phiPattern
plain-language theorem explainer
The canonical eight-tick pattern maps each tick index t in Fin 8 to φ raised to t. Gap-weight formulas and alpha-genesis pattern-forcing cite it as the forced ladder on the octave window. The body is the direct power map on the tick value; no further proof content.
Claim. The canonical $\varphi$-pattern is the map $t \mapsto \varphi^{t}$ for $t \in \{0,\ldots,7\}$, where $\varphi$ is the self-similar fixed point of Recognition Science.
background
The module fixes the canonical φ-pattern on the eight-tick register. In Recognition Science the octave period is $2^3 = 8$ (forcing chain T7), so discrete signals live on Fin 8. The golden ratio φ is the unique self-similar fixed point forced at T6; powers of φ form the natural geometric ladder for rung and gap weights.
Sibling objects convert this real pattern to complex values for DFT, extract DFT coefficients and amplitudes, and build a geometric weight and an eight-mode DFT candidate weight. Downstream pattern-forcing packages treat every admissible eight-tick ladder as forced to equal φ^n, so this pattern is the restriction of that ladder to the octave window rather than an independent choice.
proof idea
Pure definition: the function sends each t : Fin 8 to Constants.phi raised to the natural number t.val. No lemmas or tactics; noncomputable only because real exponentiation is.
why it matters
This is the time-domain seed for GapWeight and AlphaGenesis. PatternForcingCert bundles M2 closure around it: every admissible ladder equals φ^n, the GapWeight pattern is that forced ladder on Fin 8, and the pattern is reciprocal to the T9 forced measure (φ^t · w(t) = 1). The theorem phiPattern_is_forced states equality with any EightTickLadder on the window; pattern_mul_forced_measure is the reciprocity identity. Locally it feeds phiPatternComplex and the positivity proof for the DFT-based candidate weight. Framework landmarks: T6 (φ forced), T7 (eight-tick octave), and the mass/gap ladder built from φ-powers.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.