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4aa7bc6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | """Run deterministic exact and numerical release checks; write JSON/CSV results."""
from __future__ import annotations
from fractions import Fraction as F
from pathlib import Path
import random, json, csv
import numpy as np
import networkx as nx
from scipy.linalg import null_space, eigh
from exact_graph import Edge, graph, parameters, classify, exact_halfcell_spectrum, threshold_multiplicity
from port_bound import theta_bound
from fem_check import eigenvalues
ROOT=Path(__file__).resolve().parents[1]
rng=random.Random(260925)
cases=[]
def add(name, data, D=()):
es=[Edge(a,b,F(str(l))) for a,b,l in data]
cases.append((name,es,tuple(D)))
add('DD_interval',[(0,1,2)],(0,1))
add('DN_interval',[(0,1,2.5)],(0,))
add('NN_interval',[(0,1,2)])
add('mixed_star',[(0,1,1.5),(0,2,1),(0,3,1)],(2,3))
add('theta_odd',[(0,1,1),(0,1,1),(0,1,3)])
add('theta_even',[(0,1,2),(0,1,2),(0,1,4)])
add('theta_wrong_parity',[(0,1,1),(0,1,1),(0,1,2)])
add('figure_eight',[(0,0,2),(0,0,4)])
add('figure_eight_bad',[(0,0,1),(0,0,3)])
add('Dirichlet_lasso',[(0,0,2),(0,1,1.5)],(1,))
add('Neumann_lasso',[(0,0,2),(0,1,1)])
add('barbell',[(0,0,2),(0,1,1),(1,1,2)])
add('branch_lasso_tree',[(0,1,1),(0,2,.5),(0,3,2.5),(2,2,2)],(1,))
add('loop_at_degree_four',[(0,0,2.5),(0,1,.5),(0,2,1.5)])
for p in range(3,7):
add(f'bouquet_{p}',[(0,0,2)]*p)
for p in range(2,8):
add(f'pumpkin_{p}',[(0,1,1)]*p)
# Connected simple graph atlas, with all degree-two labels harmlessly retained.
atlas=[g for g in nx.graph_atlas_g() if 2<=len(g)<=6 and nx.is_connected(g)]
for j,g in enumerate(atlas):
for repetition in range(3):
leaves=[v for v in g if g.degree(v)==1]
D=[v for v in leaves if rng.random()<.5]
es=[(a,b,F(rng.randint(1,4),2)) for a,b in g.edges()]
add(f'atlas_{j}_{repetition}',es,D)
# Explicit compatible stars and loop-decorated trees guarantee substantial
# positive coverage, rather than a suite consisting mostly of strict cases.
for i in range(60):
branches=rng.randint(3,6)
D=[]; es=[]
for v in range(1,branches+1):
virtualN=rng.random()<.6
m=rng.randint(0,1) if virtualN else rng.randint(1,2)
es.append((0,v,F(m)+F(int(virtualN),2)))
if virtualN and rng.random()<.5:
es.append((v,v,2))
elif not virtualN:
D.append(v)
add(f'compatible_star_{i}',es,D)
# Compatible nonsymmetric tree skeletons, including multiple branch vertices.
for order in range(2,9):
for tree_index,tree in enumerate(nx.nonisomorphic_trees(order)):
for rep in range(2):
leaves=[v for v in tree if tree.degree(v)==1]
virtualN={v for v in leaves if rng.random()<.55}
D=[v for v in leaves if v not in virtualN]
es=[]
for a,b in tree.edges():
nu=int(a in virtualN)+int(b in virtualN)
m=rng.randint(0,1) if nu else rng.randint(1,2)
es.append((a,b,F(m)+F(nu,2)))
for v in virtualN:
if rng.random()<.5:
es.append((v,v,2))
add(f'compatible_tree_{order}_{tree_index}_{rep}',es,D)
results=[]
for name,es,D in cases:
g=graph(es,D)
d,n,beta,L=parameters(g); B=n+beta
candidate=L+F(B,2)
inregime=candidate.denominator==1 and int(candidate)>=max(B,1 if d else 2)
# Circle is excluded from the inequality and classifier.
circle=all(g.degree(v)==2 for v in g)
certificate=exact_halfcell_spectrum(es,D)
classification=classify(es,D)
exact_sharp=bool(inregime and not circle and certificate['multiplicity']>0 and certificate['top_index']==int(candidate))
assert exact_sharp == classification['saturated'], (name,classification,certificate)
ode_multiplicity=threshold_multiplicity(es,D)
assert certificate['multiplicity']==ode_multiplicity,(name,certificate,ode_multiplicity)
if exact_sharp:
assert certificate['multiplicity']==d+n+2*beta-1,(name,certificate)
results.append(dict(name=name,edges=[[e.u,e.v,str(e.length)] for e in es],Dirichlet=list(D),classification=classification,exact=certificate,ODE_multiplicity=ode_multiplicity,passed=True))
# General abstract quantitative inheritance: independent random matrix tests.
np_rng=np.random.default_rng(260925)
abstract=[]
for case in range(200):
size=12; k=5; m=int(np_rng.integers(1,4)); lower=k-m
lam=2.0; g=float(np_rng.uniform(.2,2.0))
values=np.r_[np.linspace(.3,1.3,lower),np.full(m,lam),lam+g,lam+g+np.arange(1,size-k)]
A=np.diag(values)
ports=int(np_rng.integers(1,4))
C=np_rng.normal(size=(ports,size))
S=C@np.diag(1/values)@C.T
R=C[:,lower:k]@C[:,lower:k].T
rho=max(0.,float(eigh(R,S,eigvals_only=True)[-1]))
bound=g*rho/(lam+g+rho)
Q=null_space(C)
delta=float(eigh(Q.T@A@Q,eigvals_only=True)[k-1]-lam)
assert delta+1e-10>=bound,(delta,bound)
abstract.append(dict(case=case,delta=delta,bound=bound,passed=True))
# Numerical convergence and the concrete strict theta gap certificate.
numerical=[]
for name in ['theta_odd','theta_wrong_parity','figure_eight','Dirichlet_lasso','branch_lasso_tree','barbell']:
_,es,D=next(c for c in cases if c[0]==name)
d,n,beta,L=parameters(graph(es,D)); k=int(L+F(n+beta,2))
for density in (30,60,120):
vals=eigenvalues(es,D,count=max(k+3,12),density=density)
row=dict(name=name,density=density,k=k,lambda_k=float(vals[k-1]),excess=float(vals[k-1]-np.pi**2))
numerical.append(row)
strict=theta_bound((1,1,2),5)
strict['exact_excess_expression']='4*(pi-atan(sqrt(5)))**2-pi**2'
strict['exact_excess_decimal']=float(4*(np.pi-np.arctan(np.sqrt(5)))**2-np.pi**2)
strict['numerical_finest_excess']=next(x['excess'] for x in numerical if x['name']=='theta_wrong_parity' and x['density']==120)
assert strict['exact_excess_decimal']>strict['gap_lower_bound']
theta_fem=[x['excess'] for x in numerical if x['name']=='theta_wrong_parity']
assert all(x>strict['exact_excess_decimal'] for x in theta_fem)
assert all(a>b for a,b in zip(theta_fem,theta_fem[1:]))
assert strict['numerical_finest_excess']>strict['gap_lower_bound']
# Port-coordinate invariance is structural; verify a nonorthogonal example.
U=np.array([[2.,1.],[0.,3.]])
R=np.array(strict['residue']);S=np.array(strict['path_gram'])
assert np.allclose(eigh(U@R@U.T,U@S@U.T,eigvals_only=True),eigh(R,S,eigvals_only=True))
out=ROOT/'data';out.mkdir(exist_ok=True)
(out/'exact_cases.json').write_text(json.dumps(results,indent=2))
(out/'abstract_bound_checks.json').write_text(json.dumps(abstract,indent=2))
(out/'theta_gap_certificate.json').write_text(json.dumps(strict,indent=2))
with (out/'fem_convergence.csv').open('w',newline='') as f:
writer=csv.DictWriter(f,fieldnames=numerical[0].keys());writer.writeheader();writer.writerows(numerical)
summary=dict(exact_graph_cases=len(results),exact_graph_passed=len(results),exact_saturated_cases=sum(r['classification']['saturated'] for r in results),exact_ODE_nullity_crosschecks=len(results),abstract_inequality_cases=len(abstract),abstract_inequality_passed=len(abstract),FEM_runs=len(numerical),failures=0,seed=260925,proof_role='Regression and finite-instance certificates only; the general theorem is proved in the manuscript.',theta_gap_bound=strict['gap_lower_bound'],theta_actual_excess_FEM=strict['numerical_finest_excess'],theta_exact_excess_decimal=strict['exact_excess_decimal'])
(out/'check_summary.json').write_text(json.dumps(summary,indent=2))
print(json.dumps(summary,indent=2))
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