to progressively stronger feasibility conditions: 1. Row sums: Q satisfies (30)–(32). 2. W -subset consistency: for every subset S of W symbols, the pairwise entries Q[s][t] for s, t ∈ S must be jointly achievable by some set of triangles on S. 22 3. 3-gram linear system: there exists a non-negative integer assignment of counts to each of the n(n − 1)(n − 2) ordered 3-grams satisfying flow conservation, symbol counts, and the Q constraints. 4. Connectivity: the support of the 3-gram assignment has connected support (is a single connected component). Proposition 6.26 (Feasibility Ladder (Computational)). Each condition is strictly stronger than the previous one. Exhaustive enumeration at n = 5, L = 10 yields: Condition Matrices admitted Row sums only + W -subset consistency + 3-gram linear system + Connectivity 10,577 158 148 87 The final condition admits exactly the achievable matrices. Soundness (achievable ⇒ passes all four) holds because any physical strip’s Q satisfies all four conditions by construction. Completeness (passes all four ⇒ achievable) follows from the Euler–Hierholzer theorem [11, 8]: at W = 3, windows are the 3-grams of the cyclic strip. A non-negative integer 3-gram count vector is realizable as a cyclic sequence if and only if it satisfies flow conservation on the 2-gram de Bruijn graph and has connected support — an Eulerian circuit on the directed multigraph. Conditions 3 (the 3-gram linear system) and 4 (connectivity) are precisely these two requirements. This holds at all (n, L), not only at the enumerated parameters. At general W , the same argument applies with (W −1)-gram nodes and W -gram edges. 6.11 Constructibility from Q The feasibility ladder (Proposition 6.26) operates at the 3-gram level. In practice, the designer holds Q — the pair matrix — not the full 3-gram census. The question is whether a given (L, n, Q) admits some strip realizing it, without specifying the census. Proposition 6.27 (Constructibility Test). A specification (L, n, Q) is realizable by a cyclic strip if and only if there exists a non-negative integer W -gram count vector y satisfying: (1) the Q constraints (pairwise sums over y reproduce Q), (2) flow conservation on the (W −1)-gram de Bruijn graph, and (3) strong connectivity of the support. This is an integer feasibility problem over nW variables and nW −1 nodes, with L appearing only in the right-hand sides. A YES returns y, and Hierholzer’s algorithm reads the strip off in O(L) time. Remark 6.28 (The Gap Is Real). Comparing constraint-legal Q matrices (satisfying margin laws, integrality, and non-negativity) against those realizable by a strip, by exhaustive enumeration at small parameters: n L Constraint-legal Realizable Not realizable 39 3 12 48 249 606 4489 54 78.3% 261 56.9% 157 96.5% 23 At n = 4, roughly 97 times in 100 a constraint-legal Q is unrealizable. The constructibility test (Proposition 6.27) classifies all 801 cases with zero false positives and zero false negatives, validating the implementation against exhaustive ground truth. The correctness of the test itself follows from the Euler–Hierholzer theorem (Proposition 6.26), which holds at all (n, L). Rejection costs ∼60 ms; acceptance with the built strip costs ∼0.4–2.6 s. Rejection is ∼20× cheaper than acceptance, which is the right asymmetry for a design loop that proposes many candidates. Remark 6.29 (The Realizable Set Is Connected). Once on the realizable set, every Q with the same counts is reachable from every other via coordinated moves (two pair counts up, two down). The set is a single connected component under L1-radius-4 moves, with diameter 4–8 hops, at n = 3, 4, 5 and L = 8..12. This means the design system can navigate the realizable set freely without leaving it, provided moves are coordinated rather than single-entry. Remark 6.30 (Q-Space Design Loop). With constructibility decidable from Q in milliseconds, the design loop operates entirely in matrix space: 1. Evaluate RTP and volatility from (L, n, Q): O(n2), ∼10 ms. 2. Navigate Q via coordinated moves: O(n2) per step. 3. Check constructibility at each candidate: ∼60 ms (reject) or ∼2.6 s (accept). 4. Build the strip once, from the final Q: O(L) via Hierholzer. The strip is never touched during design. Q is the order-2 design object (complete for RTPinvisibility and the hit-rate oracle; the full design additionally carries the W -gram census). Full pipeline from specification to verified strips: ∼17 s. 6.12 Reachability by Mixing The co-location matrix Q specifies the arrangement. This subsection shows that the achievable set of (H, CVwin) pairs at fixed counts is reachable by mixing, and that any interior point is achieved by splicing rather than searching. Proposition 6.31 (Exact Moment Preservation under Concatenation). A cyclic strip concatenated with itself has identical normalized moments: m/L, Q[s][t]/L, p, c, and all higher-order statistics are bit-identical. Concatenation scales the integer lattice without changing any game metric. Proof. The K-fold concatenation of a strip of length L is a strip of length KL. Every window of the concatenated strip is a window of the original strip, and each original window appears exactly K times. Therefore every normalized statistic (computed as a sum over windows divided by total windows) is identical. Proposition 6.32 (Splice Linearity). Let A and B be two strips of equal length L sharing the same skeleton and the same per-symbol counts. The spliced strip An−kBk ((n−k) copies of A followed by k copies of B) has moment vector: µ(An−kBk) = n − k µ(A) + k µ(B) + O(1/n), (40) n n where µ includes all per-reel presences, co-locations, and payout moments. The O(1/n) term is a junction correction arising from the W −1 windows that straddle block boundaries. When the skeleton pins the first and last W −1 cells of each block, all homogeneous junctions (A|A and B|B) 24 produce windows identical to the original strip’s wrap, contributing zero correction. The single heterogeneous cyclic wrap (the last B block returning to the first A block) contributes at most W −1 windows out of nL total — a correction of O(1/(nL)), negligible at production lengths. Proof. Each block contributes L windows (cyclic within the block) plus W −1 junction windows at each boundary. With n blocks there are n boundaries contributing n(W −1) junction windows and n(L−W +1) interior windows, totalling nL. The interior moments are the weighted average ((n−k)µ(A) + kµ(B))/n by linearity. The junction fraction per block is (W −1)/L, independent of n. The O(1/n) term in (40) is the resolution of the mixing ratio k/n, not the junction size. When the skeleton pins the boundary cells, both A and B produce the same junction windows (e.g., both end with H1 filler and begin with filler 10), so the junction correction is zero. Corollary 6.33 (Reachability by Mixing). Let S be the set of strips sharing a given skeleton and count vector. The achievable set of moment vectors {(H(S), E[Y 2](S)) : S ∈ S} is convex, since both H and E[Y 2] are affine along the splice path. Since CVw2 in = H · E[Y 2]/RTP2 − 1 is a product of two affine quantities at fixed RTP, the achievable (H, CVwin) pairs trace a parabolic arc under a two-block splice, not a segment. The arc bulges outward (decreasing H increases E[Y 2], so the quadratic term is negative), meaning the achievable set in (H, CVwin) is a smooth image of a convex set and the interior is reachable. Note: these statements hold for single-reel splices (one reel varied, others fixed). Since H is a product across reels, a simultaneous multi-reel splice at common ratio t produces a degree-R polynomial in t, not an affine function. Multi-reel targets are reached by sequential single-reel splices, each preserving the previous reel’s statistics. Proof. For any two achievable points µ(A) and µ(B), Proposition 6.32 shows that every rational convex combination k n µ(A) + n−k n µ(B ) is achieved by the splice An−k B k . Both H and E[Y 2] are sums over windows, hence affine in the splice ratio. The derived coordinate CVwin is a smooth function of these affine quantities, so the achievable set in (H, CVwin) is a smooth curve. Three blocks suffice to cover a two-dimensional region (Proposition 6.35). Precision costs strip length, linearly. Remark 6.34 (Construction by Mixing). Reachability by mixing reduces construction to three steps: (1) build extreme-point strips at the boundary of the achievable set, (2) choose the splice ratio that hits the target, (3) concatenate. No Monte Carlo, no fitness function, no convergence hope. The extreme points are found by maximizing and minimizing the hit rate over the filler arrangement freedom (the off-diagonal block of Q). The interior is reached by mixing. The RTP is preserved exactly throughout because all blocks share the same counts. The hit-rate resolution is O(1/n); the CVwin resolution is O(1/n). Both are made arbitrarily fine by increasing n. Proposition 6.35 (Two-Simplex Targeting). A two-block splice traces a line through (H, CVwin) space. To hit both coordinates simultaneously, three blocks suffice: let A, B, C be strips at distinct operating points. The splice AaBbCc with a + b + c = n achieves any point in the triangle conv{(µ(A), µ(B), µ(C))} at resolution O(1/n). 6.13 The E[c2] Lattice The per-reel second moment of the count distribution determines the diagonal contribution to CVwin. At W = 3, a symbol placed as k runs totalling n cells, of which k1 are singletons, has: E[c2 | c ≥ 1] = 9n − 8k + 2k1 , n + 2k k1 ∈ [max(0, 2k − n), k − 1]. (41) 25 The upper bound k1 ≤ k −1 is strict: k1 = k forces n = k (every run is a singleton). The maximum- variance layout at fixed (n, k) is determined : k − 1 singletons plus one run of length r = n − k + 1, giving: T −n k= , r = n − k + 1, (42) W −1 where T is the total coverage (number of windows containing the symbol). This layout is not searched — it is the unique maximizer of (41) at given (n, T ). Remark 6.36 (The k1 Residual). At fixed (n, p, k, c) and fixed hit rate, the remaining freedom is k1 per symbol per reel. This is the fine CVwin lever: closed-form range ×1.12, operating at frozen RTP and frozen hit rate. Remark 6.37 (The Clustering Profile as a CV Lever). The per-(symbol, reel) clustering vector c = (c(fi)) is a higher-dimensional freedom than the k1 residual. At fixed global RTP and fixed global hit rate, different per-symbol cf vectors can produce the same aggregate H (because H is a function of the per-symbol presences pf = mf /cf , and many c vectors map to the same hit rate via the inclusion–exclusion). Each such vector gives a different CVwin, because the per-reel second moments m2,f,i = τf2,i + c2f,i depend on cf . Treating c as a scalar (uniform clustering across all fillers) sweeps a one-dimensional line through this space; the achievable CVwin range along that line is a lower bound on the full range. With F fillers on Rfree non-blocker reels, the profile vector has F · Rfree free coordinates. The residual decomposition is: fix counts (RTP), fix the profile c and arrangement (hit rate), and k1 per symbol is all that remains. If the profile itself is allowed to vary at fixed globals, the reachable CVwin range is strictly larger. 7 Hit-Rate Targeting via Co-location The skeleton (Section 4) fixes the designed symbols’ placements. The filler allocation (Section 5) fixes the filler counts at an operating point on the RTP iso-surface. The choice of operating point is the coarse hit-rate control: different count allocations place different amounts of filler mass on each reel, changing per-reel m values and hence the achievable presence range. At the chosen operating point, the remaining freedom is the arrangement of fillers among the free positions — specifically, the clustering cf per filler per reel on non-blocker reels (Definition 5.1). Clustering moves pf = mf /cf at fixed mf , which is the fine hit-rate control. The splice construction mixes two arrangements to hit the target exactly. The co-location matrix Q is both the analysis and construction object: it characterizes the design space, proves RTP-invisibility, and provides the per-symbol clustering targets that the construction realizes. Splice linearity (Proposition 6.32) fills lattice gaps between nearby Q-targeted strips. This section shows that the arrangement freedom is sufficient to realize the target hit rate at the chosen operating point, with RTP preserved exactly. 7.1 Filler Presences: Fixed and Free Once the filler allocation from Section 5 is committed, each filler f has a known stop count nf on each reel. On blocker reels (reel df + 1), fillers are singles (cf = 1), so the presence is fixed by counts: pf = mf = W nf L . (43) On non-blocker reels (1, . . . , df ), the clustering cf ≥ 1 is a design parameter (Definition 5.1), giving pf = mf /cf . Higher cf means fewer windows contain the symbol, reducing pf and hence reducing 26 the total hit rate. This is the primary arrangement-level hit-rate control, operating at frozen mf (and hence frozen RTP). 7.2 The Off-Diagonal as the Hit-Rate Handle The game hit rate (Proposition 6.17) is: H= s d i=1 p(si) − s