Chapter 15
Continuous ↔ Discrete Equivalence

Flow Algebra

flow "Continuous_Discrete_Equivalence"

# ---------- STATES ----------
state:
  field.ψ: complex                      # coherence amplitude field
  field.phase.Φ: float                  # coherence phase
  field.amplitude.A: float              # coherence magnitude
  coherence.integrity: float            # 0..1; measure of continuous overlap
  coherence.scale.λ_coh: float          # coherence wavelength (phase loop length)
  lattice.spacing.Δx: float             # sampling / discretisation scale
  continuum.limit: bool                 # true if Δx << λ_coh
  quantisation.condition: float         # ∮ p·dx = n h  (discrete condition)
  curvature.tension.Σ: float            # boundary curvature tension density
  density.ρ: float                      # |ψ|²
  derivative.discrete: float            # finite difference operator
  derivative.continuous: float          # continuous derivative
  operator.H: functional                # Hamiltonian or curvature operator
  eigenstates.ψn: list[complex]         # discrete eigenmodes
  energy.levels.En: list[float]         # quantised energy values
  time.t: float

# ---------- PARAMETERS ----------
params:
  h: 6.626e-34
  ħ: 1.054e-34
  Δx_threshold: 0.01                    # ratio threshold for continuum validity (Δx/λ_coh)
  dt: float

# ---------- INVARIANTS ----------
invariant:
  - discrete form:  D_Δx ψ = (ψ(x+Δx) - ψ(x-Δx)) / (2Δx)
  - continuous form:  D ψ = ∂ψ/∂x = lim_{Δx→0} D_Δx ψ
  - continuum valid iff Δx / λ_coh ≪ 1
  - quantisation arises when boundary enforces closure: ∮ p·dx = n h
  - discrete spectrum {E_n} ↔ boundary resonance nodes
  - continuous spectrum → boundary open or λ_coh → 0 limit
  - same governing equation holds:  Hψ = Eψ  (discrete eigenbasis ↔ continuous function basis)
  - coherence.integrity = overlap_integral(ψ_i, ψ_{i+1})  (1 = continuous, <1 = discrete)

# ---------- TRANSFORMS ----------
transform:

  initialise_field:
    do:
      field.amplitude.A = 1.0
      field.phase.Φ = 0.0
      coherence.scale.λ_coh = 1.0
      lattice.spacing.Δx = 0.1
      coherence.integrity = exp(-lattice.spacing.Δx / coherence.scale.λ_coh)
      continuum.limit = (lattice.spacing.Δx / coherence.scale.λ_coh < params.Δx_threshold)
      time.t = 0

  compute_derivatives:
    do:
      derivative.discrete = (ψ(x+Δx) - ψ(x-Δx)) / (2*Δx)
      derivative.continuous = limit(Δx→0, derivative.discrete)
      if continuum.limit: derivative = derivative.continuous
      else: derivative = derivative.discrete

  apply_quantisation_condition:
    do:
      quantisation.condition = n * params.h
      if boundary_closed():
        eigenstates.ψn = standing_wave_modes(boundary_length=L)
        energy.levels.En = [n² h² / (8mL²)]
      else:
        eigenstates.ψn = continuous_plane_waves()
        energy.levels.En = continuous_spectrum()

  update_coherence_integrity:
    do:
      ratio = lattice.spacing.Δx / coherence.scale.λ_coh
      coherence.integrity = exp(-ratio)
      if ratio < params.Δx_threshold:
        continuum.limit = true
      else:
        continuum.limit = false

  unify_representation:
    do:
      # connect discrete eigenbasis to continuous function via Fourier synthesis
      ψ_cont(x) = Σ_n ψn * e^{i k_n x}
      assert H ψ_cont = E ψ_cont  (same operator structure)
      mark("continuous_discrete_equivalence")

  tick:
    do:
      time.t += params.dt

# ---------- MONITORS ----------
monitor:
  coherence_ratio:
    observe: [lattice.spacing.Δx, coherence.scale.λ_coh, continuum.limit]
    report: "Shows when Δx/λ_coh ≪ 1 (continuous) or comparable (discrete quantisation)."

  spectral_transition:
    observe: [energy.levels.En]
    report: "Tracks discrete → continuous spectrum transition as λ_coh/Δx increases."

  integrity_health:
    observe: [coherence.integrity]
    report: "Integrity near 1 implies overlapping coherence (continuum); lower values indicate discrete isolation."

# ---------- NOTES (coherence view) ----------
note:
  - The discrete and continuous forms of physics describe the same coherence field at different sampling resolutions.
  - Quantisation emerges when the coherence field’s boundary is finite and closed: standing curvature waves form discrete nodes.
  - Continuity emerges when λ_coh ≫ Δx, and adjacent phases overlap strongly — the curvature field appears smooth.
  - The Schrödinger and wave equations are both linear coherence laws; their discrete spectra arise from boundary conditions, not separate principles.
  - In HST, discreteness = **phase isolation**; continuity = **phase overlap**.
  - The transition between them is governed by the coherence ratio Δx/λ_coh.
  - Both regimes obey the same underlying operator: Hψ = Eψ.
  - Therefore, discrete and continuous descriptions are **two resolutions of one coherence geometry** — not different kinds of physics.