"""Tammes spherical-code separations -- numberdb.org/T479. Run with SageMath and the attached sources.json: sage -pip install numberdb sage -python generate.py sage -python checks.py NUMBERDB_API_KEY=... sage -python generate.py --fill The default verifies every stored entry. --fill writes entries and attachments on the existing draft; it does not publish it or offer it for review. The source snapshot was fetched on 2026-10-25. It uses Cohn's 3-dimensional codes for 7..100 and Hardin, Sloane and Smith for 4..6. Published minimal polynomials give 100-digit algebraic-cosine rows. Other rows retain ten significant digits, with the printed source cosine enclosed by +/- 1e-12. The 768-bit field gives at least 100 reliable decimal digits after root isolation and conversion; checks.py measures the margin. Rounded coordinates are independent checks, never the basis for extra digits or optimality claims. """ import os import sys import json import re from math import comb from functools import lru_cache from pathlib import Path import numberdb.sage as numberdb from sage.rings.rational_field import QQ from sage.rings.real_arb import RealBallField from numberdb._write import to_text DATA = json.loads(Path(__file__).with_name('sources.json').read_text()) WORKING_BITS = 768 ROOT_STEPS = 400 def rational(s): from decimal import Decimal a,b=Decimal(str(s)).as_integer_ratio() return QQ(a)/QQ(b) def coefficients(text): out={} for term in re.findall(r'[+-]?[^+-]+',text): if 'x' not in term: c,k=int(term),0 else: left,right=term.split('x') c=int(left.rstrip('*')) if left.rstrip('*') not in ('','+','-') else (-1 if left=='-' else 1) k=int(right[1:]) if right else 1 out[k]=c return [QQ(out.get(i,0)) for i in range(max(out)+1)] def evaluate(cs,x): ans=0 for c in reversed(cs): ans=ans*x+c return ans def isolate(cs,center): """Exact bisection, preceded by a rational uniqueness certificate.""" radius=QQ(1)/10**10 lo,hi=center-radius,center+radius # Shift p(center+y) exactly to control its derivative without cancellation. shifted=[sum(cs[j]*comb(j,k)*center**(j-k) for j in range(k,len(cs))) for k in range(len(cs))] tail=sum(abs(shifted[k])*k*radius**(k-1) for k in range(2,len(cs))) assert abs(shifted[1])>tail, 'initial interval does not certify uniqueness' flo,fhi=evaluate(cs,lo),evaluate(cs,hi) assert flo*fhi<0, 'source interval does not bracket a root' for _ in range(ROOT_STEPS): mid=(lo+hi)/2 fm=evaluate(cs,mid) if fm==0:return mid,mid if flo*fm<0: hi=mid else: lo,flo=mid,fm assert evaluate(cs,lo)*evaluate(cs,hi)<0 return lo,hi @lru_cache(None) def cosine(n): R=RealBallField(WORKING_BITS) if n==4:return R(-QQ(1)/3) if n in (5,6):return R(0) if n==9:return R(QQ(1)/3) row=DATA['rows'][str(n)] c=rational(row['cosine']) if 'polynomial' in row: a,b=isolate(coefficients(row['polynomial']),c) return R((a+b)/2,(b-a)/2) return R(c,QQ(1)/10**12) def entry_comment(n): if n==4:return r'The regular tetrahedron has $\cos(\pi\theta_4/180)=-1/3$ and $d_4=\sqrt{8/3}$. Optimality is proven CITE{FejesToth}.' if n in (5,6):return r'$\theta_n=90$ and $d_n=\sqrt2$. Optimality is proven CITE{Schutte} CITE{FejesToth}.' row=DATA['rows'][str(n)] parts=[] if 'polynomial' in row: poly=row['polynomial'].replace('*','') # Long algebraic descriptions remain in the attached source data. if len(poly)<=180: poly=re.sub(r'\^(\d+)',r'^{\1}',poly) c=row['cosine'] parts.append(r'With $c_n=\cos(\pi\theta_n/180)$, $c_n$ is the unique zero of $'+poly+r'$ in $('+c+r'-10^{-10},'+c+r'+10^{-10})$ CITE{Cohn}.') else:parts.append(r'The cosine $c_n=\cos(\pi\theta_n/180)$ is algebraic. Its polynomial and isolating interval are specified in the attached sources.json and generate.py CITE{Cohn}.') if n in (7,8,9):parts.append('Optimality is proven CITE{Schutte}.') elif n in (10,11):parts.append('Optimality is proven CITE{Danzer}.') elif n==12:parts.append('Optimality is proven CITE{FejesToth}.') elif n==13:parts.append('Optimality is proven CITE{Musin13}.') elif n==14:parts.append('Optimality is proven CITE{Musin14}.') elif n==24:parts.append('Optimality is proven CITE{Robinson}.') elif n==15:parts.append('Cohn marks this configuration as optimal, citing the 2026 preprint CITE{Tammes15}; that proof claim is distinguished here from the established published cases.') return ' '.join(parts) class Tammes(numberdb.Generator): table='T479' parameters=('n','normalisation') type='R' digits=100 rigour='heuristic (agreement-checked)' files=('generate.py','checks.py','check.py','sources.json','checks-result.json','source-notes.md') def enumerate(self): for n in range(4,101): for normalisation in ('angle','chord'): yield {'n':n,'normalisation':normalisation} def value(self,params,digits): n=int(params['n']);norm=params['normalisation'] R=RealBallField(WORKING_BITS);c=cosine(n) if norm=='angle': value=90 if n in (5,6) else c.arccos()*180/R.pi() elif norm=='chord':value=(2-2*c).sqrt() else:raise ValueError(norm) precise=n<=6 or 'polynomial' in DATA['rows'][str(n)] if not precise: # A ball straddling a rounding boundary can make to_text shorten # several digits. Round its midpoint, then certify the WHOLE input # ball fits the decimal's stated +/- one-last-place interval. from decimal import Decimal, localcontext mid=QQ(value.mid()) with localcontext() as context: context.prec=80 text=format(Decimal(int(mid.numerator()))/Decimal(int(mid.denominator())),'.10g') ulp=QQ(10)**Decimal(text).as_tuple().exponent assert abs(value-R(rational(text)))