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e3bdcca | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | """Finite checks for coherent sparse-repair refusal. Synthetic, not hardware."""
from itertools import product, combinations
from fractions import Fraction
from math import comb, gcd
import json
from pathlib import Path
import numpy as np
def vectors(p,n):
return np.array(list(product(range(p),repeat=n)),dtype=int)
def rank_mod(A,p):
A=np.array(A,dtype=int).copy()%p
m,n=A.shape
row=0
for j in range(n):
pivot=next((i for i in range(row,m) if A[i,j]),None)
if pivot is None: continue
A[[row,pivot]]=A[[pivot,row]]
A[row]=(A[row]*pow(int(A[row,j]),-1,p))%p
for i in range(m):
if i!=row and A[i,j]: A[i]=(A[i]-A[i,j]*A[row])%p
row+=1
if row==m: break
return row
def rank_q(A):
A=[[Fraction(int(x)) for x in row] for row in A]
m=len(A); n=len(A[0]) if m else 0; row=0
for j in range(n):
pivot=next((i for i in range(row,m) if A[i][j]),None)
if pivot is None: continue
A[row],A[pivot]=A[pivot],A[row]
den=A[row][j]
A[row]=[x/den for x in A[row]]
for i in range(row+1,m):
if A[i][j]:
a=A[i][j]; A[i]=[x-a*y for x,y in zip(A[i],A[row])]
row+=1
if row==m: break
return row
def rank_z(A):
"""Exact integer elimination, normalizing rows by their gcd."""
A=[[int(x) for x in row] for row in A]
m=len(A); n=len(A[0]) if m else 0; row=0
for j in range(n):
pivot=next((i for i in range(row,m) if A[i][j]),None)
if pivot is None: continue
A[row],A[pivot]=A[pivot],A[row]
den=A[row][j]
for i in range(row+1,m):
if A[i][j]:
z=A[i][j]
A[i]=[den*x-z*y for x,y in zip(A[i],A[row])]
g=0
for x in A[i]: g=gcd(g,abs(x))
if g>1: A[i]=[x//g for x in A[i]]
row+=1
if row==m: break
return row
def repair_distance(B,s,p):
for k in range(B.shape[1]+1):
for S in combinations(range(B.shape[1]),k):
C=B[:,S]
if rank_mod(C,p)==rank_mod(np.column_stack([C,s]),p): return k
return None
def run():
out={}
B=np.array([[1,0,1,0,0,0],[0,1,1,0,0,0],
[0,0,0,1,0,1],[0,0,0,0,1,1]],dtype=int)
G=vectors(2,4)
legal=np.array(sorted({(0,0,0,0),*[tuple(B[:,j]) for j in range(6)]}),dtype=int)
evals=(-1.)**(legal@G.T%2)
counts={}
for D in range(1,5):
deficient=0
total=0
for inds in combinations(range(16),D):
total+=1
if rank_q(evals[:,inds])<D: deficient+=1
counts[str(D)]={'total_alphabets':total,'nonzero_refusal_alphabets':deficient}
out['exact_rational_alphabet_scan']=counts
A=np.array([[0,0,0,0],[1,0,0,0],[0,0,1,0],[1,0,1,0]])
mu=np.full(4,.25)
c=np.array([1.,-1.,-1.,1.])
v=c/2
probs=[]
for s in G:
state=np.sqrt(mu)*((-1.)**(A@s%2))
prob=abs(np.vdot(v,state))**2
probs.append({'s':s.tolist(),'distance':repair_distance(B,s,2),'prob':prob})
out['binary_block_dictionary']={'B':B.tolist(),'rank':rank_mod(B,2),'legal_count':len(legal),'probabilities':probs}
rng=np.random.default_rng(20261024)
mu=rng.dirichlet(np.ones(4))
Z=np.sum(1/mu)
v=(c/np.sqrt(mu))/np.sqrt(Z)
maxerr=0
for s in G:
state=np.sqrt(mu)*((-1.)**(A@s%2))
prob=abs(np.vdot(v,state))**2
predicted=(abs(np.prod(1-(-1.)**(np.array([s[0],s[2]]))))**2)/Z
maxerr=max(maxerr,abs(prob-predicted))
out['nonuniform_cube']={'mu':mu.tolist(),'Z':float(Z),'success_on_1111':float(16/Z),'uniform_success':1.0,'formula_max_error':maxerr}
# Every nonzero 2-bit covector in each block: exact average success 4/9.
nonzero=[a for a in vectors(2,2) if np.any(a)]
alt=[s for s in G if np.any(s[:2]) and np.any(s[2:])]
rates=[]
for s in alt:
rates.append(sum(int((a@s[:2])%2)*int((b@s[2:])%2) for a in nonzero for b in nonzero)/9)
out['random_block_queries']={'alternatives':len(alt),'min_success':min(rates),'max_success':max(rates),'exact_rate':'4/9'}
# General p uniform cube formula and one-sided soundness.
p=3;n=3;t=2
A=np.array([[e[0],e[1],0] for e in product((0,1),repeat=t)])
v=np.array([(-1.)**sum(e) for e in product((0,1),repeat=t)])/2
maxerr=0; maxlegal=0
for s in vectors(p,n):
state=np.exp(2j*np.pi*(A@s%p)/p)/2
prob=abs(np.vdot(v,state))**2
pred=float(np.prod(np.sin(np.pi*s[:t]/p)**2))
maxerr=max(maxerr,abs(prob-pred))
if np.count_nonzero(s)<=t-1: maxlegal=max(maxlegal,prob)
out['ternary_cube']={'max_formula_error':maxerr,'max_legal_refusal_roundoff':maxlegal,'alternative_success_when_both_selected_nonzero':9/16}
# A target-dependent optimum based on a general signed witness is Cauchy exact.
c=np.array([1+2j,-2+1j,0.5-1j,-1j])
mu=np.abs(c)/np.sum(np.abs(c))
out['l1_witness_amplitudes']={'cauchy_gap':float(np.sum(np.abs(c)**2/mu)-np.sum(np.abs(c))**2)}
identity_scans={}
chars=vectors(2,4)
for k,max_D in [(1,3),(2,7)]:
legal=chars[np.count_nonzero(chars,axis=1)<=k]
full=(-1)**((legal@chars.T)%2)
by_D={}
for D in range(1,max_D+1):
deficient=0
for inds in combinations(range(16),D):
deficient+=int(rank_z(full[:,inds])<D)
by_D[str(D)]={'total':comb(16,D),'rank_deficient':deficient}
cube=np.array([list(e)+[0]*(4-k-1) for e in product((0,1),repeat=k+1)])
cube_matrix=(-1)**((legal@cube.T)%2)
identity_scans[str(k)]={'n':4,'k':k,'alphabets':by_D,'threshold_cube_modes':len(cube),'threshold_cube_rank':rank_z(cube_matrix)}
out['identity_sparse_ball_exact_scans']=identity_scans
prime_checks=[]
for p in [2,3,5,7]:
n=3;t=2
A=np.array([[e[0],e[1],0] for e in product((0,1),repeat=t)])
c=np.array([(-1.)**sum(e) for e in product((0,1),repeat=t)])
mu=rng.dirichlet(np.ones(4)); Z=np.sum(1/mu)
v=(c/np.sqrt(mu))/np.sqrt(Z)
maxerr=0;maxlegal=0;tested=0
for s in vectors(p,n):
state=np.sqrt(mu)*np.exp(2j*np.pi*(A@s%p)/p)
prob=abs(np.vdot(v,state))**2
pred=float(np.prod(4*np.sin(np.pi*s[:t]/p)**2))/Z
maxerr=max(maxerr,abs(prob-pred));tested+=1
if np.count_nonzero(s)<=1:maxlegal=max(maxlegal,prob)
prime_checks.append({'p':p,'n':n,'tested_all_syndromes':tested,'formula_max_error':maxerr,'legal_refusal_max_roundoff':maxlegal})
out['all_small_prime_syndromes']=prime_checks
assert all(counts[str(D)]['nonzero_refusal_alphabets']==0 for D in [1,2,3])
assert counts['4']['nonzero_refusal_alphabets']==36
assert all(row['rank_deficient']==0 for scan in identity_scans.values() for row in scan['alphabets'].values())
assert all(scan['threshold_cube_rank']==scan['threshold_cube_modes']-1 for scan in identity_scans.values())
assert all(row['prob']==0 for row in probs if row['distance']<=1)
assert max(abs(rate-4/9) for rate in rates)<1e-12
assert out['nonuniform_cube']['formula_max_error']<1e-12
assert all(row['formula_max_error']<1e-12 and row['legal_refusal_max_roundoff']<1e-12 for row in prime_checks)
assert abs(out['l1_witness_amplitudes']['cauchy_gap'])<1e-12
Path('research/refusal_test_results.json').write_text(json.dumps(out,indent=2))
print(json.dumps({k:v for k,v in out.items() if k!='binary_block_dictionary'},indent=2))
if __name__=='__main__': run()
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