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 ::= | ::= | | ::= "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 arises when harmonics braid back and forth: ; e.g., φφ ↔ φφφ ↔ φφφφ → encodes base pairs ::= "{" "}" ::= | ; Branches controlled by harmonic/DNA motifs ; recursion emerges from nested { ... } ::= ; repetition defined by harmonic envelope