discrete_continuous_duality
The duality theorem equates the electron-muon transition contribution of 1 over 4 pi to the muon-tau transition contribution of 6 over 4, with the discrete face measure fixed at 4 vertices. Researchers deriving lepton mass ladders from cube geometry cite it to establish the dimension-dependent correction without empirical fitting. The proof proceeds by direct term construction, reflexivity on the continuous and vertex definitions, and numerical reduction after unfolding the face-count and vertex-count auxiliaries.
claimThe continuous-discrete duality asserts that the electron-to-muon contribution equals $1/(4pi)$, the muon-to-tau contribution equals $6/4$, and the discrete measure for a two-dimensional face equals 4.
background
The module derives the dimension-dependent correction Delta(D) = D/2 from cube geometry without calibration to observed masses. It contrasts the e to mu step, which uses the continuous solid-angle measure 4 pi for the active-edge contribution 1 over 4 pi, with the mu to tau step, which uses the discrete vertex count V as the normalization for the facet contribution F over V. The vertex count is the discrete analog of the solid angle: each facet contributes 1 over V where V equals 2 to the power D minus 1, yielding Delta equals 3 over 2 at D equals 3. Upstream structures supply the active-edge count A equals 1 and the J-cost definitions that anchor the underlying forcing chain.
proof idea
The term proof constructs the three-way conjunction by reflexivity on the first and third conjuncts. For the middle conjunct it unfolds the definitions of muTauContribution, discreteMeasure2DFace, faceCount and faceVertexCount, then applies numerical normalization to reduce the ratio to 6 over 4.
why it matters in Recognition Science
This theorem anchors the first-principles derivation of Delta(D) equals D over 2 inside the lepton-generations module. It supplies the discrete-continuous duality that parallels the continuous solid-angle factor 1 over 4 pi with the discrete vertex factor 1 over 4, thereby supporting the Recognition Science claim that lepton masses arise from phi-ladder steps. The result connects to the T8 forcing of three spatial dimensions and the Recognition Composition Law through the J-cost structures that generate the underlying ledger.
scope and limits
- Does not compute explicit numerical lepton masses.
- Does not generalize the duality beyond D equals 3.
- Does not incorporate quantum-field or radiative corrections.
- Does not address direct comparison with measured mass ratios.
formal statement (Lean)
236theorem discrete_continuous_duality :
237 -- e→μ uses 1/(continuous measure)
238 eMuContribution = 1 / (4 * Real.pi) ∧
239 -- μ→τ uses F/(discrete measure)
240 muTauContribution = (6 : ℝ) / 4 ∧
241 -- The discrete measure is the vertex count
242 discreteMeasure2DFace = 4 := by
proof body
Term-mode proof.
243 constructor
244 · rfl
245 constructor
246 · unfold muTauContribution discreteMeasure2DFace faceCount faceVertexCount
247 norm_num
248 · rfl
249
250/-! ## Why Vertices Specifically?
251
252The vertex count is forced as the normalization factor because:
253
2541. **Discrete lattice**: The RS ledger is Z³, not continuous R³.
2552. **Anchoring**: A face's contribution must be localized to lattice points.
2563. **Vertices as anchors**: The vertices of a face are exactly the
257 lattice points that define that face.
2584. **Uniform distribution**: Each vertex receives 1/V of the face's
259 total contribution (by symmetry).
260
261The vertex count is the unique natural normalization for a discrete
262face on a discrete lattice.
263-/
264
265/-- The number of vertices equals the number of lattice anchor points
266 for a 2D face. (This is definitional in the discrete setting.) -/