TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/geometric_source_return/check_source_boundary_response.py Original file SHA-256: 6486986571b2f1cab226a404d3e6365e6c75627ac365a4f1bddba543221dc80f Inclusive original lines: 105-200 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000105 | need(zero(sp.I*sigma3*sp.diff(phi,w)-lam*phi),'Original spatial operator') 000106 | for endpoint in [-1,1]: 000107 | need(zero((sigma1-b*identity)*phi.subs(w,endpoint)), 'Fixed boundary domain') 000108 | 000109 | sigma2=sp.Matrix([[0,-sp.I],[sp.I,0]]) 000110 | need(zero(sigma1*sigma2*sigma1+sigma2), 'Normal metric variation preserves Pin covariance') 000111 | # The existing transverse zero branch of the PURE D-sea realizes the old 000112 | # quarter circle. It is a different source quantization from thermal C. 000113 | x=sp.symbols('x', real=True) 000114 | scale=sp.symbols('a', positive=True) 000115 | pure_bridge_checks=0 000116 | for b in [-1,1]: 000117 | phi0=sp.Matrix([1,b])/2 000118 | need(zero(sp.integrate((phi0.T*phi0)[0],(w,-1,1))-1),'Zero branch isometry') 000119 | need(zero(sp.integrate(w*(phi0.T*phi0)[0],(w,-1,1))),'Zero mean normal coordinate') 000120 | need(zero(sp.integrate(w*w*(phi0.T*phi0)[0],(w,-1,1))-sp.Rational(1,3)), 'Normal nonlocality witness') 000121 | for pin in [sp.I,-sp.I]: 000122 | quarter=pin*b 000123 | alpha=sp.Rational(1,4) if quarter==sp.I else sp.Rational(3,4) 000124 | for nlong in [-1,0,1]: 000125 | f=sp.exp(2*sp.pi*sp.I*(nlong+alpha)*x) 000126 | psi=phi0*f 000127 | applied=-sp.I*(sigma1*sp.diff(psi,x)+scale*sigma2*sp.diff(psi,w)) 000128 | need(zero(applied-phi0*(-sp.I*b*sp.diff(f,x))), 'Pure source same-time intertwiner') 000129 | need(zero(psi.subs(x,x+1)-pin*sigma1*psi.subs(w,-w)), 'Actual quarter Pin domain') 000130 | need(zero(sp.conjugate(psi.subs(x,-x))-phi0*sp.conjugate(f.subs(x,-x))), 'Same longitudinal reflection') 000131 | need(zero(sp.diff(applied,scale)), 'No exterior response to uniform normal scale') 000132 | pure_bridge_checks+=1 000133 | # First response of an actual, deck-even conformal metric variation. 000134 | # D_eps=e^(-eps*w^2/2) D_0 e^(-eps*w^2/2), in the original L2 density. 000135 | kk=sp.symbols('k0', real=True) 000136 | eps=sp.symbols('epsilon', real=True) 000137 | for b in [-1,1]: 000138 | phi0=sp.Matrix([1,b])/2 000139 | orth=sp.Matrix([1,-b])/2 000140 | psi=phi0*sp.exp(sp.I*kk*x) 000141 | u=sp.exp(-eps*w*w/2) 000142 | direct=u*(-sp.I)*(sigma1*sp.diff(u*psi,x)+sigma2*sp.diff(u*psi,w)) 000143 | predicted=b*sp.exp(-eps*w*w)*(kk*phi0+eps*w*orth)*sp.exp(sp.I*kk*x) 000144 | need(zero(direct-predicted),'Actual conformal Dirac first response') 000145 | variation=sp.diff(direct,eps).subs(eps,0)*sp.exp(-sp.I*kk*x) 000146 | compression=sp.integrate((phi0.T*variation)[0],(w,-1,1)) 000147 | need(zero(compression+b*kk/3),'Geometric input compression') 000148 | exterior=sp.simplify(variation-phi0*compression) 000149 | exterior_norm=sp.integrate((sp.conjugate(exterior).T*exterior)[0],(w,-1,1)) 000150 | need(zero(exterior_norm-(4*kk*kk/45+sp.Rational(1,3))), 'All-mode geometric return form') 000151 | spectral_sum=4/sp.pi**4*(2*kk**2*sp.zeta(4)+sp.pi**2*sp.zeta(2)/2) 000152 | need(zero(spectral_sum-(4*kk**2/45+sp.Rational(1,3))),'Independent full Fourier return sum') 000153 | # The coupling law holds for any smooth real even conformal profile. 000154 | # Check distinct profiles directly; the accompanying proof uses no ansatz. 000155 | phi0=sp.Matrix([1,1])/2;orth=sp.Matrix([1,-1])/2 000156 | for fprofile in [sp.Integer(1),w*w,w**4,sp.cos(sp.pi*w)]: 000157 | average=sp.integrate(fprofile,(w,-1,1))/2 000158 | exterior=-kk*(fprofile-average)*phi0+sp.diff(fprofile,w)*orth/2 000159 | actual=sp.integrate((sp.conjugate(exterior).T*exterior)[0],(w,-1,1)) 000160 | variance=sp.integrate((fprofile-average)**2,(w,-1,1))/2 000161 | gradient=sp.integrate(sp.diff(fprofile,w)**2,(w,-1,1))/2 000162 | need(zero(actual-kk*kk*variance-gradient/4),'General conformal profile return law') 000163 | # Independent Gibbs computation of the neutral negative-frequency witness. 000164 | d=[annihilator(i,2) for i in range(2)] 000165 | n=[x.T*x for x in d] 000166 | c1=sp.symbols('c1', positive=True) 000167 | rho=sp.diag((1-c1)/2,(1-c1)/2,c1/2,c1/2) 000168 | neutral=d[0].T*d[1] 000169 | ham=energy*n[1] 000170 | need(zero(ham*neutral-neutral*ham+energy*neutral),'Neutral energy frequency') 000171 | need(zero((rho*neutral.T*neutral).trace()-c1/2),'Neutral witness from Gibbs density') 000172 | # Actual spectral values: finite numerical validation of the analytic 000173 | # infinite-mode formulas. No claim of an infinite cutoff proof here. 000174 | finite_modes=[] 000175 | for n in range(-8,9): 000176 | lam=float(sp.pi*n/2) 000177 | rr=np.exp(-lam) 000178 | p=np.array([[1,rr],[rr,rr*rr]])/(1+rr*rr) 000179 | need(np.linalg.norm(p@p-p)<1e-14,'Numeric original projection') 000180 | c=rr*rr/(1+rr*rr) 000181 | eps=1e-5 000182 | cp=1/(1+np.exp(2*np.exp(-eps)*lam)) 000183 | cm=1/(1+np.exp(2*np.exp(eps)*lam)) 000184 | need(abs((cp-cm)/(2*eps)-2*lam*c*(1-c))<1e-9,'Original geometric response') 000185 | finite_modes.append(dict(n=n,energy=lam,covariance=c,cross_response=float(p[0,1]), 000186 | geometric_derivative=float(2*lam*c*(1-c)))) 000187 | 000188 | # A neutral energy-lowering operator already detects that a thermal state 000189 | # is not a positive-energy ground state for the SAME H time. 000190 | # B=a_0^* a_m: =C_m(1-C_0)>0, frequency=-E_m. 000191 | witnesses=[] 000192 | for m in [1,2]: 000193 | lam=sp.pi*m/2 000194 | norm=1/(2*(1+sp.exp(2*lam))) 000195 | witnesses.append(dict(m=m,energy_frequency=str(-lam),norm_squared=str(norm), 000196 | norm_squared_numeric=float(norm), 000197 | spatial_J_invariant=(m%2==0))) 000198 | result=dict( 000199 | analytic_identities='Graph, purity, restriction, NS complement, full return and first geometric response', 000200 | exact_four_point_checks=4**4,