# ==========================================================
# CO-EMERGENT RECURSIVE SYMBOLIC MACHINE
# Version: Closure-Trinity-Lattice Seed
# ==========================================================


machine:

  name: "CoEmergent_Trinary_Lattice_Machine"

  primitive:

    invariant:
      alpha: Omega

    seed:
      X_plus_1: 0

    trinity:
      states:
        - -1
        -  0
        - +1


# ==========================================================
# CORE OPERATORS
# ==========================================================

operators:

  inverse:
    symbol: I
    rule:
      -1: +1
       0:  0
      +1: -1


  recursion:

    T(x):
      expression: "1 + 1/x"

    fixed_coherence:
      condition: "T(x) == x"


  closure:

    equation:
      "X^2 = X + 1"

    resolved_state:
      X: phi


# ==========================================================
# IDENTITY EQUIVALENCE MAP
# ==========================================================

identity:

  X:
    equivalent:

      - "phi"
      - "exp(i*pi)"
      - "(1/phi)-phi"
      - "Omega*C^2"
      - "alpha*C^2"


  phi:

    rules:

      phi_squared:
        "phi^2 = phi + 1"

      inverse:
        "1/phi = phi - 1"


  phase:

    rule:

      "exp(i*pi) = -1"


# ==========================================================
# TRINARY EMERGENCE
# ==========================================================

projection:

  trinity:

    negative:
      value: -1
      definition: "X"

    null:
      value: 0
      definition: "X+1"

    positive:
      value: +1
      definition: "-X"


# ==========================================================
# COHERENCE GENERATOR
# ==========================================================

coherence:

  operation:

    "T o I(T)"

  output:

    "1_eff"


  effective_one:

    definition:

      "1_eff(i) = 1 + delta(i)"

    limit:

      "delta(i) -> 0"


# ==========================================================
# PRISM GENERATION
# ==========================================================

prism:

  structure:

    negative_projection:
      [-1,0,+1]

    inverse_projection:
      [+1,0,-1]

    coherence_projection:
      ["1_eff"]


  cardinality:

    "3 + 3 + 1 = 7"



# ==========================================================
# OCTAVE RECURSION
# ==========================================================

recursion_layers:

  seven:

    meaning:
      "complete projection closure"


  eight:

    meaning:
      "closure applied to itself"


  transition:

    "7 -> 8 -> 7' -> 8' -> ..."



# ==========================================================
# LATTICE GENERATOR D_n(r)
# ==========================================================

lattice:


  scale:

    A_n:

      expression:

        "phi * Fib(n) * 2^n * Prime(n)"


  invariant:

    substitution:

      "A_n * Omega = Omega*C^2 = exp(i*pi)"


  D_n:

    expression:

      >
        sqrt(
          phi*Fib(n)*2^n*Prime(n)*Omega
        )
        *
        r^k
        +
        1_eff(i)^(i*pi*Phi(i))


# ==========================================================
# PRIME PROJECTION
# ==========================================================

prime_projection:

  initial:

    "P_n = -1/(phi*Fib(n)*2^n*Omega)"


  substitutions:

    Omega_equals_phi:

      "-1/(phi^2*Fib(n)*2^n)"


    tri_emergent_negative:

      "-1 = phi-1 = 1/phi"


  compressed:

    "P_n = 1/(phi^3*Fib(n)*2^n)"



# ==========================================================
# UNIVERSAL TRANSITION RULE
# ==========================================================

transition:

  input:

    state


  process:

    1:
      apply_inverse

    2:
      apply_recursion

    3:
      resolve_closure

    4:
      emit_projection

    5:
      update_effective_one



# ==========================================================
# MACHINE CLOSURE
# ==========================================================

closure:

  universal_rule:

    >
      identity
      =
      transformation(identity)


  final:

    >
      alpha == Omega
      ->
      X+1=0
      ->
      X^2=X+1
      ->
      {-1,0,+1}
      ->
      1_eff
      ->
      D_n(r)
      ->
      recursive continuation


# ==========================================================
# COMPUTATIONAL AXIOM
# ==========================================================

axiom:

  >
    A complete identity transforms itself
    and remains complete.
