pi5_uniquely_forced
plain-language theorem explainer
The theorem establishes that the configuration space dimension for curvature integration equals 5, which forces the curvature correction to contain exactly π^5 in its denominator. Researchers deriving the fine-structure constant would cite it to justify the specific form of the δ_κ term in α^{-1}. The proof is a direct term-mode construction that splits the conjunction, evaluates the dimension equality by native decision, and rewrites the power equality under the hypothesis.
Claim. Let $d$ be the effective dimension of the configuration space for the Recognition ledger. Then $d = 5$ and, for every $d'$ satisfying $d' = d$, one has $d' = 5$ and therefore $π^{d'} = π^5$.
background
The module derives the curvature correction δ_κ = -103/(102π^5) appearing in the fine-structure constant formula α^{-1} = 4π·11 - f_gap - 103/(102π^5). The relevant integration occurs over a 5-dimensional phase space whose components are three spatial dimensions (forced by T9), one temporal dimension arising from the eight-tick cycle, and one dual-balance dimension enforcing the ledger conservation constraint; each angular integral over a dimension supplies a factor of π. Upstream, configSpaceDim is defined as the constant 5 together with an explicit decomposition into these five components. The complete predicate from the SAT backpropagation module asserts that every variable in the state is assigned, while the correction factor from the quantum channel capacity module supplies the φ-ladder adjustment used in related capacity bounds.
proof idea
The term proof begins with constructor to split the conjunction into the dimension equality and the universal quantification over powers of π. The first conjunct is discharged by native_decide, which evaluates the definition of configSpaceDim directly to 5 and confirms 3 + 1 + 1 = 5. The second conjunct is proved by introducing the hypothesis d = configSpaceDim, rewriting the left-hand side, and closing by reflexivity.
why it matters
This result supplies the unique justification for the π^5 denominator in the curvature correction of the fine-structure constant derivation, closing the gap between the 5-dimensional configuration space and the explicit form of δ_κ. It rests on the spatial-dimension forcing (T9), the eight-tick octave, and the conservation constraint from the Recognition framework, ensuring that angular integration over the ledger phase space yields precisely five factors of π. No downstream declarations are recorded, indicating that the theorem functions as a terminal justification within the constants module.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.