code/orynthra.py

"""ORYNTHRA-H6: reference computations for branch-covariant compression.

This is a small research implementation, not a general-purpose file compressor,
a quantum hardware controller, or a certified formal proof system.
Finite-field routines support prime fields only; the manuscript permits F_q.
"""
from __future__ import annotations
from dataclasses import dataclass
from functools import lru_cache
from itertools import permutations, product
from math import comb, ceil, log2, sqrt, log1p, log, isfinite
from typing import Iterable, Sequence, Hashable
import numpy as np


def _positive_int(value: int, name: str) -> int:
    if isinstance(value, bool) or not isinstance(value, (int, np.integer)) or value < 1:
        raise ValueError(f"{name} must be a positive integer")
    return int(value)


def _prime(p: int) -> int:
    p = _positive_int(p, "p")
    if p < 2 or any(p % d == 0 for d in range(2, int(sqrt(p)) + 1)):
        raise ValueError("Only prime fields are implemented")
    return p


def rref_mod(rows: np.ndarray, p: int = 2) -> tuple[np.ndarray, tuple[int, ...]]:
    """Reduced row basis over F_p, with pivot-column indices."""
    p = _prime(p)
    a = np.asarray(rows, dtype=np.int64).copy() % p
    if a.ndim != 2:
        raise ValueError("rows must be a two-dimensional array")
    pivot_columns: list[int] = []
    r = 0
    for c in range(a.shape[1]):
        nz = np.flatnonzero(a[r:, c])
        if not len(nz):
            continue
        j = r + int(nz[0])
        a[[r, j]] = a[[j, r]]
        a[r] = (a[r] * pow(int(a[r, c]), -1, p)) % p
        for j in range(a.shape[0]):
            if j != r and a[j, c]:
                a[j] = (a[j] - a[j, c] * a[r]) % p
        pivot_columns.append(c)
        r += 1
        if r == a.shape[0]:
            break
    return a[:r], tuple(pivot_columns)


def inverse_mod(matrix: np.ndarray, p: int = 2) -> np.ndarray:
    a = np.asarray(matrix, dtype=np.int64)
    if a.ndim != 2 or a.shape[0] != a.shape[1]:
        raise ValueError("matrix must be square")
    n = a.shape[0]
    reduced, pivots = rref_mod(np.hstack((a, np.eye(n, dtype=int))), p)
    if pivots[:n] != tuple(range(n)):
        raise ValueError("matrix is singular over the supplied field")
    return reduced[:, n:]


@dataclass(frozen=True)
class LinearEdge:
    source: int
    target: int
    matrix: np.ndarray
    name: str = ""


def observable_closure(observations: Sequence[np.ndarray],
                       edges: Sequence[LinearEdge], p: int = 2,
                       horizon: int | None = None) -> tuple[list[np.ndarray], list[list[int]]]:
    """Simultaneous pullback closure, or paths of length <= horizon.

    A row at the target of T pulls back to row @ T at its source.
    The trace contains ranks at initialization and after strict changes.
    Matrices need not be invertible for this routine.
    """
    p = _prime(p)
    if not observations:
        raise ValueError("At least one domain is required")
    bases = [rref_mod(np.asarray(a), p)[0] for a in observations]
    dimensions = [a.shape[1] for a in bases]
    if horizon is not None and (not isinstance(horizon, int) or horizon < 0):
        raise ValueError("horizon must be a nonnegative integer or None")
    for e in edges:
        if not (0 <= e.source < len(bases) and 0 <= e.target < len(bases)):
            raise ValueError("edge domain index out of range")
        if np.shape(e.matrix) != (dimensions[e.target], dimensions[e.source]):
            raise ValueError("edge matrix shape is incompatible with domains")
    trace = [[len(a) for a in bases]]
    for _ in range(sum(dimensions) + 1 if horizon is None else horizon):
        candidates = [[a] for a in bases]
        for e in edges:
            candidates[e.source].append((bases[e.target] @ e.matrix) % p)
        newer = [rref_mod(np.vstack(parts), p)[0] for parts in candidates]
        if all(np.array_equal(a, b) for a, b in zip(bases, newer)):
            break
        bases = newer
        trace.append([len(a) for a in bases])
    return bases, trace


def induced_linear_map(c_source: np.ndarray, c_target: np.ndarray,
                       transition: np.ndarray, p: int = 2) -> np.ndarray:
    """Return B with C_target T = B C_source; C_source must be in RREF."""
    c, pivots = rref_mod(c_source, p)
    if not np.array_equal(c, np.asarray(c_source) % p):
        raise ValueError("c_source must be an independent RREF basis")
    pulled = (np.asarray(c_target) @ transition) % p
    b = pulled[:, list(pivots)]
    if not np.array_equal((b @ c) % p, pulled):
        raise ValueError("The proposed source quotient is not transition-closed")
    return b


def canonical_labels(values: Sequence[Hashable]) -> tuple[int, ...]:
    labels: dict[Hashable, int] = {}
    return tuple(labels.setdefault(v, len(labels)) for v in values)


def finite_partition(observation: Sequence[Hashable],
                     transitions: Sequence[Sequence[int]]) -> tuple[int, ...]:
    """Minimal deterministic future-observation partition of one finite domain."""
    n = len(observation)
    if not n:
        raise ValueError("At least one state is required")
    for t in transitions:
        if len(t) != n or any(not 0 <= int(j) < n for j in t):
            raise ValueError("Invalid transition map")
    labels = canonical_labels(observation)
    for _ in range(n):
        newer = canonical_labels([(labels[x], tuple(labels[t[x]] for t in transitions))
                                  for x in range(n)])
        if newer == labels:
            return labels
        labels = newer
    raise RuntimeError("Partition refinement exceeded its finite bound")


def permutation_matrix(perm: Sequence[int]) -> np.ndarray:
    """Column convention: P|j> = |perm[j]>."""
    p = tuple(int(x) for x in perm)
    if sorted(p) != list(range(len(p))):
        raise ValueError("perm is not a permutation of 0,...,n-1")
    result = np.zeros((len(p), len(p)))
    result[np.array(p), np.arange(len(p))] = 1.0
    return result


def six_wing_actions(n: int, wings: int = 6) -> list[np.ndarray]:
    """Involutions swapping equally colored edges of a path of n coordinates."""
    n = _positive_int(n, "n")
    wings = _positive_int(wings, "wings")
    if wings < 2:
        raise ValueError("At least two colors are needed for disjoint matchings")
    maps = [np.arange(n) for _ in range(wings)]
    for i in range(n - 1):
        color = i % wings
        maps[color][i], maps[color][i + 1] = maps[color][i + 1], maps[color][i]
    return [permutation_matrix(a).astype(int) for a in maps]


def six_wing_rank_formula(n: int, active: Iterable[int], wings: int = 6) -> int:
    active = set(active)
    if not active.issubset(set(range(wings))):
        raise ValueError("Invalid wing index")
    for i in range(n - 1):
        if i % wings not in active:
            return i + 1
    return n


def photon_sector(modes: int, photons: int) -> int:
    modes = _positive_int(modes, "modes")
    if not isinstance(photons, (int, np.integer)) or photons < 0:
        raise ValueError("photons must be a nonnegative integer")
    return comb(modes + int(photons) - 1, int(photons))


def photon_capacity(modes: int, cutoff: int) -> int:
    modes = _positive_int(modes, "modes")
    if not isinstance(cutoff, (int, np.integer)) or cutoff < 0:
        raise ValueError("cutoff must be a nonnegative integer")
    return comb(modes + int(cutoff), int(cutoff))


def _first_true(predicate) -> int:
    if predicate(0):
        return 0
    high = 1
    while not predicate(high):
        high *= 2
    low = 0
    while low + 1 < high:
        mid = (low + high) // 2
        if predicate(mid):
            high = mid
        else:
            low = mid
    return high


def storage_cutoff(messages: int, modes: int) -> int:
    messages = _positive_int(messages, "messages")
    modes = _positive_int(modes, "modes")
    return _first_true(lambda k: photon_capacity(modes, k) >= messages)


def symmetric_cutoff(messages: int, modes: int) -> int | None:
    """Exact phase-registered S_M action, arbitrary number-preserving gates.

    None means impossible for a single nondegenerate oscillator and M >= 3.
    """
    messages = _positive_int(messages, "messages")
    modes = _positive_int(modes, "modes")
    if messages == 1:
        return 0
    if messages == 2:
        return 1
    if modes == 1:
        return None
    return max(1, _first_true(lambda k: photon_sector(modes, k) >= messages - 1))


def minimum_storage_mean(messages: int, modes: int) -> float:
    """Mean photons for uniform orthogonal messages, without gate constraints."""
    k = storage_cutoff(messages, modes)
    if k == 0:
        return 0.0
    lower_count = photon_capacity(modes, k - 1)
    lower_sum = 0 if k == 1 else modes * comb(modes + k - 1, k - 2)
    return (lower_sum + k * (messages - lower_count)) / messages


def pack_items(items: Sequence[int], capacities: Sequence[int]) -> tuple[int, ...] | None:
    """Exact indivisible packing. Return the bin index for each original item.

    This exponential reference solver is intended only for small instances.
    Equal residual capacities are symmetry-pruned. Bin identity is retained.
    """
    original = tuple(_positive_int(v, "item size") for v in items)
    cap = tuple(int(v) for v in capacities)
    if any(v < 0 for v in cap):
        raise ValueError("capacities must be nonnegative")
    order = sorted(range(len(original)), key=lambda i: original[i], reverse=True)
    ordered = tuple(original[i] for i in order)
    if sum(ordered) > sum(cap):
        return None

    @lru_cache(None)
    def recurse(i: int, residual: tuple[int, ...]):
        if i == len(ordered):
            return ()
        if sum(ordered[i:]) > sum(residual):
            return None
        tried = set()
        for b, room in enumerate(residual):
            if room in tried or room < ordered[i]:
                continue
            tried.add(room)
            nxt = list(residual)
            nxt[b] -= ordered[i]
            tail = recurse(i + 1, tuple(nxt))
            if tail is not None:
                return (b,) + tail
        return None

    assignment = recurse(0, cap)
    if assignment is None:
        return None
    result = [0] * len(order)
    for i, b in zip(order, assignment):
        result[i] = b
    return tuple(result)


def maximum_packed_dimension(items: Sequence[int], capacities: Sequence[int]) -> int:
    """Largest total dimension of a subset of irrep copies that fits."""
    values = tuple(sorted((_positive_int(v, "item size") for v in items), reverse=True))
    capacities = tuple(sorted((int(x) for x in capacities), reverse=True))
    if any(x < 0 for x in capacities):
        raise ValueError("capacities must be nonnegative")

    @lru_cache(None)
    def recurse(i: int, residual: tuple[int, ...]) -> int:
        if i == len(values):
            return 0
        best = recurse(i + 1, residual)
        tried = set()
        for b, room in enumerate(residual):
            if room < values[i] or room in tried:
                continue
            tried.add(room)
            nxt = list(residual)
            nxt[b] -= values[i]
            candidate = values[i] + recurse(i + 1, tuple(sorted(nxt, reverse=True)))
            best = max(best, candidate)
        return best
    return recurse(0, capacities)


def representation_cutoff(irrep_dimensions: Sequence[int], modes: int,
                          max_cutoff: int = 100) -> int | None:
    """Find exact cutoff for a supplied list of complex irrep-copy dimensions.

    The group decomposition is supplied, not inferred. None can mean the
    requested finite search range was insufficient (except single-mode no-go).
    """
    dims = tuple(_positive_int(d, "irrep dimension") for d in irrep_dimensions)
    modes = _positive_int(modes, "modes")
    if not dims:
        raise ValueError("At least one irrep copy is required")
    if modes == 1 and max(dims) > 1:
        return None
    for k in range(storage_cutoff(sum(dims), modes), max_cutoff + 1):
        if pack_items(dims, [photon_sector(modes, n) for n in range(k + 1)]) is not None:
            return k
    return None


def helmert_matrix(messages: int) -> np.ndarray:
    """Orthogonal matrix with constant first row; remaining rows span a simplex."""
    m = _positive_int(messages, "messages")
    h = np.zeros((m, m))
    h[0] = 1 / sqrt(m)
    for j in range(1, m):
        h[j, :j] = 1 / sqrt(j * (j + 1))
        h[j, j] = -j / sqrt(j * (j + 1))
    return h


def simplex_code(messages: int, modes: int = 2):
    """Return code matrix in its occupied subspace, photon number, and cutoff.

    First physical basis vector is vacuum. Remaining M-1 vectors are chosen
    within the same K-photon sector. Full unused Fock levels are omitted.
    """
    k = symmetric_cutoff(messages, modes)
    if k is None:
        raise ValueError("The requested coherent branch representation is impossible")
    v = helmert_matrix(messages)
    number = np.diag([0] + [k] * (messages - 1))
    return v, number, k


def bosonic_entropy(mean_per_mode: float) -> float:
    """Thermal entropy in bits, evaluated without large-n cancellation."""
    n = float(mean_per_mode)
    if not isfinite(n) or n < 0:
        raise ValueError("mean_per_mode must be finite and nonnegative")
    if n == 0:
        return 0.0
    if n >= 1:
        # g(n) = log(1+n) + n*log(1+1/n), converted from nats to bits.
        return (log1p(n) + n * log1p(1 / n)) / log(2)
    # This form also handles subnormal n without evaluating 1/n.
    return (1 + n) * log1p(n) / log(2) - n * log2(n)


def binary_entropy(error: float) -> float:
    if not 0 <= error <= 1:
        raise ValueError("error must lie in [0,1]")
    if error in (0, 1):
        return 0.0
    return -error * log2(error) - (1 - error) * log2(1 - error)


def mean_photon_lower_bound(bits: float, modes: int, error: float = 0.0) -> float:
    """Numerical value of a necessary mean-energy bound; not certified rounding.

    This does not claim that a code attains the entropy-based lower bound.
    """
    modes = _positive_int(modes, "modes")
    if not isfinite(bits) or bits < 0 or not 0 <= error <= 0.5:
        raise ValueError("bits must be finite and nonnegative and error in [0,0.5]")
    target = bits * (1 - binary_entropy(error))
    if target == 0:
        return 0.0
    hi = 1.0
    while modes * bosonic_entropy(hi / modes) < target:
        hi *= 2
        if hi > 1e12:
            raise ValueError("Target is too large for this floating-point demo")
    lo = 0.0
    for _ in range(90):
        mid = (lo + hi) / 2
        if modes * bosonic_entropy(mid / modes) >= target:
            hi = mid
        else:
            lo = mid
    return hi