lemma proved term proof high

`phi_zpow_zero`

The lemma records that the golden ratio raised to the zero integer power equals one. Workers manipulating integer exponents on phi in Recognition Science calculations would invoke it to simplify expressions. The proof is a one-line term wrapper that applies the standard zero-exponent rule for real numbers.

claim$phi^0 = 1$ in the real numbers, where $phi$ is the golden ratio.

background

Phi is the self-similar fixed point forced in the Recognition Science chain at step T6. The Support.Powers module supplies basic exponentiation facts for constants and ladders used throughout the framework. The declaration imports Mathlib for the underlying real-number power rules.

proof idea

The proof is a one-line term wrapper that invokes zpow_zero on the real number phi, with simpa performing the simplification.

why it matters in Recognition Science

The identity supplies the base case for integer powers of phi, which appears in the phi-ladder mass formula and the eight-tick octave. It supports arithmetic steps inside the unified forcing chain (T0-T8) and the Recognition Composition Law. No downstream theorems are listed, so the lemma functions as a local convenience.

scope and limits

Does not treat non-zero integer exponents.
Does not extend to fractional or real exponents.
Applies only inside the real numbers.
Does not define or compute the numerical value of phi.

formal statement (Lean)

  10lemma phi_zpow_zero : phi ^ (0 : ℤ) = (1 : ℝ) := by

proof body

Term-mode proof.

  11  simpa using (zpow_zero (phi : ℝ))
  12
  13end Support
  14end IndisputableMonolith
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