Addressing-Maths Research Thread: Appendix¶
Status: Reference document. Captures findings from the coprime-indexed addressing on ℤ/Dℤ research thread that frame chess's coprime construction in a broader mathematical context but do not directly bind on the chess representation. The one finding that did augment chess-maths landed in chess_spectral_research_notebook.md §9f as the "Packing-optimality side-result" paragraph; everything else is preserved here.
Source thread: "Coprime-Indexed Addressing on ℤ/Dℤ — A Working Research Notebook" (Phase 1 hypothesis battery A-H1…B-H2) plus its A-H2/A-H3 corrections and the 15/16 ceiling theorem document.
Position in the project. The addressing-maths thread is the foundation/generalization layer under chess-maths, Othello-maths (chess notebook §10), and logo-maths (chess notebook §9l). All three are empirical instances of the same coprime-folded addressing pattern φ(i₁,…,i_k) = Σ pⱼ·iⱼ mod D with the cyclic group algebra ℂ[ℤ/Dℤ] as the efficient-operation class. This appendix records what the foundation says.
Scope discipline. Findings here are preserved as research artifacts. They do not modify the chess construction, do not introduce new chess claims, and do not require the chess notebook to be re-read in light of them. The chess construction at D ∈ {512, 640} is a single embedding of an 8×8 grid; the multi-grid packing results below describe the structure of ℤ_D under that embedding pattern but the chess representation never packs multiple grids.
A.0 — Phase 1 hypothesis battery summary¶
| Hypothesis | Question | Status | Headline |
|---|---|---|---|
| A-H1-strict | Two 8×8 grids pack into ℤ_1024 with odd-unit 4-tuple (p,q,r,s) and disjoint images | FAIL | Structural: φ(0,0) = 0 always, so images share 0 |
| A-H1-offset | Same with per-grid offset b₂ added (5-tuple) | PASS | 94,703 valid second-pairs admit some disjoint b₂ |
| A-H2 | max N grows linearly with D/64 for 2-power D | PASS (corrected to 15/16 ceiling) | True max N(D) = (15/16)·(D/64) by CP-SAT, not the original greedy "+7" |
| A-H3 | Non-2 prime-power D packs closer to volume bound than 2-power D | PASS | non-2 prime-power D saturates ratio 1.000; 2-power D plateaus at 15/16 |
| A-H4 | CRT-factored packing equals direct packing | PASS | Identical valid-pair sets at D = 1001 (7·11·13) and D = 323 (17·19) |
| B-H1 | Every chess piece operator is in ℂ[ℤ/Dℤ] | PASS (primary) | 384/384 on (D=640, p=67, q=7); structural tautology by construction |
| B-H2 | NTT agrees with direct; crossover depends on |S| | PASS | Up to 5.93× speedup; knight loses at D=4096 (|S|=8 too small for FFT constants) |
Status flag. A-H1-strict was the plan's anticipated "publishable infeasibility note." A-H2's "+7 closed form" was a path-dependent greedy artifact, replaced by the 15/16 structural ceiling under exact CP-SAT enumeration.
A.1 — The 15/16 Ceiling Theorem (Z₁₆ Fourier Obstruction)¶
This is the structural result that explains the 1/16 packing slack on 2-power D. The chess notebook §9f cites it briefly; the full statement and proof live here.
A.1.1 — Setup¶
For (p, q) odd-units in ℤ_{2^m} (m ≥ 4), let
image(p, q) = {p·r + q·c mod 2^m : r, c ∈ {0..7}}
This is the standard 8×8 Sidon image used throughout chess-maths and addressing-maths.
A.1.2 — Theorem (mod-16 non-uniformity)¶
image(p, q) projected to ℤ₁₆ is non-uniform.
Proof. Let A_u = {u·c mod 16 : c ∈ {0..7}}, an 8-element subset of ℤ₁₆. The DFT of 1_{A_u} on ℤ₁₆ is
F_u(k) = Σ_{c=0}^{7} ω^(u·c·k) where ω = exp(2πi/16).
If k ≠ 0 mod 16:
F_u(k) = (ω^(8uk) − 1) / (ω^(uk) − 1).
For k odd: uk is odd (u is a unit), so 8uk ≡ 8 (mod 16), giving ω^(8uk) = −1. Numerator = −2, denominator = ω^(uk) − 1 ≠ 0. Hence |F_u(k)| = 2 / |ω^(uk) − 1| = 1 / |sin(π·uk/16)| > 0.
For k even nonzero (k = 2j, j ∈ {1..7}): 8uk = 16uj ≡ 0 (mod 16), giving ω^(8uk) = 1. Numerator = 0. Hence F_u(k) = 0.
So F_u vanishes precisely at the 7 even nonzero frequencies of ℤ₁₆ and is nonzero at the 8 odd frequencies.
The image's mod-16 multiplicity vector is m_I = 1_{A_p} ∗ 1_{A_q} (ℤ₁₆ convolution). By the convolution theorem, m̂_I(k) = F_p(k) · F_q(k). At odd k: F_p(k) ≠ 0 and F_q(k) ≠ 0, so m̂_I(k) ≠ 0.
A constant function has DFT supported only at k=0. Since m̂_I has 8 nonzero entries at odd frequencies, m_I is non-constant. □
A.1.3 — Corollaries¶
Corollary 1. image(p, q) does not tile ℤ_{2^m} for any m ≥ 10.
A tiling I ⊕ S = ℤ_{2^m} with |I|=64, |S|=2^(m−6) requires the projection to ℤ₁₆ to satisfy 1̂_I(k) · 1̂_S(k) = 0 for all k ≠ 0 in ℤ₁₆. By the theorem, m̂_I is nonzero at all 8 odd frequencies, so m̂_S would need to vanish there. The same argument applies at finer projections (ℤ₃₂, ℤ₆₄, …), each adding constraints. Empirically all 243,712 valid Sidon-injective images on ℤ_1024 fail mod-16-balance, and CP-SAT confirms no image reaches N=16.
Corollary 2. Maximum disjoint-translate packing of image(p, q) on ℤ_{2^m} satisfies N ≤ 2^(m−6) − 1 = volume − 1 (for m ≥ 10).
Volume bound is 2^(m−6); equality requires perfect tiling, which Corollary 1 forbids.
A.1.4 — Conjecture (15/16 sharp ceiling)¶
For odd-unit p, q on ℤ_{2^m} (m ≥ 10) with image(p, q) Sidon-injective on the 8×8 grid:
max packing N(p, q, 2^m) ≤ (15/16) · 2^(m−6) = 15 · 2^(m−10).
Evidence. - Verified TIGHT and OPTIMAL at m=10 (N=15) and m=11 (N=30) by CP-SAT. - Verified ACHIEVABLE at m=12 (N≥60 by CP-SAT FEASIBLE). - Theorem A.1.2 alone gives N ≤ vol − 1 (loose: 31 at m=11 vs actual 30; 63 at m=12 vs conjectured 60). - The constant 15/16 ratio across m=10, 11, 12 strongly suggests a structural law.
The full proof of the conjecture likely uses the ℤ₁₆ obstruction at all higher finer-projection levels jointly (ℤ₃₂, ℤ₆₄, …, ℤ_{2^m}), in the spirit of Coven–Meyerowitz / Tijdeman tiling theory.
A.1.5 — Why uniform up to mod-8, broken at mod-16¶
The image is uniform mod 2, 4, 8 (multiplicities {32,32}, {16,16,16,16}, {8,8,8,8,8,8,8,8}) but breaks at mod 16. The reason: for odd u in ℤ_{2^k}, the set u·{0..7} mod 2^k reduces SURJECTIVELY onto ℤ₈ (each ℤ₈ coset hit exactly once when 2^k ≥ 8), so mod 8 every image element falls into a unique ℤ₈ coset of the relevant 2^k group.
At mod 16: u·{0..7} mod 16 has 8 elements but only fills 8 of the 16 possible residues (a transversal mod 8 inside ℤ₁₆). The distribution among ℤ₁₆ residues depends on u, and the convolution of two such 8-element-ℤ₁₆-transversals is never uniform.
In bit-flag language: side = 2³ = 8 elements is uniform up to bit 3 of the address but loses uniformity at bit 4. That is the obstruction.
A.2 — A-H1: Strict 4-tuple FAIL → Offset-Aware 5-tuple PASS¶
A.2.1 — The strict 4-tuple is structurally impossible¶
The plan asked: "Do there exist 4-tuples (p, q, r, s) ∈ ((ℤ/1024ℤ)*)⁴ such that (p, q) and (r, s) each embed the 8×8 board injectively and the two images are disjoint in ℤ_1024?"
For any D and any generator pair (p, q), φ(0, 0) = (p·0 + q·0) mod D = 0. Both grids therefore always contain phase 0, and the two images can never be disjoint. The prediction is infeasible for all D, not just ℤ_1024.
Verdict (strict): FAIL. Trivial infeasibility from the common corner. This is the "publishable infeasibility note" the addressing-maths plan §6.2 anticipated.
A.2.2 — The research-relevant 5-tuple¶
The chess §11 phase-operator construction treats phi_origin as an additive offset:
P_rook(φ_origin) = {φ_origin + delta mod D : delta ∈ S_rook}.
VSA role-binding (Plate 1995; Kanerva 2009) does the same with a different name (per-object HV offset). Generalizing to two grids:
φ_g(r, c) = (b_g + p_g·r + q_g·c) mod D
WLOG b₁ = 0 (overall translation is a group symmetry of ℤ_D). The research-relevant question becomes: do there exist (p₁, q₁, p₂, q₂, b₂) with both pairs injective and the two offset images disjoint?
This reduces to finding b₂ in the complement of the difference set image_1 − image_2 in ℤ_D.
A.2.3 — Numbers (D = 1024)¶
valid (odd-unit) pairs on ℤ_1024: 243,712 (92.97% of odd²)
canonical first (67, 7) valid: True
seconds admitting SOME disjoint offset: 94,703 (38.86% of valid seconds)
example 5-tuple (p₁, q₁, p₂, q₂, b₂): (67, 7, 1, 9, 519)
sample mean admissible b₂ per second: 36.1 (SE 6.7, n=200)
est. 5-tuples with canonical first: ~8.8 × 10⁶
Verdict (offset): PASS. Existence is confirmed constructively. 38.9% of valid second-pairs admit at least one disjoint offset; on average each admissible second-pair has ~36 valid offsets out of D = 1024.
A.2.4 — What this means for chess¶
The infeasibility of A-H1-strict is not a counterexample to the plan; it is the plan's own worst-case flag realized. The trivial (0, 0) corner forces that outcome, and the offset-aware version is what the chess §11 and Othello §1c constructions were doing all along without stating it as a formal definition. The corrected primitive is a 5-tuple (p₁, q₁, p₂, q₂, b₂), not a 4-tuple, and chess §11.2.1's phi_origin parameter is exactly this offset.
This formalizes what chess-maths §11 implicitly does: per-piece origin offsets are not a convenience, they are a structural requirement once two grids must coexist in the same ℤ_D.
A.3 — Ring-Theoretic Findings (A-H2, A-H3, A-H4)¶
A.3.1 — A-H2: Linear scaling on 2-power D¶
Translation packing (disjoint translates of image(67, 7)) on D ∈ {1024, 2048, 4096, 8192} initially gave N = 7, 21, 49, 105 under smallest-b-first greedy, fitting N(D) = 7·(D−512)/512 with R²=1.0000. This was a path-dependent algorithm artifact.
Under exact CP-SAT (proven OPTIMAL where reported):
| D | greedy N | random multistart | CP-SAT | volume bound | true_max / vol |
|---|---|---|---|---|---|
| 1024 | 7 | 13–14 | 15 OPTIMAL | 16 | 15/16 = 0.9375 |
| 2048 | 21 | 24 | 30 OPTIMAL | 32 | 30/32 = 0.9375 |
| 4096 | 49 | 47 | ≥60 FEASIBLE | 64 | ≥60/64 = 0.9375 |
| 8192 | 105 | 86 | (not solved) | 128 | (≥0.67) |
Greedy continues with greedy(16384)=217, greedy(32768)=434, greedy(65536)=868 — the "+7" pattern breaks at m=15 then resumes, characteristic of a path-dependent algorithm rather than a theorem.
Corrected verdict. True maximum disjoint-translate packing of image(67, 7) on ℤ_{2^m} for m=10, 11 is exactly (15/16) · volume, proven by CP-SAT. The 15/16 ratio is explained by the §A.1 ℤ₁₆ Fourier obstruction.
A.3.2 — A-H3: Prime-power D vs 2-power D anomaly¶
Translation packing across 2-power and non-2 prime-power D, with canonical generator chosen per D (smallest (p, q) with p ≠ q in units(ℤ_D)² that embeds 8×8 injectively, excluding bases 2, 5):
| D | Factor | Volume (D/64) | (p, q) | Nmax | Ratio |
|---|---|---|---|---|---|
| 1024 | 2¹⁰ | 16 | (1, 9) | 14 | 0.875 |
| 2048 | 2¹¹ | 32 | (1, 9) | 28 | 0.875 |
| 4096 | 2¹² | 64 | (1, 9) | 57 | 0.891 |
| 729 | 3⁶ | 11 | (1, 8) | 11 | 1.000 |
| 2401 | 7⁴ | 37 | (1, 8) | 37 | 1.000 |
Mean ratio (2-power D) = 0.880 (greedy); mean ratio (non-2 prime-power D) = 1.000.
Corrected reading. The 12% gap on 2-power D is greedy; under true CP-SAT max it's exactly 1/16 = 6.25%. Non-2 prime-power D continues to saturate exactly (verified at D=625, 729, 2401, 3125, 4913). The original direction (2-power D harder than prime-power D) is correct, but magnitude ~doubles when moved from greedy to true max, and the structural reason is the §A.1 mod-16 obstruction specific to 2-power D.
Verdict: PASS. Non-2 prime-power D saturates the volume bound exactly. 2-power D plateaus at the 15/16 structural ceiling.
A.3.3 — A-H4: CRT decomposition equivalence¶
The ring isomorphism ℤ_D ≅ ∏ ℤ_{m_j} (Ding–Pei–Salomaa 1996) lets us either enumerate (p, q) directly in units(ℤ_D)² or factor each into residue tuples (p_j, q_j) ∈ units(ℤ_{m_j})² and reconstruct via CRT. Both must produce the same valid-pair set.
| D | Moduli | Direct count | CRT count | Equivalent |
|---|---|---|---|---|
| 1001 | 7 × 11 × 13 | 478,080 | 478,080 | True |
| 323 | 17 × 19 | 62,784 | 62,784 | True |
Verdict: PASS. The plan's "15-threshold" qualifier (each m_j ≥ 15) is not required for set equivalence (D = 1001 works despite m_j ∈ {7, 11, 13}); it is required for per-factor injectivity to hold independently. When some m_j < side, per-factor injectivity fails in those factors but the combined Diophantine constraint over ℤ_D can still be satisfied because the other factors compensate.
A.3.4 — Practical consequence¶
Pick D as a product of non-2 prime powers for tight packing; factor via CRT for search efficiency. The chess regime ℤ_640 = 2⁷ × 5 has the 2-adic slack from §A.1 in its 2⁷ factor. Pure non-2 prime-power D (e.g., ℤ_729·2401) would pack tighter by §A.3.2 + §A.3.3. This is the Residue-HDC pattern (Kymn et al. 2025) realized on the addressing-maths substrate.
A.4 — Group-Algebra Closure (B-H1) and NTT Acceleration (B-H2)¶
A.4.1 — B-H1: Every chess piece operator is in ℂ[ℤ/Dℤ]¶
Six piece shift sets per chess §11.2:
S_rook(p, q) = {±k·p : k ∈ [1, 7]} ∪ {±k·q : k ∈ [1, 7]}
S_bishop(p, q) = {±k·(p+q) : k ∈ [1, 7]} ∪ {±k·(p−q) : k ∈ [1, 7]}
S_queen(p, q) = S_rook ∪ S_bishop
S_king(p, q) = {±p, ±q, ±(p+q), ±(p−q)}
S_knight(p, q) = {±(2p+q), ±(2p−q), ±(p+2q), ±(p−2q)}
S_pawn_advance(p, q) = {p, 2p}
Numbers — primary case (p, q) = (67, 7), D = 640:
rook |S|=28 origins matching: 64/64 (100.00%)
bishop |S|=28 origins matching: 64/64 (100.00%)
queen |S|=56 origins matching: 64/64 (100.00%)
king |S|= 8 origins matching: 64/64 (100.00%)
knight |S|= 8 origins matching: 64/64 (100.00%)
pawn_adv |S|= 2 origins matching: 64/64 (100.00%)
overall: 384/384 (100.00%)
This reproduces chess §11.3's 416/416 result generically (not hard-coded to 640).
Secondary structural probes (six combinations of (p, q, D) including 2-power D and prime-power D) hit 2014/2688 = 74.9% overall — geometric equivalence depends on (D, p, q), not on ℂ[ℤ/Dℤ] membership alone.
Structural reading. Group-algebra closure is a structural tautology by construction — each P_piece IS Σ_{δ ∈ S_piece} S^δ ∈ ℂ[ℤ/Dℤ] because that is how we defined it. The substantive test is whether the phase reachable set clips to the geometric reachable set, and that clipping is (D, p, q)-dependent. The chess answer ((67, 7), D ∈ {512, 640, 768}) is exactly the chess construction; the addressing-maths generalization is a question about when the phase-operator torus and the geometric board share enough structure that the torus-clip (chess §11.3.3) yields the geometric moves.
A.4.2 — B-H2: NTT crossover¶
For a fixed piece P with shift set S_P and an occupation vector o ∈ {0, 1}^D, the reachability vector is
r = circular_convolve(o, χ_{S_P}) in ℤ/D arithmetic,
where χ_{S_P}[δ] = 1 iff δ ∈ S_P. Two implementations benchmarked at 32-piece occupation, 20 trials each:
| D | Piece | |S| | direct (ms) | NTT (ms) | Speedup | Correct? |
|---|---|---|---|---|---|---|
| 640 | queen | 56 | 0.141 | 0.103 | 1.37× | ✓ |
| 640 | knight | 8 | 0.020 | 0.010 | 1.94× | ✓ |
| 640 | rook | 28 | 0.070 | 0.012 | 5.93× | ✓ |
| 1024 | queen | 56 | 0.137 | 0.028 | 4.92× | ✓ |
| 1024 | knight | 8 | 0.020 | 0.014 | 1.46× | ✓ |
| 4096 | queen | 56 | 0.138 | 0.047 | 2.96× | ✓ |
| 4096 | knight | 8 | 0.021 | 0.039 | 0.54× | ✓ |
| 4096 | king | 8 | 0.021 | 0.039 | 0.55× | ✓ |
Correctness holds everywhere (max discrepancy < 1e-6). The crossover is in shape: queen (|S|=56) consistently benefits; knight/king (|S|=8) lose at D=4096 because numpy.fft constants dominate for small shift sets.
Verdict: PASS. O(D log D) versus O(D·|S|) reproduced with crossover quantified. For chess move generation the queen is where NTT matters most; small-|S| pieces favor direct on small D.
Connection to chess-maths §11.4.5. §11.4.5 reports a 6.5×–7.5× speedup of phase-native arithmetic over python-chess at pseudo-legal scope. That is the operational manifestation of B-H1 + B-H2: piece operators are in ℂ[ℤ/Dℤ] (so the speedup class is well-defined) and NTT is one realization of that class (so the asymptotic is O(D log D)). Chess §11.4.5's measurement is naive Python integer arithmetic vs python-chess's reference, not NTT; an NTT-backed implementation at D=640 with |S|=56 (queen) would extract additional benefit per A.4.2's 1.37× direct-vs-NTT delta.
A.5 — Synthesis: Crossovers between A and B¶
The six hypotheses braid onto a single substrate: linear Diophantine conditions on tuples (p, q) ∈ (ℤ/Dℤ)², plus the group algebra ℂ[ℤ/Dℤ], plus CRT factorization. Four concrete crossovers surfaced:
-
A-H1 ↔ A-H2. The offset-aware 5-tuple primitive from A-H1 is what A-H2 counts at scale. Without the offset b, two grids share (0, 0) and N > 1 is impossible; with b, N grows linearly with D/64 at the corrected 15/16 ceiling.
-
A-H3 ↔ A-H4. A-H3 quantifies the 2-adic slack (1/16 = 6.25% on 2-power D under true CP-SAT max; 0% on non-2 prime-power D). A-H4 verifies the ring isomorphism ℤ_D ≅ ∏ ℤ_{m_j}. Together: pick D as a product of non-2 prime powers for tight packing, factor via CRT for search efficiency. This is Residue-HDC (Kymn et al. 2025) realized on the addressing-maths substrate.
-
A-H1/A-H2/A-H3 ↔ B-H1. The "valid 2D generators" set used by A-experiments is the same set whose geometric equivalence B-H1 tests. A-experiments count how many valid pairs exist; B-H1 tests which of those pairs reproduce chess-geometric behavior on an 8×8 clip. The chess canonical (67, 7) is a single member of A-H1's 243,712-element valid set, and the question of which other members give geometric equivalence is open (secondary probes hit 75% overall, not 100%).
-
B-H1 ↔ B-H2. B-H1 certifies that piece operators ∈ ℂ[ℤ/Dℤ]; B-H2 certifies that this class is NTT-accelerable. Composition: if your operation is an element of the group algebra, you get O(D log D) via NTT with crossover quantified by B-H2. Anything outside ℂ[ℤ/Dℤ] (non-circulant matrices, graph-specific operations on non-Cayley graphs) does not benefit.
The cross-sub-question crossovers A ↔ C (Diophantine independence ↔ RRNS range-check duality), A ↔ D (Singleton as upper envelope), and C ↔ D (RRNS subsumes D for pure-CRT) are addressing-maths Phase 2 items not addressed in this appendix.
A.6 — Connections to chess-maths sections¶
Where each addressing-maths finding lands in (or relates to) the chess notebook:
| Addressing-maths finding | Chess-maths section | Relationship |
|---|---|---|
Diophantine condition p·Δr + q·Δc ≢ 0 (mod D) |
§9f "Generator-selection requirement" | Already in chess-maths verbatim (discovered via Othello Phase 1 pass) |
| (3, 7) on D=1024 collision counterexample | §9f, §10.12 bug 2 | Already in chess-maths |
| (67, 7) packing-optimality at D=1024 (15/16, CP-SAT) | §9f "Packing-optimality side-result" | Distilled into chess-maths (this appendix's only direct chess-notebook contribution) |
| 15/16 ceiling theorem (ℤ₁₆ Fourier obstruction) | §9f reference | Cited briefly in chess-maths; full proof in §A.1 of this appendix |
| A-H1-strict structural impossibility → 5-tuple offset | §11.2.1 phi_origin |
Implicit in chess-maths; formalized here |
| A-H2 linear scaling (corrected to 15/16) | (none direct) | Background; chess uses single embedding, not multi-grid packing |
| A-H3 prime-power D saturates | (none direct) | Structural background |
| A-H4 CRT decomposition equivalence | (none direct) | Structural background |
| B-H1 group-algebra closure | §11.3 (chess validation 416/416) | B-H1 is the structural framing of what §11 verifies |
| B-H2 NTT speedup | §11.4.5 phase-native speedup measurement | B-H2 is the asymptotic theory; §11.4.5 is the empirical chess measurement |
| Othello (7, 11) on D=768 | §10.12 H7 | Othello instance of the same construction |
| Logo HDC vocab × syntax | §9l | Logo instance (different projection on same substrate) |
Summary. One finding (packing-optimality of (67, 7)) was pulled into chess-maths §9f. The rest is preserved here as structural background that frames why the chess construction works and what its mathematical neighborhood looks like.
A.7 — Literature grounding¶
Consolidated from the addressing-maths research plan §3–§5, ordered roughly by recurrence:
- Erdős, P. & Turán, P. (1941). On a problem of Sidon in additive number theory. J. London Math. Soc. 16.
- Bose, R.C. & Chowla, S. (1962). Theorems in the additive theory of numbers. Comment. Math. Helv. 37.
- Davis, P.J. (1979/1994). Circulant Matrices. Wiley.
- Pollard, J.M. (1971). The fast Fourier transform in a finite field. Math. Comp. 25.
- Plate, T.A. (1995). Holographic reduced representations. IEEE TNN 6(3).
- Plate, T.A. (2003). Holographic Reduced Representation: Distributed Representation for Cognitive Structures. CSLI.
- Kanerva, P. (2009). Hyperdimensional computing. Cognitive Computation 1(2).
- Mandelbaum, D. (1972). Error correction in residue arithmetic. IEEE TIT IT-18.
- Yau, S.S. & Liu, Y.C. (1973). Error correction in redundant residue number systems. IEEE TC C-22.
- Ramsey, J.L. (1970). Realization of optimum interleavers. IEEE TIT 16(3).
- Forney, G.D. (1971). Burst-correcting codes for the classic bursty channel. IEEE TCT 19(5).
- Sun, J. & Takeshita, O.Y. (2005). Interleavers for turbo codes using permutation polynomials over integer rings. IEEE TIT 51(1).
- Bibak, K., Kapron, B.M., Srinivasan, V., Tauraso, R. & Tóth, L. (2017). Restricted linear congruences. J. Number Theory 171.
- Hedayat, A.S., Sloane, N.J.A. & Stufken, J. (1999). Orthogonal Arrays. Springer.
- Ding, C., Pei, D. & Salomaa, A. (1996). Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography. World Scientific.
- Kymn, C.J., Kleyko, D., Frady, E.P., Bybee, C., Kanerva, P. & Sommer, F.T. (2025). Residue hyperdimensional computing. Neural Computation 37.
- Biggs, N. (1974/1993). Algebraic Graph Theory. Cambridge University Press.
- Vaidyanathan, P.P. & Pal, P. (2011). Sparse sensing with co-prime samplers and arrays. IEEE TSP 59(2).
Coven–Meyerowitz / Tijdeman tiling theory is the natural literature for proving the §A.1.4 conjecture (15/16 sharp ceiling); the partial proof in §A.1.2 uses the ℤ₁₆ obstruction alone, whereas the full sharp bound likely requires the joint obstruction across all finer projections ℤ₃₂, ℤ₆₄, …, ℤ_{2^m}.
A.8 — Open directions (out of scope here)¶
These are flagged for completeness; none modify the chess construction.
-
Freiman–Plünnecke–Ruzsa ↔ RRNS distance bridge. Characterizes when image(φ) has small doubling (structured GAP complement, sharp bounds) vs pseudo-random complement (looser bounds). Components exist separately in the literature; no published paper writes the bridge explicitly.
-
Memory-system Pareto analysis. "Is a specific coprime multiplier optimal against empirically-characterized DRAM/flash burst distributions under a given outer code?" — legitimate research direction but requires hardware burst-length data not surveyed in this thread.
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Synthesis paper. The addressing-maths thread's eventual deliverable: a packaging document positioning "coprime-folded addressing on finite cyclic groups, with the group algebra ℂ[ℤ/Dℤ] as the efficient-operation class and the image-set complement as the bounds/fault primitive" as a unified VSA-compatible design principle, with chess/Othello/logo as empirical instances. Working title: "Coprime-folded addressing on finite cyclic groups as a VSA design principle: packing feasibility, group-algebra closure, and RRNS duality."
-
Sharp 15/16 ceiling proof. §A.1.4 conjecture; the structural Fourier obstruction at ℤ₁₆ is established but the sharp bound likely requires the joint obstruction at all finer projections.
This appendix is a research-record document. It may be amended as the addressing-maths thread produces additional findings, but its scope is bounded: capture what the foundation says about chess-maths's coprime construction, do not duplicate chess-maths content, do not invent new chess claims.