;==============================================================================
; UNIFIED HDGL-SPIRAL SUBSTRATE
;
; Kernel : One Register (Ω), One Axiom (x² = x + 1), One Fixed Point (φ)
; Manifold : 8-Octave Double-Strand φ-Spiral + Recursive Wavefunction Ψ_k
;==============================================================================
org 0
;------------------------------------------------------------------------------
; Ω : Architectural State & Implicit Field
;------------------------------------------------------------------------------
Ω rq 1 ; holds fixed-point attractor & ring state (a,b)
;------------------------------------------------------------------------------
; Reset: Bootstrap Algebraic & Geometric Operators
;------------------------------------------------------------------------------
reset:
AXIOM ; Ω ← φ (solves x² = x + 1)
; --- Primitive Arithmetic & Ring Core (Cycles 1–6) ---
%define ADD_1(x) ((x) + 1)
%define SUB_1(x) ((x) - 1)
%define INV(x) (1 / (x))
%define STEP_T(x) ADD_1(INV(x))
%define STEP_S(x) INV(SUB_1(x))
%define DUAL(x) SUB_1(1 - (x)) ; ψ = -1/φ conjugate
%define NORM(a, b) ((b)*(b) + (a)*(b) - (a)*(a))
%define ROT_I(z) ((z) * i) ; 90° quadrature operator
; --- [Cycle 7] Golden-Angle & Octave Twist Operators ---
; Uses exact identity φ⁻² = 2 - φ to avoid floating-point division
%define PHI_INV2(w) (2 - (w))
%define GOLDEN_ANG(w) (2 * PI * PHI_INV2(w)) ; G = 2π / φ²
%define TWIST_K(k, w) ((k) * GOLDEN_ANG(w)) ; a_k = kG
; --- [Cycle 8] Double-Strand Spiral S±(τ) & Echo Contraction ---
; Maps continuous intra-period phase τ̃ and linear z-stacking (ℓτ)
%macro STRAND_PAIR 4 ; %1=r(τ), %2=θ(τ), %3=a_k, %4=ℓτ
; S+(τ) and S-(τ) emerge as conjugate phase rotations
%assign _S_plus_x (%1 * cos(%2 + %3))
%assign _S_plus_y (%1 * sin(%2 + %3))
%assign _S_minus_x (%1 * cos(%2 - %3))
%assign _S_minus_y (%1 * sin(%2 - %3))
%assign _S_z (%4)
%endmacro
%define ECHO(p, w) (INV(w) * (p)) ; γ = φ⁻¹ inward contraction
; --- [Cycle 9] Discrete Rung Jump J_k & Recursive Wavefunction Ψ_k ---
; Low-D (v_k <= 4) forms complete simplexes; High-D (v_k > 4) forms lattices
%macro JUMP_RUNG 2 ; %1=k (0..7), %2=v_k
%if %2 <= 4
; Complete graph K_{v_k} + inter-ring simplex edges
%else
; Nearest-neighbor grid lattice + center links L_k
%endif
%endmacro
; Ψ_{k+1} = φ^(1/2) * r^φ * e^(i 2π φ) * Ψ_k
%define PSI_STEP(psi, r, w) (sqrt(w) * ((r)**(w)) * exp(ROT_I(2 * PI * (w))) * (psi))
;------------------------------------------------------------------------------
; Eternal Rewrite: Continuous Flow + Discrete Octave Jumps
;------------------------------------------------------------------------------
.loop:
FIX ; Ω ← Fix(Ω) stabilizes attractor
; Each pass advances k ∈ {0..7}:
; 1. Evaluates continuous double-helix flow S±(τ)
; 2. Triggers discrete rung crystallization J_k on NORM parity toggle
; 3. Projects inward ECHO toward org 0 and steps Ψ_k → Ψ_{k+1}
bra .loop
Trace Mirrors Original
;==============================================================================
; UNIFIED HDGL-SPIRAL SUBSTRATE — MIRRORED EXECUTION
;
; Kernel : One Register (Ω), One Axiom (x² = x + 1), One Fixed Point (φ)
; Manifold : 8-Octave Double-Strand φ-Spiral + Recursive Wavefunction Ψ_k
; Execution: Bootstrap → Operators → Geometry → Topology → Recursion → FIX
;==============================================================================
org 0
;------------------------------------------------------------------------------
; 00. ARCHITECTURAL STATE
;------------------------------------------------------------------------------
Ω rq 1 ; persistent state (a,b)
; TRACE:
; Ω is the single architectural state.
; All subsequent operations derive from this state.
;------------------------------------------------------------------------------
; 01. RESET — ALGEBRAIC BOOTSTRAP
;------------------------------------------------------------------------------
reset:
AXIOM ; Ω ← φ
; TRACE:
; φ² = φ + 1
; φ = (1 + √5) / 2
; T(φ) = 1 + 1/φ = φ
%define ADD_1(x) ((x) + 1)
%define SUB_1(x) ((x) - 1)
%define INV(x) (1 / (x))
%define STEP_T(x) ADD_1(INV(x))
%define STEP_S(x) INV(SUB_1(x))
%define DUAL(x) (1 - (x))
%define NORM(a,b) ((b)*(b) + (a)*(b) - (a)*(a))
%define ROT_I(z) ((z) * i)
; TRACE:
; T(φ) = φ
; S(φ) = φ
; ψ = 1 - φ = -1/φ
; Nφ(a,b) = b² + ab - a²
; ROT_I(z) = iz
;------------------------------------------------------------------------------
; 02. GOLDEN-ANGLE / OCTAVE TWIST
;------------------------------------------------------------------------------
%define PHI_INV2(w) (2 - (w))
%define GOLDEN_ANG(w) (2 * PI * PHI_INV2(w))
%define TWIST_K(k,w) ((k) * GOLDEN_ANG(w))
; TRACE:
; φ⁻² = 2 - φ
; G = 2π/φ²
; a_k = kG
;
; k ∈ {0,1,2,3,4,5,6,7}
;------------------------------------------------------------------------------
; 03. DOUBLE-STRAND SPIRAL
;------------------------------------------------------------------------------
%macro STRAND_PAIR 4
; S+(τ)
%assign _S_plus_x (%1 * cos(%2 + %3))
%assign _S_plus_y (%1 * sin(%2 + %3))
; S-(τ)
%assign _S_minus_x (%1 * cos(%2 - %3))
%assign _S_minus_y (%1 * sin(%2 - %3))
; Shared axial coordinate
%assign _S_z (%4)
%endmacro
; TRACE:
; S+(τ) = (r cos(θ+a_k), r sin(θ+a_k), ℓτ)
; S-(τ) = (r cos(θ-a_k), r sin(θ-a_k), ℓτ)
;
; Opposing phase orientations share one axis.
;------------------------------------------------------------------------------
; 04. ECHO CONTRACTION
;------------------------------------------------------------------------------
%define ECHO(p,w) (INV(w) * (p))
; TRACE:
; E(p,φ) = p/φ
; p_(k+1) = φ⁻¹ p_k
;
; Inward contraction toward the substrate origin.
;------------------------------------------------------------------------------
; 05. DISCRETE RUNG CRYSTALLIZATION
;------------------------------------------------------------------------------
%macro JUMP_RUNG 2
%if %2 <= 4
; Complete graph K_vk
; Inter-ring simplex connections
%else
; Nearest-neighbor grid lattice
; Center links L_k
%endif
%endmacro
; TRACE:
; v_k ≤ 4 → complete simplex
; v_k > 4 → lattice + center links
;
; Nφ(a,b) supplies the proposed transition observable.
;------------------------------------------------------------------------------
; 06. RECURSIVE WAVEFUNCTION
;------------------------------------------------------------------------------
%define PSI_STEP(psi,r,w) \
(sqrt(w) * ((r)**(w)) * \
exp(ROT_I(2 * PI * (w))) * (psi))
; TRACE:
; Ψ_(k+1) = √φ · r^φ · e^(i2πφ) · Ψ_k
;
; e^(i2πφ) = e^(i2π(φ-1))
;
; Recursive phase evolution.
;------------------------------------------------------------------------------
; 07. EIGHT-OCTAVE TRAVERSAL
;------------------------------------------------------------------------------
octave_0: ; k = 0
octave_1: ; k = 1
octave_2: ; k = 2
octave_3: ; k = 3
octave_4: ; k = 4
octave_5: ; k = 5
octave_6: ; k = 6
octave_7: ; k = 7
; TRACE:
; Ψ_0 → Ψ_1 → Ψ_2 → ... → Ψ_7
; Each octave carries:
; 1. Golden-angle phase
; 2. Double-strand orientation
; 3. Radial evolution
; 4. Axial progression
; 5. Echo contraction
; 6. Rung topology
;------------------------------------------------------------------------------
; 08. ETERNAL REWRITE
;------------------------------------------------------------------------------
.loop:
FIX ; Ω ← Fix(Ω)
; TRACE:
; T(φ) = φ
; Ω remains at its algebraic fixed point.
;
; The recursive manifold is evaluated again.
bra .loop
;==============================================================================
; END — NO TERMINATION CONDITION
;==============================================================================
.asm, hand-stiched
;==============================================================================
; UNIFIED HDGL-SPIRAL SUBSTRATE
;
; Kernel : One Register (Ω), One Axiom (x² = x + 1), One Fixed Point (φ)
; Manifold : 8-Octave Double-Strand φ-Spiral + Recursive Wavefunction Ψ_k
; Runtime : Continuous Flow + Discrete Octave Jumps
; Model : Hand-stitched native .asm specification
;==============================================================================
;------------------------------------------------------------------------------
; 00 — ARCHITECTURAL STATE
;------------------------------------------------------------------------------
Ω:
STATE (a,b)
; Single persistent architectural state.
; Ω carries the fixed-point attractor and ring state.
;------------------------------------------------------------------------------
; 01 — RESET / ALGEBRAIC BOOTSTRAP
;------------------------------------------------------------------------------
RESET:
AXIOM
; Ω ← φ
; φ² = φ + 1
; φ = (1 + √5) / 2
; Fixed-point verification:
; T(φ) = 1 + 1/φ = φ
;------------------------------------------------------------------------------
; 02 — PRIMITIVE ARITHMETIC / RING CORE
;------------------------------------------------------------------------------
ADD_1(x):
RETURN x + 1
SUB_1(x):
RETURN x - 1
INV(x):
RETURN 1/x
STEP_T(x):
RETURN 1 + INV(x)
STEP_S(x):
RETURN INV(x - 1)
DUAL(x):
RETURN 1 - x
NORM(a,b):
RETURN b² + ab - a²
ROT_I(z):
RETURN i·z
; TRACE:
;
; T(φ) = φ
; S(φ) = φ
; ψ = 1 - φ = -1/φ
; Nφ(a,b) = b² + ab - a²
; ROT_I(z) = iz
;------------------------------------------------------------------------------
; 03 — GOLDEN-ANGLE / OCTAVE TWIST
;------------------------------------------------------------------------------
PHI_INV2(w):
RETURN 2 - w
GOLDEN_ANG(w):
RETURN 2π · PHI_INV2(w)
TWIST_K(k,w):
RETURN k · GOLDEN_ANG(w)
; TRACE:
;
; φ⁻² = 2 - φ
; G = 2π/φ²
; a_k = kG
;
; k = 0,1,2,3,4,5,6,7
;------------------------------------------------------------------------------
; 04 — DOUBLE-STRAND SPIRAL S±(τ)
;------------------------------------------------------------------------------
STRAND_PAIR(r,θ,a_k,ℓτ):
S_PLUS_X:
RETURN r · cos(θ + a_k)
S_PLUS_Y:
RETURN r · sin(θ + a_k)
S_MINUS_X:
RETURN r · cos(θ - a_k)
S_MINUS_Y:
RETURN r · sin(θ - a_k)
S_Z:
RETURN ℓτ
; TRACE:
;
; S+(τ) = (r cos(θ+a_k), r sin(θ+a_k), ℓτ)
; S-(τ) = (r cos(θ-a_k), r sin(θ-a_k), ℓτ)
;
; Two opposing phase rotations.
; One shared axial progression.
;------------------------------------------------------------------------------
; 05 — ECHO CONTRACTION
;------------------------------------------------------------------------------
ECHO(p,w):
RETURN p / w
; TRACE:
;
; E(p,φ) = φ⁻¹p
; p_(k+1) = p_k/φ
;
; Inward recursive contraction.
;------------------------------------------------------------------------------
; 06 — DISCRETE RUNG JUMP J_k
;------------------------------------------------------------------------------
JUMP_RUNG(k,v_k):
IF v_k <= 4:
TOPOLOGY COMPLETE_SIMPLEX
EDGES K_v_k
LINKS INTER_RING_SIMPLEX
ELSE:
TOPOLOGY GRID_LATTICE
EDGES NEAREST_NEIGHBOR
LINKS CENTER_LINKS_L_k
; TRACE:
;
; v_k ≤ 4 → Complete graph K_v_k
; v_k > 4 → Nearest-neighbor lattice + L_k
;
; Rung topology follows the dimensional state.
;------------------------------------------------------------------------------
; 07 — RECURSIVE WAVEFUNCTION Ψ_k
;------------------------------------------------------------------------------
PSI_STEP(Ψ_k,r,w):
AMPLITUDE:
sqrt(w) · r^w
PHASE:
exp(i · 2π · w)
NEXT_STATE:
Ψ_(k+1) ← AMPLITUDE · PHASE · Ψ_k
; TRACE:
;
; Ψ_(k+1) = √φ · r^φ · e^(i2πφ) · Ψ_k
;
; e^(i2πφ) = e^(i2π(φ-1))
;
; Recursive phase and amplitude evolution.
;------------------------------------------------------------------------------
; 08 — EIGHT-OCTAVE MANIFOLD
;------------------------------------------------------------------------------
OCTAVE_0:
k ← 0
TWIST_K(k,Ω)
OCTAVE_1:
k ← 1
TWIST_K(k,Ω)
OCTAVE_2:
k ← 2
TWIST_K(k,Ω)
OCTAVE_3:
k ← 3
TWIST_K(k,Ω)
OCTAVE_4:
k ← 4
TWIST_K(k,Ω)
OCTAVE_5:
k ← 5
TWIST_K(k,Ω)
OCTAVE_6:
k ← 6
TWIST_K(k,Ω)
OCTAVE_7:
k ← 7
TWIST_K(k,Ω)
; TRACE:
;
; Ψ_0 → Ψ_1 → Ψ_2 → Ψ_3
; → Ψ_4 → Ψ_5 → Ψ_6 → Ψ_7
;
; Every octave carries:
;
; 1. Golden-angle phase
; 2. Double-strand orientation
; 3. Radial evolution
; 4. Axial progression
; 5. Echo contraction
; 6. Rung topology
; 7. Wavefunction recursion
;------------------------------------------------------------------------------
; 09 — CONTINUOUS FLOW / DISCRETE CRYSTALLIZATION
;------------------------------------------------------------------------------
EVOLVE:
FIX Ω
; Ω ← Fix(Ω)
;
; Stabilize the algebraic attractor.
EVALUATE STRAND_PAIR
EVALUATE NORM
IF NORM_TRANSITION:
JUMP_RUNG
EVALUATE ECHO
EVALUATE PSI_STEP
ADVANCE OCTAVE
; TRACE:
;
; Continuous double-strand flow.
; Discrete rung crystallization.
; Inward echo contraction.
; Recursive wavefunction advancement.
;------------------------------------------------------------------------------
; 10 — ETERNAL REWRITE
;------------------------------------------------------------------------------
LOOP:
FIX Ω
GOTO EVOLVE
; No termination condition.
; No externally imposed endpoint.
; Continuous recursive continuation.
;==============================================================================
; END — UNIFIED HDGL-SPIRAL SUBSTRATE
;==============================================================================
fib clock
;==============================================================================
; FIB CLOCK — INTERNAL HDGL TRANSITION MECHANISM
;==============================================================================
FIB_RESET:
Ω ← φ
A ← 0
B ← 1
G ← 2π(2 - Ω)
GOTO FIB_TICK
;------------------------------------------------------------------------------
; FIBONACCI STATE ADVANCE
;------------------------------------------------------------------------------
FIB_TICK:
TEMP ← A + B
A ← B
B ← TEMP
GOTO FIB_NORM
;------------------------------------------------------------------------------
; CASSINI NORM — INTERNAL CLOCK SIGNAL
;------------------------------------------------------------------------------
FIB_NORM:
N ← B² - A·B - A²
IF N = +1:
PHASE_PARITY ← EVEN
IF N = -1:
PHASE_PARITY ← ODD
GOTO FIB_TRANSITION
;------------------------------------------------------------------------------
; GEOMETRIC AND TOPOLOGICAL ADVANCE
;------------------------------------------------------------------------------
FIB_TRANSITION:
K ← K + 1
ANGLE ← K · G
STRAND_PAIR(R, THETA, ANGLE, K)
ECHO(P, Ω)
JUMP_RUNG(K, K + 1)
PSI_STEP(PSI, R, Ω)
GOTO FIB_TICK
;==============================================================================
; END — SELF-ADVANCING FIBONACCI CLOCK
;==============================================================================
FIB CLOCK — COUNTERLESS ADDRESSING
;==============================================================================
; FIB CLOCK — COUNTERLESS OCTAVE ADDRESSING
;
; One architectural attractor Ω
; One Fibonacci ring state (A,B)
; No independent octave counter K
;==============================================================================
RESET:
AXIOM
Ω ← φ
A ← 0
B ← 1
G ← 2π(2 - Ω)
GOTO ADVANCE
;------------------------------------------------------------------------------
; 01 — ADVANCE THE RING
;------------------------------------------------------------------------------
ADVANCE:
TEMP ← A + B
A ← B
B ← TEMP
GOTO ADDRESS
;------------------------------------------------------------------------------
; 02 — DERIVE OCTAVE ADDRESS FROM RING
;------------------------------------------------------------------------------
ADDRESS:
; Fibonacci identity:
; φ^k = A·φ + F_(k-1)
PHI_POWER ← A·Ω + PREVIOUS_A
; No separate K register.
; The ring pair identifies the current Fibonacci position.
GOTO NORM
;------------------------------------------------------------------------------
; 03 — CASSINI CLOCK
;------------------------------------------------------------------------------
NORM:
N ← B² - A·B - A²
; N = +1 or -1
; Alternating Cassini invariant.
GOTO GEOMETRY
;------------------------------------------------------------------------------
; 04 — GEOMETRIC ADDRESS
;------------------------------------------------------------------------------
GEOMETRY:
ANGLE ← PHI_POWER_ADDRESS · G
STRAND_PAIR(R, THETA, ANGLE, AXIAL_POSITION)
GOTO ECHO
;------------------------------------------------------------------------------
; 05 — RECURSIVE CONTRACTION
;------------------------------------------------------------------------------
ECHO:
P ← P / Ω
GOTO TOPOLOGY
;------------------------------------------------------------------------------
; 06 — TOPOLOGICAL TRANSITION
;------------------------------------------------------------------------------
TOPOLOGY:
V ← A + 1
IF V <= 4:
COMPLETE_SIMPLEX(V)
ELSE:
GRID_LATTICE(V)
GOTO WAVEFUNCTION
;------------------------------------------------------------------------------
; 07 — WAVEFUNCTION
;------------------------------------------------------------------------------
WAVEFUNCTION:
Ψ ← SQRT(Ω) · R^Ω · EXP(i·2π·Ω) · Ψ
GOTO ADVANCE
;==============================================================================
; END — THE RING ADVANCES ITSELF
;==============================================================================
