src/continuation.py

"""Exact conditional-memory certificates for finite positive branch models.

The model is known.  A compact state receives externally supplied branch labels;
it does not, unless separately certified, predict their probabilities.
"""
from __future__ import annotations
from dataclasses import dataclass
from itertools import combinations
from typing import Sequence
import numpy as np
import sympy as sp
from scipy.optimize import linprog


def pairs(n: int) -> list[tuple[int, int]]:
    if n < 2:
        raise ValueError("Ambient dimension must be at least two.")
    return list(combinations(range(n), 2))


def wedge(a: sp.Matrix, b: sp.Matrix) -> sp.Matrix:
    a, b = sp.Matrix(a), sp.Matrix(b)
    if a.rows != 1 or b.shape != a.shape:
        raise ValueError("Wedge arguments must be equal-length rows.")
    return sp.Matrix([[a[i]*b[j] - a[j]*b[i] for i,j in pairs(a.cols)]])


def compound2(T: sp.Matrix) -> sp.Matrix:
    T = sp.Matrix(T)
    if T.rows != T.cols or T.rows < 2:
        raise ValueError("A square matrix of order at least two is required.")
    ij = pairs(T.rows)
    return sp.Matrix([[T[i,k]*T[j,l]-T[i,l]*T[j,k]
                       for k,l in ij] for i,j in ij])


def row_basis(F: sp.Matrix) -> sp.Matrix:
    F = sp.Matrix(F)
    if F.rows == 0:
        return F
    inds = F.T.rref()[1]
    return F[list(inds), :] if inds else sp.zeros(0, F.cols)


@dataclass(frozen=True)
class Ratio:
    numerator: sp.Matrix
    denominator: sp.Matrix
    word: tuple[int, ...]


@dataclass
class ExteriorCertificate:
    normalizer: sp.Matrix
    readouts: tuple[sp.Matrix, ...]
    seeds: tuple[Ratio, ...]
    branches: tuple[sp.Matrix, ...]
    ratios: list[Ratio]
    R: sp.Matrix
    transports: list[sp.Matrix]
    depth: int

    def signature_exact(self, x: sp.Matrix) -> sp.Matrix:
        x = sp.Matrix(x)
        self._validate_state(x)
        out = []
        for r in self.ratios:
            den = (r.denominator*x)[0]
            if den <= 0:
                raise ValueError("A stored denominator is nonpositive.")
            out.append((r.numerator*x)[0]/den)
        return sp.Matrix(out)

    def _validate_state(self, x: sp.Matrix) -> None:
        if x.shape != (self.normalizer.cols, 1):
            raise ValueError("Incorrect state shape.")
        if (self.normalizer*x)[0] != 1 or any(v < 0 for v in x):
            raise ValueError("Expected an exactly normalized simplex state.")

    def fiber_matrix(self, z: Sequence) -> sp.Matrix:
        if len(z) != len(self.ratios):
            raise ValueError("Incorrect signature dimension.")
        rows = [r.numerator-sp.sympify(v)*r.denominator
                for r,v in zip(self.ratios,z)]
        return sp.Matrix.vstack(*rows) if rows else sp.zeros(0, self.normalizer.cols)

    def reconstruct_fiber_float(self, z: Sequence[float]) -> np.ndarray:
        """Find any simplex representative; numerical, not an exact certificate."""
        L = self.fiber_matrix(z)
        eq = np.array(sp.Matrix.vstack(self.normalizer,L),dtype=float)
        rhs = np.zeros(eq.shape[0]); rhs[0] = 1.0
        res = linprog(np.zeros(eq.shape[1]), A_eq=eq, b_eq=rhs,
                      bounds=(0,None), method="highs")
        if not res.success or np.max(np.abs(eq@res.x-rhs))>1e-7:
            raise ValueError("No numerically certified feasible representative.")
        return res.x

    def update_float(self, z: Sequence[float], branch: int) -> np.ndarray:
        if branch < 0 or branch >= len(self.branches):
            raise ValueError("Unknown branch.")
        x = self.reconstruct_fiber_float(z)
        T = np.array(self.branches[branch], dtype=float)
        y = T@x
        if np.sum(y) <= 0:
            raise ValueError("Impossible branch.")
        y /= np.sum(y)
        return np.array([float(np.array(r.numerator,dtype=float).ravel()@y /
                               (np.array(r.denominator,dtype=float).ravel()@y))
                         for r in self.ratios])

    def verify(self) -> bool:
        """Check both directions: seed coverage, genuine orbit rows, closure.

        Instrument probability normalization is an additional model obligation;
        this checker permits a library containing outcomes of several actions.
        """
        try:
            n = self.normalizer.cols
            if self.normalizer != sp.ones(1,n) or n < 2:
                return False
            if len(self.branches) != len(self.transports) or not self.branches:
                return False
            if self.R.shape != (len(self.ratios), n*(n-1)//2):
                return False
            if self.R.rank() != self.R.rows:
                return False
            if any(T.shape != (n,n) or any(v <= 0 for v in T)
                   for T in self.branches):
                return False
            for seed in self.seeds:
                if (seed.numerator.shape != (1,n) or
                    seed.denominator.shape != (1,n) or seed.word or
                    any(v <= 0 for v in seed.denominator)):
                    return False
                if sp.Matrix.vstack(self.R,wedge(seed.numerator,
                                                seed.denominator)).rank() != self.R.rows:
                    return False
            for i,r in enumerate(self.ratios):
                if (r.numerator.shape != (1,n) or
                    r.denominator.shape != (1,n) or
                    any(v <= 0 for v in r.denominator) or
                    wedge(r.numerator,r.denominator) != self.R[i,:]):
                    return False
                if any(a < 0 or a >= len(self.branches) for a in r.word):
                    return False
                genuine = False
                for seed in self.seeds:
                    beta, alpha = seed.numerator, seed.denominator
                    # Words are chronological; row products are reverse ordered.
                    for a in reversed(r.word):
                        beta, alpha = beta*self.branches[a], alpha*self.branches[a]
                    if beta == r.numerator and alpha == r.denominator:
                        genuine = True
                        break
                if not genuine:
                    return False
            for T,H in zip(self.branches,self.transports):
                if H.shape != (self.R.rows,self.R.rows):
                    return False
                if self.R*compound2(T) != H*self.R:
                    return False
            return True
        except (ValueError, TypeError, IndexError):
            return False


def compile_exterior(branches: Sequence[sp.Matrix],
                     readouts: Sequence[sp.Matrix]) -> ExteriorCertificate:
    """Wrapper for ordinary affine mandatory readouts."""
    if not branches:
        raise ValueError("At least one branch is required.")
    n=sp.Matrix(branches[0]).rows
    seeds=[Ratio(sp.Matrix(f),sp.ones(1,n),()) for f in readouts]
    return compile_ratios(branches,seeds)


def compile_ratios(branches: Sequence[sp.Matrix],
                   seeds: Sequence[Ratio]) -> ExteriorCertificate:
    """Exact closure for linear-fractional mandatory readouts.

    All denominators must be strictly positive on the entire simplex. Branches
    are known strictly positive matrices. Instrument normalization is separate.
    """
    if not branches:
        raise ValueError("At least one branch is required.")
    Ts = tuple(sp.Matrix(T) for T in branches)
    n = Ts[0].rows
    for T in Ts:
        if T.shape != (n,n) or any(x <= 0 for x in T):
            raise ValueError("Branches must be square and strictly positive.")
    fs = tuple(sp.Matrix(r.numerator) for r in seeds)
    for r in seeds:
        if r.word:
            raise ValueError("Initial readouts must have empty provenance words.")
        if r.numerator.shape != (1,n) or r.denominator.shape != (1,n):
            raise ValueError("Readout rows have wrong dimensions.")
        if any(x<=0 for x in r.denominator):
            raise ValueError("Denominator must be positive on the full simplex.")
    e = sp.ones(1,n)
    ratios: list[Ratio] = []
    R = sp.zeros(0,n*(n-1)//2)
    def admit(beta,alpha,word):
        nonlocal R
        r=wedge(beta,alpha)
        if sp.Matrix.vstack(R,r).rank() > R.rows:
            R=sp.Matrix.vstack(R,r)
            ratios.append(Ratio(beta,alpha,word))
    for r in seeds:
        admit(r.numerator,r.denominator,r.word)
    cursor=0
    while cursor < len(ratios):
        r=ratios[cursor]; cursor+=1
        for a,T in enumerate(Ts):
            admit(r.numerator*T,r.denominator*T,(a,)+r.word)
    Hs=[]
    for T in Ts:
        target=R*compound2(T)
        if R.rows:
            hrows=[R.T.gauss_jordan_solve(target[i,:].T)[0].T
                   for i in range(R.rows)]
            H=sp.Matrix.vstack(*hrows)
        else:
            H=sp.zeros(0,0)
        Hs.append(H)
    cert=ExteriorCertificate(e,fs,tuple(seeds),Ts,ratios,R,Hs,
                             max((len(r.word) for r in ratios),default=0))
    if not cert.verify():
        raise ArithmeticError("Internal closure verification failed.")
    return cert


def snapshot_space(cert: ExteriorCertificate) -> sp.Matrix:
    """Least affine initial-information space for the conditional contract.

    Unlike affine_closure, these coordinates need not update from themselves.
    """
    rows=[cert.normalizer]
    for r in cert.ratios:
        rows.extend([r.numerator,r.denominator])
    return row_basis(sp.Matrix.vstack(*rows))


def factor_two_fixture(r: int):
    """r independent ratios, exactly 2r affine snapshot scalars, reset future."""
    if r<1: raise ValueError("Positive rank required.")
    n=2*r+1; e=sp.ones(1,n); E=sp.eye(n)
    beta=[E[2*j,:]-E[n-1,:] for j in range(r)]
    alpha=[e+sp.Rational(1,4)*(E[2*j+1,:]-E[n-1,:]) for j in range(r)]
    seeds=[Ratio(b,a,()) for b,a in zip(beta,alpha)]
    return compile_ratios([sp.ones(n)/n],seeds)


def affine_closure(branches: Sequence[sp.Matrix], initial: sp.Matrix) -> sp.Matrix:
    """Least fixed affine self-updating space, using reset-or-invariance."""
    W=row_basis(initial)
    n=W.cols
    if sp.Matrix.vstack(W,sp.ones(1,n)).rank()!=W.rows:
        raise ValueError("Normalizer is missing.")
    while True:
        extra=[W*T for T in branches if (W*T).rank()>1]
        new=row_basis(sp.Matrix.vstack(W,*extra))
        if new.rows==W.rows:
            return W
        W=new


def hadamard_exact(n: int) -> sp.Matrix:
    if n<4 or n & (n-1):
        raise ValueError("Use a power of two n >= 4.")
    H=sp.Matrix([[1]])
    while H.rows<n:
        H=sp.Matrix.vstack(sp.Matrix.hstack(H,H),sp.Matrix.hstack(H,-H))
    return H


@dataclass
class Family:
    n: int
    F: sp.Matrix
    LA: sp.Matrix
    LB: sp.Matrix
    A: sp.Matrix
    B: sp.Matrix
    delta: sp.Rational
    c: sp.Rational = sp.Rational(1,2)
    tau: sp.Rational = sp.Rational(1,4)
    gamma: sp.Rational = sp.Rational(1,8)
    kappa: sp.Rational = sp.Rational(1,8)

    def signature(self,x: sp.Matrix) -> sp.Matrix:
        z=self.F*sp.Matrix(x)
        if z[0]!=1:
            raise ValueError("Not normalized.")
        return sp.Matrix([z[1], z[2]/(1+self.tau*z[3])])

    def compact_step(self, state: Sequence, label: int) -> sp.Matrix:
        if label==0:
            return sp.Matrix([self.gamma*state[1],self.kappa*state[1]])
        if label==1:
            return sp.zeros(2,1)
        raise ValueError("Unknown label.")


def separation_family(n: int, delta=0) -> Family:
    F=hadamard_exact(n)
    c,tau,g,k=map(sp.Rational,[sp.Rational(1,2),sp.Rational(1,4),
                              sp.Rational(1,8),sp.Rational(1,8)])
    delta=sp.Rational(delta)
    if not (0<=delta<sp.Rational(1,8)):
        raise ValueError("Require 0 <= delta < 1/8.")
    lam=sp.Rational(1,8*n)
    LA=sp.zeros(n); LB=sp.zeros(n)
    LA[0,0]=c; LA[0,3]=c*tau
    LA[1,2]=c*g; LA[2,2]=c*k
    LB[0,0]=1-c; LB[0,3]=-c*tau
    LB[1,3]=delta
    for j in range(3,n-1): LB[j,j+1]=lam
    A=F.T*LA*F/n; B=F.T*LB*F/n
    return Family(n,F,LA,LB,A,B,delta,c,tau,g,k)


def normalize(T: sp.Matrix,x: sp.Matrix) -> sp.Matrix:
    y=T*x; p=sum(y)
    if p<=0: raise ValueError("Nonpositive branch mass.")
    return y/p


def jacobian_rank(cert: ExteriorCertificate, x: sp.Matrix) -> int:
    z=cert.signature_exact(x)
    return cert.fiber_matrix(z).rank()


def reconstruct_hidden_from_B_readouts(n: int, observations: Sequence,
                                        delta) -> sp.Matrix:
    """Recover initial coordinates z3..z(n-1) after the rank-one rewrite.

    observation[t] = mandatory f after t+1 consecutive B outcomes.
    Exact arithmetic; small delta/late observations can be ill-conditioned.
    """
    delta=sp.sympify(delta)
    if delta<=0 or len(observations)!=n-3:
        raise ValueError("Need positive delta and n-3 successive B readouts.")
    d0=sp.Rational(1,2); d1=sp.Rational(1,8); lam=sp.Rational(1,8*n)
    P=sp.Integer(1); out=[]
    for t,y in enumerate(observations):
        a=sp.simplify(d0*y/(delta+d1*y))
        out.append(sp.simplify(a*P/lam**t))
        P=sp.simplify(P*(d0-d1*a))
    return sp.Matrix(out)