deltaStructural_eq_half_D3
The structural correction for the tau lepton generation step equals 3/2 when spatial dimension is fixed at three. Lepton mass derivations on the Recognition Science phi-ladder cite this to fix the facet-mediated term without empirical fitting. The proof is a one-line wrapper applying the general deltaStructural_D3 result to the D = 3 case.
claimIn three spatial dimensions the structural correction for the tau step satisfies $Δ_{struct}(3) = 3/2$.
background
Recognition Science derives the dimension-dependent correction Δ(D) = D/2 from cube geometry without calibration to observed masses. The function deltaStructural encodes the facet-vertex ratio: F = 2D faces each contribute 1/V where V = 2^{D-1} is the discrete vertex count per facet, so Δ(D) = F/V. The module contrasts this facet-mediated μ→τ step with the edge-mediated e→μ step that uses the continuous solid angle 4π. Upstream results include the voxel definition as the fundamental length quantum and the Physical structure requiring positive c, ħ, G in RS-native units.
proof idea
The proof is a one-line wrapper that applies the general deltaStructural_D3 theorem directly to the D = 3 case.
why it matters in Recognition Science
This theorem specializes the general structural delta to D = 3, confirming Δ(3) = 3/2 as required by the tau step derivation. It supports the module claim that the correction follows from face geometry without calibration and aligns with the forced D = 3 in the unified forcing chain (T8). No downstream uses appear yet.
scope and limits
- Does not derive the general formula for arbitrary D.
- Does not compute the numerical tau mass value.
- Does not include higher-order radiative corrections beyond the structural term.
- Does not address the e→μ edge-mediated step.
formal statement (Lean)
165theorem deltaStructural_eq_half_D3 : deltaStructural 3 = (3 : ℝ) / 2 := deltaStructural_D3
proof body
Term-mode proof.
166
167/-! ## The Face-Vertex Interpretation
168
169**Physical interpretation of the derivation**:
170
171The tau transition is mediated by the faces of the cubic voxel.
172Each face is a 2D object (square in D=3) with V = 4 vertices.
173
174The radiative correction receives a contribution from each face,
175but the contribution is "spread" over the face's vertices.
176
177Contribution per face-vertex pair: 1
178Total contribution: F faces × 1 / V vertices per face = F/V = D/2
179
180This is NOT a fit — it follows from the face geometry.
181-/
182
183/-- The face-vertex ratio F/V equals D/2 when V = 4 (the 2D case).
184 Verified specifically for D = 3. -/