code/verify_primary.py

#!/usr/bin/env python3
"""Exact contact-collar certificate construction and verification.
Python standard library only. No optimizer or earlier verifier is imported.
The output certifies the listed templates, not all unconditional bodies.
"""
from __future__ import annotations
import argparse, itertools as it, json, math, time
from collections import Counter
from fractions import Fraction as F
from functools import lru_cache
from pathlib import Path
ROOT=Path(__file__).resolve().parents[1]
ST=Counter()
def require(ok:bool,msg:str)->None:
    if not ok: raise ValueError(msg)
def dot(a,b): return sum((x*y for x,y in zip(a,b)),F(0))
def solve(a,b):
    """Gauss-Jordan solve over Q; None means a singular square system."""
    n=len(b);z=[[F(x) for x in r]+[F(y)] for r,y in zip(a,b)]
    for k in range(n):
        p=next((j for j in range(k,n) if z[j][k]),None)
        if p is None:return None
        z[k],z[p]=z[p],z[k];h=z[k][k];z[k]=[x/h for x in z[k]]
        for j in range(n):
            if j!=k and z[j][k]:
                h=z[j][k];z[j]=[x-h*y for x,y in zip(z[j],z[k])]
    return tuple(z[j][-1] for j in range(n))
@lru_cache(None)
def restricted_vertices(rows,k):
    ans=set()
    for inds in it.combinations(range(len(rows)),k):
        ST['restricted_linear_systems']+=1
        x=solve([rows[i] for i in inds],[1]*k)
        if x and all(t>0 for t in x) and all(dot(r,x)<=1 for r in rows):ans.add(x)
    return tuple(sorted(ans))
def orthant_vertices(rows):
    """Complete vertices of {x>=0 : rows*x<=1}, including cut-plane vertices."""
    n=len(rows[0]);ans={(F(0),)*n}
    for support_mask in range(1,1<<n):
        s=[i for i in range(n) if support_mask>>i&1];k=len(s)
        rr=tuple(sorted({tuple(r[i] for i in s) for r in rows if any(r[i] for i in s)}))
        if len(rr)<k:continue
        for x in restricted_vertices(rr,k):
            v=[F(0)]*n
            for i,t in zip(s,x):v[i]=t
            ans.add(tuple(v))
    return tuple(sorted(ans))
def orbit(v):
    s=[i for i,x in enumerate(v) if x]
    for signs in it.product((-1,1),repeat=len(s)):
        w=list(v)
        for i,t in zip(s,signs):w[i]*=t
        yield tuple(w)
def signed_rows(rows):
    return tuple(sorted({z for a in rows for z in orbit(a)}))
def expand_pairs(pairs):
    ans=[]
    for d in pairs:
        h=max(map(abs,d));require(h>0,'zero direction')
        for s in (1,-1):ans.append(tuple(F(s*x,h) for x in d))
    require(len(set(ans))==len(ans),'duplicate normalized direction')
    return tuple(ans)
def make_certificate(label,rows,positive_vertices,pairs,save_landings=False):
    n=len(rows[0]);D=expand_pairs(pairs);rows=tuple(tuple(map(F,r)) for r in rows)
    vertices=tuple(positive_vertices)
    require(all(dot(a,v)<=1 for a in rows for v in vertices),'infeasible input point')
    require(all(any(v[i]>=1 for v in vertices) for i in range(n)),'axis intercept lower bound')
    g_candidates=[]
    for v in vertices:
        for a in rows:
            value=dot(a,v)
            if value<1:g_candidates.append(1-value)
            else:g_candidates.extend(2*a[i]*v[i] for i in range(n) if a[i]*v[i]>0)
    g=min(g_candidates);require(g>0,'nonpositive slack gap')
    A=max(sum(a,F(0)) for a in rows)
    B=max(dot(a,tuple(map(abs,d))) for a in rows for d in D)
    witness=[];margins=[];point_count=0
    for v0 in vertices:
        active=[a for a in rows if dot(a,v0)==1]
        if not active:continue
        for v in orbit(v0):
            best=None;bm=None
            for di,d in enumerate(D):
                margin=min(-sum((ai*(dj if vj>0 else -dj if vj<0 else abs(dj)) for ai,vj,dj in zip(a,v,d)),F(0)) for a in active)
                ST['active_row_margin_tests']+=len(active)
                if bm is None or margin>bm:bm=margin;best=di
            require(bm is not None and bm>0,f'uncovered point in {label}: {v}')
            margins.append(bm);witness.append((v,best));point_count+=1
    a=min(margins)
    eta=min(F(1,2),g/F(2*(n+1)))
    tau=eta/(a+B);radius=tau*a/(3*A);beta_lower=3*radius
    core=1-2*A*radius
    require(core>0,'invalid inner core')
    saved=[]
    for v,di in witness:
        w=tuple(x+tau*y for x,y in zip(v,D[di]))
        require(all(dot(r,tuple(map(abs,w)))<=core for r in rows),'uniform-step landing failed')
        ST['uniform_core_landings']+=1;ST['absolute_row_core_checks']+=len(rows)
        if save_landings:saved.append({'point':list(map(str,v)),'direction_index':di,'landing':list(map(str,w))})
    cert={'label':label,'dimension':n,'positive_rows':[[str(x) for x in r] for r in rows],
          'positive_generators':[[str(x) for x in v] for v in vertices],
          'directions':[[str(x) for x in d] for d in D],
          'direction_count':len(D),'signed_boundary_points_checked':point_count,
          'g':str(g),'a':str(a),'A':str(A),'B':str(B),'eta':str(eta),'tau':str(tau),
          'closed_Hausdorff_radius':str(radius),'blocking_slack_lower_bound':str(beta_lower),
          'open_Hausdorff_radius':str(beta_lower/2),'inner_core_factor':str(core)}
    if save_landings:cert['landings']=saved
    ST['certificates']+=1
    return cert

def model_certificate():
    j=json.loads((ROOT/'data/cyclic_discovery.json').read_text())
    rows=tuple(tuple(F(x,j['rhs']) for x in r) for r in j['positive_rows'])
    pos=orthant_vertices(rows)
    require(set(pos)=={tuple(map(F,v)) for v in j['positive_orthant_vertices']},'discovery vertex list mismatch')
    cert=make_certificate('irregular cyclic five-body',rows,pos,j['opposite_pair_representatives'],True)
    # Verify all extreme perturbations of every saved boundary point. This is
    # a stress check; the continuum theorem, not this finite set, covers all L.
    D=[tuple(map(F,d)) for d in cert['directions']];h=F(cert['closed_Hausdorff_radius']);t=F(cert['tau']);core=F(cert['inner_core_factor'])
    for rec in cert['landings']:
        v=tuple(map(F,rec['point']));d=D[rec['direction_index']]
        for s in it.product((-1,1),repeat=5):
            w=tuple(v[i]+h*s[i]+t*d[i] for i in range(5))
            require(all(dot(r,tuple(map(abs,w)))<=core for r in rows),'perturbed landing failed')
            ST['perturbed_vertex_landings']+=1;ST['perturbed_absolute_row_checks']+=len(rows)
    # A deliberate corruption must be rejected at a recorded witness.
    rec=next(r for r in cert['landings'] if any(F(x) for x in r['point']))
    v=tuple(map(F,rec['point']));d=D[rec['direction_index']]
    bad=tuple(v[i]-t*d[i] for i in range(5))
    require(any(dot(r,tuple(map(abs,bad)))>1 for r in rows),'mutation was not rejected')
    cert['opposite_direction_mutation_rejected']=True
    (ROOT/'data/cyclic_certificate.json').write_text(json.dumps(cert,indent=2)+'\n')
    return cert

def legacy_atlas():
    j=json.loads((ROOT/'data/legacy_catalogue.json').read_text());out=[]
    require(len(j['representatives'])==180 and len(j['certificates'])==360,'catalogue length')
    for k,c in enumerate(j['certificates']):
        h=tuple(c['facets']);unit=tuple(tuple(F((m>>i)&1) for i in range(5)) for m in h)
        pv=orthant_vertices(unit)
        if c['body']=='P':rows=unit;points=pv
        else:
            rows=tuple(v for v in pv if any(v));points=unit
        cert=make_certificate(f"{c['body']}_{c['index']:03d}",rows,points,c['pairs'])
        cert.update({'legacy_index':c['index'],'legacy_body':c['body'],'support_masks':list(h),'orbit_size':c['orbit_size']})
        out.append(cert)
        if (k+1)%60==0:print(f'Exact atlas: {k+1}/360',flush=True)
    (ROOT/'data/robust_atlas_360.json').write_text(json.dumps(out,indent=2)+'\n')
    return out

def integer_cube_check():
    # Complete n=2,N=14 cube representative of the integer finite-level test.
    n,N=2,14;threshold=(4*n-3)*N;margin=4*(n-1)*N
    normals=[z for z in it.product(range(-N,N+1),repeat=n) if sum(map(abs,z))==N]
    directions=list(it.product((-N,N),repeat=n));count=0
    for y in it.product(range(-N,N+1),repeat=n):
        near=[z for z in normals if N*N-sum(a*b for a,b in zip(z,y))<=threshold]
        require(any(all(sum(a*b for a,b in zip(z,d))<=-margin for z in near) for d in directions),'integer cube certificate failed')
        count+=1
    return {'dimension':n,'N':N,'grid_points':count,'normal_grid_points':len(normals),'point_normal_incidence_tests':count*len(normals),'status':'PASS','scope':'one cube template, not the universal finite statement'}

def main():
    p=argparse.ArgumentParser();p.add_argument('--skip-atlas',action='store_true');args=p.parse_args();start=time.perf_counter()
    model=model_certificate();atlas=[] if args.skip_atlas else legacy_atlas();ic=integer_cube_check()
    report={'status':'PASS','arithmetic':'integers and fractions.Fraction only','stats':dict(ST),'model_direction_count':model['direction_count'],
            'model_closed_radius':model['closed_Hausdorff_radius'],'model_open_radius':model['open_Hausdorff_radius'],
            'model_margin':model['a'],'model_positive_generators':len(model['positive_generators']),
            'atlas_certificates':len(atlas),'integer_cube_test':ic,'unrestricted_5d_theorem_proved':False,
            'universal_integer_level_completed':None,'elapsed_seconds':round(time.perf_counter()-start,3)}
    if atlas:
        h=min(F(c['closed_Hausdorff_radius']) for c in atlas)
        report.update({'atlas_minimum_closed_radius':str(h),'atlas_direction_count_distribution':dict(sorted(Counter(c['direction_count'] for c in atlas).items())),
                       'atlas_minimum_radius_labels':[c['label'] for c in atlas if F(c['closed_Hausdorff_radius'])==h],
                       'atlas_labeled_instances':sum(c['orbit_size'] for c in atlas)})
    (ROOT/'verification/primary_report.json').write_text(json.dumps(report,indent=2)+'\n');print(json.dumps(report,indent=2))
if __name__=='__main__':main()