Ὁ Πενταμετρικὸς Κύκλος: A Nested Palindrome Encoding the Pythagorean Consonances

We present the Pentametric Cycle (Ὁ Πενταμετρικὸς Κύκλος), a structure generated by an integer-valued quadratic formula 𝔓(n), the Pentametric Formula: 𝔓(n) = (n² + 4C(m))/5, with m = 4n² mod 10 and C(m) = (26m − 5m²)/24. We show that the correction term C(m), the Interval Map, coincides on every attained residue with the Legendre symbol (n/5), so that 𝔓(n) = (n² + 4(n/5))/5. We collect ten equivalent forms of 𝔓 found by the author. They use decimal digits, floors, a quintic polynomial, cosines (a quadratic Gauss sum), rotations by the golden ratio and pentagonal angles. We prove that each one computes the same quadratic character, and we derive the general laws 𝔓(n) − n²/5 = ±4/5 and ±2π/5 for square and circle areas. Reducing the doubled value 2𝔓(n) modulo 10 and halving gives a digit H(n) ∈ {0,1,2,3,4}. The single rule H : (H−1) then turns it into silence or one of the four Pythagorean consonances: unison 1:1, octave 2:1, fifth 3:2 and fourth 4:3. The sequence H has minimal period 25 and forms a 51-term palindrome mirrored at n = 25. In each period, 9 terms are silent and each consonance occurs exactly 4 times. We prove the Pentametric–Lucas Theorem: 𝔓(Lₖ) = Fₖ² for every Lucas number Lₖ. We also prove its converse: for n ≥ 1, 𝔓(n) is a perfect square if and only if n is a Lucas number. We confirm the author's conjecture that the areas at Lucas positions admit a form as simple as the distances: Fₖ = round(Lₖ·√0.2) for k ≥ 2. Finally, we relate the theorem to the classical result of H. L. Holden (1975): the centres of the squares in the Fibonacci spiral lie on two orthogonal lines, at distances Lₖ·√0.1 from their intersection. We call this the Geometric Lucas Meter and fully credit it to Holden. From it we derive geometric identities that make 𝔓 visible in the Fibonacci tiling. GeoGebra constructions, spreadsheets and verification scripts: https://github.com/lluisgarcia/The-Pentametric-Cycle