{ "_about": "Machine-readable discrete matching system of Chair44 (R44), the seven-cube-chair aperiodic monotile of Ioannis Tsiokos (2026), plus the eight-chair substitution. Extracted and re-derived from the author's repository by an independent reader on 2026-09-17.", "source": { "paper": "Ioannis Tsiokos, 'A Strongly Aperiodic Monotile in Three Dimensions', v2, 16 September 2026, 62 pp.", "zenodo_v2": "https://doi.org/10.5281/zenodo.22792358", "zenodo_v1": "https://zenodo.org/records/22734468", "zenodo_concept_doi": "10.5281/zenodo.22734467", "repository": "https://github.com/ioannist/six-birds-tiles", "release_checkpoint_commit": "137e46b15d36266c37879478cfc62af6e4469147", "mathematical_baseline_commit": "d90313a717", "files_read": [ "solid/r44_solid.json", "solid/native_panels.csv", "certificates/candidate_certificate.json", "verify/packets/einstein_macrostate/src/framed_chair.py" ], "r44_solid_json_sha256": "f320d7a0c2d784a3eb29f001dc035808f67591d9ea8d3dd45949e66442f0ed55", "candidate_certificate_json_sha256": "44e9b3f6048497de02d1be72d0c80ddc94aaaf235382138c7468ffdbc5de5ff2", "license": "CC BY 4.0 (paper); see repository LICENSE for code" }, "coordinate_convention": { "carrier": "P = union over a in {0,1}^3 \\ {(1,1,1)} of (a + [0,1]^3): the 2x2x2 cube of unit cells minus the corner cell [1,2]^3.", "axes": "x = axis 0, y = axis 1, z = axis 2. Unit = one carrier cell edge.", "panels": "24 unit squares of the boundary of P: 21 on the six outer faces (4+4+4+3+3+3) and 3 inset notch faces (panels 4 at x=1, 14 at y=1, 23 at z=1).", "panel_ids": "0..23 as in solid/native_panels.csv; identical to the IDs in Figure 3 of the paper; they are the sorted order of (normal_axis, doubled centre) used by framed_chair.py NATIVE.", "in_panel_axes": "for a panel with outward normal along axis n, the in-panel coordinates (u,v) are the two remaining axes in increasing index order: n=x -> (y,z); n=y -> (x,z); n=z -> (x,y). This matches the six diagram headings of Figure 3.", "feature_positions": "each panel carries exactly 8 features at in-panel offsets (+-1/8,+-1/4) and (+-1/4,+-1/8) from the panel centre. The offset index k=0..7 enumerates them as [(s*u/8, t*v/8) for (u,v) in ((1,2),(2,1)) for (s,t) in ((-1,-1),(-1,1),(1,-1),(1,1))], i.e. k=0:(-1/8,-1/4) 1:(-1/8,+1/4) 2:(+1/8,-1/4) 3:(+1/8,+1/4) 4:(-1/4,-1/8) 5:(-1/4,+1/8) 6:(+1/4,-1/8) 7:(+1/4,+1/8).", "role_index": "role = 8*panel + k, 0..191, the indexing used by the solid file's 'profile' array.", "feature_shape": "axis-parallel square pyramid, base side 1/50 (half-width eta = 1/100), apex displaced along the OUTWARD normal by a/10000. a>0 is a protrusion (bump), a<0 a recess (dent). |a| in 1..12.", "verified": "the 'profile' array of solid/r44_solid.json is entrywise equal to the a_0..a_7 columns of solid/native_panels.csv, and both agree with Figure 3 of the paper (spot-checked on panels 0, 3, 13)." }, "matching_rule": { "statement": "There is NO imposed matching rule: Chair44 is a bare solid and the only rule is that copies must not overlap. The discrete rule below is a THEOREM about the shape, valid for registered (integer-lattice, cubic-frame) placements.", "discrete_form": "When two panels meet face to face their outward normals are opposite. At every point of space where a feature of one coincides with a feature of the other, the two signed coefficients must sum to zero: a_u + a_v = 0, i.e. a bump of magnitude m meets a dent of the same magnitude m. Because each of the twelve magnitudes 1..12 is carried by exactly one signed group, the profiles across a shared panel are compared centre by centre with no tolerance.", "consequence": "two panels can meet only if, after the change of in-panel coordinates induced by the relative pose, their 8-vectors of coefficients are exact entrywise negatives." }, "panels": [ { "panel": 0, "normal_axis": 0, "outward_sign": -1, "panel_centre": [ "0", "1/2", "1/2" ], "in_panel_axes": [ 1, 2 ], "is_notch_panel": false, "coefficients_by_offset_index": [ 1, 2, 3, 4, 5, 6, 7, 8 ], "features": [ { "k": 0, "offset_eighths": [ -1, -2 ], "offset": [ "-1/8", "-1/4" ], "centre": [ "0", "3/8", "1/4" ], "a": 1, "signed_height": "1/10000", "apex": [ "-1/10000", "3/8", "1/4" ], "role": 0 }, { "k": 1, "offset_eighths": [ -1, 2 ], "offset": [ "-1/8", "1/4" ], "centre": [ "0", "3/8", "3/4" ], "a": 2, "signed_height": "1/5000", "apex": [ "-1/5000", "3/8", "3/4" ], "role": 1 }, { "k": 2, "offset_eighths": [ 1, -2 ], 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"k": 2, "offset_eighths": [ 1, -2 ], "offset": [ "1/8", "-1/4" ], "centre": [ "13/8", "5/4", "1" ], "a": -10, "signed_height": "-1/1000", "apex": [ "13/8", "5/4", "999/1000" ], "role": 186 }, { "k": 3, "offset_eighths": [ 1, 2 ], "offset": [ "1/8", "1/4" ], "centre": [ "13/8", "7/4", "1" ], "a": -9, "signed_height": "-9/10000", "apex": [ "13/8", "7/4", "9991/10000" ], "role": 187 }, { "k": 4, "offset_eighths": [ -2, -1 ], "offset": [ "-1/4", "-1/8" ], "centre": [ "5/4", "11/8", "1" ], "a": 12, "signed_height": "3/2500", "apex": [ "5/4", "11/8", "2503/2500" ], "role": 188 }, { "k": 5, "offset_eighths": [ -2, 1 ], "offset": [ "-1/4", "1/8" ], "centre": [ "5/4", "13/8", "1" ], "a": 10, "signed_height": "1/1000", "apex": [ "5/4", "13/8", "1001/1000" ], "role": 189 }, { "k": 6, "offset_eighths": [ 2, -1 ], "offset": [ "1/4", "-1/8" ], "centre": [ "7/4", "11/8", "1" ], "a": 11, "signed_height": "11/10000", "apex": [ "7/4", "11/8", "10011/10000" ], "role": 190 }, { "k": 7, "offset_eighths": [ 2, 1 ], "offset": [ "1/4", "1/8" ], "centre": [ "7/4", "13/8", "1" ], "a": 9, "signed_height": "9/10000", "apex": [ "7/4", "13/8", "10009/10000" ], "role": 191 } ], "motif_class": "C" } ], "motif_classes": { "_note": "Independently computed here: under the dihedral group D4 acting on the 8 in-panel feature positions, the 24 panels fall into exactly THREE classes of eight. This reproduces the A/B/C 'three face motifs' reported in yamaton/aperiodic-chair-lab (strong/audit/face_motifs.json) for its own curved-cap realization of the same discrete system.", "classes": { "A": { "canonical_word_by_offset_index": [ 1, 2, 3, 4, 5, 6, 7, 8 ], "panels": [ 0, 2, 7, 9, 11, 14, 20, 21 ] }, "B": { "canonical_word_by_offset_index": [ -8, -6, -7, -5, -4, -2, -3, -1 ], "panels": [ 1, 4, 6, 8, 12, 15, 18, 19 ] }, "C": { "canonical_word_by_offset_index": [ -12, -11, -10, -9, 12, 10, 11, 9 ], "panels": [ 3, 5, 10, 13, 16, 17, 22, 23 ] } }, "relations": [ { "claim": "motif C (panels 3,5,10,13,16,17,22,23) is the entrywise negative of itself under the in-panel reflection that swaps the two in-panel axes (the diagonal reflection). It is the only D4 element with that property. In CGS's words at the meeting, 'the blue marking matches itself'.", "element": "swap_uv (rot=0, ref=1)" }, { "claim": "motif A (panels 0,2,7,9,11,14,20,21) and motif B (panels 1,4,6,8,12,15,18,19) are entrywise negatives of each other after an in-panel reflection: A = -(B o g) with g = swap_uv then rot270. In CGS's words, 'the empty purple marking with the full purple marking'.", "element": "swap_uv then rot270" }, { "claim": "A uses only the magnitudes 1..8 with + sign and B only 1..8 with - sign; C uses 9..12 with four + and four - (so C is 'half full').", "element": null } ], "plain_language": "Three marking types, eight panels each. Two of them (A, all bumps of magnitudes 1..8; B, all dents of magnitudes 1..8) are complementary to each other; the third (C, magnitudes 9..12 with mixed polarity) is complementary to itself under the diagonal reflection of the panel.", "d4_convention": "g=(rot,ref) acts on an in-panel offset (u,v) by: if ref, swap u,v first; then apply (u,v)->(-v,u) rot times.", "cross_validation_yamaton": { "_what": "Independent confirmation that the three-class partition computed here from Tsiokos's raw panel data is the same partition that yamaton/aperiodic-chair-lab computed from its own, geometrically different (curved polynomial-cap) realization.", "source_file": "aperiodic-chair-lab/strong/audit/chair44_comparison.json, key face_motif_correspondence (24 entries)", "class_map": { "A_here": "B_yamaton", "B_here": "C_yamaton", "C_here": "A_yamaton" }, "warning": "The letters A/B/C are assigned differently in the two projects. Do not identify them by name.", "yamaton_rule_table_verbatim": { "A and A": "U2 = V1", "B and C": "U2 = -U1", "C and B": "U2 = -U1", "note": "'In particular, C cannot meet C.' (strong/MOTIF_GROUPING.md)" }, "translated_into_the_labels_used_here": [ "yamaton 'A meets A' == C_here is self-complementary (matches the diagonal-reflected negative of itself)", "yamaton 'B meets C' == A_here meets B_here (the mutually complementary pair)", "yamaton 'C cannot meet C' == B_here cannot meet B_here (two all-dent panels cannot face each other)" ], "matches_CGS_description": "CGS at the meeting: 'the two purple markings match, so you have to have the empty purple marking with the full purple marking, and the blue marking matches itself.' Purple pair = A_here/B_here (all-bump / all-dent, magnitudes 1..8); blue = C_here (magnitudes 9..12, mixed polarity)." } }, "notch_translation_identity": [ { "notch_panel": 4, "opposite_outer_panel": 0, "entrywise_negation_holds": true, "note": "the notch panel is the exact entrywise negative of the outer panel with the same in-panel coordinates; hence a copy translated by (1,1,1) plugs the notch and all three notch panels match" }, { "notch_panel": 14, "opposite_outer_panel": 8, "entrywise_negation_holds": true, "note": "the notch panel is the exact entrywise negative of the outer panel with the same in-panel coordinates; hence a copy translated by (1,1,1) plugs the notch and all three notch panels match" }, { "notch_panel": 23, "opposite_outer_panel": 16, "entrywise_negation_holds": true, "note": "the notch panel is the exact entrywise negative of the outer panel with the same in-panel coordinates; hence a copy translated by (1,1,1) plugs the notch and all three notch panels match" } ], "substitution": { "name": "universal eight-chair grouping / eight-child dissection", "statement": "Eight properly rotated chair carriers partition the doubled chair 2P exactly (56 unit cells), so the chair is a rep-8 rep-tile. Seven children sit at the corners 4a, a in {0,1}^3 \\ {(1,1,1)}, and one 'central' child sits at (1,1,1) in the coordinates of 2P.", "frame_action": "a frame is (p,s) with p a permutation of the axes and s a sign vector, acting by G(x)_i = s_i * x_{p_i}; a pose is a frame followed by a translation. Refinement: (G,t) -> {(G*H, 2t + G*u) : (H,u) a child pose}.", "children": [ { "child": "000", "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 0, 0, 0 ] }, { "child": "001", "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "child": "010", "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "child": "011", "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ 0, 4, 4 ] }, { "child": "100", "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "child": "101", "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 4, 0, 4 ] }, { "child": "110", "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 4, 4, 0 ] }, { "child": "central", "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 1, 1, 1 ] } ], "children_raw_from_certificate": [ [ [ [ 0, 1, 2 ], [ 1, 1, 1 ] ], [ 0, 0, 0 ] ], [ [ [ 1, 0, 2 ], [ 1, 1, -1 ] ], [ 0, 0, 4 ] ], [ [ [ 0, 2, 1 ], [ 1, -1, 1 ] ], [ 0, 4, 0 ] ], [ [ [ 2, 0, 1 ], [ 1, -1, -1 ] ], [ 0, 4, 4 ] ], [ [ [ 2, 1, 0 ], [ -1, 1, 1 ] ], [ 4, 0, 0 ] ], [ [ [ 1, 2, 0 ], [ -1, 1, -1 ] ], [ 4, 0, 4 ] ], [ [ [ 0, 1, 2 ], [ -1, -1, 1 ] ], [ 4, 4, 0 ] ], [ [ [ 0, 1, 2 ], [ 1, 1, 1 ] ], [ 1, 1, 1 ] ] ], "all_child_frames_are_proper_rotations": true, "paper_reference": "Table 1 and Figure 5, page 13 of the v2 PDF" }, "contact_atlas_A44": { "meaning": "the 44 legal relative poses of a neighbouring chair, expressed in the coordinate frame of a root chair normalized to the identity pose at the origin. Each entry is ((permutation, signs), translation).", "how_obtained": "a root chair in the identity pose has 22 grid cells adjacent to it across its 24 panels; a neighbour in one of the 48 signed frames covering a shell cell gives 2388 candidate poses after deduplication and overlap rejection; exactly 44 of them have matching feature profiles on every shared panel.", "distinct_relative_rotations": 19, "rotation_group_generated": 24, "improper_relative_contacts": 0, "contacts": [ { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, -2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, -1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, 1, -1 ], "translation": [ 2, -2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, 1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, -1, -1 ], "translation": [ -2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, -1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ -1, -1, -1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 1, 1, 1 ] }, { "perm": [ 0, 2, 1 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ -1, 3, -1 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] }, { "perm": [ 1, 0, 2 ], "signs": [ -1, 1, 1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ -1, -1, 3 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, -2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 2, -2, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 3, -1, 3 ] }, { "perm": [ 1, 2, 0 ], "signs": [ 1, -1, -1 ], "translation": [ -2, 2, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ 1, -1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, -2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, 1, -1 ], "translation": [ 2, -2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, 1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ -2, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ -1, 3, 3 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 3, -1, -1 ] }, { "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] } ], "paper_reference": "Figure 7, page 15 of the v2 PDF", "pure_translation_contacts": { "_note": "The only two contacts in A44 whose relative frame is the identity are the pure translations by +-(1,1,1): a second, unrotated chair plugging the notch (or the tile itself sitting in a neighbour's notch). Verified here by scanning certificates/candidate_certificate.json 'legal_contacts'.", "indices_zero_based": { "minus": 8, "plus": 9 }, "entries": [ [ [ [ 0, 1, 2 ], [ 1, 1, 1 ] ], [ -1, -1, -1 ] ], [ [ [ 0, 1, 2 ], [ 1, 1, 1 ] ], [ 1, 1, 1 ] ] ] } }, "closed_contact_set_30": { "meaning": "the closure of the 21 internal face contacts of the eight-child dissection under refinement; the profile (the height table) was solved from the 372 sign equations these impose, whose solution space has dimension 12 (the twelve magnitudes).", "counts": { "internal_face_contacts": 21, "closed_states": 30, "sign_equations": 372, "balanced_components": 12 }, "contacts": [ { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, -2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, -1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ -1, -1, -1 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 1, 1, 1 ] }, { "perm": [ 0, 2, 1 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ -1, 3, -1 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] }, { "perm": [ 1, 0, 2 ], "signs": [ -1, 1, 1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ -1, -1, 3 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 2, -2, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 3, -1, 3 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 3, 3, 1 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ -2, 2, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ -1, 3, 3 ] }, { "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 3, -1, -1 ] }, { "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 4, 0, 0 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 4, 0 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 4 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 1, 1, 3 ] } ] }, "parent_atlas_halving": { "meaning": "the decisive finite identity: enumerate 8*44*8 parent-contact candidates, deduplicate to 697, keep the 116 with disjoint parent bodies, test every fine contact across the two parents -> 44 legal parent contacts, all with even translation; halving the translations returns exactly A44.", "candidates": 697, "nonoverlapping": 116, "legal": 44, "all_even_translations": true, "half_of_parent_atlas_equals_fine_atlas": true, "macro_contacts": [ { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 4, 4, -4 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 6, 6, -2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, -1, 1 ], "translation": [ 6, 6, 2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, 1, -1 ], "translation": [ 4, -4, 4 ] }, { "perm": [ 0, 1, 2 ], "signs": [ -1, 1, -1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, -1, -1 ], "translation": [ -4, 4, 4 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, -1, -1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ -2, -2, -2 ] }, { "perm": [ 0, 1, 2 ], "signs": [ 1, 1, 1 ], "translation": [ 2, 2, 2 ] }, { "perm": [ 0, 2, 1 ], "signs": [ -1, 1, 1 ], "translation": [ 8, 0, 0 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ -2, 6, -2 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 8, 0 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 8 ] }, { "perm": [ 0, 2, 1 ], "signs": [ 1, 1, -1 ], "translation": [ 2, 2, 6 ] }, { "perm": [ 1, 0, 2 ], "signs": [ -1, 1, 1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ -1, 1, 1 ], "translation": [ 8, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 8, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ -2, -2, 6 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 0 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 8 ] }, { "perm": [ 1, 0, 2 ], "signs": [ 1, 1, -1 ], "translation": [ 2, 2, 6 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 4, 4, -4 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, -1, 1 ], "translation": [ 6, 6, 2 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 4, -4, 4 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 1, 2, 0 ], "signs": [ -1, 1, -1 ], "translation": [ 6, -2, 6 ] }, { "perm": [ 1, 2, 0 ], "signs": [ 1, -1, -1 ], "translation": [ -4, 4, 4 ] }, { "perm": [ 1, 2, 0 ], "signs": [ 1, -1, -1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 4, 4, -4 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, -1, 1 ], "translation": [ 6, 6, 2 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, 1, -1 ], "translation": [ 4, -4, 4 ] }, { "perm": [ 2, 0, 1 ], "signs": [ -1, 1, -1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ -4, 4, 4 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ -2, 6, 6 ] }, { "perm": [ 2, 0, 1 ], "signs": [ 1, -1, -1 ], "translation": [ 4, 4, 4 ] }, { "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 6, -2, -2 ] }, { "perm": [ 2, 1, 0 ], "signs": [ -1, 1, 1 ], "translation": [ 8, 0, 0 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, -1, 1 ], "translation": [ 0, 8, 0 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 0, 0, 8 ] }, { "perm": [ 2, 1, 0 ], "signs": [ 1, 1, -1 ], "translation": [ 2, 2, 6 ] } ] }, "shell_cells_22": [ [ -1, 0, 0 ], [ -1, 0, 1 ], [ -1, 1, 0 ], [ -1, 1, 1 ], [ 0, -1, 0 ], [ 0, -1, 1 ], [ 0, 0, -1 ], [ 0, 0, 2 ], [ 0, 1, -1 ], [ 0, 1, 2 ], [ 0, 2, 0 ], [ 0, 2, 1 ], [ 1, -1, 0 ], [ 1, -1, 1 ], [ 1, 0, -1 ], [ 1, 0, 2 ], [ 1, 1, -1 ], [ 1, 1, 1 ], [ 1, 2, 0 ], [ 2, 0, 0 ], [ 2, 0, 1 ], [ 2, 1, 0 ] ], "independent_checks_run_here": [ "python3 verify/packets/einstein_macrostate/src/replay.py : PASS in 2.6 s (verify.py + edge_audit.py), reproducing 21->30 contact closure, 372 equations, kernel dimension 12, 2388->44 atlas, 19 rotations generating the 24-element group, 33 first shells, 14/18 completions, 28 parent conflicts, 697->116->44 parent census with coarse==fine, mesh V=2138 E=6408 F=4272 chi=2, volume 7.", "verify/boundary_sphere.py, verify/tube_formula_controls.py, verify/substitution_modular_coincidence.py, verify/seed_language_closure.py : PASS.", "profile array == native_panels.csv == Figure 3 of the paper.", "24 panels fall into exactly 3 D4-classes of 8 (computed here).", "notch panels 4/14/23 are exact entrywise negatives of outer panels 0/8/16 (computed here)." ], "known_reproducibility_gap": "verify/replay.py (the single top-level entry point) fails: the packet verify/packets/r44_unrestricted_alignment/src/replay.py needs upstream/einstein_r44_proof_submission.zip, which .gitignore line 47 (/verify/packets/*/upstream/*.zip) excludes from the repository. The einstein_macrostate packet replays fine on its own.", "magnitude_census": { "note": "each of the 12 magnitudes is carried by exactly 16 of the 192 features, 8 as bumps (+) and 8 as dents (-). Verified here from solid/r44_solid.json.", "per_magnitude": { "1": { "plus": 8, "minus": 8 }, "2": { "plus": 8, "minus": 8 }, "3": { "plus": 8, "minus": 8 }, "4": { "plus": 8, "minus": 8 }, "5": { "plus": 8, "minus": 8 }, "6": { "plus": 8, "minus": 8 }, "7": { "plus": 8, "minus": 8 }, "8": { "plus": 8, "minus": 8 }, "9": { "plus": 8, "minus": 8 }, "10": { "plus": 8, "minus": 8 }, "11": { "plus": 8, "minus": 8 }, "12": { "plus": 8, "minus": 8 } } }, "figure_numbering": { "panel_recipe_six_grids": { "v2": "Figure 3, PDF page 10", "v1": "Figure 4, PDF page 11" }, "solid_render": { "v2": "Figure 1, PDF page 3", "v1": "Figure 1, PDF page 3" }, "44_contact_atlas_table": { "v2": "Figure 7, PDF page 15", "v1": "Figure 8" }, "eight_child_dissection": { "v2": "Figure 5, PDF page 13", "v1": "Figure 6" } }, "shell_structure_from_paper": { "quote": "A root chair in the identity pose has 22 grid cells adjacent to it across its 24 panels (the 21 outer panels face distinct exterior cells, while all three notch panels face the missing cell (1,1,1), giving 22 distinct shell cells)", "where": "Tsiokos v2 PDF page 51 (Appendix C, census provenance)", "why_it_matters": [ "It is the paper's own statement of the '21 outer panels' count that CGS half-remembered as 'all those crazy 21 markings'.", "It is also the paper's own statement that all three notch panels face the single cell (1,1,1) - the geometric fact behind the pure translation contact." ], "arithmetic": "22 shell cells x 48 signed frames x 7 cells of the neighbour = 7392 raw placements; after deduplication and overlap rejection 2388 candidate contact poses remain, of which 44 are legal." } }