"""Independent checks for numberdb.org/T689. sage -python checks.py sage -python checks.py --stored # authenticated readback, no writes Download generate.py, check.py and skill-manifest.json alongside this file. The independent computation uses Python Fraction, nilpotent coefficient products for the second Frobenius period, Lagrange inversion, and Mobius inversion. It calls no generator arithmetic. Primary values are computed only for comparison. This is independent code, not an independent mirror theorem. The external geometric controls are the Macaulay2 documentation's localization outputs, plus a Grassmannian top-Chern intersection calculation for lines. """ import json import os import shutil import sys from fractions import Fraction as F from math import prod, gcd from pathlib import Path from itertools import combinations_with_replacement import numberdb.sage as numberdb from sage.rings.rational_field import QQ from sage.env import SAGE_VERSION import generate from check import exactness, agrees_with, measure, names_its_rings TYPES = ((5,), (2, 4), (3, 3), (2, 2, 3), (2, 2, 2, 2)) # rationalCurve documentation o3, o7, o11, o14, o15, o16; independently # read from the source during this run. Listed numbers are the GV controls. CONTROLS = { '5': [2875, 609250, 317206375, 242467530000], '2,4': [1280, 92288, 15655168, 3883902528], '3,3': [1053, 52812, 6424326, 1139448384], '2,2,3': [720, 22428, 1611504], '2,2,2,2': [512, 9728, 416256], } GW_CONTROLS = { '5': ['2875', '4876875/8', '8564575000/27', '15517926796875/64'], '2,4': ['1280', '92448', '422690816/27', '3883914084'], '3,3': ['1053', '423549/8', '6424365'], '2,2,3': ['720', '22518', '4834592/3'], '2,2,2,2': ['512', '9792', '11239424/27'], } def convolution(a, b, n): c = [F(0)]*(n+1) for i, x in enumerate(a): for j, y in enumerate(b[:max(0, n+1-i)]): c[i+j] += x*y return c def quotient(a, b): c = [] for k in range(len(a)): c.append((a[k] - sum(b[k-j]*c[j] for j in range(k))) / b[0]) return c def exp_by_powers(a, n): assert a[0] == 0 result = [F(1)] + [F(0)]*n term = result[:] for k in range(1, n+1): term = [v/F(k) for v in convolution(term, a, n)] result = [x+y for x, y in zip(result, term)] return result def mobius(n): primes = 0 p = 2 while p*p <= n: if n % p == 0: n //= p primes += 1 if n % p == 0: return 0 while n % p == 0: n //= p p += 1 if n > 1: primes += 1 return (-1)**primes def independent(target, n): # A_m(epsilon)=prod (j+l epsilon) / prod (j+epsilon)^(r+4), # reduced modulo epsilon^3. No harmonic-sum formula is used here. columns = [[], [], []] for m in range(n+1): a = [F(1), F(0), F(0)] for l in target: for j in range(1, l*m+1): a = convolution(a, [F(j), F(l)], 2) for j in range(1, m+1): for _ in range(len(target)+4): a = convolution(a, [F(1,j), -F(1,j*j), F(1,j**3)], 2) for k in range(3): columns[k].append(a[k]) h = quotient(columns[1], columns[0]) b = quotient(columns[2], columns[0]) hsquared = convolution(h, h, n) e = [v-w/F(2) for v, w in zip(b, hsquared)] derivative = [F(j)*e[j] for j in range(1,n+1)] # y2/y0 = (log q)^2/2 + E(x(q)); K=kappa*(q d/dq)^2(y2/y0). gw = [F(0)] for d in range(1,n+1): exponential = exp_by_powers([-F(d)*v for v in h[:d]], d-1) coefficient = sum(derivative[j]*exponential[d-1-j] for j in range(d))/F(d) gw.append(F(prod(target),d)*coefficient) gv = [F(0)] + [sum(F(mobius(k),k**3)*gw[d//k] for k in range(1,d+1) if d%k==0) for d in range(1,n+1)] return gw, gv def line_intersection(target, ambient=None): # Integrate top Chern of direct sum Sym^l S^* on Gr(2,r+4). # For symmetric f(u,v) of top degree, integral is # [u^(r+3) v^(r+2)] (u-v) f(u,v). poly = {(0,0): 1} for l in target: for j in range(l+1): new = {} for (a,b), value in poly.items(): new[a+1,b] = new.get((a+1,b),0)+j*value new[a,b+1] = new.get((a,b+1),0)+(l-j)*value poly = new ambient = len(target)+3 if ambient is None else ambient return poly.get((ambient-1,ambient-1),0)-poly.get((ambient,ambient-2),0) def template_check(target, n): f,g = generate.periods(target,n) alpha = [F(j,l) for l in target for j in range(1,l)] c=prod(l**l for l in target) a=F(1); h=F(0) for m in range(n+1): assert F(str(f[m])) == c**m*a assert F(str(g[m])) == c**m*a*h a *= prod(F(m)+v for v in alpha)/F((m+1)**4) h += sum(1/(v+m) for v in alpha)-F(4,m+1) keymap = {'5':'5','2,4':'2.2.4','3,3':'3.3','2,2,3':'2.2.3','2,2,2,2':'2.2.2.2'} key=','.join(map(str,target)); template=keymap[key] cyclotomic=[F(j,l) for l in map(int,template.split('.')) for j in range(1,l) if gcd(j,l)==1] assert sorted(alpha)==sorted(cyclotomic) return {'template':template,'C':c,'kappa':prod(target),'coefficients_checked':n+1} def run(stored=False): # Known-answer controls precede comparisons, including nonzero geometry # outside the Calabi–Yau class: 27 lines on a cubic surface. assert line_intersection((3,), ambient=3) == 27 # A known CY answer controls both a computational and geometric route. assert line_intersection((5,)) == 2875 assert independent((5,),2)[1][1:] == [F(2875),F(609250)] assert [mobius(n) for n in range(1,11)] == [1,-1,-1,0,-1,1,-1,0,0,1] assert exp_by_powers([F(0),F(1),F(0),F(0)],3) == [F(1),F(1),F(1,2),F(1,6)] assert convolution([F(1),F(2)],[F(3),F(4)],2)==[F(3),F(10),F(8)] assert quotient([F(1),F(1),F(1)],[F(1),F(-1),F(0)])==[F(1),F(2),F(3)] assert generate.TARGETS==TYPES partition_types=[] for r in range(1,5): partition_types.extend(t for t in combinations_with_replacement(range(2,6),r) if sum(l-1 for l in t)==4) assert set(partition_types)==set(TYPES) generator=generate.GenusZeroGV() params=list(generator.enumerate()) assert len(params)==100 and len({tuple(p.items()) for p in params})==100 wanted=[{'multidegree':','.join(map(str,t)), 'd':str(d)} for t in TYPES for d in range(1,21)] assert params==wanted source_values={} independent_values={} evidence={'table':'T689','skill_manifest':'skill-manifest.json','skill_manifest_sha256':__import__('hashlib').sha256(Path(__file__).with_name('skill-manifest.json').read_bytes()).hexdigest(),'software':{'sage':SAGE_VERSION,'python':sys.version.split()[0],'numberdb':numberdb.__version__,'Macaulay2_available':bool(shutil.which('M2'))},'control_tests':'passed','targets':{}} for target in TYPES: key=','.join(map(str,target)) k,gw,gv=generate.coefficients(target,24) k2,gw2,gv2=generate.coefficients(target,28) assert k[:21]==k2[:21] and gw[:21]==gw2[:21] and gv[:21]==gv2[:21] igw,igv=independent(target,24) for d in range(1,21): ident=(key,str(d));source_values[ident]=gv[d];independent_values[ident]=QQ(igv[d].numerator)/QQ(igv[d].denominator) assert F(str(gw[d]))==igw[d] assert sum(F(str(gv[d//j]))/F(j**3) for j in range(1,d+1) if d%j==0)==igw[d] assert sum(F(e**3)*igv[e] for e in range(1,d+1) if d%e==0)==F(str(k[d])) assert igv[d].denominator==1 assert gv[1:len(CONTROLS[key])+1]==CONTROLS[key] assert [F(str(v)) for v in gw[1:len(GW_CONTROLS[key])+1]]==list(map(F,GW_CONTROLS[key])) assert line_intersection(target)==gv[1] evidence['targets'][key]={'series_order_comparison':[24,28],'degrees_checked':20,'GV_localization_controls':len(CONTROLS[key]),'GW_localization_controls':len(GW_CONTROLS[key]),'Grassmannian_line_count':line_intersection(target),'template':template_check(target,24),'largest_value':str(gv[20])} assert not agrees_with(source_values,lambda key:independent_values[key]) assert not exactness(source_values) assert not names_its_rings(Path(__file__).with_name('generate.py')) evidence['measure']=measure(source_values) evidence['all_values']={key:{str(d):str(source_values[key,str(d)]) for d in range(1,21)} for key in CONTROLS} if stored: document=numberdb.table('T689') sibling=numberdb.table('T688') actual=document['Numbers']; other=sibling['Numbers'] assert list(actual)==list(CONTROLS) for target in TYPES: key=','.join(map(str,target)) assert list(actual[key])==list(map(str,range(1,21))) for d in range(1,21): row=actual[key][str(d)];value=row['number'] if isinstance(row,dict) else row assert F(value)==F(str(independent_values[key,str(d)])) conv=sum(F(str(actual[key][str(d//j)]))/F(j**3) for j in range(1,d+1) if d%j==0) assert conv==F(other[key][str(d)]) report=generator.verify(sample=None) assert report.ok evidence['readback']={'independent_values':100,'GW_companion_relations':100,'generator_verify':str(report)} else: sibling_path=Path(__file__).with_name('T688.json') if sibling_path.exists(): other=json.loads(sibling_path.read_text())['Numbers'] for target in TYPES: key=','.join(map(str,target)) igw,_=independent(target,20) assert all(F(other[key][str(d)])==igw[d] for d in range(1,21)) evidence['archived_T688_comparisons']=100 print(json.dumps(evidence,indent=2,default=str)) return evidence if __name__=='__main__': if os.environ.get('NUMBERDB_KEY_FROM_STDIN')=='1': token=sys.stdin.read().strip() if '=' in token and token.split('=',1)[0].replace('_','').isalnum():token=token.split('=',1)[1].strip().strip('\"\'') numberdb.configure(api_key=token) run(stored='--stored' in sys.argv or os.environ.get('NUMBERDB_KEY_FROM_STDIN')=='1')