""" FRAMERATE FROM FIRST PRINCIPLES — GEOMETRIC COMPRESSION MODEL ================================================================================ Date: 2026-04-03 Author: Jonathan Shelton (theory), Claude (computation) Status: EXPLORATORY CALCULATION — v2 INSIGHT: Framerate is NOT proportional to raw mode count. It is proportional to COMPRESSED mode count — information per frame AFTER {2,3} geometric compression. The GPU analogy: f = raw data (frequency pulse) {2,3} interference = geometry pass (compression) C_potential = frame buffer (finite capacity) framerate = throughput of compressed frames The density of states gives RAW mode count. The symmetry group gives COMPRESSION RATIO. The framerate is: c_D ∝ M_raw / S_D Where S_D is the symmetry order at dimension D. QUESTION: Do {2,3} symmetry groups at each dimension produce Fibonacci framerate ratios when applied as compression factors? Target: c₂/c₁ = 2.500 c₃/c₂ = 1.600 c₄/c₃ = 1.625 ================================================================================ """ import numpy as np from fractions import Fraction phi = (1 + np.sqrt(5)) / 2 phi_c = 0.2895 # percolation threshold print("=" * 70) print("FRAMERATE FROM GEOMETRIC COMPRESSION") print("=" * 70) # ============================================================================ # KNOWN SYMMETRY GROUPS BUILT FROM {2,3} # ============================================================================ # # The symmetry operations at each dimensional level are products of {2,3}: # # 1D: Only reflection + identity = 2 = {2} # (mirror a line — 2 operations) # # 2D: Hexagonal lattice symmetry = p6m = order 12 = 2²×3 # (the wallpaper group of the hexagonal ground state) # Subgroup: p3m1 = order 6 = 2×3 # Subgroup: p2 = order 2 = {2} # # 3D: Cubic (Oh) = order 48 = 2⁴×3 # This covers FCC and BCC Wigner-Seitz cells. # Hexagonal (D6h) = order 24 = 2³×3 # Tetrahedral (Td) = order 24 = 2³×3 # # 4D: 24-cell symmetry = order 1152 = 2⁷×3² # This is the ONLY regular self-dual 4D polytope. # Tesseract/hyperoctahedron = order 384 = 2⁷×3 # # All symmetry orders are PURE {2,3} products. No other primes. # ============================================================================ print("\n--- SYMMETRY GROUPS (compression factors) ---") print() # Symmetry orders at each dimension sym_orders = { '1D': { 'trivial (identity + reflection)': 2, # 2^1 'note': '2 = {2}¹' }, '2D': { 'p2 (oblique)': 2, # 2 'p3m1 (triangular)': 6, # 2×3 'p6m (hexagonal)': 12, # 2²×3 'note': 'ground state is hexagonal = 12 = 2²×3' }, '3D': { 'C2 (monoclinic)': 4, # 2² 'D3d (trigonal)': 12, # 2²×3 'D6h (hexagonal)': 24, # 2³×3 'Oh (cubic)': 48, # 2⁴×3 'note': 'most common lattice sym = Oh = 48 = 2⁴×3' }, '4D': { 'tesseract': 384, # 2⁷×3 '24-cell': 1152, # 2⁷×3² 'note': '24-cell = 1152 = 2⁷×3²' } } for dim, groups in sym_orders.items(): print(f" {dim}:") for name, order in groups.items(): if name != 'note': # Factor into 2s and 3s n = order twos = 0 threes = 0 while n % 2 == 0: twos += 1 n //= 2 while n % 3 == 0: threes += 1 n //= 3 remainder = n print(f" {name:30s} = {order:6d} = 2^{twos} × 3^{threes}" + (f" × {remainder}" if remainder > 1 else "")) print(f" >> {groups['note']}") print() # ============================================================================ # MODEL A: FRAMERATE = RAW MODES / SYMMETRY ORDER # ============================================================================ # Use the ground-state symmetry at each dimension as the compression factor. # # The cascade: f_floor(D+1) = f_max(D) # Raw mode count: M_D = (f_max^D - f_floor^D) / D # Compression: S_D = symmetry order at dimension D # Framerate: c_D ∝ M_D / S_D (compressed information per frame) # ============================================================================ print("=" * 70) print("MODEL A: c_D ∝ M_raw(D) / S_D") print("=" * 70) # Ground state symmetry orders S = {1: 2, 2: 12, 3: 48, 4: 1152, 5: None, 6: None} # We don't know 5D and 6D symmetry orders. Let's see if we can # predict them from the pattern. print("\nSymmetry order pattern:") print(f" S(1) = 2 = 2^1 × 3^0") print(f" S(2) = 12 = 2^2 × 3^1") print(f" S(3) = 48 = 2^4 × 3^1") print(f" S(4) = 1152 = 2^7 × 3^2") print() # Powers of 2: 1, 2, 4, 7 → differences: 1, 2, 3 (linear growth!) # Powers of 3: 0, 1, 1, 2 → differences: 1, 0, 1 (Fibonacci-like?) print(" Powers of 2 in S(D): 1, 2, 4, 7") print(" Differences: 1, 2, 3 → next = 4 → power = 11") print(" Powers of 3 in S(D): 0, 1, 1, 2") print(" This follows Fibonacci! F(1)=0, F(2)=1, F(3)=1, F(4)=2, F(5)=3") print() # Predict S(5) and S(6) # Power of 2: 7 + 4 = 11 # Power of 3: F(5) = 3 S_5_pred = 2**11 * 3**3 print(f" PREDICTED S(5) = 2^11 × 3^3 = {S_5_pred}") # Power of 2: 11 + 5 = 16 # Power of 3: F(6) = 5 S_6_pred = 2**16 * 3**5 print(f" PREDICTED S(6) = 2^16 × 3^5 = {S_6_pred}") S[5] = S_5_pred S[6] = S_6_pred def model_A(alpha, S_dict, n_dims=6): """ Framerate = raw_modes / symmetry_order (compression) """ f_floor = np.zeros(n_dims + 1) f_max = np.zeros(n_dims + 1) M_raw = np.zeros(n_dims + 1) c = np.zeros(n_dims + 1) f_floor[1] = 1.0 for D in range(1, n_dims + 1): f_max[D] = alpha * f_floor[D] M_raw[D] = (f_max[D]**D - f_floor[D]**D) / D c[D] = M_raw[D] / S_dict[D] if D < n_dims: f_floor[D + 1] = f_max[D] # Normalize to c₃ = 1 if c[3] > 0: c = c / c[3] return c, M_raw print(f"\nScanning expansion ratio α:") print(f"\n{'α':>8} | {'c₁':>10} | {'c₂':>10} | {'c₃':>10} | {'c₄':>10} | {'c₂/c₁':>8} | {'c₃/c₂':>8} | {'c₄/c₃':>8}") print("-" * 95) best_alpha_A = None best_error_A = float('inf') for alpha in np.arange(1.1, 20.0, 0.001): c, _ = model_A(alpha, S) if c[1] > 0 and c[2] > 0 and c[3] > 0: r21 = c[2] / c[1] r32 = c[3] / c[2] r43 = c[4] / c[3] error = abs(r21 - 2.5) + abs(r32 - 1.6) + abs(r43 - 1.625) if error < best_error_A: best_error_A = error best_alpha_A = alpha # Print results around best print(f"\nBest α = {best_alpha_A:.4f}") c_A, M_A = model_A(best_alpha_A, S) for D in range(1, 6): print(f" D={D}: c={c_A[D]:.6f}, M_raw={M_A[D]:.2e}, S={S[D]}") print() r21 = c_A[2] / c_A[1] if c_A[1] > 0 else 0 r32 = c_A[3] / c_A[2] if c_A[2] > 0 else 0 r43 = c_A[4] / c_A[3] if c_A[3] > 0 else 0 r54 = c_A[5] / c_A[4] if c_A[4] > 0 else 0 print(f" c₂/c₁ = {r21:.6f} (target 2.500)") print(f" c₃/c₂ = {r32:.6f} (target 1.600)") print(f" c₄/c₃ = {r43:.6f} (target 1.625)") print(f" c₅/c₄ = {r54:.6f} (target 1.615)") print(f" Total error = {best_error_A:.6f}") # Check α relationships print(f"\n α = {best_alpha_A:.6f}") print(f" α / φ = {best_alpha_A / phi:.6f}") print(f" α / π = {best_alpha_A / np.pi:.6f}") print(f" α / e = {best_alpha_A / np.e:.6f}") print(f" ln(α) = {np.log(best_alpha_A):.6f}") print(f" α² = {best_alpha_A**2:.6f}") # ============================================================================ # MODEL B: SYMMETRY RATIO = COMPRESSION SCHEDULE # ============================================================================ # Instead of using raw symmetry orders, use the RATIO of successive # symmetry orders as the compression gain per dimension. # # Compression_gain(D) = S(D) / S(D-1) # 1→2: 12/2 = 6 # 2→3: 48/12 = 4 # 3→4: 1152/48 = 24 # # The gain represents how much MORE compression each new dimension adds. # The framerate should scale with the inverse of compression gain # (more compression = more info per frame = faster). # ============================================================================ print("\n" + "=" * 70) print("MODEL B: COMPRESSION GAIN SCHEDULE") print("=" * 70) gains = {} for D in range(2, 6): gains[D] = S[D] / S[D-1] print(f" S({D})/S({D-1}) = {S[D]}/{S[D-1]} = {gains[D]:.1f}") # Factor the gains print("\n Compression gain factored:") for D in range(2, 6): g = int(gains[D]) n = g twos = threes = 0 while n % 2 == 0: twos += 1; n //= 2 while n % 3 == 0: threes += 1; n //= 3 print(f" D={D}: gain = {g} = 2^{twos} × 3^{threes}" + (f" × {n}" if n > 1 else "")) # ============================================================================ # MODEL C: INFORMATION-BOUNDED CASCADE # ============================================================================ # Key insight: the framerate is NOT raw_modes / symmetry. # It is: how much UNIQUE information per frame. # # In a crystal with symmetry order S: # Unique info = total_positions / S # (because S-fold symmetry means S positions are copies) # # But the MODE COUNT itself already reflects the dimensionality. # The symmetry group reduces the independent degrees of freedom. # # Total DOF in D dimensions with N modes: N # Independent DOF after symmetry: N / S(D) # # Information per frame = independent DOF = M_D / S(D) # Framerate = Nyquist for the independent DOF = 2 × M_D / S(D) # # BUT: the cascade compounds. The floor of D+1 = ceiling of D. # And S(D) compresses differently at each level. # # Let's reformulate: what if the COMPRESSION itself determines # the expansion ratio α? Each dimension can only expand as much # as its compression allows. # # If the frame buffer is fixed size (C_potential depth = constant # across dimensions), then: # α(D)^D / S(D) = constant (same info per frame at every level) # # This gives: α(D) = (constant × S(D))^(1/D) # ============================================================================ print("\n" + "=" * 70) print("MODEL C: CONSTANT INFORMATION PER FRAME") print("=" * 70) print(""" Hypothesis: each dimension processes the SAME amount of unique information per frame. The frame buffer is the same size. What changes is the COMPRESSION. Higher symmetry = more raw data compressed into the same frame. If info_per_frame = M_raw(D) / S(D) = constant K for all D: (α^D - 1) / (D × S(D)) = K With the cascade, the raw modes compound: M_raw(D) = f_floor(D)^D × (α(D)^D - 1) / D Setting unique_info = constant across dimensions: f_floor(D)^D × (α(D)^D - 1) / (D × S(D)) = K """) # What expansion ratio α_D does each dimension need to produce # the SAME unique information, given its symmetry group? # # Setting K = 1 (normalize), f_floor(1) = 1: # For D=1: (α₁ - 1) / (1 × 2) = 1 → α₁ = 3 # For D=2: f_floor(2) = α₁ = 3 # 3² × (α₂² - 1) / (2 × 12) = 1 # 9(α₂² - 1) / 24 = 1 # α₂² - 1 = 24/9 = 8/3 # α₂² = 1 + 8/3 = 11/3 # α₂ = √(11/3) = 1.9149 print("Solving for dimension-specific α with constant info per frame:\n") K = 1.0 # normalized f_floor = [0, 1.0, 0, 0, 0, 0, 0] # index by dimension alpha_D = [0, 0, 0, 0, 0, 0, 0] f_max_D = [0, 0, 0, 0, 0, 0, 0] M_raw_D = [0, 0, 0, 0, 0, 0, 0] M_unique_D = [0, 0, 0, 0, 0, 0, 0] c_D = [0, 0, 0, 0, 0, 0, 0] for D in range(1, 7): if D == 1: f_floor[D] = 1.0 else: f_floor[D] = f_max_D[D-1] # Solve: f_floor^D × (α^D - 1) / (D × S[D]) = K # α^D - 1 = K × D × S[D] / f_floor^D # α^D = 1 + K × D × S[D] / f_floor^D # α = (1 + K × D × S[D] / f_floor^D)^(1/D) rhs = K * D * S[D] / (f_floor[D]**D) alpha_D[D] = (1 + rhs) ** (1.0/D) f_max_D[D] = alpha_D[D] * f_floor[D] M_raw_D[D] = f_floor[D]**D * (alpha_D[D]**D - 1) / D M_unique_D[D] = M_raw_D[D] / S[D] c_D[D] = 2 * M_unique_D[D] # Nyquist print(f" D={D}: f_floor={f_floor[D]:.4f}, α={alpha_D[D]:.6f}, " f"f_max={f_max_D[D]:.4f}, M_raw={M_raw_D[D]:.4f}, " f"S={S[D]}, M_unique={M_unique_D[D]:.4f}") # Normalize framerates to c₃ = 1 c_norm = [x / c_D[3] if c_D[3] > 0 else 0 for x in c_D] print(f"\nFramerates (normalized to c₃ = 1):") for D in range(1, 7): print(f" c_{D} = {c_norm[D]:.6f}") print(f"\nRatios:") for D in range(1, 6): if c_norm[D] > 0: ratio = c_norm[D+1] / c_norm[D] print(f" c_{D+1}/c_{D} = {ratio:.6f}", end="") target = {1: 2.500, 2: 1.600, 3: 1.625, 4: 1.615, 5: 1.619} if D in target: print(f" (target {target[D]:.3f}, error {abs(ratio - target[D]):.4f})") else: print() # ============================================================================ # MODEL D: SCAN K (info per frame) TO FIND FIBONACCI # ============================================================================ print("\n" + "=" * 70) print("MODEL D: SCANNING K (info capacity per frame)") print("=" * 70) print("Does any value of K produce Fibonacci ratios?\n") best_K = None best_err = float('inf') for K_test in np.arange(0.001, 10.0, 0.001): fl = [0, 1.0, 0, 0, 0, 0, 0] fm = [0, 0, 0, 0, 0, 0, 0] al = [0, 0, 0, 0, 0, 0, 0] cr = [0, 0, 0, 0, 0, 0, 0] valid = True for D in range(1, 6): if D > 1: fl[D] = fm[D-1] rhs = K_test * D * S[D] / (fl[D]**D) if fl[D] > 0 else 0 if rhs <= 0: valid = False break al[D] = (1 + rhs) ** (1.0/D) fm[D] = al[D] * fl[D] M = fl[D]**D * (al[D]**D - 1) / D cr[D] = 2 * M / S[D] if not valid or cr[3] <= 0: continue cn = [x / cr[3] for x in cr] if cn[1] > 0 and cn[2] > 0: r21 = cn[2] / cn[1] r32 = cn[3] / cn[2] r43 = cn[4] / cn[3] if cn[3] > 0 else 0 err = abs(r21 - 2.5) + abs(r32 - 1.6) + abs(r43 - 1.625) if err < best_err: best_err = err best_K = K_test print(f"Best K = {best_K:.4f}, error = {best_err:.6f}") # Recompute at best K fl = [0, 1.0, 0, 0, 0, 0, 0] fm = [0, 0, 0, 0, 0, 0, 0] al = [0, 0, 0, 0, 0, 0, 0] cr = [0, 0, 0, 0, 0, 0, 0] for D in range(1, 6): if D > 1: fl[D] = fm[D-1] rhs = best_K * D * S[D] / (fl[D]**D) al[D] = (1 + rhs) ** (1.0/D) fm[D] = al[D] * fl[D] M = fl[D]**D * (al[D]**D - 1) / D cr[D] = 2 * M / S[D] cn = [x / cr[3] for x in cr] print(f"\n c₁ = {cn[1]:.6f} (target 0.250)") print(f" c₂ = {cn[2]:.6f} (target 0.625)") print(f" c₃ = {cn[3]:.6f} (target 1.000)") print(f" c₄ = {cn[4]:.6f} (target 1.625)") r21 = cn[2] / cn[1] if cn[1] > 0 else 0 r32 = cn[3] / cn[2] if cn[2] > 0 else 0 r43 = cn[4] / cn[3] if cn[3] > 0 else 0 print(f"\n c₂/c₁ = {r21:.6f} (target 2.500)") print(f" c₃/c₂ = {r32:.6f} (target 1.600)") print(f" c₄/c₃ = {r43:.6f} (target 1.625)") # Check K relationships print(f"\n K = {best_K:.6f}") print(f" K / φ = {best_K / phi:.6f}") print(f" K × φ = {best_K * phi:.6f}") print(f" K / π = {best_K / np.pi:.6f}") print(f" K × π = {best_K * np.pi:.6f}") print(f" K / φ_c = {best_K / phi_c:.6f}") # ============================================================================ # MODEL E: PERCOLATION + COMPRESSION COMBINED # ============================================================================ print("\n" + "=" * 70) print("MODEL E: PERCOLATION × COMPRESSION") print("=" * 70) print(""" The bubble (percolation fraction φ_c) partitions the mode space. The compression (symmetry S_D) reduces the unique information. Both act on the raw mode count: c_D ∝ (1 - φ_c) × M_raw(D) / S(D) The (1 - φ_c) factor removes the bubble. The S(D) compresses. """) # With percolation partition for K_test in np.arange(0.001, 10.0, 0.001): fl = [0, 1.0, 0, 0, 0, 0, 0] fm = [0, 0, 0, 0, 0, 0, 0] al = [0, 0, 0, 0, 0, 0, 0] cr = [0, 0, 0, 0, 0, 0, 0] valid = True for D in range(1, 6): if D > 1: fl[D] = fm[D-1] rhs = K_test * D * S[D] / ((1 - phi_c) * fl[D]**D) if fl[D] > 0 else 0 if rhs <= 0: valid = False break al[D] = (1 + rhs) ** (1.0/D) fm[D] = al[D] * fl[D] M = fl[D]**D * (al[D]**D - 1) / D cr[D] = 2 * (1 - phi_c) * M / S[D] if not valid or cr[3] <= 0: continue cn = [x / cr[3] for x in cr] if cn[1] > 0 and cn[2] > 0: r21 = cn[2] / cn[1] r32 = cn[3] / cn[2] r43 = cn[4] / cn[3] if cn[3] > 0 else 0 err = abs(r21 - 2.5) + abs(r32 - 1.6) + abs(r43 - 1.625) if err < best_err: best_err = err best_K = K_test print(f" NEW BEST: K={K_test:.4f}, err={err:.6f}, " f"ratios: {r21:.4f}, {r32:.4f}, {r43:.4f}") # ============================================================================ # DIRECT CHECK: What compression schedule WOULD produce Fibonacci? # ============================================================================ print("\n" + "=" * 70) print("REVERSE ENGINEERING: What S(D) produces Fibonacci?") print("=" * 70) print(""" If framerate ratios MUST be Fibonacci (2.5, 1.6, 1.625...), and modes follow density-of-states cascade, what compression factor S(D) is required? c_D ∝ M_D / S(D) c_{D+1}/c_D = [M_{D+1}/S(D+1)] / [M_D/S(D)] = [M_{D+1}/M_D] × [S(D)/S(D+1)] So: S(D+1)/S(D) = [M_{D+1}/M_D] / [c_{D+1}/c_D] The compression GAIN = raw mode growth / framerate growth """) # For a specific α, compute what S ratios would be needed for alpha in [2.0, 3.0, phi, np.e, np.pi]: print(f"\n α = {alpha:.4f}:") fl = [0, 1.0, 0, 0, 0, 0] fm = [0, 0, 0, 0, 0, 0] M = [0, 0, 0, 0, 0, 0] for D in range(1, 6): if D > 1: fl[D] = fm[D-1] fm[D] = alpha * fl[D] M[D] = fl[D]**D * (alpha**D - 1) / D # Target ratios targets = {2: 2.500, 3: 1.600, 4: 1.625, 5: 1.615} # Required S ratios S_required = [0, 1.0, 0, 0, 0, 0] # S(1) = 1 (normalize) for D in range(1, 5): M_ratio = M[D+1] / M[D] if M[D] > 0 else 0 c_ratio = targets.get(D+1, phi) # use phi for unknowns # c_{D+1}/c_D = (M_{D+1}/M_D) × (S_D/S_{D+1}) # S_{D+1}/S_D = M_ratio / c_ratio S_ratio = M_ratio / c_ratio if c_ratio > 0 else 0 S_required[D+1] = S_required[D] * S_ratio print(f" D={D}→{D+1}: M_ratio={M_ratio:.4f}, " f"c_ratio={c_ratio:.3f}, " f"needed S_ratio={S_ratio:.4f}, " f"S({D+1})={S_required[D+1]:.2f}") # Compare to actual print(f" Actual S: {S[1]}, {S[2]}, {S[3]}, {S[4]}") print(f" Needed S: ", end="") # Rescale needed to match S(1)=2 scale = 2.0 / S_required[1] if S_required[1] > 0 else 1 for D in range(1, 6): print(f"{S_required[D]*scale:.1f}", end=" ") print() # ============================================================================ # FINAL: WHAT IF FIBONACCI IS THE COMPRESSION SCHEDULE ITSELF? # ============================================================================ print("\n" + "=" * 70) print("HYPOTHESIS: FIBONACCI *IS* THE COMPRESSION SCHEDULE") print("=" * 70) print(""" What if the Fibonacci budget (2, 5, 8, 13, 21...) doesn't describe the framerate directly, but describes the COMPRESSION EFFICIENCY at each dimension? Budget(D) = how many independent modes can be packed into one frame at dimension D, given the {2,3} symmetry available. 1D: 2 independent modes per frame (just {2}: mirror) 2D: 5 independent modes per frame ({2,3}: hex compression) 3D: 8 independent modes per frame ({3,5}: crystal compression) 4D: 13 independent modes per frame ({5,8}: 24-cell compression) Framerate = modes_per_frame × frame_writing_speed If frame_writing_speed is CONSTANT (same for all dimensions): c_D ∝ Budget(D) = Fibonacci This would mean the speed of light is NOT the frame-writing speed. It is Budget(3D) × frame_writing_speed = 8 × v_write. v_write = c / 8 And the base pulse rate would be: c_1D = 2 × v_write = 2c/8 = 0.250c ✓ c_2D = 5 × v_write = 5c/8 = 0.625c ✓ c_3D = 8 × v_write = 8c/8 = 1.000c ✓ (calibration) c_4D = 13 × v_write = 13c/8 = 1.625c ✓ """) print("The Fibonacci budget IS the number of independent modes") print("that {2,3} compression makes available per frame.") print() print("The frame-writing speed v_write = c/8 is the CONSTANT.") print("What changes is HOW MUCH fits in each frame.") print() print("This is exactly the GPU analogy:") print(" v_write = the bus speed (fixed hardware)") print(" Budget = how many compressed primitives fit per transfer") print(" FPS = bus_speed × primitives_per_transfer") print() # Verify: do the Fibonacci numbers relate to the symmetry orders? print("Fibonacci budget vs symmetry orders:") fib_budget = [0, 2, 5, 8, 13, 21, 34] for D in range(1, 6): ratio = S[D] / fib_budget[D] if fib_budget[D] > 0 else 0 print(f" D={D}: Budget={fib_budget[D]}, S={S[D]}, " f"S/Budget={ratio:.2f}") print() print("S/Budget: 1.00, 2.40, 6.00, 88.62, ...") print("These are NOT constant — the symmetry group grows MUCH faster") print("than the Fibonacci budget.") print() print("This means: the Fibonacci budget is NOT S(D).") print("The budget captures something more fundamental —") print("the number of INDEPENDENT geometric modes, not the total") print("symmetry operations.") print() print("The budget might be: the number of IRREDUCIBLE REPRESENTATIONS") print("of the symmetry group that participate in {2,3} geometry.") print("Not all symmetry operations compress — only the ones that") print("align with the {2,3} interference pattern.")