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427 lines
14 KiB
Python
427 lines
14 KiB
Python
"""
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finsler_metric.py -- Randers Metric F = α + β computation
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Computes the Finsler-Randers metric on the Hachimoji 8-state simplex.
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The metric is the UNIQUE geometry on Δ⁷ (proven by ChentsovFinite.lean):
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F(a→b) = α(a,b) + β(a,b)
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α: Fisher information metric (symmetric, from Chentsov uniqueness theorem)
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β: Drift 1-form (asymmetric, encodes torsion / wind field)
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The Fisher metric is canonical: Chentsov's theorem proves it is the ONLY
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Riemannian metric on the probability simplex that is invariant under all
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Markov embeddings (stochastic refinements).
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Reference: CoreFormalism/ChentsovFinite.lean -- chentsov_hachimoji theorem
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"""
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from __future__ import annotations
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import math
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from dataclasses import dataclass, field
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from typing import Any, Optional
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import numpy as np
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# =========================================================================
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# Q16_16 Fixed-Point Constants (from CoreFormalism/FixedPoint.lean)
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# =========================================================================
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Q16_SCALE: int = 65536 # 2^16 -- number of subdivisions per unit
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def to_q16(value: float) -> int:
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"""Convert float to Q16_16 raw integer with canonical rounding."""
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scaled = value * Q16_SCALE
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rounded = round(scaled)
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return max(-2147483648, min(2147483647, rounded))
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def from_q16(raw: int) -> float:
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"""Convert Q16_16 raw integer to float."""
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return raw / Q16_SCALE
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# =========================================================================
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# Hachimoji 8-State System
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# =========================================================================
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# Greek states with phases (45° steps, from HachimojiBase.lean §5)
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GREEK_STATES: list[str] = ["\u03a6", "\u039b", "\u03a1", "\u039a", "\u03a9", "\u03a3", "\u03a0", "\u0396"]
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# Phase angles in degrees (0, 45, 90, 135, 180, 225, 270, 315)
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GREEK_PHASE: dict[str, float] = {
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"\u03a6": 0.0, "\u039b": 45.0, "\u03a1": 90.0, "\u039a": 135.0,
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"\u03a9": 180.0, "\u03a3": 225.0, "\u03a0": 270.0, "\u0396": 315.0,
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}
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# Latin ↔ Greek bijection (from HachimojiBase.lean §2)
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LATIN_TO_GREEK: dict[str, str] = {
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"A": "\u03a6", "T": "\u039b", "G": "\u03a1", "C": "\u039a",
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"B": "\u03a9", "S": "\u03a3", "P": "\u03a0", "Z": "\u0396",
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}
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GREEK_TO_LATIN: dict[str, str] = {v: k for k, v in LATIN_TO_GREEK.items()}
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# =========================================================================
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# State Descriptors (4D HachimojiState)
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# =========================================================================
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@dataclass
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class HachimojiState4D:
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"""4D descriptor for a Hachimoji state on the probability simplex.
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Components:
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- symbol: Greek letter (Φ Λ Ρ Κ Ω Σ Π Ζ)
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- phase: angle on S¹ in degrees [0, 360)
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- probability: point on Δ⁷ (8 probabilities, sum to 1)
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- fisher_curvature: local Fisher information scalar at this state
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- drift: β component (wind field) at this state
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"""
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symbol: str
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phase: float
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probability: np.ndarray # 8-element, sums to 1
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fisher_curvature: float = 0.0
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drift: np.ndarray = field(default_factory=lambda: np.zeros(8))
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def __post_init__(self):
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self.probability = np.asarray(self.probability, dtype=float)
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self.drift = np.asarray(self.drift, dtype=float)
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# Normalize probability
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s = self.probability.sum()
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if s > 0:
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self.probability /= s
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def to_dict(self) -> dict:
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return {
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"symbol": self.symbol,
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"phase": self.phase,
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"probability": self.probability.tolist(),
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"fisher_curvature": self.fisher_curvature,
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"drift": self.drift.tolist(),
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}
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@classmethod
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def from_dict(cls, d: dict) -> "HachimojiState4D":
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return cls(
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symbol=d["symbol"],
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phase=d["phase"],
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probability=np.array(d["probability"]),
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fisher_curvature=d.get("fisher_curvature", 0.0),
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drift=np.array(d.get("drift", [0.0] * 8)),
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)
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def make_uniform_hachimoji_states() -> list[HachimojiState4D]:
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"""Create the 8 canonical Hachimoji states at the uniform distribution.
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Each state corresponds to one vertex of the Greek alphabet on Δ⁷.
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At the uniform distribution p_i = 1/8, the Fisher metric is:
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g_Fisher(X, X) = 8 * Σ_i X_i² (since 1/p_i = 8)
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"""
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states = []
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for sym in GREEK_STATES:
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phase = GREEK_PHASE[sym]
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# Uniform distribution on Δ⁷
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prob = np.ones(8) / 8.0
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# Fisher curvature at uniform: g_ii = 1/p_i = 8
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fisher_curvature = 8.0
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# Drift (β) points in the direction of increasing phase
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# This encodes the circular topology of the Hachimoji states
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drift = np.zeros(8)
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idx = GREEK_STATES.index(sym)
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# Wind field: stronger drift toward adjacent states on the circle
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drift[(idx + 1) % 8] = 0.3 # forward neighbor
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drift[(idx - 1) % 8] = -0.1 # backward neighbor
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states.append(HachimojiState4D(
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symbol=sym,
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phase=phase,
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probability=prob,
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fisher_curvature=fisher_curvature,
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drift=drift,
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))
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return states
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# =========================================================================
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# Fisher Information Metric (α component -- symmetric, canonical)
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# =========================================================================
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def fisher_information_metric(
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p: np.ndarray,
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X: np.ndarray,
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Y: np.ndarray,
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) -> float:
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"""Compute the Fisher information metric g_Fisher(X, Y) at point p.
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g_Fisher(X, Y) = Σ_i X_i · Y_i / p_i
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This is the UNIQUE Riemannian metric on the probability simplex
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that is invariant under all Markov embeddings (Chentsov's theorem).
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Args:
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p: probability distribution (positive, sums to 1)
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X, Y: tangent vectors (components sum to 0)
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Returns:
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Fisher inner product
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"""
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p = np.asarray(p, dtype=float)
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X = np.asarray(X, dtype=float)
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Y = np.asarray(Y, dtype=float)
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eps = 1e-12
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result = 0.0
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for i in range(len(p)):
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if p[i] > eps:
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result += X[i] * Y[i] / p[i]
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return float(result)
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def fisher_metric_matrix(p: np.ndarray) -> np.ndarray:
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"""Compute the Fisher metric matrix G_ij = δ_ij / p_i at point p.
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Returns:
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8×8 diagonal matrix with G_ii = 1/p_i
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"""
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p = np.asarray(p, dtype=float)
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eps = 1e-12
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G = np.zeros((len(p), len(p)))
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for i in range(len(p)):
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if p[i] > eps:
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G[i, i] = 1.0 / p[i]
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return G
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# =========================================================================
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# Randers Finsler Metric: F = α + β
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# =========================================================================
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def compute_alpha_component(
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state_a: HachimojiState4D | dict,
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state_b: HachimojiState4D | dict,
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) -> float:
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"""Compute the α (Fisher) component: symmetric Riemannian distance.
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α(a,b) = arccosh(1 + ½ · g_Fisher(v, v))
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where v = b - a (tangent vector at the midpoint)
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For the uniform distribution, this simplifies to the
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Fisher-Rao distance on Δ⁷.
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"""
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if isinstance(state_a, dict):
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state_a = HachimojiState4D.from_dict(state_a)
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if isinstance(state_b, dict):
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state_b = HachimojiState4D.from_dict(state_b)
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# Tangent vector: difference of probability distributions
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v = state_b.probability - state_a.probability
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# Use midpoint for metric evaluation
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p_mid = (state_a.probability + state_b.probability) / 2.0
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p_mid = np.maximum(p_mid, 1e-12)
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p_mid /= p_mid.sum()
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# Fisher norm: g_Fisher(v, v) = Σ_i v_i² / p_i
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fisher_norm_sq = fisher_information_metric(p_mid, v, v)
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# α = sqrt(g_Fisher(v, v)) -- the Riemannian length
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alpha = math.sqrt(max(0.0, fisher_norm_sq))
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return alpha
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def compute_beta_component(
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state_a: HachimojiState4D | dict,
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state_b: HachimojiState4D | dict,
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) -> float:
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"""Compute the β (drift) component: antisymmetric 1-form.
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β(a,b) = ½ · (drift_a + drift_b) · (b - a)
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β is a 1-form: β(-v) = -β(v), so β(b,a) = -β(a,b).
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This encodes the torsion / wind field on the Hachimoji manifold.
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"""
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if isinstance(state_a, dict):
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state_a = HachimojiState4D.from_dict(state_a)
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if isinstance(state_b, dict):
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state_b = HachimojiState4D.from_dict(state_b)
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# Average drift field
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avg_drift = (state_a.drift + state_b.drift) / 2.0
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# Displacement vector
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v = state_b.probability - state_a.probability
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# β = drift · v
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beta = float(np.dot(avg_drift, v))
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return beta
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def compute_finsler_metric(
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state_a: HachimojiState4D | dict,
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state_b: HachimojiState4D | dict,
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) -> float:
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"""Compute Randers metric F(a→b) = α(a,b) + β(a,b).
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α: Fisher information metric (symmetric, from Chentsov uniqueness)
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β: Drift 1-form (asymmetric, encodes torsion)
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The metric is the UNIQUE geometry on the Hachimoji simplex
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(proven by ChentsovFinite.lean: chentsov_hachimoji theorem).
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Args:
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state_a: HachimojiState4D for source (or dict)
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state_b: HachimojiState4D for target (or dict)
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Returns:
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F(a→b): positive float, direction-dependent distance
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"""
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alpha = compute_alpha_component(state_a, state_b)
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beta = compute_beta_component(state_a, state_b)
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# F = α + β, but ensure positivity
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# For a valid Finsler metric, we need α > |β|
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F = alpha + beta
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return max(F, 1e-10) # clamp to positive
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def compute_finsler_distance_matrix(
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states: list[HachimojiState4D | dict],
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) -> np.ndarray:
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"""Compute the full 8×8 Finsler distance matrix.
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D[i,j] = F(states[i] → states[j])
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Note: D is NOT symmetric because β is antisymmetric:
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D[i,j] ≠ D[j,i] when drift ≠ 0
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"""
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n = len(states)
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D = np.zeros((n, n))
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for i in range(n):
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for j in range(n):
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if i != j:
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D[i, j] = compute_finsler_metric(states[i], states[j])
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return D
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# =========================================================================
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# Phase Distance on S¹ (circular topology)
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# =========================================================================
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def phase_distance_s1(phase_a: float, phase_b: float) -> float:
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"""Compute the shortest distance between two phases on S¹.
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d_phase(a,b) = min(|a-b|, 360 - |a-b|) · π/180
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The phases live on a circle (0° = 360°), so the distance
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is the arc length along the shorter arc.
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"""
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diff = abs(phase_a - phase_b)
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diff = min(diff, 360.0 - diff)
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# Convert to radians for geometric distance
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return diff * (math.pi / 180.0)
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def circular_phase_matrix(states: list[HachimojiState4D | dict]) -> np.ndarray:
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"""Compute the 8×8 phase distance matrix on S¹.
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P[i,j] = d_phase(phase_i, phase_j)² -- squared circular distance
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"""
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n = len(states)
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P = np.zeros((n, n))
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for i in range(n):
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pa = states[i].phase if hasattr(states[i], "phase") else states[i]["phase"]
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for j in range(n):
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pb = states[j].phase if hasattr(states[j], "phase") else states[j]["phase"]
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d = phase_distance_s1(pa, pb)
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P[i, j] = d * d
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return P
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# =========================================================================
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# Fisher Metric (canonical) factory
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# =========================================================================
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def build_fisher_metric(states: list[HachimojiState4D]) -> dict:
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"""Build the canonical Fisher metric dict for the state simplex.
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Returns dict compatible with compute_finsler_metric's
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fisher_metric parameter:
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{
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"type": "Fisher",
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"metric_matrix": 8×8 array,
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"chentsov_constant": c, # positive constant from theorem
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"at_uniform": True, # at p_i = 1/8
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}
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"""
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# At the uniform distribution, the Fisher metric is 8·I
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G = fisher_metric_matrix(np.ones(8) / 8.0)
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return {
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"type": "Fisher",
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"metric_matrix": G.tolist(),
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"chentsov_constant": 1.0, # c = 1 at uniform
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"at_uniform": True,
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}
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# =========================================================================
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# Convenience: full pipeline from states to distance matrix
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# =========================================================================
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def build_full_finsler_pipeline(
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states: Optional[list[HachimojiState4D]] = None,
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) -> dict:
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"""Run the full Finsler metric computation pipeline.
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Returns:
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{
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"states": [state dicts],
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"finsler_matrix": 8×8 distance matrix,
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"alpha_matrix": 8×8 symmetric α component,
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"beta_matrix": 8×8 antisymmetric β component,
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"phase_matrix": 8×8 phase distance on S¹,
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"fisher_metric": Fisher metric dict,
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"is_anisotropic": bool, # True if β ≠ 0
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}
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"""
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if states is None:
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states = make_uniform_hachimoji_states()
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n = len(states)
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F_mat = np.zeros((n, n))
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A_mat = np.zeros((n, n))
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B_mat = np.zeros((n, n))
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for i in range(n):
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for j in range(n):
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if i != j:
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A_mat[i, j] = compute_alpha_component(states[i], states[j])
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B_mat[i, j] = compute_beta_component(states[i], states[j])
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F_mat[i, j] = A_mat[i, j] + B_mat[i, j]
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P_mat = circular_phase_matrix(states)
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fisher = build_fisher_metric(states)
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# Check anisotropy: max |B_ij + B_ji| should be ~0 (antisymmetric)
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# but the Finsler matrix has asymmetric off-diagonals
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anisotropic = np.any(np.abs(B_mat) > 1e-9)
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return {
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"states": [s.to_dict() for s in states],
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"finsler_matrix": F_mat.tolist(),
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"alpha_matrix": A_mat.tolist(),
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"beta_matrix": B_mat.tolist(),
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"phase_matrix": P_mat.tolist(),
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"fisher_metric": fisher,
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"is_anisotropic": bool(anisotropic),
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}
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if __name__ == "__main__":
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result = build_full_finsler_pipeline()
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print(f"Finsler metric computed for 8 Hachimoji states")
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print(f"Anisotropic: {result['is_anisotropic']}")
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print(f"Finsler matrix (first row): {result['finsler_matrix'][0]}")
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print(f"Alpha matrix (first row): {result['alpha_matrix'][0]}")
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print(f"Beta matrix (first row): {result['beta_matrix'][0]}")
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