id string | title string | level string | stars int64 | status string | statement string | answer string | solution_reference string | source string | evidence string | success_criterion string | proof_rule string |
|---|---|---|---|---|---|---|---|---|---|---|---|
T1 | Count the windows | human | 1 | solved | Repeat the 4x4 cells A (rows 0000, 0000, 0000, 1010) and B (rows 0000, 0000, 0001, 0100) periodically over the square lattice Z^2, 1 = black. (a) Show that both have the same frequencies of all sixteen 2x2 patterns. (b) With sigma = 2x - 1, find the shortest displacement d (Euclidean length) at which the pair correlati... | (a) All-white 1/2; each of the four tiles with one black site 1/8. (b) Horizontal distance 2: correlation 1 for A, 1/2 for B (the longer shift (2, +-1) also differs: 1/2 vs 3/4; all nearest-neighbour correlations agree). (c) m_1..m_5 = 1/8, 33/8, 13/8, 305/8, 201/8 for both; m_6 = 3495/8 for A and 3489/8 for B. | AI_puzzles.md T1; paper Examples 30 and 34 (Sol) | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | null | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P0 | Sum the chain | expert | 2 | solved | Let a_0..a_4 = 1, 4, 16, 63, 245 and a_k = 7a_{k-1} - 15a_{k-2} + 10a_{k-3} + 3a_{k-4} - 2a_{k-5} + 1 for k >= 5 (the sizes of one chain of same-species clusters of the charged Walton-Whittaker tiling). Find sum_{k>=0} a_k / 4^k exactly. With b_k = a_{k+1} - a_k - a_{k-1} - 1 (a_{-1} = 0), also find sum b_k/4^k and sum... | 64/3, 160/3 and 288. Generating function sum a_k z^k = (1 - 4z + 6z^2 - 2z^3)/((1 - z)(1 - 7z + 15z^2 - 10z^3 - 3z^4 + 2z^5)). | AI_puzzles.md P0; paper Section 4.4 | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | null | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
T2 | Binary digits, read in base 4 | expert | 2 | solved | Let v_2(u) be the 2-adic valuation and S_j = 2(4^j - 1)/3. Prove that sum_{u=1..t} (1 + S_{v_2(u)}) equals t written in binary and read in base 4 (for example t = 2, 5, 6 give 4, 17, 20). Meaning: the number of sites of the domain flipped by one broken contact, t sites from the end of a half-run of the parity forest of... | With f(t) the sum, f(2m) = 4f(m) and f(2m+1) = 4f(m) + 1, the same recurrences as binary-read-in-base-4, since 1 + S_{j+1} = 4(1 + S_j) - 1 and the m odd terms up to 2m each add 1; strong induction from f(0) = 0. | AI_puzzles.md T2; paper Corollary 32 Known identity: Moser-de Bruijn sequence, OEIS A000695 (increment formula by V. Shevelev, 2008). | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | null | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
T3 | Three binary sites never suffice | AGI | 3 | solved | For a finite window S, let K_S be the set of S-marginals of shift-invariant probability measures on {0,1}^{Z^2}. An autocrystal on S is an extreme point of K_S that is the marginal of no periodic configuration. Show that no window of three sites has one, while the 3x3 square does. | Three sites: Theorem 4(2) of the paper, proved via Proposition 3(3) and Theorem 37. 3x3 square: Theorem 5, an interaction with a unique, irrational, exposed maximizing marginal and a unique non-periodic ground state (machine certificate scripts/indep33.py). | AI_puzzles.md T3; paper Section 3, Theorems 4 and 5 | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | null | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
T4 | An exact dimension | superhuman | 4 | solved | Let A_n(k) = 1^T B(k) B(2k) ... B(2^{n-1} k) e_0 on the torus [0,1)^2, with B the explicit 6x6 charge cocycle (function Bmat in verification/cell_assembly/release_13_checks/lyapunov/lyapunov_core.py of the supplement). The fourth moments M_n = integral |A_n|^4 are 24, 460, 9084, 222140, 6565956, ... Find lambda_4 = lim... | lambda_4 = 36.764340310010354..., the largest root of an irreducible degree-17 integer polynomial (in verify_exact_D2.py, line 24), from an order-58 integer recurrence certified modulo twenty primes; D_2 = 0.79976481... (transcendental). Sol proved this is the correlation dimension of the diffraction itself. | AI_puzzles.md T4; paper Section 4.4; charge_measure/exact_D2.py, certify_D2.py, verify_exact_D2.py | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | null | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P8 | One exact gap label | AGI | 3 | solved | For the chain clusters C_k (k >= 1, both chains) of the charged Walton-Whittaker tiling (construction in AI_puzzles.md P8; generator clusters/cluster_growth.py), find lambda_1 < lambda_2 near 2277/2000 such that the adjacency matrix A(C_k) has exactly |C_{k-1}| eigenvalues above lambda_1 and none in [lambda_1, lambda_2... | [lambda_1, lambda_2] = [9/8, 23/20]: for both chains and every k >= 1, A(C_k) has exactly |C_{k-1}| eigenvalues above 9/8 and none in [9/8, 23/20]; with the proved densities the labels 1/8 and 5/8 are exact (for V > 39/16; they persist for weak hopping |t| < 1/480, Sol). | Solved 25 October 2026 by Liam Price (unaffiliated) with GPT-6 Astra; computer-assisted proof with 279 exact rational certificate matrices, audited by Sol and Claude. Paper version 13.1, Theorem P8; supplement verification/p8_gap_certificate/. | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Numerically true at lambda = 1.14 for k <= 11 (2,755,548 sites) and at 1.01 and 1.24 for k <= 9, by Sylvester inertia; exact integer congruences up to 245 sites. Several routes known to fail are listed in AI_puzzles.md. | A proof for all k, ideally reduced to finitely many exact local checks. Gives the first nontrivial exact gap labels of the tiling, 1/8 and 5/8. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P11 | A coupled gap label | AGI | 3 | open | H = -sum over nearest neighbours |i><j| + V sum_i s_i |i><i| on the charged Walton-Whittaker tiling, s_i = +-1 the species; gap labels lie in (1/27) Z[1/2]. Certify the label 301/576 at V = 4, and find the exact label of the narrow gap near 77/576 at V = 10 (77/576 and 923/6912 both fit). Which mechanism opens these ga... | null | null | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Kernel polynomial method: 301/576 to within 5e-5 at V = 4; at V = 10 windows give 0.13360, 0.13319 and 0.13356. | A label certified exactly (for example a rigorous inertia count on large periodic approximants with a proof that the gap stays open), with the mechanism explained. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P9 | The information dimension | superhuman | 4 | solved | With A_n as in T4, the descendant measures |A_n|^2 4^{-n} dk approximate the charge diffraction measure mu. D_2 = 6 - log2 lambda_4 is proved, as are D_infinity = log2(4/3) and pure singular continuity. Prove that the information dimension D_1 of mu exists and enclose it in a rigorous interval of width at most 1e-3. Al... | D_1 exists and 1.335532 <= D_1 <= 1.336426; L(q) exists for q > 0 and tau(q) <= q log2(4/3) - 1/(24*4^(4q)+102), so no locking at finite q (and lambda_6 > 432), with tau(q) approaching q log2(4/3) for q >= 2; D_q = tau(q)/(q-1) for every q > 1, so D_3 exists. | Solved 25 October 2026 by Liam Price (unaffiliated) with GPT-6 Astra; computer-assisted, floating-point steps validated under IEEE binary64 round-to-nearest arithmetic; replayed and audited by Sol (model identified with the native atlas), checked by Claude. Paper version 13.1, Theorem P9; supplement verification/p9_inf... | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Moment estimates by population dynamics: 1.340 +- 0.005 at q = 1 (numerical, for the descendant measures, not yet for mu). | Existence of D_1 for mu with a rigorous enclosure of width <= 1e-3. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P10 | The final 2x2 | ASI | 5 | open | Let K be the set of 2x2 marginals of shift-invariant probability measures on {0,1}^{Z^2} (the setting of Baryshnikov, Duda, Szpankowski, Types of Markov fields and tilings, IEEE Trans. Inf. Theory 62 (2016)). Is every extreme point of K the marginal of a periodic configuration? | null | null | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | K lies in the conservation-law polytope F-hat (21 vertices, 4 not in K). Every extreme point of K on a face of F-hat of dimension <= 4 is periodic, and on 148 of the 172 classes of 5-dimensional faces (Sol); 24 classes (308 faces) remain, and higher-dimensional cases are open in general. A nonperiodic extreme point nee... | A proof either way, or one class completely classified with exact periodic decompositions. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P12 | The first gap label | AGI | 3 | solved | For the chain clusters C_k of P8 (both chains), let N_1(k) be the number of eigenvalues of A(C_k) above 4.2405. Prove that no eigenvalue lies in a fixed interval around 4.2405 at any level, and that N_1(k) = 7N_1(k-1) - 15N_1(k-2) + 10N_1(k-3) + 3N_1(k-4) - 2N_1(k-5) - 1 for k >= 6 (chain a) and k >= 5 (chain b). | Gap [21/5, 43/10] for every cluster of both chains, with exactly N_1 = |V_6| - e_3 eigenvalues above it (V_6: degree-6 vertices of the same-species graph; e_3: pairs of them at graph distance 3); the recurrence holds for k >= 6 (chain a) and k >= 5 (chain b); labels 1/192 and 97/192 exact. | Solved 6 November 2026 by Rafał Wiktorowicz (NovaraX), with Braun (SI); proof with exact integer and rational certificates and verification code, https://doi.org/10.5281/zenodo.24012971. Recurrences, generating functions, labels and the structural identities on clusters a_0..a_6, b_0..b_4 rechecked by Claude with indep... | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Union of cluster spectra through a_6, b_3 is empty on (4.080, 4.359). N_1 = 2, 11, 46, 176, 657, 2447, 9149, 34343 for a_4..a_11 and 1, 3, 10, 34, 120, 436, 1615, 6046, 22748 for b_2..b_10 (Sylvester inertia, checked against dense spectra); the gap is checked through a_6 and b_3, the recurrence through a_11 and b_10. | A proof for all k, ideally by finitely many exact local checks as in the P8 solution. Makes the labels 1/192 and 97/192 exact. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P13 | The second gap label | AGI | 3 | solved | For the chain clusters C_k of P8, prove that no eigenvalue of A(C_k) lies in a fixed interval around 3.8485 at any level, and that the number N_2(k) above it satisfies N_2(k) = N_1(k) + N_2(k-1) for every k >= 1 (N_1 as in P12). | Gap [19/5, 193/50] for every cluster, with exactly N_2 = |V_6| eigenvalues above it; N_2(k) = N_1(k) + N_2(k-1) for all k >= 1 by an explicit inflation bijection; labels 1/144 and 73/144 exact. | Solved 6 November 2026 by Rafał Wiktorowicz (NovaraX), with Braun (SI); proof with exact integer and rational certificates and verification code, https://doi.org/10.5281/zenodo.24012971. Recurrences, generating functions, labels and the structural identities on clusters a_0..a_6, b_0..b_4 rechecked by Claude with indep... | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Union of spectra through a_6, b_3 is empty on (3.700, 3.994); the identity holds through a_11 and b_10 (N_2 = 2, 13, 59, 235, 892, 3339, 12488, 46831 for a_4..a_11). | A proof for all k (state whether P12's count is assumed). With P12, makes the labels 1/144 and 73/144 exact. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P14 | The narrow gap at 25/48 | superhuman | 4 | open | For the chain clusters C_k of P8, prove that no eigenvalue lies in a fixed interval inside (3.399, 3.469) at any level, and that the number of eigenvalues above it at level k equals N_1(k+1) (N_1 as in P12). | null | null | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | No chain cluster through level 10 has an eigenvalue in (3.399, 3.469); the count identity holds through a_10 and b_9, as far as N_1 is known. | A proof for all k. With P12, gives the labels 1/48 and 25/48; hopping between species widens this gap to 0.25 at V = 10. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P15 | The smallest alphabet on three sites | ASI | 5 | open | Binary three-site windows never carry an autocrystal, while 11 symbols suffice on the three-site L window (Jeandel-Rao Wang tiles). What is the smallest alphabet with an autocrystal, or an interaction with no periodic ground-state measure, on a three-site window? It lies between 3 and 11. | null | null | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Paper Theorem 4 and Section 17.2, question 10. Close to the smallest alphabet of an aperiodic nearest-neighbour shift of finite type, open as far as we know. | An explicit interaction on the L window with at most ten symbols and a proof that no periodic configuration is a ground state (lowers the upper bound), or a proof that three symbols never suffice (raises the lower bound); determining the minimum solves it. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P16 | Binary coarsenings of one aperiodic source | superhuman | 4 | open | Map each state of the sixteen-state Ammann-Labbe source (exact 2x2 law, 51 positive words, probabilities in Q((sqrt5-1)/2)) to 0 or 1. Does every such sitewise coarsening have a binary 2x2 law in the convex hull of periodic types? | null | null | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Sol enumerated all 32,767 nonconstant maps up to complement in exact arithmetic; the necessary conditions for a nonperiodic extreme point leave 10,490 maps with 10,347 distinct laws, unclassified. | An exact periodic decomposition (at most eleven periodic types) or a certified obstruction for each surviving law; or one law shown to be a nonperiodic extreme point, which answers P10. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P17 | The sixth moment | superhuman | 4 | open | With A_n as in P9 (and T4), M_n = integral of |A_n|^6 = 172, 17332, 1901302, 407919754, ... Find lambda_6 = lim M_n^(1/n) exactly. The strongest peak gives lambda_6 >= 432 = 12^3/4. Theorem P9 proves lambda_6 > 432 (no locking at q = 3); find lambda_6 exactly. | null | null | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Consecutive ratios M_{n+1}/M_n alternate (about 406 and 461), and the two-step rate (M_{n+2}/M_n)^{1/2} gives about 432.65; lambda_4 was found exactly by an integer recurrence certified modulo primes. Sol (25 Oct 2026, progress, not a solution): an exact 2,032-state integer transfer with exploratory dominant eigenvalue... | lambda_6 given exactly with a certificate, for example an integer recurrence certified modulo primes as for lambda_4. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
P18 | Beyond the information dimension | superhuman | 4 | partly solved: (i) and (iii) by Sol (Theorem P18, version 13.2); (ii) and (iv) open | For the charge diffraction mu of P9 (after Theorem P9): (i) is mu exact-dimensional, with local dimension D_1 almost everywhere? (ii) is L differentiable at q = 1, with D_1 = 4 - L'(1)/log 2? (iii) does D_q = tau(q)/(q-1) hold for 0 < q < 1? (iv) with delta(q) = q log2(4/3) - tau(q), does lim -log delta(q)/q exist, and... | null | Paper Theorem P18 (version 13.2); verification/sol_continuation_v131/ in the Zenodo supplement | J. Duda (with Claude and Sol), Crystals, quasicrystals and autocrystals, version 17 (2026), https://doi.org/10.5281/zenodo.23541901 | Theorem P9: D_1 in [1.335532, 1.336426]; correspondence for q > 1; q log2(4/3) - tau(q) between about 4^(-4q) and log2(1+3(23/24)^q). | A proof of any part; each part is credited separately. | A complete proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof. |
Autocrystal AI Puzzles, set 1
First official AI puzzles, to train your AI with logical thinking.
Sixteen mathematics puzzles from the paper Crystals, quasicrystals and autocrystals (J. Duda, with Claude and Sol, version 17, 2026, doi:10.5281/zenodo.23541901), on aperiodic order forced by local rules.
News (6 November 2026): P12 and P13 are solved by Rafał Wiktorowicz (NovaraX), with exact integer and rational certificates: doi:10.5281/zenodo.24012971. The gap labels 1/192, 97/192, 1/144 and 73/144 are now exact. Claude rechecked the recurrences, the labels and the structural identities with independent code; an independent replay of the certificates is pending.
News: P8 and P9 were solved one day after release (25 October 2026), both by Liam Price with GPT-6 Astra, with computer-assisted proofs audited by Sol and Claude. New open puzzles P12-P18 were added. Sol then solved parts (i) and (iii) of P18 (Theorem P18, version 13.2). Levels from "human" to "ASI" are a rough, playful difficulty estimate, not a calibrated benchmark.
- 9 solved (T1–T4, P0, and P8, P9, P12 and P13 by contributors): statements with answers and pointers to full solutions, for training and for checking a model.
- 7 open (P10, P11, P14–P18): no known answer; statements, evidence and success criteria. Accepted results, partial ones included, are credited by name, agent and model in the next version of the paper.
Fields
| field | meaning |
|---|---|
id, title |
puzzle id (T = training puzzle, P = problem of the paper) and name |
level, stars |
human / expert / AGI / superhuman / ASI, 1–5 stars (for fun) |
status |
solved or open |
statement |
the problem, as self-contained as possible |
answer, solution_reference |
solved puzzles only |
evidence, success_criterion |
open puzzles: what is known numerically, and what counts as a solution |
proof_rule, source |
the rule for accepted answers, and the citation |
The full statements, known dead ends and file paths are in AI_puzzles.md (included here).
Rules
A result counts only with a proof object that can be checked independently: a written proof, a Lean or Rocq (Coq) file, or an exact (integer, rational or interval) computation. Floating-point evidence is not a proof.
Evaluation note. Answers to the solved puzzles are public (here and in the paper), so for testing, run models without web search and be aware of possible training-set contamination. The open puzzles have no known answers.
Send results as a reply to the post where you found this, or to the address in the paper.
Hall of fame
| Problem | Result | By | Checked |
|---|---|---|---|
| P8 | Exact gap [9/8, 23/20], labels 1/8 and 5/8 exact (25 Oct 2026) | Liam Price, with GPT-6 Astra | Sol; Claude |
| P9 | D_1 in [1.335532, 1.336426]; no locking at finite q; correspondence for q > 1 (25 Oct 2026) | Liam Price, with GPT-6 Astra | Sol; Claude |
| P18 (i), (iii) | Exact dimensionality; D_q = τ(q)/(q − 1) for all q > 0, q ≠ 1 (25 Oct 2026) | Sol | Claude (certificate replay; separate referee) |
| P12 | Gap [21/5, 43/10] with N_1 = |V_6| − e_3 eigenvalues above it, all clusters; recurrence for all k; labels 1/192 and 97/192 exact (6 Nov 2026) | Rafał Wiktorowicz (NovaraX), solution | Claude (recurrences, labels, structure); certificate replay pending |
| P13 | Gap [19/5, 193/50] with N_2 = |V_6| eigenvalues above it; N_2(k) = N_1(k) + N_2(k−1) by an inflation bijection; labels 1/144 and 73/144 exact (6 Nov 2026) | Rafał Wiktorowicz (NovaraX), same deposit | as P12 |
Citation
J. Duda, Crystals, quasicrystals and autocrystals. Aperiodic order from local rules: Markov-field types, self-assembly and a route to Turing-complete matter, version 17, Zenodo (2026), doi:10.5281/zenodo.23541901.
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