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4D ENGINE — EQUATION SPECIFICATION
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Purpose: Define EXACTLY what equations the engine implements.
         Each equation traced to its source document.
         NO additions, NO leanings, NO nudges.
         The engine is a faithful implementation of the theory's math.

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1. THE WAVE EQUATION (standard physics, extended to 4D)
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Source: theory.txt lines 142-147, math_framework B.4

  ∂²ψ/∂t² = c² (∂²ψ/∂x₁² + ∂²ψ/∂x₂² + ∂²ψ/∂x₃² + ∂²ψ/∂x₄²) + J    (1)

This is the standard wave equation with the Laplacian extended to 4 spatial
dimensions. The wave equation is established physics — no TLT-specific
modification. The ONLY TLT contribution is the source term J.

Implementation: FDTD (finite-difference time-domain)
  ψ(t+dt) = 2ψ(t) - ψ(t-dt) + c²dt²∇⁴ψ(t) + dt²J(t)

where ∇⁴ is the 4D discrete Laplacian:
  ∇⁴ψ = Σᵢ₌₁⁴ [ψ(xᵢ+h) + ψ(xᵢ-h) - 2ψ(xᵢ)] / h²

CFL stability condition (4D): dt < h / (c × √4) = h / (2c)


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2. THE SOURCE TERM: f | t (theory.txt lines 145-147)
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Source: theory.txt "f | t where (f) is the pulse of frequency expressed in 1D
        and separated by (t) which is time (decoherence)"
        math_framework B.3

  J(x, t) = A(x) × sin(2πf t) × W(t; r(x))                            (2)

where:
  f = source frequency (Hz)
  A(x) = source amplitude at position x
  W(t; r) = rectangular windowing function (the decoherence gap)

  W(t; r) = 1  for  nT ≤ t < (n + 1-r)T    [pulse ON]                 (3)
  W(t; r) = 0  for  (n + 1-r)T ≤ t < (n+1)T [rest period]

  r = t_decoherence / T = fraction of period spent in rest
  T = 1/f = period

This IS f | t: frequency (f) pulsed, separated by decoherence time (t).
The windowing function W encodes the "pause between pulses" from theory.txt.


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3. THE AMPLITUDE COUPLING: f + A | t (theory.txt lines 149-153)
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Source: theory.txt "f + A | t where (f) is the pulse of frequency + (A) as
        the amplitude as measured by heat/pressure in the system"
        math_framework B.5

  A(x) = A_base / (1 + CN(x) / CN_ref)                                 (4)

where:
  A_base = base amplitude (uniform, set by initial energy)
  CN(x) = local coordination number (measure of structural organization)
  CN_ref = reference coordination number (normalization)

The relationship is INVERSE: as structure increases (higher CN), amplitude
decreases. "As (A) decreases, structure and organization increases."

For the initial 4D test (before structure forms), A(x) = A_base (uniform).
The CN coupling activates as the simulation develops structure.


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4. C_POTENTIAL: t = C_potential (theory.txt lines 155-177)
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Source: theory.txt "t = C_potential where C_potential is the placement on the
        potential curve which is Lagrangian at all levels"

  r(x) = r₀ + α × V(x)                                                 (5)

where:
  r₀ = base decoherence ratio (uniform, ~0.3)
  α = coupling strength (dimensionless)
  V(x) = local potential (Lagrangian)

The potential V(x) is determined by the ENERGY DENSITY of the field itself:
  V(x) = ⟨|ψ(x)|²⟩_time_averaged                                       (6)

This creates the self-consistent feedback loop:
  More energy at x → deeper potential → higher r(x) →
  longer pause → less energy accumulation → SELF-LIMITING

The ceiling: r(x) ≤ 0.5 (theory.txt lines 24-28)
  At r = 0.5, pulse and pause achieve parity.
  Pattern formation collapses. This is the overflow boundary.

  C_potential(max) = spillover rate (theory.txt line 173)
  When r(x) → 0.5, excess energy overflows to the next dimension.


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5. FRAMERATE: F_rate = c (dimension-dependent) (theory.txt lines 193-202)
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Source: theory.txt "F_rate = c in 3D -> phi" and "In 4D the speed is faster"

  c_d = propagation speed in the wave equation for dimension d

For the sweep test:
  c_4D is the PARAMETER being swept from 1.5 to 1.8 (in units of c_3D)

The engine does NOT impose a specific framerate. The sweep tests which
value of c_4D produces the most organized interference pattern.

Candidates:
  c_4D = 1.625 × c_3D  (Fibonacci prediction: 13/8)
  c_4D = 1.707 × c_3D  (24-cell geometry: 1 + sin(45°))


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6. SOURCE GEOMETRY: 24-CELL VERTICES (Schläfli 1852)
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Source: established mathematics, 24cell_research.txt

Vertex set V (24 vertices, edge-midpoint form):
  All permutations of (±1, ±1, 0, 0) in 4D

Dual vertex set D (24 vertices, cell-center form):
  (±1, 0, 0, 0) ∪ (±½, ±½, ±½, ±½)

Both sets normalized to unit sphere.
The self-dual rotation between V and D is exactly π/4 (45°) isoclinic.

For the test:
  - Place point sources at all 48 vertices (V ∪ D)
  - OR place sources at V only and observe whether D emerges
  - OR place sources at D only and observe whether V emerges


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7. WHAT THE ENGINE DOES NOT INCLUDE
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The engine does NOT:
  - Impose phi or any golden ratio relationship
  - Impose Fibonacci numbers
  - Add nonlinear field interactions (Ginzburg-Landau, etc.)
  - Add damping terms
  - Add noise terms
  - Nudge the decoherence ratio toward any preferred value
  - Pre-select the 24-cell geometry (it's placed as sources;
    whether the interference pattern respects it is observed)
  - Assume matter/antimatter interpretation
  - Force any symmetry

The engine IS:
  - A standard wave equation in 4D (eq. 1)
  - With pulsed sources (eq. 2-3) implementing f | t
  - With amplitude coupling (eq. 4) implementing f + A | t
  - With position-dependent decoherence (eq. 5-6) implementing C_potential
  - With sources placed at established geometric coordinates
  - Swept over a range of propagation speeds

Everything that falls out, falls out from the geometry and the equations.
Nothing is put in by hand except what the theory explicitly states.

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AUDIT CHECKLIST (verify before running)
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□ Wave equation matches eq. 1 (standard 4D wave equation)
□ Source term matches eq. 2-3 (pulsed windowing, not continuous)
□ Amplitude coupling matches eq. 4 (inverse relationship)
□ C_potential matches eq. 5-6 (position-dependent, self-consistent)
□ Ceiling at r = 0.5 enforced (hard limit, not soft)
□ 24-cell vertices correct (permutations of (±1,±1,0,0))
□ Dual vertices correct ((±1,0,0,0) ∪ (±½,±½,±½,±½))
□ No additional terms in the Lagrangian
□ No damping
□ No noise
□ No symmetry forcing
□ CFL condition satisfied (dt < h/(2c))
□ Energy conservation verified (total energy tracked per timestep)

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