pi_power_eq_pi5_iff
plain-language theorem explainer
The uniqueness result states that for a natural number d, pi to the power d equals pi to the fifth power precisely when d equals 5. Derivations of the fine structure constant cite this to fix the exponent in the curvature correction term of the alpha formula. The forward direction takes the logarithm, cancels using positivity of log pi, and injects the real equality back to naturals; the converse substitutes directly.
Claim. For every natural number $d$, $pi^d = pi^5$ if and only if $d = 5$.
background
The Curvature Space Derivation module establishes that the curvature correction delta_kappa = -103/(102 pi^5) arises because the phase-space integration runs over a five-dimensional configuration space. This space combines three spatial dimensions forced by D=3, one temporal dimension from the eight-tick cycle, and one dual-balance dimension from the ledger conservation constraint; each angular integral contributes a factor of pi. Upstream definitions supply the canonical arithmetic object and the power spectrum at wavenumber k, framing the integration that selects the exponent 5.
proof idea
The tactic proof opens with constructor to split the biconditional. The forward direction assumes the equality, invokes pi greater than 1 to obtain a positive logarithm, applies log to both sides, simplifies the power rule, cancels the positive factor, and uses cast injectivity from reals back to naturals. The reverse direction performs direct substitution of d equals 5.
why it matters
This lemma supplies the uniqueness step that justifies the specific power in the curvature term of the alpha derivation and feeds directly into the canonical curvature-family uniqueness theorem. It closes the accounting for why the integration yields pi^5, consistent with the five-dimensional space built from the eight-tick octave and D=3. No open scaffolding remains; the result is a finished supporting uniqueness fact.
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