full_turn
The equality of eight times forty-five to three hundred sixty is recorded as the full-turn cell in the cross-pattern matrix. Researchers assembling Recognition Science meta-theorems cite it when verifying products of the eight-tick factor with the gap. The proof is a direct term reference to the twoCube_times_gap lemma.
claim$2^3 times 45 = 360$
background
The CrossPatternMatrix module builds a matrix of products among five RS patterns: D=5, the eight-tick factor 2^3, J-cost at zero, the golden ratio phi, and gap=45. The full-turn entry is the product of the eight-tick with the gap, listed as 360 in the module table. The upstream twoCube_times_gap lemma supplies the arithmetic identity by decision procedure.
proof idea
The proof is a one-line term wrapper that directly invokes the twoCube_times_gap theorem.
why it matters in Recognition Science
This theorem supplies the full_turn field of the CrossPatternMatrixCert structure, which is used by crossPatternMatrixCert and feeds into cardinalitySpectrumCert. It realizes the eight-tick octave (T7) multiplied by gap-45 to close the corresponding cell in the C26 cross-domain matrix.
scope and limits
- Does not derive the gap value of 45 from the phi-ladder.
- Does not interpret the equality in terms of physical rotations or angles.
- Does not connect to the alpha band or mass formula.
- Does not prove non-degeneracy of the full matrix beyond the listed entries.
Lean usage
example : (2 : ℕ)^3 * 45 = 360 := full_turnformal statement (Lean)
49theorem full_turn : (2 : ℕ)^3 * 45 = 360 := twoCube_times_gap
proof body
Term-mode proof.
50
51/-- Each entry corresponds to a unique integer (no two non-trivial entries
52 coincide). -/