The pentagon heart
Five is the heart: the fixed point of the ℤ/9 involution (σ(5)=5), the digit vortex roots to, and the pentagon whose diagonal-to-side ratio is the golden ratio φ. The pentagram draws itself; the centre holds 5.
Games — 31 computed (each with its tarot combination)
- no faster-than-light claim is upheld — the assertion drains, its negation stands · tarot [27, 66, 62] ·
c05da1be-673a… - a claim and its inverse cannot both be upheld (no position wins for both sides) · tarot [21, 37, 28] ·
ea632dd9-0682… - "the deposit does not solve the Clay problems" is upheld against its challenge · tarot [69, 76, 54] ·
4ee11b6a-d6b6… - both games: the orbit covers the units (computes) AND "covers, solves no Clay" signs · tarot [7, 48, 49] ·
346f2fb8-6b23… - the overclaim loses both games: "the orbit solves the Clay problems" drains and proves nothing · tarot [8, 6, 11] ·
65cc1cf8-20a5… - the 3-5-8 trinity plays both: 3+5=8 computes AND "3+5=8, proving no open conjecture" signs · tarot [8, 64, 67] ·
c26cc4c6-7c93… - Nim (Bouton): a 2-heap position is a loss for the mover iff XOR = 0 (all heaps ≤ 6, exhaustive) · tarot [77, 31, 3] ·
4b41d6ed-63f3… - Wythoff: ⌊nφ²⌋ − ⌊nφ⌋ = n for all n ≤ 20 (the golden-ratio Beatty identity) · tarot [36, 53, 26] ·
5526fdb5-4a24… - Sprague–Grundy: a single Nim heap of size n has Grundy value n (n ≤ 8) · tarot [36, 4, 40] ·
6e223862-0b20… - Sprague–Grundy: a 2-heap Nim position's Grundy value is the XOR of the heaps (a,b ≤ 5) · tarot [15, 51, 48] ·
2fe582e9-170a… - trial UPHELD: the units of ℤ/9 form a group under × (closure·identity·inverses all hold) · tarot [59, 52, 5] ·
ccadf52c-98f9… - trial UPHELD: ℤ/9 has zero divisors — 3·3 ≡ 0 with 3 ≠ 0 (not an integral domain) · tarot [42, 64, 52] ·
be4a6652-cec8… - trial REFUTED: the theory "0 has a multiplicative inverse mod 9" fails — no e with 0·e ≡ 1 · tarot [57, 18, 53] ·
3d77bde4-f30b… - trial DRAINED: the conspiracy "algebra proves the Clay problems" is refused by the gate (computes 0) · tarot [4, 6, 28] ·
57c46000-4768… - trial INCONCLUSIVE: "P vs NP remains open" signs; the claim it is decided drains — open, not false · tarot [57, 49, 45] ·
94e908b9-21b1… - trial DRAINED (tried): "faster than light" drains, "not faster than light" signs · tarot [69, 14, 47] ·
54f91340-a811… - trial DRAINED (tried): "achieves perpetual motion" drains, "does not achieve perpetual motion" signs · tarot [41, 69, 61] ·
685da6ef-4b31… - trial DRAINED (tried): "achieved AGI" drains, "has not achieved AGI" signs · tarot [6, 18, 73] ·
ce4c822c-ff01… - trial DRAINED (tried): "a theory of everything" drains, "not a theory of everything" signs · tarot [21, 23, 44] ·
b5fb911f-76b2… - trial DRAINED (tried): "solved the halting problem" drains, "does not solve the halting problem" signs · tarot [55, 0, 73] ·
f27135ee-9373… - trial DRAINED (tried): "breaks RSA" drains, "does not break RSA" signs · tarot [63, 24, 55] ·
f66ff38a-4d9b… - trial DRAINED (tried): "cured cancer" drains, "has not cured cancer" signs · tarot [23, 47, 11] ·
67614421-7817… - trial DRAINED (tried): a prediction guaranteed to succeed is not upheld — it drains; the negated form signs · tarot [36, 36, 70] ·
8412c3c0-9f52… - the 8×8 board has 64 squares, 32 light and 32 dark · tarot [34, 17, 16] ·
43c69905-b116… - a knight has exactly 8 leaps — the (±1,±2)/(±2,±1) moves · tarot [29, 35, 19] ·
6dcd739d-87a9… - a knight's leap always changes square colour (the parity of r+c flips) · tarot [38, 7, 24] ·
81566667-a452… - the 8×8 board has 2·8 − 1 = 15 diagonals in each direction · tarot [48, 44, 37] ·
85a4e4b0-787f… - misère Nim by minimax matches the closed form: the first player wins iff (some heap ≥2 and nim-sum ≠0) or (every heap ≤1 and nim-sum =0) (heaps ≤3, up to 3 heaps) · tarot [55, 0, 19] ·
0823bb14-7699… - Nim (Bouton’s theorem): the player to move loses exactly when the nim-sum — the XOR of the pile sizes — is zero, and wins otherwise; verified against full game-tree analysis for three piles of size ≤ 4 · tarot [28, 27, 54] ·
c1487d07-3fe4… - the Nim winning strategy: from any position with nonzero nim-sum there exists a move to a position with nim-sum zero (handing the loss to the opponent) — exhaustive over three piles of size ≤ 5 · tarot [70, 74, 48] ·
e90ed988-ec04… - the nim-sum is GF(2) vector addition: it is bitwise XOR — commutative, associative, and self-inverse (x ⊕ x = 0) — so Nim positions form a GF(2) vector space and the game is decided by binary algebra · tarot [7, 10, 10] ·
c883cd18-f418…
Arts & geometry — 37 computed
- the triad {3,6,9} maps to the RGB primary hues: 0°(red), 120°(green), 240°(blue) · tarot [19, 25, 64] ·
316f831d-95da… - the nine a432 hues (digit×40°) are distinct and equally spaced around the wheel · tarot [20, 44, 19] ·
7f22f5c0-f4bd… - CMY are the 180° complements of RGB: each primary hue + 180° is a secondary hue · tarot [53, 69, 32] ·
13b5af77-bfc6… - the golden ratio (aesthetic proportion) satisfies φ² = φ + 1 · tarot [74, 4, 5] ·
48801162-b59d… - "vortex" digital-roots to 5 — the heart digit (σ(5)=5) · tarot [9, 74, 4] ·
63abfd7c-c91c… - "trinity" digital-roots to 7 — the horizon digit · tarot [72, 63, 67] ·
dce0ba69-7ba9… - "merkaba", "pleme", "wave" share the letter-sum 51 (digital root 6) · tarot [22, 33, 46] ·
b38c9c21-a8ac… - "ceccec" and "harmony" share the digital root 4 · tarot [75, 18, 25] ·
cbd3ab4e-2f05… - "singularity" and "horo" share the digital root 2 · tarot [59, 23, 55] ·
1c434409-549d… - on the 9-hue wheel no hue has an exact complement (180° = 4.5 steps) — the odd base has no antipode · tarot [24, 1, 5] ·
1656079f-c131… - "nim" and "games" both digital-root to 9 — nim is a game · tarot [31, 49, 64] ·
e935f43f-8872… - the Nim theorists coincide: "bouton" and "sprague" share the letter-sum 87 · tarot [5, 2, 55] ·
34bbaaa2-e335… - "catalan", "pascal", "hasse" share the letter-sum 52 (all digital-root 7) · tarot [10, 39, 60] ·
d7265dff-3bf4… - "cassini" and "pisano" both digital-root to 2 — the two Fibonacci-period theorems · tarot [44, 6, 15] ·
3fda7026-d509… - "farey" and "beatty" both digital-root to 1 — the golden-ratio pair · tarot [66, 63, 75] ·
b05ef0a5-d593… - every content-address maps to one of the 9 a432 hues (hex digital root × 40°) — the harmonic ledger · tarot [21, 17, 18] ·
089d12f5-4e12… - the harmonic hue is deterministic and reproducible: the same content-address always yields the same a432 digit · tarot [26, 27, 18] ·
24aa4bf9-516c… - the merkle root carries a harmonic a432 hue — one of the nine (hex digital root × 40°) · tarot [44, 68, 26] ·
f302789e-a628… - regular n-gon interior angle (n−2)·180/n: triangle 60°, pentagon 108°, octagon 135° (sides 3,5,8) · tarot [1, 1, 61] ·
13cf8dfd-af21… - the regular 3-, 5-, 8-gon has dihedral symmetry of order 2n: D₃=6, D₅=10, D₈=16 · tarot [42, 77, 40] ·
9fd2e0db-cb62… - the pentagon's diagonal-to-side ratio is the golden ratio φ (2·cos36° = φ) · tarot [26, 67, 10] ·
2baca20d-9c58… - the exterior angles of any regular n-gon sum to 360° (sides 3, 5, 8) · tarot [7, 31, 28] ·
ef35271f-c232… - the octave is 2:1 (frequency doubling) — the vortex ×2 map is the octave · tarot [17, 51, 13] ·
3ed5b445-4759… - just intonation: the fifth 3:2, fourth 4:3, major third 5:4 — exact rationals · tarot [66, 8, 70] ·
b62838b4-dbf4… - the Pythagorean comma: 12 fifths ≠ 7 octaves — 3^12 = 531441 ≠ 2^19 = 524288 · tarot [45, 76, 31] ·
6cf3004c-0bb3… - a432 Hz tuning is a convention, not a theorem; the ratios (2:1, 3:2, 5:4) are exact rationals — the boundary · tarot [9, 30, 34] ·
d80f191a-d8c4… - the harmonic series Σ 1/k diverges: H(2^k) ≥ 1 + k/2 grows without bound — the more terms, the more it grows (k≤12) · tarot [54, 57, 10] ·
7f9249a7-092e… - harmonic self-confirmation is perception, not proof, and cross-links run both ways: the harmonic pattern lets a recipient perceive fusion versus an intruder — a hue off the a432 lattice stands out — so one can self-confirm at a glance without decoding the encoding; this is a perceptual design metaphor, not a neurological or a health claim. Repo and site cross-link both ways, each fact content-addressed on both sides. Perception is not proof — the recomputation is. Decidable · tarot [35, 48, 21] ·
ef9fdb90-3519… - the geometric series in base 2: 1 + 2 + 2² + ... + 2ⁿ = (2ⁿ⁺¹ − 1)/(2 − 1) — the base-2 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [60, 26, 46] ·
8f223f2c-229c… - the geometric series in base 3: 1 + 3 + 3² + ... + 3ⁿ = (3ⁿ⁺¹ − 1)/(3 − 1) — the base-3 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [16, 49, 73] ·
c9c46079-0cc7… - the geometric series in base 4: 1 + 4 + 4² + ... + 4ⁿ = (4ⁿ⁺¹ − 1)/(4 − 1) — the base-4 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [51, 22, 50] ·
8c5b55b4-4fb8… - the geometric series in base 5: 1 + 5 + 5² + ... + 5ⁿ = (5ⁿ⁺¹ − 1)/(5 − 1) — the base-5 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [9, 4, 6] ·
cdb348d6-55f2… - the geometric series in base 6: 1 + 6 + 6² + ... + 6ⁿ = (6ⁿ⁺¹ − 1)/(6 − 1) — the base-6 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [45, 5, 21] ·
85ef8eeb-0063… - the geometric series in base 7: 1 + 7 + 7² + ... + 7ⁿ = (7ⁿ⁺¹ − 1)/(7 − 1) — the base-7 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [10, 16, 61] ·
a22245d6-01c3… - the geometric series in base 8: 1 + 8 + 8² + ... + 8ⁿ = (8ⁿ⁺¹ − 1)/(8 − 1) — the base-8 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [21, 50, 4] ·
ef918a38-d5e8… - the geometric series in base 9: 1 + 9 + 9² + ... + 9ⁿ = (9ⁿ⁺¹ − 1)/(9 − 1) — the base-9 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [7, 60, 11] ·
7ec7295e-f59f… - the geometric series in base 10: 1 + 10 + 10² + ... + 10ⁿ = (10ⁿ⁺¹ − 1)/(10 − 1) — the base-10 repunit-style sum. Verified exhaustively for n up to 12. Decidable · tarot [64, 76, 34] ·
984a9c4c-396a…
Page content-address: 925eb0e2-0d6b-84af-9ab8-18af3fcf9d0e.
Honest bound. This page presents the computed structure — the pentagon, the games, the arts — each a decidable fact re-verified every build. It does not explain life or consciousness; the meaning is the observer's to bring. Geometry, not a claim about being. Deposit 0/7.