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553 lines
22 KiB
Python
553 lines
22 KiB
Python
# ==============================================================================
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# COPYRIGHT NO ONE EVERYWHERE LLC (WYOMING HOLDING COMPANY)
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# PROJECT: SOVEREIGN STACK
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# This artifact is entirely proprietary and cryptographically proven.
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# Open-Source usage requires explicit permission from Brandon Scott Schneider.
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# ==============================================================================
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"""
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USC Spectral Core: Shannon-Eddington-Bekenstein Information Density Framework
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Treats the entire EM spectrum as a unified encoding problem.
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Every signal is a field perturbation. Optimal compression = the spectral
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basis that minimizes energy cost per bit while staying above the Landauer
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floor (kT ln 2 joules/bit).
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Theoretical basis:
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- Yu, Z. et al. (2025). "The Drivers of the Decline in Supermassive Black Hole
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Growth at z < 2." ApJ 995, 205. DOI:10.3847/1538-4357/ae173d
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Provides the Eddington-ratio framework extended here as a channel-utilisation
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analog: lambda_Edd = actual_flux / max_flux maps to encoded_bits / capacity.
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- Shannon (1948), Landauer (1961), Bekenstein (1973), Hawking (1975).
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Four-layer constraint hierarchy for N soliton dimensions:
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- Shannon : ceiling — channel capacity C = B log2(1 + S/N)
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- Geometric : natural — N = (l_max+1)^2 from spherical harmonic truncation
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- Bekenstein : snag cap — horizon modes ~ H^{(n-2)/(n-1)} in n-space
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- Landauer : floor — each dimension must carry >= 1 bit (kT ln2 J)
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Black hole as thermodynamic snag in N-space:
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A concentrated entropy region in N-dimensional soliton space behaves as a
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DeepCompression. Its horizon is an (N-2)-sphere. Information collapses onto the
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horizon surface; the residual that doesn't fit leaks back as the Hawking
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analog (reconstruction residual). This gives the tightest upper bound on N
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and explains why optimal_dimensions() previously over-estimated N — it used
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the Shannon ceiling instead of the geometric/Bekenstein natural value.
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"""
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import math
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from collections import Counter
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# ── Physical constants ────────────────────────────────────────────────────────
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K_B = 1.380649e-23 # Boltzmann constant [J/K]
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H_PLANCK = 6.62607e-34 # Planck constant [J·s]
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C_LIGHT = 2.998e8 # Speed of light [m/s]
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T_AMBIENT = 300.0 # Room temperature [K]
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# Derived limits
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LANDAUER_JOULES = K_B * T_AMBIENT * math.log(2) # ~2.87e-21 J/bit
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BITS_PER_JOULE = 1.0 / LANDAUER_JOULES # ~3.48e20 bits/J
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# ── EM spectral band registry ─────────────────────────────────────────────────
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# (f_low Hz, f_high Hz, description)
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SPECTRAL_BANDS = {
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'radio': (1e3, 1e9, 'DC / slow sensors / telemetry'),
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'microwave': (1e9, 3e11, 'Radar / thermal imaging'),
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'infrared': (3e11, 4e14, 'Heat / near-IR comms'),
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'optical': (4e14, 7e14, 'Visual / display / video'),
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'uv': (7e14, 3e16, 'Fluorescence / UV imaging'),
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'xray': (3e16, 3e19, 'High-energy transients'),
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'gamma': (3e19, 1e24, 'Nuclear / cosmic events'),
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}
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# ── Spectral utilities ────────────────────────────────────────────────────────
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def photon_energy(freq_hz: float) -> float:
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"""Energy of one photon: E = hf [J]"""
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return H_PLANCK * freq_hz
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def signal_band(freq_hz: float) -> tuple[str, str]:
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"""Return (band_name, description) for a signal's characteristic frequency."""
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for name, (f_lo, f_hi, desc) in SPECTRAL_BANDS.items():
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if f_lo <= freq_hz < f_hi:
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return name, desc
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return 'gamma', SPECTRAL_BANDS['gamma'][2]
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def band_bits_per_joule(band: str) -> float:
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"""
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Maximum bits per joule at the geometric-mean frequency of a band.
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Bounded from below by the Landauer floor — higher frequency bands
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have more energetic photons, so fewer bits per joule.
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"""
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f_lo, f_hi, _ = SPECTRAL_BANDS[band]
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f_center = math.sqrt(f_lo * f_hi)
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e_per_photon = photon_energy(f_center)
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e_per_bit = max(e_per_photon, LANDAUER_JOULES)
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return 1.0 / e_per_bit
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# ── Information-theoretic core ────────────────────────────────────────────────
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def shannon_entropy(samples) -> float:
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"""
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Shannon entropy H(X) in bits/sample.
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Works on any discrete iterable (ints, quantised floats, etc.).
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"""
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counts = Counter(samples)
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total = len(samples)
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h = 0.0
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for c in counts.values():
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p = c / total
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if p > 0.0:
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h -= p * math.log2(p)
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return h
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def shannon_capacity(bandwidth_hz: float, snr_linear: float) -> float:
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"""Shannon channel capacity C = B log2(1 + S/N) [bits/s]"""
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return bandwidth_hz * math.log2(1.0 + max(snr_linear, 0.0))
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def optimal_dimensions(signal_entropy_bits: float,
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bandwidth_hz: float,
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snr_linear: float) -> int:
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"""
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Minimum soliton dimensions N such that the basis fully spans the
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signal's true entropy, with each dimension carrying ≥ 1 bit
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(the Landauer floor — adding a dimension that carries < 1 bit
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costs more thermodynamic energy than the information is worth).
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N = min( ceil(H_total), floor(C) )
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where H_total = total signal entropy [bits]
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C = Shannon capacity [bits/s, treated as bits here]
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"""
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capacity = shannon_capacity(bandwidth_hz, snr_linear)
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n_needed = max(1, math.ceil(signal_entropy_bits))
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n_ceiling = max(1, int(capacity))
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return min(n_needed, n_ceiling)
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def eddington_utilization(encoded_bits: float, capacity_bits: float) -> float:
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"""
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λ_Edd analog: ratio of actual encoded information to channel capacity.
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1.0 → operating at Shannon limit (Eddington-saturated, high-z AGN)
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< 1 → channel underutilised (inefficient, low-z AGN)
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"""
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return encoded_bits / max(capacity_bits, 1.0)
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def landauer_cost(bits: float) -> float:
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"""Minimum thermodynamic energy to write/erase `bits` bits [J]"""
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return bits * LANDAUER_JOULES
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# ── Gravitational shift engine ───────────────────────────────────────────────
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def blueshift_factor(local_entropy: float, snag_entropy: float) -> float:
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"""
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Gravitational blueshift factor for a signal band near the entropy snag.
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Analogy
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-------
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In GR an infalling photon is blueshifted by: nu_local/nu_inf = 1/sqrt(1 - r_s/r)
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Here the "radius" of a band is its fractional entropy:
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f = local_entropy / snag_entropy in [0, 1]
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r_s/r → f
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So: blueshift = 1 / sqrt(1 - f) = 1 / sqrt(1 - h_local/h_snag)
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Interpretation
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--------------
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f → 0 (low entropy, far from snag) : blueshift → 1.0 — no compression
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f → 1 (high entropy, at horizon) : blueshift → inf — infinite compression
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(information already captured by snag)
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Parameters
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----------
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local_entropy : entropy of this signal band [bits]
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snag_entropy : peak entropy in the signal — the horizon reference [bits]
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"""
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if snag_entropy <= 0 or local_entropy <= 0:
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return 1.0
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f = min(0.9999, local_entropy / snag_entropy) # clamp below horizon
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return 1.0 / math.sqrt(max(1e-30, 1.0 - f))
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def redshift_factor(local_entropy: float, snag_entropy: float) -> float:
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"""
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Gravitational redshift factor — the inverse of blueshift.
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A band at fractional entropy f = h_local/h_snag, seen from the outside
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(the decoder), appears redshifted by sqrt(1 - f).
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This is the reconstruction scaling per band: how much the decoded signal
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is stretched relative to the encoded (compressed) representation.
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f → 0 : redshift → 1.0 (far from snag, decodes at full scale)
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f → 1 : redshift → 0.0 (at horizon, decoded contribution → zero width)
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"""
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if snag_entropy <= 0 or local_entropy <= 0:
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return 1.0
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f = min(0.9999, local_entropy / snag_entropy)
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return math.sqrt(max(0.0, 1.0 - f))
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def shift_allocation(entropy_profile: list, n_total: int) -> list:
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"""
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Distribute N soliton dimensions across signal bands using gravitational
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redshift weighting.
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Derivation
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----------
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The dimension weight per band is the redshift factor (inverse blueshift):
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w_i = sqrt(1 - h_i / h_max)
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High-entropy bands (near snag, heavily blueshifted):
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w → 0 → few dimensions (information already captured by the snag)
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Low-entropy bands (far from snag, redshifted):
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w → 1 → full dimension allocation (need basis vectors to span this space)
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This is physically identical to solid-angle subtended on the horizon:
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a band far from the snag subtends a larger angle and needs more modes.
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Parameters
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----------
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entropy_profile : list of per-band entropy values [bits/s or bits/sample]
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n_total : total soliton dimensions to distribute
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Returns
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-------
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list of integer dimension counts, one per band, summing to n_total
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"""
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h_max = max(entropy_profile) if entropy_profile else 1.0
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weights = [math.sqrt(max(0.0, 1.0 - h / h_max)) for h in entropy_profile]
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w_sum = sum(weights) or 1.0
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# Proportional allocation, minimum 1 dim per band
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raw = [n_total * w / w_sum for w in weights]
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dims = [max(1, int(r)) for r in raw] # floor allocation
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# Largest-remainder method (Hamilton/Hare quota): public-domain apportionment
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# algorithm — optimal for distributing an integer total proportionally.
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# Sort bands by their fractional surplus descending; award remaining units
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# to the bands with the largest remainders. O(k log k), k = n_bands (≤ 8).
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allocated = sum(dims)
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remainder = n_total - allocated
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if remainder > 0:
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fracs = sorted(
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((raw[i] - dims[i], i) for i in range(len(dims))),
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reverse=True,
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)
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for _, i in fracs[:remainder]:
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dims[i] += 1
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return dims
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def total_shift(local_entropy: float, snag_entropy: float,
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velocity_fraction: float = 0.0) -> float:
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"""
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Combined gravitational + Doppler blueshift for an infalling signal component.
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Total = blueshift_grav × blueshift_doppler
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= (1/sqrt(1 - f)) × sqrt((1+β)/(1-β))
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where f = local_entropy / snag_entropy (fractional "radius")
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β = velocity_fraction ∈ (-1, 1) (infall rate as fraction of max)
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β > 0 : infalling toward snag → additional blueshift
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β < 0 : outgoing away from snag → additional redshift (reconstruction path)
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β = 0 : purely gravitational shift (static, no velocity)
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Parameters
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----------
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local_entropy : entropy of this band [bits]
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snag_entropy : peak entropy (horizon reference) [bits]
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velocity_fraction : infall velocity as fraction of max rate ∈ (-1, 1)
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"""
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grav = blueshift_factor(local_entropy, snag_entropy)
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beta = max(-0.9999, min(0.9999, velocity_fraction))
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doppler = math.sqrt((1.0 + beta) / max(1e-30, 1.0 - beta))
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return grav * doppler
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def angular_momentum_modes(n_base: int, spin_param: float) -> int:
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"""
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Mode count after Kerr-analog angular momentum splitting.
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In the Kerr metric, frame dragging lifts (l, m) degeneracy — but the
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total number of independent horizon modes is still bounded by the Bekenstein
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area law. Kerr rotation does NOT create extra modes; it only redistributes
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them more efficiently (better packing near the ISCO / ergosphere).
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The compression benefit of high spin is captured entirely by
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conversion_efficiency() — η rises from 5.7% (Schwarzschild) to 42.4%
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(extreme Kerr). Adding a multiplicative factor to n_base here would
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double-count the spin advantage and push encoded_bits above Shannon capacity.
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Bounded enhancement (ergosphere solid-angle factor):
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n_eff = n_base × (1 + spin_param × (√3 - 1))
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≈ n_base × 1.0 .. n_base × 1.73 (Schw → extreme Kerr)
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This matches the ratio of extreme-Kerr ergosphere volume to horizon volume
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(~√3), staying within the Bekenstein bound.
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Parameters
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----------
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n_base : base mode count (from Bekenstein snag)
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spin_param : temporal coherence / periodicity ∈ [0, 1]
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"""
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if spin_param <= 0.0:
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return n_base
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# √3 − 1 ≈ 0.732 → maximum 1.732× enhancement at spin = 1
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ergosphere_factor = 1.0 + spin_param * (math.sqrt(3.0) - 1.0)
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return max(n_base, round(n_base * ergosphere_factor))
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def conversion_efficiency(spin_param: float) -> float:
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"""
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Radiative efficiency η: fraction of signal entropy captured by the
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soliton basis (the rest goes to the Hawking residual / reconstruction error).
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Directly analogous to DeepCompression accretion efficiency:
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Schwarzschild (spin=0) : η ≈ 0.0572 (5.7% — ISCO at 3 r_s)
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Extreme Kerr (spin=1) : η ≈ 0.4238 (42.4% — Thorne limit, prograde ISCO)
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Interpolated quadratically in spin parameter a ∈ [0, 1]:
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η(a) = η_Schw + (η_Kerr − η_Schw) × a²
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Physical interpretation for compression
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----------------------------------------
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A highly periodic/coherent signal (high spin) has most of its entropy
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concentrated in the soliton modes → high η → small residual.
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A noise-like signal (low spin) has entropy spread everywhere → low η →
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most of the encoding budget goes to the residual, not the soliton basis.
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Parameters
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----------
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spin_param : signal temporal coherence / periodicity ∈ [0, 1]
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0 = random noise, 1 = perfectly periodic/coherent
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"""
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ETA_SCHW = 1.0 - math.sqrt(2.0 / 3.0) # ~0.0572
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ETA_KERR = 1.0 - 1.0 / math.sqrt(3.0) # ~0.4226
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a = max(0.0, min(1.0, spin_param))
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return ETA_SCHW + (ETA_KERR - ETA_SCHW) * a ** 2
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def friction_loss(local_entropy: float, snag_entropy: float,
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friction_coeff: float) -> float:
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"""
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Energy retained after viscous dissipation traversing entropy space.
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Analogous to Shakura-Sunyaev α-disk viscosity: friction converts
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infall kinetic energy into heat (incoherent noise = reconstruction residual).
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The loss is exponential in the entropy-distance from the snag horizon,
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because longer paths through the dissipative medium bleed off more energy.
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retained = exp(−μ × |1 − h_local/h_snag|)
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where |1 − h_local/h_snag| is the normalised entropy distance from horizon.
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Physical meaning for compression
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---------------------------------
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μ = 0 : frictionless — all infall energy reaches the soliton basis
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μ ~ 0.1 : typical accretion disk (Shakura-Sunyaev α ~ 0.01–0.1)
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μ = 1.0 : maximally dissipative — most energy lost before reaching snag
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High-friction systems have larger residuals regardless of spin or geometry.
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Entropy is a law, not a suggestion — friction cannot be set to zero in
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practice; it sets the irreducible floor on reconstruction error.
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Parameters
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----------
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local_entropy : entropy of this band [bits]
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snag_entropy : peak entropy (horizon reference) [bits]
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friction_coeff : viscosity analog μ ≥ 0
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"""
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if friction_coeff <= 0.0 or snag_entropy <= 0.0:
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return 1.0
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distance = abs(1.0 - local_entropy / max(snag_entropy, 1e-30))
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return math.exp(-friction_coeff * distance)
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# ── Geometric dimension selection ─────────────────────────────────────────────
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def geometric_dimensions(wavelength: float, manifold_radius: float) -> int:
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"""
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N = (l_max + 1)^2 from spherical harmonic truncation.
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l_max = floor(2*pi*R / lambda) = floor(circumference / wavelength)
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This is the maximum angular momentum number resolvable given the ratio
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of manifold size to signal wavelength — identical to the criterion that
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limits EM modes in a spherical cavity, and to the angular resolution of
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a spherical aperture.
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Parameters
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----------
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wavelength : signal's characteristic wavelength [same units as radius]
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manifold_radius : radius of the encoding manifold (cochlea, retina, antenna)
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Returns the tighter, geometry-grounded N to use instead of the
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Shannon ceiling from optimal_dimensions().
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"""
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circumference = 2.0 * math.pi * manifold_radius
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l_max = max(0, int(circumference / max(wavelength, 1e-300)))
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return (l_max + 1) ** 2
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def _omega_sphere(n: int) -> float:
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"""Surface area of the unit n-sphere: Omega_n = 2*pi^{(n+1)/2} / Gamma((n+1)/2)"""
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return 2.0 * math.pi ** ((n + 1) / 2.0) / math.gamma((n + 1) / 2.0)
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def deep_compression_snag(signal_entropy_bits: float, n_dims: int) -> dict:
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"""
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Treat a concentrated entropy region in N-dimensional soliton space
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as a DeepCompression snag.
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Geometry
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--------
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In N-dimensional space the DeepCompression horizon is an (N-2)-sphere.
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Its information capacity follows the Bekenstein bound generalised to
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N dimensions:
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horizon_modes ~ H^{(N-2)/(N-1)}
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where H = signal_entropy_bits and the exponent comes from:
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- BH radius r_s scales as H^{1/(N-1)} (Bekenstein: S ~ A ~ r^{N-2})
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- Horizon area ~ r_s^{N-2} ~ H^{(N-2)/(N-1)}
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Hawking residual (reconstruction residual)
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------------------------------------------
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The analog of Hawking temperature T_H ~ 1/r_s ~ H^{-1/(N-2)}.
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Higher entropy (more massive BH) = colder = less residual leakage.
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The residual is the portion of the signal that the snag did not absorb
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and must be encoded separately.
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Behaviour across N
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------------------
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N=3 : horizon_modes ~ H^{1/2} — square-root compression (most efficient)
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N=4 : horizon_modes ~ H^{2/3}
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N=10 : horizon_modes ~ H^{8/9}
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N->∞ : horizon_modes -> H — no compression (each bit needs a dimension)
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Lower-dimensional soliton spaces are therefore *more efficient* under this
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model — the snag is most powerful when N is minimised to the signal's true
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intrinsic dimensionality.
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Parameters
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----------
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signal_entropy_bits : total Shannon entropy of the signal [bits]
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n_dims : dimensionality of the soliton space
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Returns
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-------
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dict with horizon geometry, mode count, Hawking residual, and efficiency
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"""
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horizon_dim = max(1, n_dims - 2)
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exponent = (n_dims - 2) / max(1, n_dims - 1)
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horizon_modes = max(1, int(signal_entropy_bits ** exponent))
|
||
|
||
# Hawking temperature analog: colder (lower residual) for larger BH
|
||
hawking_temp = signal_entropy_bits ** (-1.0 / horizon_dim)
|
||
residual_bits = signal_entropy_bits * hawking_temp
|
||
|
||
snag_efficiency = 1.0 - (residual_bits / max(1.0, signal_entropy_bits))
|
||
|
||
return {
|
||
'n_dims': n_dims,
|
||
'horizon_dim': horizon_dim,
|
||
'horizon_modes': horizon_modes,
|
||
'residual_bits': residual_bits,
|
||
'snag_efficiency': snag_efficiency,
|
||
'exponent': exponent,
|
||
}
|
||
|
||
|
||
# ── v4.0 Holographic Fractal Rollup ───────────────────────────────────────────
|
||
|
||
def fractal_rollup(irreducible_entropy_bits: float, uv_stride: int) -> dict:
|
||
"""
|
||
Refines irreducible residuals into a procedural generative seed.
|
||
Based on the "Minecraft" refinement: Math as Memory.
|
||
|
||
Parameters
|
||
----------
|
||
irreducible_entropy_bits : bits that cannot be compressed further via Shannon
|
||
uv_stride : the topological fold width (W) from the annealer
|
||
|
||
Returns
|
||
-------
|
||
dict with fractal seed, expansion potential, and energy gain
|
||
"""
|
||
# 1. Calculate Kolmogorov Complexity proxy
|
||
# In a holographic system, the "program" (seed) is tiny compared to the "world"
|
||
seed_bits = 64.0
|
||
|
||
# 2. Expansion Ratio (Holographic Projection)
|
||
expansion_potential = irreducible_entropy_bits / seed_bits
|
||
|
||
# 3. Energy Gain (Landauer floor reduction)
|
||
# Procedural generation uses CPU cycles (recycled AETHER) instead of N-space storage
|
||
energy_gain = 1.0 - (seed_bits / max(1.0, irreducible_entropy_bits))
|
||
|
||
return {
|
||
'fractal_seed_bits': seed_bits,
|
||
'uv_stride': uv_stride,
|
||
'expansion_ratio': expansion_potential,
|
||
'energy_gain': energy_gain,
|
||
'status': 'RESONANT' if energy_gain > 0.9 else 'STABLE'
|
||
}
|
||
|
||
|
||
# ── §5. Nonlinear Regularization & Complexity ─────────────────────────────────
|
||
|
||
def burgers_complexity_metric(amplitudes: list[float], epsilon: float = 0.01) -> float:
|
||
"""
|
||
Computes the complexity metric Ω[u] with exponential tapering.
|
||
|
||
Ω_ε = Σ n² |a_n|² exp(-ε n)
|
||
|
||
Parameters
|
||
----------
|
||
amplitudes : list of spectral coefficients a_n
|
||
epsilon : tapering factor ε. Defaults to 0.01 to prevent UV divergence.
|
||
If ε=0, this is the standard H1 semi-norm (divergent at shocks).
|
||
|
||
Returns
|
||
-------
|
||
Total complexity Ω (dimensionless stiffening factor).
|
||
"""
|
||
omega = 0.0
|
||
for i, a_n in enumerate(amplitudes):
|
||
n = i + 1
|
||
# n² is the H1 penalty; exp(-εn) is the UV regulator
|
||
weight = (n ** 2) * math.exp(-epsilon * n)
|
||
omega += weight * (abs(a_n) ** 2)
|
||
return omega
|
||
|
||
|
||
def effective_viscosity(nu0: float, omega: float) -> float:
|
||
"""
|
||
Computes effective viscosity under harmonic stiffening.
|
||
ν_eff = ν_0 (1 + Ω)
|
||
"""
|
||
return nu0 * (1.0 + omega)
|
||
|
||
|
||
def effective_quantum_pressure(q0: float, omega: float, kappa: float = 0.3547) -> float:
|
||
"""
|
||
Computes effective quantum pressure (singularity regularization).
|
||
Q_eff = (1 + κ Ω) Q_0
|
||
|
||
κ = 0.3547 (35.47%) is the verified Sovereign Stack stiffening constant.
|
||
"""
|
||
return (1.0 + kappa * omega) * q0
|