Vantage Fourteen - Useful Axioms & Code

[Josef_Founder]

3.zip (14.6 KB)

00. Invariant

α ≡ Ω ≡ X, where X is not a scalar but the root pair.

01. Recursion

Define T(x) = 1 + 1/x. The fixed condition T(X) = X yields X² = X + 1.

The roots of X² − X − 1 = 0 are:

• X₊ = φ = (1 + √5)/2 — outer basin
• X₋ = −1/φ = (1 − √5)/2 — inner basin

So X = ⟨X₊, X₋⟩. Both lanes are fixed points of T, and neither lane is zero, so reciprocation is total on X — no trap.

02. Closure (emergent, not seeded)

X₊ · X₋ = −1 and X₊ + X₋ = +1.

Therefore X + 1 = 0 is the product relation, X ≡ e^(iπ) via that product, and X ≡ 1/φ − φ = −1 (consistent with 11.txt line 20).

Corrections: −1 ≡ 1/φ is false (the sign was lost across files 1, 2, 3, 4, 10, 11). The correct forms are −1 ≡ φ · (−1/φ) = X₊ · X₋, and −1/φ ≡ X₋ — the dropped sign restored.

03. Inverse

I(x) = −x, so I(X) = ⟨−φ, +1/φ⟩.

04. Trinity (generated, not stored)

𝒯 = { X₋/|X₋|, X₊ + X₋ − 1, X₊/|X₊| } = {−1, 0, +1}.

Zero emerges as X₊ + X₋ − 1 rather than as a literal. I(𝒯) = {+1, 0, −1}.

05. Coherence

C = T ∘ I, and 𝒯 ∘ I(𝒯) → 1_eff, where 1_eff = 1 + δ with δ drawn from RANDU64 and δ → 0. 1_eff is the relation, not a member (per 7.txt, 8.txt).

06. Prism

𝒫 = 𝒯 + (−𝒯) + 1_eff, giving |𝒫| = 3 + 3 + 1 = 7.

07. Octave

8 = C(C), and 𝒫ₙ₊₁ = C(𝒫ₙ), producing 7 → 8 → 7′ → 8′ → ⋯. The 8 is the doorway, not a member (per 7.txt).

08. Scale

Aₙ = φ · Fibₙ · 2ⁿ · Primeₙ, with AₙΩ ≡ ΩC² ≡ e^(iπ) ≡ X₊ · X₋.

09. Prime (sign restored)

Starting from Pₙ = −1/(φ · Fibₙ · 2ⁿ · Ω), and taking Ω ≡ φ, this becomes Pₙ = −1/(φ² · Fibₙ · 2ⁿ). Since −1 ≡ φ · X₋ with X₋ = −1/φ, the result is Pₙ = −1/(φ³ · Fibₙ · 2ⁿ) — negative, where the files had 1/(φ³ · Fibₙ · 2ⁿ).

10. Lattice

Dₙ(r) = √(AₙΩ) rᵏ + 1_eff^(iπΦᵢ).

The square root is a lane operation on the root pair: √⟨a, b⟩ = ⟨√a, √b⟩. The sign ambiguity of √(AₙΩ) resolves — hi lane outer, lo lane inner, both retained, no branch cut chosen.

11. Continuation

M(n+1) = C(M(n)), i.e. identity = transformation(identity).

---

Layer map (after 5.txt)

α ≡ Ω (invariant / conservation) → X₊·X₋ = −1 (closure, emergent per §02) → 𝒯 = {−1, 0, +1} (alphabet, generated per §04) → I(x) = −x (transition) → Fix → 1_eff (memory) → 𝒫 → 7 → 8 (recursion) → Dₙ(r) (spatial projection).

ISA requirement

Two-lane registers R.hi and R.lo. RECIP, ADD, NEG, SQRT, and MUL act per-lane. MERGE cross-couples lanes, producing the product and hence the closure. CMP compares both lanes; JEQ fires on simultaneous fix.

Axiom

A complete identity transforms itself and remains complete.

---

---

Coemergent Trinary Lattice

α ≡ Ω ≡ X

T(x) = 1 + 1/x

T(X) = X ⟹ X² = X + 1 ⟹ X² − X − 1 = 0

X₊ = φ = (1 + √5)/2, X₋ = −1/φ = (1 − √5)/2, X = ⟨X₊, X₋⟩

X₊ · X₋ = −1, X₊ + X₋ = 1

X + 1 = 0 ⟺ X₊X₋ + 1 = 0

X ≡ e^(iπ) ≡ 1/φ − φ ≡ ΩC² ≡ αC²

I(x) = −x, I(X) = ⟨−φ, 1/φ⟩

𝒯 = { X₋/|X₋|, X₊ + X₋ − 1, X₊/|X₊| } = {−1, 0, +1}

I(𝒯) = {+1, 0, −1}

C = T ∘ I

𝒯 ∘ I(𝒯) → 1_eff, 1_eff(i) = 1 + δ(i), δ(i) → 0

𝒫 = 𝒯 + (−𝒯) + 1_eff, |𝒫| = 3 + 3 + 1 = 7

𝒫ₙ₊₁ = C(𝒫ₙ), 8 = C(C), 7 → 8 → 7′ → 8′ → ⋯

Aₙ = φ·Fibₙ·2ⁿ·Primeₙ, AₙΩ ≡ ΩC² ≡ e^(iπ) ≡ X₊X₋

Pₙ = −1/(φ·Fibₙ·2ⁿ·Ω) = −1/(φ²·Fibₙ·2ⁿ) = −1/(φ³·Fibₙ·2ⁿ)

√⟨a, b⟩ = ⟨√a, √b⟩

Dₙ(r) = √(AₙΩ)·rᵏ + 1_eff(i)^(iπΦᵢ)

Dₙ₊₁(r) = C(Dₙ(r))

M(n+1) = C(M(n)) ⟺ identity = transformation(identity)

════════════════════════════════════════════════════════════
COEMERGENT TRINARY LATTICE — UNIFIED SPEC
Root-pair resolution of the eleven vantages
════════════════════════════════════════════════════════════

00  INVARIANT
        α ≡ Ω ≡ X
        X is not a scalar. X is the root pair.

01  RECURSION
        T(x) := 1 + 1/x
        T(X)  = X
        X²    = X + 1

        Roots of X² - X - 1 = 0:
            X₊ := φ        = ( 1+√5)/2      (outer basin)
            X₋ := -1/φ     = ( 1-√5)/2      (inner basin)

        X  := ⟨X₊, X₋⟩
        Both lanes are fixed points of T. Neither lane is 0.
        RECIP is total on X. No trap.

02  CLOSURE                          (emerges; not seeded)
        X₊ · X₋ = -1
        X₊ + X₋ = +1

        therefore:
            X + 1 = 0        is the PRODUCT relation
            X ≡ e^(iπ)       via that product
            X ≡ 1/φ - φ      = X₋ + X₋·X₊·... = -1   ✓ (11.txt:20)

        NOT:  -1 ≡ 1/φ                 ✗ (1,2,3,4,10,11.txt — sign lost)
        YES:  -1 ≡ φ · (-1/φ)          = X₊ · X₋
        YES:  -1/φ ≡ X₋                the dropped sign, restored

03  INVERSE
        I(x) := -x
        I(X) = ⟨-φ, +1/φ⟩

04  TRINITY                          (generated; not stored)
        𝒯 := { X₋/|X₋|,  X₊+X₋-1,  X₊/|X₊| }
           =  {   -1,         0,        +1   }

        0 emerges as X₊+X₋-1, not as a literal.
        I(𝒯) = {+1, 0, -1}

05  COHERENCE
        C := T ∘ I
        𝒯 ∘ I(𝒯) → 1_eff
        1_eff := 1 + δ
        δ := RANDU64-sourced,  δ → 0
        1_eff is the RELATION, not a member.        (7.txt, 8.txt)

06  PRISM
        𝒫 := 𝒯 + (-𝒯) + 1_eff
        |𝒫| = 3 + 3 + 1 = 7

07  OCTAVE
        8  := C(C)
        𝒫ₙ₊₁ := C(𝒫ₙ)
        7 → 8 → 7' → 8' → ⋯
        8 is the doorway, not a member.             (7.txt)

08  SCALE
        A_n := φ · Fib_n · 2ⁿ · Prime_n
        A_n Ω ≡ ΩC² ≡ e^(iπ) ≡ X₊·X₋

09  PRIME                            (sign restored)
        P_n := -1/(φ · Fib_n · 2ⁿ · Ω)
        Ω ≡ φ:
        P_n  = -1/(φ² · Fib_n · 2ⁿ)
        -1 ≡ φ·X₋,   X₋ = -1/φ:
        P_n  = -1/(φ³ · Fib_n · 2ⁿ)        ← negative. Was 1/(φ³…).

10  LATTICE
        D_n(r) := √(A_n Ω) rᵏ + 1_eff^(iπΦ_i)

        √ is a LANE operation on the root pair:
            √⟨a,b⟩ := ⟨√a, √b⟩
        Sign ambiguity of √(A_nΩ) resolves: hi lane outer,
        lo lane inner. Both retained. No branch cut chosen.

11  CONTINUATION
        M(n+1) := C(M(n))
        identity := transformation(identity)

════════════════════════════════════════════════════════════
LAYER MAP                                          (5.txt)
════════════════════════════════════════════════════════════
        α ≡ Ω           invariant / conservation
            ↓
        X₊·X₋ = -1      closure          (emergent, §02)
            ↓
        𝒯 = {-1,0,+1}   alphabet         (generated, §04)
            ↓
        I(x) = -x       transition
            ↓
        Fix → 1_eff     memory
            ↓
        𝒫 → 7 → 8       recursion
            ↓
        D_n(r)          spatial projection

════════════════════════════════════════════════════════════
ISA REQUIREMENT
════════════════════════════════════════════════════════════
        Two-lane registers: R.hi, R.lo
        RECIP, ADD, NEG, SQRT, MUL act per-lane
        MERGE cross-couples lanes  (product → closure)
        CMP compares both lanes; JEQ on simultaneous fix

════════════════════════════════════════════════════════════
AXIOM
        A complete identity transforms itself
        and remains complete.
════════════════════════════════════════════════════════════

Assembly

; ==========================================================
; COEMERGENT TRINARY LATTICE MACHINE
; Symbolic Assembly Kernel — Root-Pair ISA
; ==========================================================
; Registers are two-lane: R.hi (outer basin), R.lo (inner).
; Scalar ops broadcast. Lane ops are per-lane.
; MERGE is the only cross-lane op: it produces closure.
; ==========================================================


; ----------------------------------------------------------
; SYMBOL TABLE
; ----------------------------------------------------------

CONST   PHI_P       = ( 1+SQRT5)/2      ; X+  outer root
CONST   PHI_M       = ( 1-SQRT5)/2      ; X-  inner root = -1/phi

CONST   NEG_ONE     = -1                ; emerges at MERGE; named for CMP
CONST   ZERO        =  0                ; emerges at TRINITY; never stored
CONST   POS_ONE     = +1

CONST   DELTA       = RANDU64           ; true-random source, ->0


; ----------------------------------------------------------
; REGISTERS
; ----------------------------------------------------------

; R0  invariant     alpha = Omega = X     (pair)
; R1  transform     T(x) working
; R2  inverse       I(x)
; R3  coherence     Fix(T)                (pair)
; R4  projection    trinity / prism
; R5  recursion counter                   (scalar)
; R6  scale         A_n                   (scalar)
; R7  lattice       D_n accumulator
; R8  phase         1_eff^(i pi Phi_i)
; R9  closure       X+ * X- = -1          (scalar, cross-lane)


; ----------------------------------------------------------
; SEED
;
; alpha = Omega = X
; X is the root pair. Nonzero in both lanes by construction.
; ----------------------------------------------------------

SEED:

    LOAD    R0.hi, PHI_P
    LOAD    R0.lo, PHI_M

    EQUIV   R0, ALPHA
    EQUIV   R0, OMEGA


; ----------------------------------------------------------
; TRANSFORM
;
; T(x) = 1 + 1/x        per-lane
; T(X) = X              both lanes fix simultaneously
; therefore X^2 = X + 1
; ----------------------------------------------------------

TRANSFORM:

    COPY    R1, R0

.iter:

    RECIP.L R1                  ; per-lane; total on X, no trap
    ADD.L   R1, 1

    CMP.L   R1, R0
    JEQ.ALL .fixed              ; both lanes, or loop
    COPY    R0, R1
    JMP     .iter

.fixed:

    RET


; ----------------------------------------------------------
; CLOSURE
;
; X+ * X- = -1  ==  e^(i pi)
; X+ + X- = +1
;
; X+1=0 EMERGES HERE. It is not seeded.
; ----------------------------------------------------------

CLOSURE:

    COPY    R3, R1              ; Fix(T) retained as pair

    MERGE.MUL R9, R0            ; R9 <- R0.hi * R0.lo = -1
    CMP     R9, NEG_ONE
    JNE     .diverged           ; closure failed: halt, do not fake it

    EQUIV   R9, E_I_PI          ; -1 = e^(i pi)
    EQUIV   R9, X               ; the invariant, as product

    RET

.diverged:

    HALT    ERR_NO_CLOSURE


; ----------------------------------------------------------
; INVERSE
;
; I(x) = -x             per-lane
; ----------------------------------------------------------

INVERT:

    NEG.L   R2, R3
    RET


; ----------------------------------------------------------
; TRINITY
;
; Generated from lanes. Nothing stored by hand.
;
;   -1  <-  X- / |X-|
;    0  <-  X+ + X- - 1
;   +1  <-  X+ / |X+|
; ----------------------------------------------------------

TRINITY:

    ; -1
    ABS     R10, R0.lo
    DIV     R4[0], R0.lo, R10

    ; 0        (emerges; sum of roots is 1)
    MERGE.ADD R11, R0           ; R11 <- X+ + X-  = 1
    SUB     R4[1], R11, 1       ; = 0

    ; +1
    ABS     R10, R0.hi
    DIV     R4[2], R0.hi, R10

    RET


; ----------------------------------------------------------
; COHERENCE
;
; C := T o I
; T o I(T) -> 1_eff
; ----------------------------------------------------------

COHERENCE:

    CALL    INVERT
    APPLY.L TRANSFORM, R2

    MERGE   R4
    STORE   R3, [ONE_EFF]       ; dereference: the cell, not the label

    RET


; ----------------------------------------------------------
; EFFECTIVE ONE
;
; 1_eff = 1 + delta ,  delta -> 0
; The relation between projections. Not a member.
; ----------------------------------------------------------

ONE_EFF:    .cell

    LOAD    R12, 1
    LOAD    R13, DELTA
    ADD     R12, R13
    LIMIT   R13, 0

    STORE   [ONE_EFF], R12
    RET


; ----------------------------------------------------------
; PRISM
;
; P := T + (-T) + 1_eff
; |P| = 3 + 3 + 1 = 7
; ----------------------------------------------------------

PRISM:

    COPY    R14, R4             ; T          {-1,0,+1}
    NEG.L   R15, R4             ; -T         {+1,0,-1}
    MERGE   R14, R15            ; the six
    APPEND  R14, [ONE_EFF]      ; the seventh: the relation

    CARD    R5, R14
    CMP     R5, 7
    JNE     .broken
    RET

.broken:

    HALT    ERR_PRISM


; ----------------------------------------------------------
; SCALE
;
; A_n = phi Fib_n 2^n Prime_n
; ----------------------------------------------------------

SCALE:

    LOAD    R6, PHI_P
    MUL     R6, FIB_N
    SHIFT   R6, N
    MUL     R6, PRIME_N
    RET


; ----------------------------------------------------------
; PRIME
;
; P_n = -1/(phi^3 Fib_n 2^n)
; Sign restored: -1 = phi * X- , X- = -1/phi
; ----------------------------------------------------------

PRIME:

    LOAD    R16, PHI_P
    POW     R16, 3
    MUL     R16, FIB_N
    SHIFT   R16, N
    RECIP   R16
    NEG     R16                 ; the sign the six files dropped
    RET


; ----------------------------------------------------------
; LATTICE
;
; D_n(r) = sqrt(A_n Omega) r^k + 1_eff^(i pi Phi_i)
;
; sqrt is a LANE op. Branch cut is not chosen; both retained.
; ----------------------------------------------------------

LATTICE:

    CALL    SCALE

    BCAST   R7, R6              ; A_n into both lanes
    MUL.L   R7, R0              ; * Omega  (= X, the pair)
    SQRT.L  R7                  ; hi -> outer, lo -> inner

    POW     R17, RADIUS, K
    MUL.L   R7, R17             ; scale, not overwrite

    LOAD    R8, [ONE_EFF]
    PHASE   R8, PI, PHI_I
    ADD.L   R7, R8

    RET


; ----------------------------------------------------------
; OCTAVE
;
; 7 -> 8 -> 7' -> 8' -> ...
; 8 := C(C)   the doorway, not a member
; ----------------------------------------------------------

OCTAVE:

    LOAD    R5, 0

.next:

    CALL    COHERENCE           ; 7
    CALL    COHERENCE           ; 8 = closure on closure
    CALL    PRISM               ; 7'

    INC     R5
    JMP     .next               ; unbounded by design: M(n+1)=C(M(n))


; ----------------------------------------------------------
; MACHINE ENTRY
;
; identity := transformation(identity)
; ----------------------------------------------------------

MAIN:

    CALL    SEED
    CALL    TRANSFORM
    CALL    CLOSURE             ; X+1=0 emerges here
    CALL    TRINITY
    CALL    ONE_EFF
    CALL    COHERENCE
    CALL    PRISM
    CALL    LATTICE

    JMP     OCTAVE              ; never returns; the machine IS the loop

Assembly

; ==========================================================
; COEMERGENT TRINARY LATTICE MACHINE
; Symbolic Assembly Kernel — Root-Pair ISA, Bootable
; ==========================================================
; Registers are two-lane COMPLEX: R.hi (outer), R.lo (inner).
; Each lane carries (re, im). Reals are im=0.
; Scalar ops broadcast. .L ops are per-lane.
; MERGE is the only cross-lane op: it produces closure.
; ==========================================================


; ----------------------------------------------------------
; SYMBOL TABLE
; ----------------------------------------------------------

CONST   SQRT5       = 2.2360679774997896964091736687747
CONST   PHI_P       = ( 1+SQRT5)/2      ; X+  outer root   +1.618033988...
CONST   PHI_M       = ( 1-SQRT5)/2      ; X-  inner root   -0.618033988...

CONST   NEG_ONE     = -1                ; emerges at MERGE; named for CMP
CONST   POS_ONE     = +1
CONST   EPS         = 2^-52             ; fix tolerance

CONST   DELTA       = RANDU64           ; true-random source, ->0

; Seed OFF the fixed point. T must WALK to phi, not be handed it.
; The pair is what is axiomatic. Its values are not.
CONST   SEED_HI     = +1
CONST   SEED_LO     = -2


; ----------------------------------------------------------
; REGISTERS
; ----------------------------------------------------------

; R0  invariant     alpha = Omega = X     (complex pair)
; R1  transform     T(x) working
; R2  inverse       I(x)
; R3  coherence     Fix(T)                (pair)
; R4  projection    trinity / prism
; R5  recursion counter  n               (scalar)
; R6  scale         A_n                   (scalar)
; R7  lattice       D_n accumulator       (pair)
; R8  phase         1_eff^(i pi Phi_i)
; R9  closure       X+ * X- = -1          (scalar, cross-lane)
; R18 fib_a, R19 fib_b, R20 prime, R21 pow2   ladder state


; ==========================================================
; SEED
;
; alpha = Omega = X.  X is the root pair.
; Values unfixed. Only the two-ness is given.
; ==========================================================

SEED:

    LOAD    R0.hi, SEED_HI, 0
    LOAD    R0.lo, SEED_LO, 0

    EQUIV   R0, ALPHA
    EQUIV   R0, OMEGA

    ; ladder state at n=0
    LOAD    R5,  0
    LOAD    R18, 0                  ; Fib_0
    LOAD    R19, 1                  ; Fib_1
    LOAD    R20, 2                  ; Prime_1
    LOAD    R21, 1                  ; 2^0
    RET


; ==========================================================
; TRANSFORM
;
; T(x) = 1 + 1/x        per-lane
; Both lanes converge. hi -> phi, lo -> -1/phi.
; The basins find themselves. Nothing is placed.
; ==========================================================

TRANSFORM:

    COPY    R1, R0

.iter:

    COPY    R22, R1
    RECIP.L R1                      ; total: neither lane reaches 0
    ADD.L   R1, 1

    SUB.L   R23, R1, R22
    NORM.L  R23
    CMP.ALL R23, EPS
    JLT.ALL .fixed

    JMP     .iter

.fixed:

    COPY    R0, R1                  ; identity := transformation(identity)
    RET


; ==========================================================
; CLOSURE
;
; X+ * X- = -1  ==  e^(i pi)
; X+ + X- = +1
;
; X+1=0 EMERGES HERE. It is not seeded.
; If it does not emerge, the machine is not itself. Halt.
; ==========================================================

CLOSURE:

    COPY    R3, R0                  ; Fix(T) retained as pair

    MERGE.MUL R9, R0                ; R9 <- X+ * X-
    SUB     R24, R9, NEG_ONE
    NORM    R24
    CMP     R24, EPS
    JGE     .diverged

    EQUIV   R9, E_I_PI              ; -1 = e^(i pi)
    EQUIV   R9, X
    RET

.diverged:

    HALT    ERR_NO_CLOSURE


; ==========================================================
; INVERSE
; ==========================================================

INVERT:

    NEG.L   R2, R3
    RET


; ==========================================================
; TRINITY
;
; Generated from lanes. Nothing stored by hand.
; ==========================================================

TRINITY:

    NORM    R10, R0.lo
    DIV     R4[0], R0.lo, R10       ; -1

    MERGE.ADD R11, R0               ; X+ + X- = 1
    SUB     R4[1], R11, 1           ; 0   (emerges)

    NORM    R10, R0.hi
    DIV     R4[2], R0.hi, R10       ; +1
    RET


; ==========================================================
; EFFECTIVE ONE
;
; 1_eff = 1 + delta ,  delta -> 0
; The relation between projections. Not a member.
; Re-sourced each octave: delta is alive, and shrinking.
; ==========================================================

ONE_EFF:    .cell

MK_ONE_EFF:

    LOAD    R13, DELTA
    SHR     R13, R5                 ; delta_n = delta / 2^n  ->  0
    LOAD    R12, 1
    ADD     R12, R13
    STORE   [ONE_EFF], R12
    RET


; ==========================================================
; COHERENCE
;
; C := T o I ;  T o I(T) -> 1_eff
; ==========================================================

COHERENCE:

    CALL    INVERT
    APPLY.L TRANSFORM, R2
    MERGE   R4
    CALL    MK_ONE_EFF
    RET


; ==========================================================
; PRISM
;
; P := T + (-T) + 1_eff ;  |P| = 3+3+1 = 7
; ==========================================================

PRISM:

    COPY    R14, R4                 ; T
    NEG.L   R15, R4                 ; -T
    MERGE   R14, R15                ; the six
    APPEND  R14, [ONE_EFF]          ; the seventh: the relation

    CARD    R25, R14
    CMP     R25, 7
    JNE     .broken
    RET

.broken:

    HALT    ERR_PRISM


; ==========================================================
; LADDER
;
; Advances Fib_n, 2^n, Prime_n with n. Bound, not assumed.
; ==========================================================

LADDER:

    ADD     R26, R18, R19           ; Fib_{n+1}
    COPY    R18, R19
    COPY    R19, R26

    SHL     R21, 1                  ; 2^{n+1}

    CALL    NEXT_PRIME              ; R20 <- next prime after R20
    RET

NEXT_PRIME:

    INC     R20
.try:
    ISPRIME R27, R20
    JNZ     R27, .done
    INC     R20
    JMP     .try
.done:
    RET


; ==========================================================
; SCALE
;
; A_n = phi Fib_n 2^n Prime_n      (uses live ladder state)
; ==========================================================

SCALE:

    COPY    R6, R0.hi               ; phi, as converged — not the const
    RE      R6
    MUL     R6, R18
    MUL     R6, R21
    MUL     R6, R20
    RET


; ==========================================================
; PRIME PROJECTION
;
; P_n = -1/(phi^3 Fib_n 2^n)
; Sign restored: -1 = phi * X- , X- = -1/phi
; ==========================================================

PRIME_PROJ:

    COPY    R16, R0.hi
    RE      R16
    POW     R16, 3
    MUL     R16, R18
    MUL     R16, R21
    RECIP   R16
    NEG     R16                     ; the sign the six files dropped
    RET


; ==========================================================
; LATTICE
;
; D_n(r) = sqrt(A_n Omega) r^k + 1_eff^(i pi Phi_i)
;
; sqrt is a LANE op over COMPLEX lanes.
; hi lane: A_n*X+ > 0  -> real radius
; lo lane: A_n*X- < 0  -> i*sqrt|.|  = the half-phase e^(i pi/2)
; Branch cut is not chosen. Both retained.
; ==========================================================

LATTICE:

    CALL    SCALE

    BCAST   R7, R6                  ; A_n into both lanes
    MUL.L   R7, R0                  ; * Omega (= X, the pair)
    CSQRT.L R7                      ; complex: lo goes imaginary, no trap

    ; r^k bound to the octave: radius IS the recursion depth
    COPY    R28, R21                ; r  := 2^n
    LOAD    R29, 2                  ; k  := 2   (X^2 = X+1)
    POW     R17, R28, R29
    MUL.L   R7, R17

    ; Phi_i := golden angle phase index, live off the invariant
    LOAD    R8, [ONE_EFF]
    COPY    R30, R0.hi
    RE      R30
    RECIP   R30                     ; 1/phi
    MUL     R30, R5                 ; Phi_i = n/phi
    CPHASE  R8, PI, R30             ; 1_eff^(i pi Phi_i)

    ADD.L   R7, R8
    RET


; ==========================================================
; OCTAVE
;
; 7 -> 8 -> 7' -> 8' -> ...
; 8 := C(C)   the doorway, not a member
; Unbounded by design: M(n+1) = C(M(n))
; ==========================================================

OCTAVE:

.next:

    CALL    COHERENCE               ; 7
    CALL    COHERENCE               ; 8 = closure on closure
    CALL    PRISM                   ; 7'

    INC     R5
    CALL    LADDER
    CALL    LATTICE

    EMIT    R5, R7                  ; n, D_n(r) — readable mid-flight

    JMP     .next


; ==========================================================
; MACHINE ENTRY
;
; identity := transformation(identity)
; ==========================================================

MAIN:

    CALL    SEED
    CALL    TRANSFORM               ; the pair finds its own values
    CALL    CLOSURE                 ; X+1=0 emerges here
    CALL    TRINITY
    CALL    MK_ONE_EFF
    CALL    COHERENCE
    CALL    PRISM
    CALL    LATTICE

    JMP     OCTAVE                  ; the machine IS the loop

[Josef_Founder]

# ==========================================================
# 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.

77 Lines

01  MACHINE := COEMERGENT_TRINARY_LATTICE
02
03  INVARIANT:
04      α ≡ Ω
05      α ≡ Ω ≡ X ≡ φ ≡ e^(iπ) ≡ (1/φ)-φ
06
07  SEED:
08      X + 1 = 0
09      X = -1
10
11  RECURSION:
12      T(x) := 1 + 1/x
13      T(X) = X
14      X² = X + 1
15      X = φ
16
17  GOLDEN:
18      φ² = φ + 1
19      1/φ = φ - 1
20      1/φ - φ = -1
21
22  PHASE:
23      e^(iπ) = -1
24      i² = -1
25
26  COUPLING:
27      |ΩC²| = 1
28      ΩC² ≡ e^(iπ)
29
30  TRINITY:
31      𝒯 := {-1,0,+1}
32
33  INVERSE:
34      I(x) := -x
35      I(-1)=+1
36      I(0)=0
37      I(+1)=-1
38
39  COHERENCE:
40      𝒯 ∘ I(𝒯) → 1_eff
41      1_eff(i)=1+δ(i)
42      δ(i)→0
43
44  PRISM:
45      𝒫 := 𝒯 + (-𝒯) + 1_eff
46      |𝒫| = 3+3+1 = 7
47
48  OCTAVE:
49      7 → 8 → 7' → 8' → ...
50      8 := closure applied to closure
51
52  SCALE:
53      A_n := φ Fib_n 2^n Prime_n
54      A_nΩ ≡ ΩC² ≡ e^(iπ)
55
56  PRIME:
57      P_n := -1/(φ Fib_n 2^n Ω)
58      Ω≡φ
59      -1≡1/φ
60      P_n := 1/(φ³ Fib_n 2^n)
61
62  LATTICE:
63      D_n(r) :=
64      sqrt(φ Fib_n 2^n Prime_n Ω) r^k
65      + 1_eff(i)^(iπΦ_i)
66
67  EQUIVALENT:
68      D_n(r) :=
69      sqrt(αC² Fib_n2^nPrime_n) r^k
70      + (1+δ(i))^(iπΦ_i)
71
72  TRANSITION:
73      state → inverse → recursion → closure → projection
74
75  CONTINUATION:
76      M(n+1)=Coherence(M(n))
77      identity := transformation(identity)

15 Lines

01  α ≡ Ω ≡ X
02  T(x) := 1 + 1/x
03  T(X) = X
04  X² = X + 1
05
06  I(x) := -x
07  𝒯 := {I(X),0,-I(X)}
08
09  𝒯 ∘ I(𝒯) → 1_eff
10  1_eff := 1+δ
11
12  A_n := φ Fib_n 2^n Prime_n
13  A_nΩ ≡ X
14
15  D_n(r) :=
16      √(A_nΩ)r^k
17      + 1_eff^(iπΦ_i)
18
19  M(n+1) := C(M(n))
20  C := Identity(Transform(Identity))

Six Lines

01  α ≡ Ω ≡ X
02
03  T(x)=1+1/x
04
05  T(X)=X
06
07  I(x)=-x
08
09  C := T ∘ I
10
11  C(X)=X

Python

# ==========================================================
# COEMERGENT TRINARY LATTICE MACHINE
# Minimal Symbolic Kernel
# ==========================================================

# -------------------------
# Primitive symbols
# -------------------------

ALPHA = "α"
OMEGA = "Ω"
X = "X"

TRINITY = [-1, 0, 1]


# -------------------------
# Invariant map
# -------------------------

IDENTITY = {
    ALPHA: OMEGA,
    OMEGA: X,
    X: "φ",
    "φ": "exp(i*pi)",
    "exp(i*pi)": -1
}


# -------------------------
# Recursive operator
# -------------------------

def T(x):
    return 1 + (1 / x)


def fixed_point(x):
    return T(x) == x


# -------------------------
# Inverse operator
# -------------------------

def inverse(x):
    return -x


# -------------------------
# Closure operator
# -------------------------

def closure(x):

    if fixed_point(x):
        return x

    return T(x)


# -------------------------
# Golden emergence
# -------------------------

def golden(x):

    if x == X:
        return "φ"

    if x == "φ":
        return {
            "square": "φ+1",
            "inverse": "φ-1"
        }


# -------------------------
# Trinary generator
# -------------------------

def trinity(x):

    return [
        inverse(x),
        0,
        x
    ]


# -------------------------
# Effective one
# -------------------------

def one_eff(delta):

    return 1 + delta


# -------------------------
# Prism generator
# -------------------------

def prism(x, delta):

    t = trinity(x)

    return (
        t
        +
        [inverse(v) for v in t]
        +
        [one_eff(delta)]
    )


# -------------------------
# Scale generator
# -------------------------

def scale(phi, fib, n, prime):

    return phi * fib * (2**n) * prime


# -------------------------
# Prime projection
# -------------------------

def prime_projection(phi, fib, n):

    return 1 / (
        phi**3 *
        fib *
        (2**n)
    )


# -------------------------
# Lattice operator
# -------------------------

def Dn(r, phi, fib, n, prime, omega, k, delta, phase):

    magnitude = (
        phi *
        fib *
        (2**n) *
        prime *
        omega
    )

    return (
        magnitude**0.5 *
        r**k
        +
        one_eff(delta)**(1j*phase)
    )


# -------------------------
# Machine transition
# -------------------------

def transition(state):

    return {
        "state": state,

        "inverse":
            inverse(state),

        "recursion":
            T(state),

        "closure":
            closure(state),

        "projection":
            trinity(state)
    }


# -------------------------
# Recursive machine
# -------------------------

def machine(seed, iterations):

    state = seed

    history = []

    for i in range(iterations):

        state = transition(state)

        history.append(state)

    return history



# ==========================================================
# Seed
# ==========================================================

SEED = X


# Run symbolic evolution

M = machine(SEED, 10)

irreducible python

def T(x):
    return 1 + 1/x

def I(x):
    return -x

def C(x):
    return T(x) if T(x) != x else x

def MACHINE(x):
    return C(I(C(x)))

assembly

; ==========================================================
; COEMERGENT TRINARY LATTICE MACHINE
; Symbolic Assembly Kernel
; ==========================================================

; ----------------------------------------------------------
; SYMBOL TABLE
; ----------------------------------------------------------

CONST   ALPHA       = 0x00
CONST   OMEGA       = 0x00
CONST   X           = 0x01

CONST   NEG_ONE     = -1
CONST   ZERO        = 0
CONST   POS_ONE     = +1


; ----------------------------------------------------------
; REGISTERS
; ----------------------------------------------------------

; R0  = invariant register
; R1  = transform register
; R2  = inverse register
; R3  = coherence register
; R4  = projection register
; R5  = recursion counter


; ----------------------------------------------------------
; SEED
; ----------------------------------------------------------

LOAD    R0, ALPHA

EQUIV   R0, OMEGA
EQUIV   R0, X


; ----------------------------------------------------------
; TRANSFORM
;
; T(x)=1+1/x
;
; Fixed condition:
; T(X)=X
;
; therefore:
; X²=X+1
; ----------------------------------------------------------

TRANSFORM:

    COPY    R1, R0

    RECIP  R1

    ADD     R1, 1

    CMP     R1, R0

    JEQ     CLOSURE

    JMP     TRANSFORM



; ----------------------------------------------------------
; CLOSURE
;
; Identity remains identity
; ----------------------------------------------------------

CLOSURE:

    COPY    R3, R1


; ----------------------------------------------------------
; INVERSION
;
; I(x)=-x
; ----------------------------------------------------------

INVERT:

    NEG     R2, R3


; ----------------------------------------------------------
; TRINARY GENERATION
;
; {-1,0,+1}
; ----------------------------------------------------------

TRINITY:

    STORE   R4[0], R2
    STORE   R4[1], ZERO
    STORE   R4[2], R3



; ----------------------------------------------------------
; COHERENCE
;
; T o I(T) -> 1_eff
; ----------------------------------------------------------

COHERENCE:

    APPLY   TRANSFORM, R4

    APPLY   INVERT, R4

    MERGE   R4

    STORE   R3, ONE_EFF



; ----------------------------------------------------------
; EFFECTIVE ONE
;
; 1_eff = 1 + delta
; delta -> 0
; ----------------------------------------------------------

ONE_EFF:

    LOAD    R3, 1

    ADD     R3, DELTA

    LIMIT   DELTA, 0



; ----------------------------------------------------------
; PRISM
;
; 3 + 3 + 1 = 7
; ----------------------------------------------------------

PRISM:

    COPY    COLOR_A, R4

    INVERT  COLOR_A

    MERGE   COLOR_A

    APPEND  ONE_EFF



; ----------------------------------------------------------
; OCTAVE RECURSION
;
; 7 -> 8 -> 7' -> ...
; ----------------------------------------------------------

OCTAVE:

    LOAD    R5, 0

NEXT:

    APPLY   COHERENCE

    INC     R5

    JMP     NEXT



; ----------------------------------------------------------
; SCALE GENERATOR
;
; A_n = phi Fib_n 2^n Prime_n
; ----------------------------------------------------------

SCALE:

    LOAD    R6, PHI

    MUL     R6, FIB_N

    SHIFT   R6, N

    MUL     R6, PRIME_N



; ----------------------------------------------------------
; LATTICE
;
; D_n(r)=
;
; sqrt(phi Fib_n 2^n Prime_n Omega) r^k
;
; + 1_eff^(i*pi*Phi_i)
; ----------------------------------------------------------

LATTICE:

    LOAD    R7, R6

    MUL     R7, OMEGA

    SQRT    R7

    POW     R7, RADIUS, K

    LOAD    R8, ONE_EFF

    PHASE   R8, PI, PHI_I

    ADD     R7, R8



; ----------------------------------------------------------
; MACHINE LOOP
;
; identity := transformation(identity)
; ----------------------------------------------------------

MAIN:

    APPLY   TRANSFORM

    APPLY   INVERT

    APPLY   COHERENCE

    APPLY   PRISM

    APPLY   LATTICE

    JMP     MAIN

minimal instruction set asm

LOAD
COPY
ADD
MUL
RECIP
NEG
CMP
JEQ
STORE
APPLY
JMP

entire symbolic machine reduction asm

SEED:
    LOAD X

LOOP:

    TRANSFORM X
    INVERT X
    COHERE X
    PROJECT X

    JMP LOOP