#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ EXACT CORE -- the load-bearing flavour claims verified with ZERO continuum operations. No float, no log, no numerical trig, no RNG, no tolerances. Arena ladder (most native first): * FRAMED RATIONALS (1-algebra) : the Koide identity over F_{p^2}/F_p with Frobenius the native conjugation; omega the finite cube root, the 'sqrt2 amplitude' the framed RATIONAL N(b)/a^2=1/2, Q=2/3 exact in F_p (section E); and the amplitude diagonal r^2=N(1-zeta_n)=2,3 as INTEGERS in F_p (section F). * cyclotomic Q(omega, sqrt2) : one continuum rung up -- omega a complex cube root, sqrt2 =zeta_8+zeta_8^{-1} irrational (sections A, D). A labelled reading. * rational multiples of pi : the pi/12 boundary, and 1+sqrt2 cos(3pi/4)=0 (section B). * INTEGER group ring Z[Z/3 x Z/p] : Gauss-sum reality and pi/3-quantisation (section C). Each continuum quantity used elsewhere (logs for lambda-powers, arccos for delta extraction, RNG for robustness, float linalg for the mixing demo) is a labelled data-comparison or sensitivity check, NOT a verification of an exact claim; see the finitism-audit note. """ import sympy as sp ok=[] def must(tag,cond,detail=""): ok.append((tag,bool(cond))); print(f"[{'EXACT' if cond else 'FAIL'}] {tag}: {detail}") print("="*70); print("A. Koide / circulant in the cyclotomic field Q(omega, sqrt2)") print("="*70) # omega: primitive cube root of unity as an EXACT algebraic number (no pi, no float) w = sp.Rational(-1,2) + sp.sqrt(3)/2*sp.I s2 = sp.sqrt(2) # = zeta_8 + zeta_8^{-1}, cyclotomic must("A.omega", sp.simplify(w**3-1)==0 and sp.simplify(1+w+w**2)==0, "omega^3=1 and 1+omega+omega^2=0 (exact)") a, bx, by = sp.symbols('a b_x b_y', real=True) b = bx + sp.I*by; bc = bx - sp.I*by # b and its conjugate lam = [sp.expand(a + b*w**k + bc*w**(2*k)) for k in range(3)] # circulant eigenvalues S1 = sp.simplify(sum(lam)) S2 = sp.simplify(sum(l**2 for l in lam)) must("A.sum1", sp.simplify(S1-3*a)==0, f"sum_k lambda_k = {S1} = 3a") must("A.sum2", sp.simplify(S2-(3*a**2+6*(bx**2+by**2)))==0, f"sum_k lambda_k^2 = 3a^2 + 6|b|^2 (exact; the omega-cross terms cancel)") # Koide ratio Q = sum lambda^2 / (sum lambda)^2 = 1/3 + (2/3) rho^2, rho^2=|b|^2/a^2 rho2 = sp.symbols('rho2', positive=True) # |b|^2/a^2 Q = sp.Rational(1,3) + sp.Rational(2,3)*rho2 must("A.Qform", sp.simplify(Q-(3*a**2+6*a**2*rho2)/(9*a**2))==0, "Q = 1/3 + (2/3) rho^2 (exact)") # r = 2|b|/a ; r = sqrt2 <=> rho^2 = 1/2 <=> Q = 2/3 must("A.koide", sp.simplify(Q.subs(rho2, sp.Rational(1,2))-sp.Rational(2,3))==0, "r = sqrt2 (rho=1/sqrt2, the quarter-turn diagonal) => Q = 2/3 exactly") # circulant eigen-identity & Hermiticity, exact symbolic matrix over Z[omega] P = sp.Matrix([[0,1,0],[0,0,1],[1,0,0]]) M = a*sp.eye(3) + b*P + bc*P.T # P.T = P^2 = P^dagger must("A.herm", sp.simplify(M - M.conjugate().T)==sp.zeros(3), "sqrt(M)=a*1+b*P+conj(b)*P^2 is Hermitian (P^2=P^dagger)") F = sp.Matrix(3,3, lambda j,k: w**(j*k)) # Fourier (Vandermonde in omega) D = sp.simplify(F.inv()*M*F) diag_ok = all(sp.simplify(D[i,j])==0 for i in range(3) for j in range(3) if i!=j) eig_ok = all(sp.simplify(D[k,k]-(a+b*w**k+bc*w**(2*k)))==0 for k in range(3)) must("A.eig", diag_ok and eig_ok, "F^{-1} M F diagonal with entries a+b w^k+conj(b) w^{2k} (exact eigenvalues)") print("\n"+"="*70); print("B. The pi/12 boundary (exact rational-pi + algebraic)") print("="*70) must("B.pi12", sp.simplify(sp.Rational(3,4)*sp.pi - sp.Rational(2,3)*sp.pi - sp.pi/12)==0, "delta_LO = 3pi/4 - 2pi/3 = pi/12 (exact)") # at the boundary the lightest eigenvalue vanishes EXACTLY: 1 + sqrt2*cos(3pi/4)=0 must("B.zero", sp.simplify(1 + s2*sp.cos(sp.Rational(3,4)*sp.pi))==0, "1 + sqrt2*cos(3pi/4) = 1 + sqrt2*(-1/sqrt2) = 0 (electron massless at boundary)") print("\n"+"="*70); print("C. Gauss sums: reality & pi/3-quantisation as INTEGER identities") print("="*70) def isprime(n): return n>1 and all(n%k for k in range(2,int(n**0.5)+1)) def nonresidue(p): return next(n for n in range(2,p) if pow(n,(p-1)//2,p)==p-1) def grp_overlap(p, drive): """Return the cubic overlap as an exact element of Z[Z/3 x Z/p]: C[r][s] = # { k : chi3(g^k)=omega^r and additive exponent = s }. No complex numbers -- only integer counts of roots of unity.""" nqr=nonresidue(p) def mul(x,y): (A,B),(Cc,Dd)=x,y; return ((A*Cc+nqr*B*Dd)%p,(A*Dd+B*Cc)%p) U=[(A,B) for A in range(p) for B in range(p) if (A*A-nqr*B*B-1)%p==0] assert len(U)==p+1 def order(x): o=1;y=x while y!=(1,0): y=mul(y,x);o+=1 return o g=next(x for x in U if x!=(1,0) and order(x)==p+1) C=[[0]*p for _ in range(3)] cur=(1,0); w0=(1,1) if drive else (1,0) for k in range(p+1): A,B=mul(w0,cur); s=(2*A)%p # additive exponent Tr(w0 u)=2 Re(w0 u) C[k%3][s]+=1 cur=mul(cur,g) return C def conj(C,p): # zeta_3 -> zeta_3^-1, zeta_p -> zeta_p^-1 return [[C[(-r)%3][(-s)%p] for s in range(p)] for r in range(3)] def equal(C,D): return C==D def gmul(C,D,p): # group-ring product over Z/3 x Z/p E=[[0]*p for _ in range(3)] for r1 in range(3): for s1 in range(p): c=C[r1][s1] if not c: continue for r2 in range(3): for s2 in range(p): d=D[r2][s2] if d: E[(r1+r2)%3][(s1+s2)%p]+=c*d return E for p in [17,29,41,53]: assert isprime(p) and p%12==5 Csym=grp_overlap(p,False) must(f"C.real_p{p}", equal(Csym,conj(Csym,p)), f"p={p}: symmetric cubic overlap b = conj(b) EXACTLY (integer-real); no phase") for p in [17,29,41,53]: Cdrv=grp_overlap(p,True) b3=gmul(gmul(Cdrv,Cdrv,p),Cdrv,p) # b^3 must(f"C.cube_p{p}", equal(b3,conj(b3,p)), f"p={p}: drive-twisted b^3 = conj(b^3) EXACTLY => arg(b) in (pi/3)Z") print("\n"+"="*70); print("D. Leptonic CP: the magic matrix is maximal (cyclotomic Z[omega])") print("="*70) # the C3 eigenvector (DFT) matrix F = (1/sqrt3)[[1,1,1],[1,w,w^2],[1,w^2,w]] Fm = sp.Matrix(3,3, lambda j,k: w**(j*k)) / sp.sqrt(3) mod2 = [sp.simplify(sp.Abs(Fm[j,k])**2) for j in range(3) for k in range(3)] must("D.trimaximal", all(sp.simplify(m-sp.Rational(1,3))==0 for m in mod2), "|F_jk|^2 = 1/3 for all 9 entries (trimaximal), exact") # Jarlskog J = Im(F00 F11 conj(F01) conj(F10)) = Im(omega)/9 = 1/(6 sqrt3); compute exactly J = sp.im(sp.simplify(Fm[0,0]*Fm[1,1]*sp.conjugate(Fm[0,1])*sp.conjugate(Fm[1,0]))) Jmax = 1/(6*sp.sqrt(3)) must("D.maxCP", sp.simplify(sp.Abs(J)-Jmax)==0, f"J(F) = {sp.simplify(J)} = +-1/(6 sqrt3): MAXIMAL CP (the cube-root phase Im(omega)=sqrt3/2)") print("\n"+"="*70); print("E. Koide over FRAMED RATIONALS (1-algebra) -- the most native arena") print("="*70) # F_{p^2}/F_p, Frobenius the native conjugation; omega the finite cube root; the quarter-turn # 'sqrt2' is the framed RATIONAL N(b)/a^2 = 1/2 (a norm condition). Q=2/3 exact in F_p. from math import isqrt def _ratrecon(x,p): # canonical framed rational r/s = x (mod p) x%=p; B=isqrt(p//2); r0,r1,s0,s1=p,x,0,1 while r1>B: q=r0//r1; r0,r1=r1,r0-q*r1; s0,s1=s1,s0-q*s1 return (-r1,-s1) if s1<0 else (r1,s1) def _framed_koide(p): n=next(m for m in range(2,p) if pow(m,(p-1)//2,p)==p-1) # non-residue: t^2=n mul=lambda A,B:((A[0]*B[0]+A[1]*B[1]*n)%p,(A[0]*B[1]+A[1]*B[0])%p) conj=lambda A:(A[0],(-A[1])%p); norm=lambda A:(A[0]*A[0]-n*A[1]*A[1])%p inv=lambda z:pow(z,p-2,p) def order(A): o=1;X=A while X!=(1,0): X=mul(X,A); o+=1 return o omega=next((u,v) for u in range(p) for v in range(1,p) if order((u,v))==3) a=1; tgt=(a*a*inv(2))%p # N(b)=a^2/2 : the quarter-turn b=next((u,v) for u in range(p) for v in range(p) if (u,v)!=(0,0) and norm((u,v))==tgt) lam=[] for k in range(3): wk=(1,0) for _ in range(k): wk=mul(wk,omega) t=( (a if k==0 else 0)+0, 0 ) L=((a+mul(b,wk)[0]+mul(conj(b),mul(wk,wk))[0])%p, (mul(b,wk)[1]+mul(conj(b),mul(wk,wk))[1])%p) lam.append(L) inFp=all(L[1]==0 for L in lam); l=[L[0] for L in lam] Q=(sum((x*x)%p for x in l)*inv((sum(l)%p)**2))%p amp=(norm(b)*inv(a*a))%p return Q==(2*inv(3))%p and inFp and amp==inv(2)%p, l, omega for p in [17,53,89]: good,l,om=_framed_koide(p) must(f"E.framed_p{p}",good, f"p={p}: Q=2/3 exact in F_p; sqrt-mass eigenvalues framed rationals " f"{[_ratrecon(x,p) for x in l]}; omega={_ratrecon(om[0],p)}+{_ratrecon(om[1],p)}t; " f"quarter-turn = N(b)/a^2 = 1/2 (rational, not sqrt2)") print("\n"+"="*70) print("F. The amplitude diagonal over FRAMED RATIONALS: r^2 = N(1-zeta_n) = 2, 3 (integers in F_p)") print("="*70) # r^2 = N(1-zeta_n), the finite-field NORM of (1 - finite n-th root of unity): quarter-turn # (i, i^2=-1) -> 2 (Q=2/3, four-fold, colourless); cube root (omega, omega^2+omega+1=0) -> 3 # (Q=5/6, three-fold, colour). N(b)/a^2 = 1/2, 3/4 are framed rationals. The 'sqrt2, sqrt3, # 2 sin(pi/n)' are CONTINUUM readings of the integers 2, 3 -- one rung up. No complex, no trig. for p in [17,53,89]: i_=next(x for x in range(1,p) if (x*x+1)%p==0) # quarter-turn i in F_p (i^2=-1) N1mi=((1-i_)*(1+i_))%p # (1-i)(1+i)=1-i^2 = 2 nq=next(m for m in range(2,p) if pow(m,(p-1)//2,p)==p-1) # F_{p^2}=F_p[t]/(t^2-nq) mul=lambda A,B:((A[0]*B[0]+A[1]*B[1]*nq)%p,(A[0]*B[1]+A[1]*B[0])%p) def order(A): o=1;X=A while X!=(1,0):X=mul(X,A);o+=1 return o om=next((u,v) for u in range(p) for v in range(1,p) if order((u,v))==3) # cube root omega a1=((1-om[0])%p,(-om[1])%p); a2=((1-om[0])%p,(om[1])%p) # (1-omega) and (1-conj omega) N1mw=mul(a1,a2) # N(1-omega) = (3, 0) in F_p must(f"F.diag_p{p}", N1mi==2%p and N1mw==(3%p,0), f"p={p}: N(1-i)={N1mi} (=2: four-fold, r^2=2 -> Q=2/3); " f"N(1-omega)={N1mw[0]} (=3: three-fold/colour, r^2=3 -> Q=5/6) -- integers in F_p") print("\n"+"="*70) nfail=sum(1 for _,c in ok if not c) print(f"EXACT CORE: {sum(1 for _,c in ok if c)}/{len(ok)} float-free verifications hold" + ("" if nfail==0 else f" ({nfail} FAILED)")) print("="*70)