File size: 3,185 Bytes
0f725fa | 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 | """Exact Cantor/operator finite codes illustrating the sharp information law.
Uniform binary IFS x/4 and (x+3)/4: h=ln2, chi=ln4, D=1/2.
Directional gain 2^(-n/2), natural physical precision, gives 1/4 bit/step.
The implemented finite code is within two bits of the exact conditional-entropy
lower bound. Compensation to theta=5/4 permits an empty tag in this example.
All source points, cell boundaries, and tested noise values use Fraction.
"""
from collections import Counter
from fractions import Fraction as F
from pathlib import Path
import json
def numerator(word,length):
out=0
for i in range(length-1,-1,-1):
out=4*out+3*((word>>i)&1)
return out
def valid_cantor_numerator(value,length):
if value<0 or value>=4**length:return False
for _ in range(length):
digit=value%4
if digit not in (0,3):return False
value//=4
return True
reports=[]
for n in [4,8,12]:
d=3*n//4;k=n-d
gain=F(1,2**(n//2))
delta=F(1,4**d)
physical_radius=F(1,4**n)
assert physical_radius/gain==delta
states=[numerator(word,n) for word in range(2**n)]
bins=Counter(value//4**k for value in states)
assert len(bins)==2**d and set(bins.values())=={2**k}
checks=0
for word,value in enumerate(states):
point=F(value,4**n)
q=value//4**k
tag=(q%4,word%(2**k))
assert len(format(tag[0],'02b')+format(tag[1],f'0{k}b'))==2+k
for multiplier in [F(-1),F(-1,2),F(0),F(1,2),F(1)]:
scalar_observation=point+multiplier*delta
# Matched vector-line noise would have norm gain*abs(error).
assert gain*abs(scalar_observation-point)<=physical_radius
z=scalar_observation//delta
candidates=[j for j in [z-1,z,z+1] if j%4==tag[0]]
assert len(candidates)==1 and candidates[0]==q
recovered=F(candidates[0]*4**k+numerator(tag[1],k),4**n)
assert recovered==point
checks+=1
# theta=5/4 compensates the attenuation: pulled-back radius 4^(-n).
compensated_physical=F(1,2**(5*n//2))
assert compensated_physical/gain==F(1,4**n)
for multiplier in [F(-1),F(-1,2),F(0),F(1,2),F(1)]:
scaled_observation=F(value)+multiplier
z=scaled_observation//1
candidates=[j for j in [z-1,z,z+1]
if valid_cantor_numerator(j,n) and abs(F(j)-scaled_observation)<=1]
assert candidates==[value]
reports.append({'depth':n,'states_checked':2**n,'robust_boundary_checks':checks,
'observation_entropy_bits_exact':d,
'conditional_entropy_bits_exact':k,
'code_bits':k+2,'excess_bits_over_lower_bound':2,
'natural_metadata_rate_exact':'1/4',
'compensated_metadata_bits':0,
'arithmetic':'Fraction; no tolerance'})
Path(__file__).with_suffix('.json').write_text(json.dumps(reports,indent=2)+'\n')
print('PASS: exact conditional-entropy counts, all source states and five closed-noise boundaries.')
print('PASS: operator gain, quarter-bit rate, two-bit redundancy, and zero-tag compensation.')
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