Datasets:

PureOne
/

Nhairun-Exact-Complexity

#!/usr/bin/env python3
"""Finite exact audits for the scalar strengthening and matching approximation."""
from fractions import Fraction as F
from itertools import permutations,product
from pathlib import Path
import json,random
from scalar_ordering import (build_scalar_reduction,endpoint,raw_residual,
    interpolate_consecutive,determinant_mod,coefficient_oracle,next_prime,
    approximate_scalar_order,degree_bound)

CHECKS={}

def check(condition,label):
    if not condition:raise AssertionError(label)
    CHECKS[label]=CHECKS.get(label,0)+1

def scalar_case(values,rho):
    m=build_scalar_reduction(values,rho)
    n,N=m['n'],m['N']
    check(all(abs(t)<F(1,2) for t in m['centers']),'scalar_bounded_centers')
    sums=[]
    for mask in range(1<<N):
        sums.append(sum(m['weights'][i] for i in range(N) if mask>>i&1))
    check(len(set(sums))==len(sums),'rational_subset_weight_injectivity')
    min_abs=None;zeros=0;canonical_count=0;count=0
    selected=[]
    for order in permutations(range(N)):
        count+=1
        digits=[0]*n
        positions=[[] for _ in range(n)]
        for p,j in enumerate(order):
            digits[j//2]+=m['weights'][p]
            positions[j//2].append(p)
        control=sum(a*b for a,b in zip(m['markers'],digits))
        canonical=all(pos==[2*i,2*i+1] for i,pos in enumerate(positions))
        required=m['target']//m['Q']
        check((control==required)==canonical,'marker_control_equivalence')
        res=raw_residual(m,order)
        perturbation=sum(m['weights'][p]*(m['data'][j//2] if j%2==0 else -m['data'][j//2]) for p,j in enumerate(order))
        check(abs(perturbation)<=m['D'],'data_term_dominance_bound')
        if not canonical:
            check(abs(res)>=m['Q']-m['D'],'noncanonical_integer_separation')
        else:
            canonical_count+=1
            signs=[]
            for i in range(n):
                signs.append(1 if order[2*i+1]%2==0 else -1)
            check(res==m['H']*sum(s*a for s,a in zip(signs,values)),'canonical_scalar_partition_identity')
        zeros+=res==0
        min_abs=abs(res) if min_abs is None else min(min_abs,abs(res))
        if count%max(1,(__import__('math').factorial(N)//24))==0:selected.append(order)
    # Add all canonical orientations so perturbation checks include actual
    # zero-error YES witnesses, rather than only generic sampled orders.
    for signs in product([-1,1],repeat=n):
        order=[]
        for i,sign in enumerate(signs):
            order.extend([2*i+1,2*i] if sign==1 else [2*i,2*i+1])
        selected.append(tuple(order))
    expected=any(sum(s*a for s,a in zip(signs,values))==0 for signs in product([-1,1],repeat=n))
    check((zeros>0)==expected,'scalar_partition_decision_equivalence')
    for order in selected:
        state=F(0)
        for j in order:
            state=rho*state+(1-rho)*m['centers'][j]
            check(abs(state)<F(1,2),'scalar_all_checked_prefixes_bounded')
        check(state==F(raw_residual(m,order),rho.denominator**(N-1)*m['R']),'scalar_recurrence_integer_identity')
        # A uniform center perturbation realizes a near-worst-case total shift.
        eta=m['gap']/4
        perturbed=[t+eta for t in m['centers']]
        actual=endpoint(perturbed,order,rho)
        check(abs(actual-state)<=eta,'scalar_perturbation_lipschitz_bound')
        if state==0:
            check(actual*actual<=m['gap']**2/16,'robust_yes_margin')
        else:
            check(actual*actual>=9*m['gap']**2/16,'robust_no_margin')
    return dict(values=values,rho=str(rho),orders=count,canonical_orders=canonical_count,
        zero_orders=zeros,partition_exists=expected,minimum_absolute_integer_residual=str(min_abs),
        centers=[str(t) for t in m['centers']],normalizer=str(m['R']),
        guaranteed_endpoint_gap=str(m['gap']),maximum_center_encoding_bits=max(max(abs(t.numerator).bit_length(),t.denominator.bit_length()) for t in m['centers']))

def matching_audits():
    rng=random.Random(93811)
    interpolation=[]
    for degree in range(13):
        p=next_prime(1000+degree)
        coefficients=[rng.randrange(p) for _ in range(degree+1)]
        values=[sum(c*pow(x,j,p) for j,c in enumerate(coefficients))%p for x in range(degree+1)]
        check(interpolate_consecutive(values,p)==coefficients,'finite_field_interpolation')
    for n in range(1,5):
        for trial in range(6):
            costs=[[None if rng.randrange(8)==0 else rng.randrange(5) for _ in range(n)] for _ in range(n)]
            attainable={sum(costs[i][j] for i,j in enumerate(order)) for order in permutations(range(n)) if all(costs[i][j] is not None for i,j in enumerate(order))}
            p=next_prime(8*n**3*(degree_bound(costs)+1)+1)
            observed=coefficient_oracle(costs,p,rng)
            check(observed<=attainable,'matching_oracle_no_false_positives')
            # The fixed seed is an implementation audit, not a proof that
            # every random labeling succeeds.
            check(observed==attainable,'matching_oracle_observed_full_support')
    # Equal edge labels can erase all genuine matchings by cancellation.
    check(determinant_mod([[1,1],[1,1]],101)==0,'determinant_cancellation_witness')
    return dict(interpolation_degrees_tested=13,matching_instances=24)

def approximate_audits():
    configurations=[
      ([F(-1),F(0),F(1)],F(1,2),F(0),F(0),F(1,2)),
      ([F(-1,2),F(1,4),F(3,4)],F(2,3),F(1,4),F(-1,4),F(1,2)),
      ([F(-3,4),F(-1,4),F(1,4),F(3,4)],F(1,2),F(0),F(0),F(1,2)),
      ([F(-1),F(1,3),F(2,3),F(1)],F(9,10),F(-1,2),F(1,2),F(1,2)),
      ([F(-1,2),F(1,2),F(3,4)],F(1,3),F(1),F(-1),F(1,4)),
      ([F(0),F(0),F(0)],F(1,2),F(0),F(0),F(1,2))]
    results=[]
    for i,(centers,rho,initial,target,eps) in enumerate(configurations):
        optimum=min((endpoint(centers,order,rho,initial)-target)**2 for order in permutations(range(len(centers))))
        out=approximate_scalar_order(centers,rho,eps,F(1,16),initial,target,random.Random(451+i))
        check(sorted(out['order'])==list(range(len(centers))),'approximation_feasible_permutation')
        check(out['objective']==(endpoint(centers,out['order'],rho,initial)-target)**2,'approximation_exact_output_certificate')
        check(out['objective']<=optimum+eps,'observed_additive_approximation_guarantee')
        check(out['statistics']['degree']<=32*len(centers)/eps+len(centers),'rounded_degree_bound')
        results.append(dict(centers=list(map(str,centers)),rho=str(rho),initial=str(initial),target=str(target),epsilon=str(eps),order=out['order'],objective=str(out['objective']),exact_optimum=str(optimum),statistics=out['statistics']))
    return results

def main():
    cases=[(v,rho) for rho in [F(1,2),F(1,3),F(2,3),F(9,10)] for v in [[1,2,3],[1,1,3]]]
    cases.extend([(v,F(1,2)) for v in [[1,2,3,4],[1,2,4,8]]])
    result=dict(arithmetic='Exact integers, rational fractions, and exact finite-field arithmetic; no floating-point proof decisions.',scalar_cases=[scalar_case(v,rho) for v,rho in cases],matching_audits=matching_audits(),approximation_cases=approximate_audits())
    result['assertions']=CHECKS
    result['assertions_total']=sum(CHECKS.values())
    result['scalar_orders_total']=sum(case['orders'] for case in result['scalar_cases'])
    result['baseline_assertions_retained']=344707
    result['combined_assertions']=344707+result['assertions_total']
    Path(__file__).with_name('broadening_verification_results.json').write_text(json.dumps(result,indent=2)+'\n')
    print(json.dumps({key:result[key] for key in ['assertions_total','scalar_orders_total','baseline_assertions_retained','combined_assertions']},indent=2))

if __name__=='__main__':main()

	#!/usr/bin/env python3
	"""Finite exact audits for the scalar strengthening and matching approximation."""
	from fractions import Fraction as F
	from itertools import permutations,product
	from pathlib import Path
	import json,random
	from scalar_ordering import (build_scalar_reduction,endpoint,raw_residual,
	interpolate_consecutive,determinant_mod,coefficient_oracle,next_prime,
	approximate_scalar_order,degree_bound)

	CHECKS={}

	def check(condition,label):
	if not condition:raise AssertionError(label)
	CHECKS[label]=CHECKS.get(label,0)+1

	def scalar_case(values,rho):
	m=build_scalar_reduction(values,rho)
	n,N=m['n'],m['N']
	check(all(abs(t)<F(1,2) for t in m['centers']),'scalar_bounded_centers')
	sums=[]
	for mask in range(1<<N):
	sums.append(sum(m['weights'][i] for i in range(N) if mask>>i&1))
	check(len(set(sums))==len(sums),'rational_subset_weight_injectivity')
	min_abs=None;zeros=0;canonical_count=0;count=0
	selected=[]
	for order in permutations(range(N)):
	count+=1
	digits=[0]*n
	positions=[[] for _ in range(n)]
	for p,j in enumerate(order):
	digits[j//2]+=m['weights'][p]
	positions[j//2].append(p)
	control=sum(a*b for a,b in zip(m['markers'],digits))
	canonical=all(pos==[2i,2i+1] for i,pos in enumerate(positions))
	required=m['target']//m['Q']
	check((control==required)==canonical,'marker_control_equivalence')
	res=raw_residual(m,order)
	perturbation=sum(m['weights'][p]*(m['data'][j//2] if j%2==0 else -m['data'][j//2]) for p,j in enumerate(order))
	check(abs(perturbation)<=m['D'],'data_term_dominance_bound')
	if not canonical:
	check(abs(res)>=m['Q']-m['D'],'noncanonical_integer_separation')
	else:
	canonical_count+=1
	signs=[]
	for i in range(n):
	signs.append(1 if order[2*i+1]%2==0 else -1)
	check(res==m['H']sum(sa for s,a in zip(signs,values)),'canonical_scalar_partition_identity')
	zeros+=res==0
	min_abs=abs(res) if min_abs is None else min(min_abs,abs(res))
	if count%max(1,(__import__('math').factorial(N)//24))==0:selected.append(order)
	# Add all canonical orientations so perturbation checks include actual
	# zero-error YES witnesses, rather than only generic sampled orders.
	for signs in product([-1,1],repeat=n):
	order=[]
	for i,sign in enumerate(signs):
	order.extend([2i+1,2i] if sign==1 else [2i,2i+1])
	selected.append(tuple(order))
	expected=any(sum(s*a for s,a in zip(signs,values))==0 for signs in product([-1,1],repeat=n))
	check((zeros>0)==expected,'scalar_partition_decision_equivalence')
	for order in selected:
	state=F(0)
	for j in order:
	state=rhostate+(1-rho)m['centers'][j]
	check(abs(state)<F(1,2),'scalar_all_checked_prefixes_bounded')
	check(state==F(raw_residual(m,order),rho.denominator*(N-1)m['R']),'scalar_recurrence_integer_identity')
	# A uniform center perturbation realizes a near-worst-case total shift.
	eta=m['gap']/4
	perturbed=[t+eta for t in m['centers']]
	actual=endpoint(perturbed,order,rho)
	check(abs(actual-state)<=eta,'scalar_perturbation_lipschitz_bound')
	if state==0:
	check(actualactual<=m['gap']*2/16,'robust_yes_margin')
	else:
	check(actualactual>=9m['gap']**2/16,'robust_no_margin')
	return dict(values=values,rho=str(rho),orders=count,canonical_orders=canonical_count,
	zero_orders=zeros,partition_exists=expected,minimum_absolute_integer_residual=str(min_abs),
	centers=[str(t) for t in m['centers']],normalizer=str(m['R']),
	guaranteed_endpoint_gap=str(m['gap']),maximum_center_encoding_bits=max(max(abs(t.numerator).bit_length(),t.denominator.bit_length()) for t in m['centers']))

	def matching_audits():
	rng=random.Random(93811)
	interpolation=[]
	for degree in range(13):
	p=next_prime(1000+degree)
	coefficients=[rng.randrange(p) for _ in range(degree+1)]
	values=[sum(c*pow(x,j,p) for j,c in enumerate(coefficients))%p for x in range(degree+1)]
	check(interpolate_consecutive(values,p)==coefficients,'finite_field_interpolation')
	for n in range(1,5):
	for trial in range(6):
	costs=[[None if rng.randrange(8)==0 else rng.randrange(5) for _ in range(n)] for _ in range(n)]
	attainable={sum(costs[i][j] for i,j in enumerate(order)) for order in permutations(range(n)) if all(costs[i][j] is not None for i,j in enumerate(order))}
	p=next_prime(8n3(degree_bound(costs)+1)+1)
	observed=coefficient_oracle(costs,p,rng)
	check(observed<=attainable,'matching_oracle_no_false_positives')
	# The fixed seed is an implementation audit, not a proof that
	# every random labeling succeeds.
	check(observed==attainable,'matching_oracle_observed_full_support')
	# Equal edge labels can erase all genuine matchings by cancellation.
	check(determinant_mod([[1,1],[1,1]],101)==0,'determinant_cancellation_witness')
	return dict(interpolation_degrees_tested=13,matching_instances=24)

	def approximate_audits():
	configurations=[
	([F(-1),F(0),F(1)],F(1,2),F(0),F(0),F(1,2)),
	([F(-1,2),F(1,4),F(3,4)],F(2,3),F(1,4),F(-1,4),F(1,2)),
	([F(-3,4),F(-1,4),F(1,4),F(3,4)],F(1,2),F(0),F(0),F(1,2)),
	([F(-1),F(1,3),F(2,3),F(1)],F(9,10),F(-1,2),F(1,2),F(1,2)),
	([F(-1,2),F(1,2),F(3,4)],F(1,3),F(1),F(-1),F(1,4)),
	([F(0),F(0),F(0)],F(1,2),F(0),F(0),F(1,2))]
	results=[]
	for i,(centers,rho,initial,target,eps) in enumerate(configurations):
	optimum=min((endpoint(centers,order,rho,initial)-target)**2 for order in permutations(range(len(centers))))
	out=approximate_scalar_order(centers,rho,eps,F(1,16),initial,target,random.Random(451+i))
	check(sorted(out['order'])==list(range(len(centers))),'approximation_feasible_permutation')
	check(out['objective']==(endpoint(centers,out['order'],rho,initial)-target)**2,'approximation_exact_output_certificate')
	check(out['objective']<=optimum+eps,'observed_additive_approximation_guarantee')
	check(out['statistics']['degree']<=32*len(centers)/eps+len(centers),'rounded_degree_bound')
	results.append(dict(centers=list(map(str,centers)),rho=str(rho),initial=str(initial),target=str(target),epsilon=str(eps),order=out['order'],objective=str(out['objective']),exact_optimum=str(optimum),statistics=out['statistics']))
	return results

	def main():
	cases=[(v,rho) for rho in [F(1,2),F(1,3),F(2,3),F(9,10)] for v in [[1,2,3],[1,1,3]]]
	cases.extend([(v,F(1,2)) for v in [[1,2,3,4],[1,2,4,8]]])
	result=dict(arithmetic='Exact integers, rational fractions, and exact finite-field arithmetic; no floating-point proof decisions.',scalar_cases=[scalar_case(v,rho) for v,rho in cases],matching_audits=matching_audits(),approximation_cases=approximate_audits())
	result['assertions']=CHECKS
	result['assertions_total']=sum(CHECKS.values())
	result['scalar_orders_total']=sum(case['orders'] for case in result['scalar_cases'])
	result['baseline_assertions_retained']=344707
	result['combined_assertions']=344707+result['assertions_total']
	Path(__file__).with_name('broadening_verification_results.json').write_text(json.dumps(result,indent=2)+'\n')
	print(json.dumps({key:result[key] for key in ['assertions_total','scalar_orders_total','baseline_assertions_retained','combined_assertions']},indent=2))

	if __name__=='__main__':main()