C05.1 · Computations

The exact input is the chosen root of the golden polynomial and the matrix-defined Jordan product. The script does not receive a presumed Gram matrix. It constructs the adjoint and trace pairings, then checks the ratios. The two non-golden controls are explicitly defined above. Reported generator statements remain reports, not successful tests.

The command actually executed was: ```sh python CHATGPT_D1258_C05_1_CHECKS_S310.py ``` Settings: exact SymPy rational/symbolic arithmetic; no floating-point tolerances, no external numerical data files, no network calls, no optimized Python (`-O`). The coordinates and formulas above are the scientific inputs. Runtime versions are printed in stdout. The script SHA-256 is `c5ca4639869c8464fd51de433e0e44758b26e200ad7f8f80db2a6d96430f9d98`; stdout SHA-256 is `89c669560322dae32a620eed5fe08318f7525267840cc40a26d04134ef5e41bd`. An identical-process rerun is a reproducibility check, not the independent house implementation required by the contract.

Complete standalone code: ```python """D1258 C05.1. Exact chosen-vacuum identities; no physical readout."""

from itertools import product, combinations, permutations from collections import Counter import json import platform import sympy as S if not __debug__: raise SystemExit("Do not use -O: assertions are required.") def report(key, value): print(key + " = " + str(value)) def va(x,y): return tuple(a+b for a,b in zip(x,y)) def vn(x): return tuple(-a for a in x) def vs(x,y): return va(x,vn(y)) def sc(k,x): return tuple(k*a for a in x) def dot(x,y): return sum(a*b for a,b in zip(x,y)) def cross(x,y): return (x[1]*y[2]-x[2]*y[1],x[2]*y[0]-x[0]*y[2],x[0]*y[1]-x[1]*y[0]) def qm(p,r): a,b,c,d=p; e,f,g,h=r return (a*e-b*f-c*g-d*h,a*f+b*e+c*h-d*g, a*g-b*h+c*e+d*f,a*h+b*g-c*f+d*e) def qb(p): return (p[0],-p[1],-p[2],-p[3]) O_NAMES=("1","e1","e2","e3","f1","f2","f3","l") OB=[tuple(S.Integer(i==j) for i in range(8)) for j in range(8)] OZ=(S.Integer(0),)*8 def obar(x): return (x[0],)+vn(x[1:]) def omul(x,y,epsilon=1): # Same basis as D1256: f_i=-e_i*l, not +e_i*l. p=x[:4]; q=(x[7],)+vn(x[4:7]) r=y[:4]; s=(y[7],)+vn(y[4:7]) first=va(qm(p,r),sc(epsilon,qm(qb(s),q))) second=va(qm(s,p),qm(q,qb(r))) return first+vn(second[1:])+(second[0],) def onorm(x,epsilon=1): return sum(a*a for a in x[:4])-epsilon*sum(a*a for a in x[4:]) def to_zorn(x): return (x[0]+x[7],)+va(x[1:4],x[4:7])+vs(x[4:7],x[1:4])+(x[0]-x[7],) def from_zorn(z): a=z[0]; u=z[1:4]; v=z[4:7]; b=z[7] return ((a+b)/2,)+sc(S.Rational(1,2),vs(u,v))+sc(S.Rational(1,2),va(u,v))+((a-b)/2,) def zmul(z,w): a=z[0]; u=z[1:4]; v=z[4:7]; b=z[7] c=w[0]; U=w[1:4]; V=w[4:7]; d=w[7] return (a*c+dot(u,V),)+va(va(sc(a,U),sc(d,u)),cross(v,V))+vs(va(sc(c,v),sc(b,V)),cross(u,U))+(dot(v,U)+b*d,) def zero_vector(v): return all(S.expand(a)==0 for a in v) def inertia(M): """Exact rational symmetric congruence elimination; no eigenvalue tolerance.""" A=S.Matrix(M); assert A==A.T positive=negative=null=0 while A.rows: k=next((i for i in range(A.rows) if A[i,i]!=0),None) if k is None: pair=next(((i,j) for i in range(A.rows) for j in range(i+1,A.rows) if A[i,j]!=0),None) if pair is None: null+=A.rows; break i,j=pair P=S.eye(A.rows); P[j,i]=1 A=P.T*A*P k=i inds=[k]+[i for i in range(A.rows) if i!=k] A=A.extract(inds,inds); d=A[0,0] assert d.is_positive or d.is_negative positive+=int(bool(d>0)); negative+=int(bool(d<0)) v=A[1:,0]; A=A[1:,1:]-(v*v.T)/d return (positive,negative,null)

def jmat(A): a,b,c=A[:3]; z=A[3:11]; y=A[11:19]; x=A[19:27] return [[sc(a,OB[0]),z,obar(y)],[obar(z),sc(b,OB[0]),x],[y,obar(x),sc(c,OB[0])]] def jvec(M): assert all(zero_vector(M[i][i][1:]) for i in range(3)) assert all(zero_vector(vs(M[i][j],obar(M[j][i]))) for i in range(3) for j in range(i+1,3)) return tuple(M[i][i][0] for i in range(3))+tuple(M[0][1])+tuple(M[2][0])+tuple(M[1][2]) def mm(A,B): return [[va(va(omul(A[i][0],B[0][j]),omul(A[i][1],B[1][j])),omul(A[i][2],B[2][j])) for j in range(3)] for i in range(3)] def jp(A,B): X=jmat(A); Y=jmat(B); XY=mm(X,Y); YX=mm(Y,X) return jvec([[sc(S.Rational(1,2),va(XY[i][j],YX[i][j])) for j in range(3)] for i in range(3)]) JB=[tuple(S.Integer(i==j) for i in range(27)) for j in range(27)] JI=va(va(JB[0],JB[1]),JB[2]); JZ=(S.Integer(0),)*27 def tr(A): return sum(A[:3]) def sig(A): a,b,c=A[:3];z=A[3:11];y=A[11:19];x=A[19:27] return a*b+a*c+b*c-onorm(x)-onorm(y)-onorm(z) def jnorm(A): a,b,c=A[:3];z=A[3:11];y=A[11:19];x=A[19:27] return a*b*c-a*onorm(x)-b*onorm(y)-c*onorm(z)+2*omul(omul(z,x),y)[0] def sharp(A): return va(vs(jp(A,A),sc(tr(A),A)),sc(sig(A),JI)) def L(A): return S.Matrix.hstack(*[S.Matrix(jp(A,e)) for e in JB]) def brief(A): return {i:S.simplify(a) for i,a in enumerate(A) if S.simplify(a)!=0}

report("environment", {"python":platform.python_version(),"sympy":S.__version__}) report("data_basis","Positive root phi of t^2-t-1; declared H3(CD-split).") phi=(1+S.sqrt(5))/2 simp=lambda z:S.simplify(S.expand(z)) J=(phi,S.Integer(1),1/phi)+(S.Integer(0),)*24 Js=tuple(simp(v) for v in sharp(J)) def B(x,y):return simp(tr(jp(x,y))) G=S.Matrix([[B(J,J),B(J,Js)],[B(Js,J),B(Js,Js)]]) report("golden.polynomial_residual",simp(phi**2-phi-1)) report("golden.reciprocal_square_sum",simp(phi**2+phi**(-2))) report("golden.determinant",simp(jnorm(J))) report("golden.adjoint_diagonal",Js[:3]) report("golden.Gram",G.tolist()) report("golden.Gram_determinant",G.det()) report("golden.mismatch_cosine_squared",simp(G[0,1]**2/(G[0,0]*G[1,1]))) report("golden.mismatch_sine_squared",simp(G.det()/(G[0,0]*G[1,1]))) assert G==S.Matrix([[4,3],[3,4]]) assert G.det()/(G[0,0]*G[1,1])==S.Rational(7,16) rank_frame=sum(jp(e,e)==e for e in JB[:3]) report("dimensions.octonion_Jordan_frame",(len(OB),len(JB),rank_frame)) report("dimensions.ratio",S.Rational(len(OB),rank_frame)) w=JB[3] normblock=B(w,w) gen_norm=dot((1,1,1),(1,1,1)) report("normalization.off_diagonal_unit_trace_square",normblock) report("normalization.chosen_three_copy_vector_square",gen_norm) report("normalization.product",simp(1/S.sqrt(normblock)/S.sqrt(gen_norm))) report("normalization.generations","The three-copy readout is stipulated, not derived here.") # Golden-only controls within unit-determinant reciprocal triples. r=S.symbols("r",positive=True) D=(r,1,1/r)+(S.Integer(0),)*24 Ds=tuple(simp(x) for x in sharp(D)) GD=S.Matrix([[B(D,D),B(D,Ds)],[B(Ds,D),B(Ds,Ds)]]) report("reciprocal_family.determinant",simp(jnorm(D))) report("reciprocal_family.Gram_determinant",S.factor(GD.det())) report("negative_control.r3over2.Gram_determinant",simp(GD.det().subs(r,S.Rational(3,2)))) report("negative_control.r3over2.mismatch_sine_squared", simp((GD.det()/(GD[0,0]*GD[1,1])).subs(r,S.Rational(3,2)))) # Exact unit determinant plus DET-7 countermodel to the extra equal-norm shorthand. # u=((11+sqrt(105))/4)^(1/3)>0, spectrum (sqrt(u),sqrt(u),1/u). u=S.symbols("u",positive=True) P=2*u**6-11*u**3+2 Drep=(S.sqrt(u),S.sqrt(u),1/u)+(S.Integer(0),)*24 Dreps=tuple(simp(z) for z in sharp(Drep)) UU=B(Drep,Drep); VV=B(Dreps,Dreps) assert simp(UU-(2*u+u**(-2)))==0 and simp(VV-(2/u+u*u))==0 assert simp(jnorm(Drep))==1 and B(Drep,Dreps)==3 q=(11+S.sqrt(105))/4 assert simp(2*q*q-11*q+2)==0 and q.is_positive report("DET7_equal_norm_control.q_quadratic_residual",simp(2*q*q-11*q+2)) report("DET7_equal_norm_control.actual_determinant_and_pairing",(simp(jnorm(Drep)),B(Drep,Dreps))) assert S.rem(S.together(UU*VV-16).as_numer_denom()[0],P,u)==0 norm4poly=S.together(UU-4).as_numer_denom()[0] gcd=S.gcd(P,norm4poly) assert gcd==1 report("DET7_equal_norm_control.positive_u_cubed",(11+S.sqrt(105))/4) report("DET7_equal_norm_control.defining_polynomial",P) report("DET7_equal_norm_control.Gram_product_mod_polynomial", S.rem(S.together(UU*VV-16).as_numer_denom()[0],P,u)) report("DET7_equal_norm_control.gcd_with_norm_equals4",gcd) report("DET7_equal_norm_control.conclusion", "Unit determinant and Gram determinant 7 do not alone imply each squared norm is 4.") report("registered_C1_coefficient_proof","NOT VERIFIED: source formula only; A558 matrices absent.") report("registered_generator_uniqueness","NOT VERIFIED: A603/E110 construction absent.") report("finite_trace_Frobenius_proof","NOT TESTED by this script; see companion C08.2, not dim/rank arithmetic.") report("physical_attachment","NOT COMPUTED; no angle, mass or generation assignment inferred.") ```

Actual stdout (not an expected-output fixture): ```text environment = {'python': '3.13.5', 'sympy': '1.14.0'} data_basis = Positive root phi of t^2-t-1; declared H3(CD-split). golden.polynomial_residual = 0 golden.reciprocal_square_sum = 3 golden.determinant = 1 golden.adjoint_diagonal = (-1/2 + sqrt(5)/2, 1, 1/2 + sqrt(5)/2) golden.Gram = [[4, 3], [3, 4]] golden.Gram_determinant = 7 golden.mismatch_cosine_squared = 9/16 golden.mismatch_sine_squared = 7/16 dimensions.octonion_Jordan_frame = (8, 27, 3) dimensions.ratio = 8/3 normalization.off_diagonal_unit_trace_square = 2 normalization.chosen_three_copy_vector_square = 3 normalization.product = sqrt(6)/6 normalization.generations = The three-copy readout is stipulated, not derived here. reciprocal_family.determinant = 1 reciprocal_family.Gram_determinant = (r - 1)**2*(r + 1)**2*(r**4 + 4*r**2 + 1)/r**4 negative_control.r3over2.Gram_determinant = 6025/1296 negative_control.r3over2.mismatch_sine_squared = 6025/17689 DET7_equal_norm_control.q_quadratic_residual = 0 DET7_equal_norm_control.actual_determinant_and_pairing = (1, 3) DET7_equal_norm_control.positive_u_cubed = sqrt(105)/4 + 11/4 DET7_equal_norm_control.defining_polynomial = 2*u**6 - 11*u**3 + 2 DET7_equal_norm_control.Gram_product_mod_polynomial = 0 DET7_equal_norm_control.gcd_with_norm_equals4 = 1 DET7_equal_norm_control.conclusion = Unit determinant and Gram determinant 7 do not alone imply each squared norm is 4. registered_C1_coefficient_proof = NOT VERIFIED: source formula only; A558 matrices absent. registered_generator_uniqueness = NOT VERIFIED: A603/E110 construction absent. finite_trace_Frobenius_proof = NOT TESTED by this script; see companion C08.2, not dim/rank arithmetic. physical_attachment = NOT COMPUTED; no angle, mass or generation assignment inferred. ```