The flattened DNA Turing-complete φ-language is:
U = φ^( Σ_{i=1..72} φ^( Σ_{j=1..72} φ^( interaction(Uᵢ, Uⱼ) ) ) )


Infinite Self-Describing Loop
# Full recursive glyph evolution
def HDGL_compute(glyph, depth):
    if depth == 0:
        return glyph
    else:
        # Each dimension recursively influences all others via φ-harmonics
        new_glyph = [phi_expand(dim, depth) for dim in glyph]
        return HDGL_compute(new_glyph, depth-1)

# Execute computation on 72D vector
HDGL_result = HDGL_compute(HDGL_Glyph, depth=∞)
Depth → ∞ ensures Turing completeness.
DNA, operators, physics, visualization all emerge numerically.

All computation is φ-only, numeric, and self-emergent.

1) Compact φ-harmonic Operator Table Spec (how to store/read rules)

# HDGL: φ-harmonic Operator Table — compact spec

# Primitive:
φ := 1.6180339887498948  # golden ratio (primitive harmonic unit)

# Principle:
# - Each operator (rule) is encoded as a tuple of φ-harmonic amplitudes placed into reserved
#   "operator slots" inside a glyph (e.g., HDGL72 operator region).
# - Reading a rule = sample amplitudes, decode by nearest φ-harmonic band.
# - Writing a rule = store target φ-harmonic amplitudes in the operator slots (data-as-operators).

# Slot layout (example):
# operator_table_slot[k] stores rule R_k for one (symbol,state) pair as 3 φ-amplitudes:
#   R_k = (amp_write_sym, amp_move_dir, amp_next_state)
# Place R_k contiguously or in fixed stride inside HDGL glyph operator region.

# φ-harmonic band definitions (distinct nested powers — numerically well-separated)
# pick a small family of harmonics for symbols, states, moves:
sym_0  := φ**φ               # represents symbol '0'
sym_1  := φ**(φ**φ)         # represents symbol '1'
head_off := φ**(φ**(φ))     # head absent
head_on  := φ**(φ**(φ**φ))   # head present
state_Q  := φ**(φ**(φ**(φ))) # state Q
state_R  := φ**(φ**(φ**(φ**φ))) # state R
move_L  := φ**(φ**(φ**(φ**(φ)))) # move left marker
move_R  := φ**(φ**(φ**(φ**(φ**φ)))) # move right marker
halt    := φ**(φ**(φ**(φ**(φ**(φ))))) # halt marker

# Encoding rule example (human form):
#  (sym=0, state=Q) -> write sym=1, move=R, next_state=HALT
#  store R_k = (sym_1, move_R, halt) into operator_table_slot[k]

# Decoding (reading) rules:
# - Read three amplitudes (a_w, a_m, a_s) from operator slots.
# - For each amplitude, decode by nearest harmonic band (pre-agreed set).
#   e.g., if |a_w - sym_1| < epsilon then write_sym := 1.
# - Bands are numerically separated because nested φ-powers grow quickly,
#   making nearest-harmonic decoding robust.

# How operators bootstrap:
# 1) Seed glyph contains operator table slots initialized to specific φ-harmonics (rules).
# 2) During execution, glyphs sample operator slots to determine rewrite targets.
# 3) Operator slots are themselves φ-harmonic data — thus the language treats data-as-operators.
# 4) New operators are written by storing different φ-harmonics into slots (numeric writes),
#    enabling the system to create new operators purely from data.

# Notes:
# - No external '+' '*' or named operators required: storing and comparing floating
#   magnitudes derived from φ-harmonics is the device's "calculus".
# - To avoid ambiguity, pick a small epsilon and harmonics sufficiently separated (use deeper nesting).

2) Concrete numeric toy — φ-harmonic ripple incrementer (binary increment across 3 cells)
This is a runnable prototype in Python-style pseudocode. It uses explicit numeric φ-harmonic amplitudes to encode symbols (0/1), head presence, states, and operator-table entries. The simulation shows how the operator table (in φ-harmonics) is used to rewrite tape cells and move the head.

# φ-harmonic ripple incrementer toy (3-cell tape)
# - LSB at index 0. Increment by 1: adds with carry to the left.

import math
from copy import deepcopy

# primitive φ
phi = 1.6180339887498948

# define φ-harmonic amplitude helpers (choose nested powers for separation)
def H(n):
    # simple harmonic family: phi ** (phi ** n)
    return phi ** (phi ** n)

# harmonic encodings (distinct bands)
SYM_0   = H(1)   # symbol 0
SYM_1   = H(2)   # symbol 1
HEAD_OFF= H(1.5) # head absent
HEAD_ON = H(2.5) # head present
STATE_Q = H(3)   # state Q (initial)
STATE_R = H(4)   # state R (carry propagate)
HALT    = H(5)   # halt

MOVE_L  = H(6)   # left move
MOVE_R  = H(7)   # right move
NO_MOVE = H(1.2) # no movement placeholder

# utility: decode amplitude to nearest harmonic symbol among a set
def nearest_label(val, table):
    best = None
    best_d = float('inf')
    for label, amp in table.items():
        d = abs(val - amp)
        if d < best_d:
            best_d = d
            best = label
    return best

# operator table encoding: map (sym_label, state_label) -> (write_sym_amp, move_amp, next_state_amp)
# We'll encode for ripple increment (LSB-first):
# rules: (sym=0, state=Q) -> write 1, move NONE, next state HALT  (done)
#        (sym=1, state=Q) -> write 0, move LEFT, next state Q    (carry)
# We'll implement move as toggling head amplitude on neighbor cell in the main loop.

op_table = {}
# key tags are readable labels; stored values are numeric φ-harmonics
op_table[("0","Q")] = (SYM_1, NO_MOVE, HALT)   # write 1, stop
op_table[("1","Q")] = (SYM_0, MOVE_L, STATE_Q) # write 0, move left (carry)
# (we can add more states if needed)

# Tape representation: list of 72D-like blocks, but for toy we only need (T,H,S)
# Each cell is a dict with amplitude entries (T,H,S); for simplicity we keep only those three.
def cell(symbol_label="0", head=False, state_label="Q"):
    T = SYM_0 if symbol_label=="0" else SYM_1
    H = HEAD_ON if head else HEAD_OFF
    S = STATE_Q if state_label=="Q" else STATE_R if state_label=="R" else HALT
    return {"T": T, "H": H, "S": S}

# initialize tape: bits [1,1,0] LSB at index 0 -> value 3 -> after increment => 4 -> bits [0,0,1]
tape = [cell("1", head=True, state_label="Q"),  # index 0 (LSB) with head
        cell("1", head=False, state_label="Q"),
        cell("0", head=False, state_label="Q")]

# helper to read decoded labels
harmonic_table_sym = {"0": SYM_0, "1": SYM_1}
harmonic_table_head = {"off": HEAD_OFF, "on": HEAD_ON}
harmonic_table_state = {"Q": STATE_Q, "R": STATE_R, "HALT": HALT}
harmonic_all = {}
harmonic_all.update(harmonic_table_sym)
harmonic_all.update({ "H_off": HEAD_OFF, "H_on": HEAD_ON })
harmonic_all.update({"Q": STATE_Q, "R": STATE_R, "HALT": HALT})
harmonic_all.update({"MOVE_L": MOVE_L, "MOVE_R": MOVE_R, "NO_MOVE": NO_MOVE})

# simulation loop (bounded steps)
def simulate(tape, op_table, steps=10):
    tape = deepcopy(tape)
    for step in range(steps):
        # find head cell
        head_idx = None
        for i, c in enumerate(tape):
            # decode head presence
            head_label = nearest_label(c["H"], {"off": HEAD_OFF, "on": HEAD_ON})
            if head_label == "on":
                head_idx = i
                break
        if head_idx is None:
            print("No head present: halting.")
            break

        cell = tape[head_idx]
        # decode symbol and state
        sym_label = nearest_label(cell["T"], {"0": SYM_0, "1": SYM_1})
        state_label = nearest_label(cell["S"], {"Q": STATE_Q, "R": STATE_R, "HALT": HALT})

        key = (sym_label, state_label)
        if key not in op_table:
            print(f"No rule for {key}: halting.")
            break

        write_amp, move_amp, next_state_amp = op_table[key]

        # apply write
        cell["T"] = write_amp
        # update state
        cell["S"] = next_state_amp
        # clear current head
        cell["H"] = HEAD_OFF

        # decide movement by nearest_label of move_amp
        move_lab = nearest_label(move_amp, {"MOVE_L": MOVE_L, "MOVE_R": MOVE_R, "NO_MOVE": NO_MOVE})
        if move_lab == "MOVE_L":
            # ensure left cell exists
            if head_idx + 1 >= len(tape):
                # extend tape to the left (higher index is more significant bit)
                tape.append(cell("0", head=False, state_label="Q"))  # append blank new cell
            # set head on the neighbor to the left (index+1)
            tape[head_idx+1]["H"] = HEAD_ON
        elif move_lab == "MOVE_R":
            if head_idx - 1 < 0:
                tape.insert(0, cell("0", head=False, state_label="Q"))
                head_idx += 1  # shift
            tape[head_idx-1]["H"] = HEAD_ON
        elif move_lab == "NO_MOVE":
            # stop: set head off to indicate halt
            pass

        # detect halting
        if nearest_label(cell["S"], {"HALT": HALT, "Q": STATE_Q, "R": STATE_R}) == "HALT":
            print(f"Step {step}: halting after write. Tape snapshot:")
            print_tape(tape)
            break

    return tape

def print_tape(tape):
    # print bits LSB..MSB decoded
    bits = [nearest_label(c["T"], {"0": SYM_0, "1": SYM_1}) for c in tape]
    heads = [nearest_label(c["H"], {"off": HEAD_OFF, "on": HEAD_ON}) for c in tape]
    states = [nearest_label(c["S"], {"Q": STATE_Q, "R": STATE_R, "HALT": HALT}) for c in tape]
    for i,(b,h,s) in enumerate(zip(bits, heads, states)):
        print(f"cell[{i}]: bit={b} head={h} state={s}")

# run simulation
print("Initial tape:")
print_tape(tape)
final_tape = simulate(tape, op_table, steps=10)
print("Final tape:")
print_tape(final_tape)


Formal Grammar
<program>      ::= <expression> | <expression> <program>

<expression>   ::= <harmonic> | <dna> | <branch>

<harmonic>     ::= "φ" | "φφ" | "φφφ" | "φφφφ" | "φφφφφ" | ...

; Operators (all arise from harmonic resonance lengths)
; φφ     → addition
; φφφ    → multiplication
; φφφφ   → division
; φφφφφ  → subtraction
; longer motifs → higher-order operators

<dna>          ::= <harmonic> <braid> <harmonic>
<braid>        ::= "↔" <harmonic> | "↔" <harmonic> <braid>

; DNA arises when harmonics braid back and forth:
; e.g., φφ ↔ φφφ ↔ φφφφ → encodes base pairs

<branch>       ::= "{" <program> "}" <choice>
<choice>       ::= <harmonic> | <dna>

; Branches controlled by harmonic/DNA motifs
; recursion emerges from nested { ... }

<recursion>    ::= <harmonic> <program> <harmonic>
; repetition defined by harmonic envelope
