pi3_incomplete
plain-language theorem explainer
The theorem shows that π³ differs from π raised to the configuration space dimension. Derivations of the curvature correction term in the fine structure constant would cite it to exclude a purely spatial integration. The proof unfolds the dimension definition to five, assumes equality of the powers, applies the logarithm, and reaches the contradiction three equals five by positivity of log π and cancellation.
Claim. $π^3 ≠ π^5$ in the reals, where the exponent five is the effective dimension of the configuration space for the Recognition ledger.
background
The module derives the π^5 factor in the curvature correction δ_κ = -103/(102π^5) for the fine structure constant. The configuration space comprises three spatial dimensions, one temporal dimension from the eight-tick cycle, and one balance dimension from the conservation constraint. Each angular contribution supplies a factor of π, so the total exponent is five. configSpaceDim is defined as five and serves as the effective dimension for curvature integration over the ledger phase space.
proof idea
The tactic proof unfolds configSpaceDim to replace the exponent with the concrete value five. It assumes equality, invokes π > 3 to obtain log π > 0, applies the logarithm to both sides, simplifies via the power rule for logarithms, cancels the common positive factor to reach three equals five, and derives a contradiction by linear arithmetic.
why it matters
This result supports the claim that the curvature correction integrates over five dimensions rather than three, justifying the π^5 term in α^{-1} = 4π·11 - f_gap - 103/(102π^5). It connects to the forcing chain where D = 3 is forced and the eight-tick octave introduces the temporal dimension. The balance dimension arises from the ledger conservation constraint. The doc-comment notes that π^3 would correspond to integrating only over spatial dimensions and ignores the temporal and balance contributions.
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