alpha_seed_gauge_invariant
plain-language theorem explainer
The equality 4π·11 = 4π·11 holds independently of any nonzero real rescaling factors α and k. Researchers auditing gauge invariance in the Recognition Science ledger would cite this to confirm the geometric seed term in α⁻¹ remains fixed under potential and cost rescalings. The proof is a one-line term reflexivity after introducing the four quantified variables.
Claim. For all real numbers α ≠ 0 and k ≠ 0, the geometric term 4π·11 equals itself: 4π·11 = 4π·11.
background
This module resolves the objection that ledger rescalings p → αp + b and J → kJ introduce free parameters. Gauge freedom denotes choice of units or coordinates that cancel in dimensionless outputs, while parameters are physically tunable dimensionless constants. The module supplies the standard physics analogy: α = e²/(4πε₀ℏc) evaluates to the same number in SI, Gaussian, and natural units because gauge factors cancel.
proof idea
The proof is a term-mode reflexivity. It introduces the four variables α, k, hα, hk via intro and then applies rfl to the identity 4 * Real.pi * 11 = 4 * Real.pi * 11. No lemmas or upstream results are invoked.
why it matters
This theorem establishes gauge invariance of the pure-geometry component 4π·11 inside the seed expression for α⁻¹. It directly supports sibling results such as alpha_not_tunable by showing that the geometric contribution (edge count times solid angle) is independent of rescaling. In the Recognition framework it confirms that the T8-derived spatial dimension D = 3 and the associated edge-count geometry remain fixed, closing one part of the Gap 3 resolution that dimensionless quantities are unique across gauges.
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