constructive_at_zero
Two non-empty configuration paths exhibit constructive interference when their relative phase is zero. Researchers modeling possibility spaces in Recognition Science cite this base case before deriving curvature or connectivity properties of the modal manifold. The term proof substitutes phase zero into the amplitude formula, applies the cosine identity at zero together with the multiplicative unit, and confirms positivity of the resulting square-root product.
claimLet $p_1$ and $p_2$ be non-empty finite lists of configurations. Then the interference amplitude between $p_1$ and $p_2$ evaluated at phase difference zero is strictly positive.
background
The modal manifold is the two-dimensional space whose points are configurations, equipped with the modal distance metric and possibility curvature. Its tangent directions represent possible changes, and the boundary occurs where the J-cost diverges. The lemma supplies the zero-phase base case inside this manifold. It rests on the arithmetic identity that multiplication by one leaves a quantity unchanged, together with the positivity of path weights for non-empty lists.
proof idea
The term proof selects phase zero, simplifies the interference amplitude via the cosine-of-zero rule and the multiplicative identity, then applies the product of two positive square roots (each arising from a positive path weight) to obtain a positive result.
why it matters in Recognition Science
The declaration anchors interference calculations inside the modal geometry layer of Recognition Science. It precedes statements on manifold connectivity, curvature positivity, and the Nyquist limit within the same module. No immediate downstream theorems are recorded, yet the result closes the zero-phase case required for any later analysis of the eight-tick octave or D=3 forcing chain.
scope and limits
- Does not treat empty configuration lists.
- Does not cover nonzero phase differences.
- Does not incorporate explicit dimensional exponents.
- Does not address continuous rather than discrete path lists.
formal statement (Lean)
194lemma constructive_at_zero (path1 path2 : List Config)
195 (_ : path1 ≠ []) (_ : path2 ≠ []) :
196 constructive_interference path1 path2 := by
proof body
Term-mode proof.
197 use 0
198 simp only [interference_amplitude, Real.cos_zero, mul_one]
199 apply mul_pos
200 · apply Real.sqrt_pos_of_pos
201 exact mul_pos (path_weight_pos path1) (path_weight_pos path2)
202 · norm_num
203
204/-! ## The Modal Manifold -/
205
206/-- **MODAL MANIFOLD**: The space of all possibilities with its natural structure.
207
208 M_modal is the manifold whose points are configurations and whose
209 tangent space at each point represents "directions of possible change."
210
211 Key properties:
212 - Dimension = 1 (value) + 1 (time) = 2
213 - Metric = modal_distance
214 - Curvature = possibility_curvature
215 - Boundary = J → ∞ (x → 0⁺) -/