constructive_interference
The definition states that two paths through configuration space exhibit constructive interference precisely when their phase difference is an integer multiple of 2π, yielding positive interference amplitude. Modal physicists analyzing reinforcement in possibility spaces cite this when deriving wave propagation from the eight-tick phase lattice. The definition is realized directly as an existential statement over the amplitude predicate at those discrete phases.
claimPaths $p_1$ and $p_2$ through the space of configurations exhibit constructive interference if there exists an integer $n$ such that the interference amplitude evaluated at phase difference $2n$ is strictly positive.
background
In the ModalGeometry module, configuration space is assembled from the Config structure (upsilonStar, eps_r, eps_v, eps_t, eps_a) imported from Gravity.ILG. Paths are lists of these configurations, and interference_amplitude is the sibling function that evaluates amplitude from phase differences drawn from the eight-tick phases. The module imports Possibility and Actualization to equip possibility balls with path connectivity and curvature data. Upstream, the phase definition supplies the 8-tick lattice: phases are $k$ times π/4 for $k$ in Fin 8, periodic with period 2π. Related structures from SpectralEmergence and PhiForcingDerived supply the underlying J-cost and spectral content that calibrate the amplitude.
proof idea
The definition is a direct encoding. It quantifies over integers n and requires interference_amplitude path1 path2 (2 * π * n) > 0, using the amplitude predicate already defined in the same module.
why it matters in Recognition Science
This definition supplies the positive-amplitude condition invoked by the downstream lemma constructive_at_zero, which specializes it at phase zero to establish reinforcement at the identity. It fills the modal-geometry step that links the eight-tick octave (T7) to interference phenomena inside possibility spaces. The parent result constructive_at_zero demonstrates its immediate utility for proving concrete reinforcement cases.
scope and limits
- Does not compute explicit numerical values of the amplitude for concrete paths.
- Does not address destructive interference, which uses odd multiples of π.
- Does not guarantee existence of reinforcing paths for arbitrary configurations.
- Does not embed the phi-ladder or mass formula from the foundation.
Lean usage
lemma constructive_at_zero (path1 path2 : List Config) (_ : path1 ≠ []) (_ : path2 ≠ []) : constructive_interference path1 path2 := by use 0; simp only [interference_amplitude, Real.cos_zero, mul_one]; apply mul_posformal statement (Lean)
184def constructive_interference (path1 path2 : List Config) : Prop :=
proof body
Definition body.
185 ∃ n : ℤ, interference_amplitude path1 path2 (2 * Real.pi * n) > 0
186
187/-- **DESTRUCTIVE INTERFERENCE**: When paths cancel.
188
189 Occurs when phase_diff = π(2n+1) (paths out of phase). -/