code/exact_graph.py

"""Exact half-cell spectral certificates and a rational-length rigidity classifier.

No floating-point eigensolver is used by exact_halfcell_spectrum. At frequency pi,
subdivision into edges of length 1/2 reduces the eigenvalue count to rational
inertia of a vertex adjacency matrix. See MANUSCRIPT, computational appendix.
"""
from __future__ import annotations
from dataclasses import dataclass
from fractions import Fraction
from typing import Iterable
import networkx as nx
import sympy as sp

@dataclass(frozen=True)
class Edge:
    u: int
    v: int
    length: Fraction
    def __post_init__(self):
        object.__setattr__(self, 'length', Fraction(self.length))
        if self.length <= 0:
            raise ValueError('Edge lengths must be strictly positive.')


def graph(edges: Iterable[Edge], dirichlet=()) -> nx.MultiGraph:
    es = list(edges)
    if not es:
        raise ValueError('At least one edge is required.')
    g = nx.MultiGraph()
    for e in es:
        g.add_edge(e.u, e.v, length=e.length)
    ds = set(dirichlet)
    if not nx.is_connected(g):
        raise ValueError('The graph must be connected.')
    if not ds <= set(g):
        raise ValueError('Unknown Dirichlet vertex.')
    if any(g.degree(v) != 1 for v in ds):
        raise ValueError('This release allows Dirichlet conditions only at leaves.')
    nx.set_node_attributes(g, {v: v in ds for v in g}, 'D')
    return g


def suppress_degree_two(g: nx.MultiGraph) -> nx.MultiGraph:
    g = g.copy()
    while True:
        changed = False
        for v in list(g):
            if g.degree(v) != 2 or g.nodes[v].get('D', False):
                continue
            inc = list(g.edges(v, keys=True, data=True))
            if len(inc) != 2 or any(a == b for a, b, _, _ in inc):
                continue  # A single circle remains a loop and is excluded later.
            ends = [b if a == v else a for a, b, _, _ in inc]
            length = sum((z[3]['length'] for z in inc), Fraction())
            g.remove_node(v)
            g.add_edge(ends[0], ends[1], length=length)
            changed = True
            break
        if not changed:
            return g


def parameters(g):
    d = sum(bool(g.nodes[v].get('D', False)) for v in g)
    n = sum(g.degree(v) == 1 and not g.nodes[v].get('D', False) for v in g)
    beta = g.number_of_edges() - g.number_of_nodes() + 1
    length = sum((x['length'] for _, _, x in g.edges(data=True)), Fraction())
    return d, n, beta, length


def classify(edges: Iterable[Edge], dirichlet=(), k: int | None = None):
    """Classify exact high-branch saturation using rational arithmetic.

    If k=None, test the threshold pi^2 (d=1). Otherwise d=L/(k-(N+beta)/2).
    A False answer means non-saturation in the declared regime, not a numerical
    lower bound on the amount of strictness.
    """
    g = suppress_degree_two(graph(edges, dirichlet))
    D, N, beta, L = parameters(g)
    B = N + beta
    if len(g) == 1 and g.number_of_edges() == 1:
        return dict(saturated=False, family='excluded-circle', reason='The circle has a different bound.')
    if k is None:
        candidate = L + Fraction(B, 2)
        if candidate.denominator != 1:
            return dict(saturated=False, family='none', reason='The proposed index is not integral.')
        k = int(candidate)
        d = Fraction(1)
    else:
        if not isinstance(k, int):
            raise ValueError('k must be an integer.')
        c = Fraction(k) - Fraction(B, 2)
        if c <= 0:
            return dict(saturated=False, family='outside-regime', reason='Nonpositive target frequency.')
        d = L / c
    base = dict(k=k, D=D, N=N, beta=beta, d=str(d), multiplicity=D+N+2*beta-1)
    if k < max(B, 1 if D else 2):
        return dict(base, saturated=False, family='outside-regime', reason='Outside the high-index theorem.')
    q = lambda x: 2*x/d
    def positive_integer(x):
        return x.denominator == 1 and x > 0
    if len(g) == 1 and g.number_of_edges() == 2:
        good = all(positive_integer(x['length']/(2*d)) for _, _, x in g.edges(data=True))
        return dict(base, saturated=good, family='figure-eight', reason='Both loop lengths must be positive even cell counts.')
    if len(g) == 2 and g.number_of_edges() == 3 and nx.number_of_selfloops(g) == 0:
        counts = [x['length']/d for _, _, x in g.edges(data=True)]
        good = all(positive_integer(x) for x in counts) and len({int(x) % 2 for x in counts}) == 1
        return dict(base, saturated=good, family='theta', reason='Three positive integer cell counts of common parity.')
    skeleton = g.copy()
    loop_vertices = set()
    for a, b, key, x in list(g.edges(keys=True, data=True)):
        if a == b:
            if g.degree(a) != 3 or a in loop_vertices:
                return dict(base, saturated=False, family='excluded-topology', reason='A nonexceptional loop must be pendant at a degree-three vertex.')
            if not positive_integer(x['length']/(2*d)):
                return dict(base, saturated=False, family='lasso-tree', reason='An attached loop has incompatible phase.')
            loop_vertices.add(a)
            skeleton.remove_edge(a, b, key)
    if len(skeleton) < 2 or not nx.is_tree(skeleton):
        return dict(base, saturated=False, family='excluded-topology', reason='The nonloop skeleton must be a tree.')
    virtual_N = loop_vertices | {v for v in g if g.degree(v) == 1 and not g.nodes[v].get('D', False)}
    for a, b, x in skeleton.edges(data=True):
        nu = int(a in virtual_N) + int(b in virtual_N)
        m = x['length']/d - Fraction(nu, 2)
        if m.denominator != 1 or m < 0 or (nu == 0 and m < 1):
            return dict(base, saturated=False, family='lasso-tree' if beta else 'tree', reason='A skeleton edge violates the integer/half-integer phase rule.')
    return dict(base, saturated=True, family='lasso-tree' if beta else 'tree', reason='All exact topology and metric conditions hold.')


def rational_inertia(matrix):
    """Return (positive, negative, zero) using exact 1x1/2x2 congruences."""
    A = sp.Matrix(matrix)
    if A != A.T:
        raise ValueError('A symmetric matrix is required.')
    pos = neg = zero = 0
    while A.rows:
        size = A.rows
        pivot = next((i for i in range(size) if A[i,i] != 0), None)
        if pivot is not None:
            perm = [pivot] + [i for i in range(size) if i != pivot]
            A = A.extract(perm, perm)
            a = A[0,0]
            pos += int(bool(a > 0)); neg += int(bool(a < 0))
            v = A[1:,0]
            A = A[1:,1:] - v*v.T/a
        else:
            pair = next(((i,j) for i in range(size) for j in range(i+1,size) if A[i,j] != 0), None)
            if pair is None:
                zero += size
                break
            i,j = pair
            perm = [i,j] + [h for h in range(size) if h not in (i,j)]
            A = A.extract(perm, perm)
            pivot_block = A[:2,:2]
            X = A[2:,:2]
            A = A[2:,2:] - X*pivot_block.inv()*X.T
            pos += 1; neg += 1
    return pos, neg, zero


def exact_halfcell_spectrum(edges: Iterable[Edge], dirichlet=()):
    """Exact counts below/at pi^2; lengths must be positive multiples of 1/2."""
    es = list(edges)
    g = graph(es, dirichlet)
    ds = set(dirichlet)
    vertices = {v:i for i,v in enumerate(g)}
    next_node = len(vertices)
    segments = []
    for e in es:
        q = 2*e.length
        if q.denominator != 1:
            raise ValueError('All lengths must be integer multiples of 1/2.')
        count = int(q)
        nodes = [vertices[e.u]] + list(range(next_node, next_node+count-1)) + [vertices[e.v]]
        next_node += count-1
        segments += list(zip(nodes[:-1],nodes[1:]))
    A = sp.zeros(next_node)
    for a,b in segments:
        A[a,b] += 1
        A[b,a] += 1  # A loop contributes two on the diagonal.
    keep = [i for i in range(next_node) if i not in {vertices[v] for v in ds}]
    A = A.extract(keep,keep)
    pos, neg, zero = rational_inertia(A)
    return dict(below=pos, multiplicity=zero, top_index=pos+zero, adjacency_dimension=len(keep), adjacency_inertia=[pos,neg,zero])


def threshold_multiplicity(edges: Iterable[Edge], dirichlet=()):
    """Independent exact ODE coefficient-nullity check at frequency pi."""
    es = list(edges)
    g = graph(es, dirichlet)
    ds = set(dirichlet)
    traces = {v:[] for v in g}; derivatives = {v:[] for v in g}
    n = 2*len(es)
    for j,e in enumerate(es):
        q = 2*e.length
        if q.denominator != 1:
            raise ValueError('Half-cell lengths are required.')
        c = (1,0,-1,0)[int(q)%4]; s = (0,1,0,-1)[int(q)%4]
        t0=sp.zeros(1,n); t1=sp.zeros(1,n); p0=sp.zeros(1,n); p1=sp.zeros(1,n)
        t0[2*j]=1; t1[2*j]=c; t1[2*j+1]=s
        p0[2*j+1]=1; p1[2*j]=s; p1[2*j+1]=-c
        traces[e.u].append(t0); traces[e.v].append(t1)
        derivatives[e.u].append(p0); derivatives[e.v].append(p1)
    rows=[]
    for v in g:
        if v in ds:
            rows.append(traces[v][0])
        else:
            rows += [t-traces[v][0] for t in traces[v][1:]]
            rows.append(sum(derivatives[v],sp.zeros(1,n)))
    M=sp.Matrix.vstack(*rows)
    return n-M.rank()