TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ELLIPTIC-LOCAL-INSERTION-PROOF.txt Original file SHA-256: 6d009be249a02f8274350d6f2b3b27f62e6225814160ea3dd8d9deb622bb7871 Inclusive original lines: 1-194 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. 000001 | R066. Same original elliptic source: supported insertion, holonomy and causality. 000002 | Sitzung034, 1 November2026. Experiment, not a new primitive P1 quantization. 000003 | 000004 | 1. Question and exact retained input 000005 | R065 requires an actual original source insertion, not another localizer on 000006 | a changed input state. Use the existing labelled annulus CAR marginal: 000007 | D_a=-i d/dx+a on periodic H1(S1), a=1/4 or3/4, 000008 | C_a=(1+exp(b D_a))^-1, b=4pi, and its ORIGINAL sea subtraction. 000009 | The existing free quantization gives rho_a=Z_a^-1 exp(-b H_a), 000010 | H_a=sum_(n>0) J_-n J_n+Q²/2+(a-1/2)Q, [J_n,J_-m]=n delta_nm. 000011 | J(f)=f0 Q+sum_(n>0)(f_n J_-n+bar(f_n)J_n), 000012 | f(x)=sum_Z f_n exp(inx), mean integral normalized by1/(2pi). 000013 | The two quarter charge/oscillator blocks of this marginal are independent. 000014 | They are NOT the two annulus faces. The full two-face graph is retained below. 000015 | No new geometry, vacuum, current normalization, Yukawa or physical clock. 000016 | 000017 | Decision: does D_a+epsilon f for any real smooth f supported in a proper 000018 | connected interval I produce a genuine local operation on this SAME source? 000019 | Success: exact full-state/word formula, original interior AND exterior test, 000020 | and actual time response. Abort any primitive closure if the source/action 000021 | and insertion measure are merely assumed. f is a test insertion, not a 000022 | freely fitted physical profile. The entire smooth supported class is treated. 000023 | 000024 | 2. Exact operator, state, normalization and all-word transport 000025 | For real epsilon let k0=0, k_n=i epsilon f_n/n for n!=0. 000026 | Then k is real smooth, k'=-epsilon(f-f0). V=exp(ik(x)) is unitary on H1. 000027 | Direct differentiation, INCLUDING the original domain, gives 000028 | D_a+epsilon f=V D_(a+epsilon f0) V*. 000029 | Therefore C_epsilon=V C_(a+epsilon f0) V*. This is a FORWARD evaluation 000030 | of an explicit original operator insertion, not log-inversion of an input Y. 000031 | Its first derivative is i[k1,C_a]+f0 partial_a C_a, k1=k/epsilon. 000032 | In Fourier components the offdiagonal derivative is 000033 | (c_r-c_s) f_(r-s)/(r-s); diagonal is -b c_r(1-c_r) f0. 000034 | For the actual resolvent G_z=(D_a-z)^-1, 000035 | delta G_z=-G_z f G_z; if f0=0 it also equals i[k1,G_z]. 000036 | This is the same resolvent insertion rule, with a given source operator, 000037 | not a derivation of a previously missing Brel from this identity. 000038 | 000039 | The existing current implementer W(k)=exp(iJ(k)) has 000040 | W(k)J_n W(k)*=J_n+epsilon f_n. Completing every oscillator square gives 000041 | H_a+epsilon J(f)=W(k) H_(a+epsilon f0) W(k)*-epsilon² S_f, 000042 | S_f=sum_(n>0)|f_n|². Smoothness makes all displacement/energy sums finite. 000043 | This gives the selfadjoint closure and traceclass Gibbs state at every real 000044 | epsilon: the charge quadratic remains confining even outside00) f_n bar(g_n)=mean(fg)-f0 g0. 000051 | For every real smooth g, including exterior probes, 000052 | omega_epsilon(J(g))=g0 _(a+epsilon f0)-epsilon B(f,g), 000053 | omega_epsilon(W(g))=exp[-i epsilon B(f,g)] 000054 | Theta_(a+epsilon f0)(g0)/Theta_(a+epsilon f0)(0) 000055 | exp[-1/2 sum_(n>0)n coth(bn/2)|g_n|²]. 000056 | All ordered Weyl words retain the ORIGINAL BCH factor 000057 | exp[-1/2 sum_(i0)n[bar(f_n)g_n-f_n bar(g_n)]=i mean(f'g). 000059 | Use the last one-Weyl formula on sum_i g_i. This determines the full current 000060 | functional, not only its mean or covariance. Original full CAR words follow 000061 | from C_epsilon by their original Wick prescription, not a new Gaussian rule. 000062 | 000063 | In the original canonical two-face chart G(C)=[[1-C,X],[X,C]], 000064 | X=sqrt(C(1-C)), the entire graph transforms as 000065 | G(C_epsilon)=diag(V,V) G(C_(a+epsilon f0)) diag(V*,V*). 000066 | All cross-face sources are retained. Apply the existing reflection map in 000067 | its transported chart; no independent opposite-face state is inserted. 000068 | For the time-dependent operation below, the full source remains the original 000069 | state acted on in one specified algebra; do not replace it by G(C_epsilon). 000070 | 000071 | 3. Whole fixed-holonomy class gives an actual local operation 000072 | If f0=0, k'= -epsilon f. Because I is a proper connected interval and the 000073 | integral of f vanishes, choose the otherwise irrelevant constant so k=0 000074 | on the complement. Adding this constant only applies a global charge phase, 000075 | which leaves the original gauge invariant state unchanged. 000076 | Thus k is genuinely supported in I, W(k) belongs to the given local current 000077 | algebra, and rho_epsilon=W(k)rho_a W(k)* on the SAME initial source. 000078 | All original exterior CAR/current words are unchanged, all inside/outside 000079 | joint words follow this same unitary, and no product state/reset is used. 000080 | The partition scalar exp(b epsilon² S_f) remains, although normalized 000081 | observations are locally unitarily transported. This is not a new vacuum. 000082 | This supplies a source-determined local implementation in the whole 000083 | fixed-holonomy scalar insertion class. It does not select which f occurs 000084 | as a physical event, nor the original60 record probabilities or effects. 000085 | 000086 | 4. The original quarter pair fixes an exact outside susceptibility 000087 | At b=4pi denote the independent quarter charges q,r and p=q+r,d=q-r. 000088 | F_1/4(q)+F_3/4(r)=(p²+d²-d)/4, p=d mod2. 000089 | Thus the actual joint Gibbs weight is exp[-pi(p²+d²-d)]. 000090 | The d weights exp[-pi(d-1/2)²] have equal sums on even and odd d, 000091 | by d->1-d. Hence the COMPLETE total-charge law is exactly 000092 | P(p)=exp(-pi p²)/sum_l exp(-pi l²), not a continuous Gaussian. 000093 | Poisson summation Z(t)=t^-1/2 Z(1/t) for Z(t)=sum_p exp(-pi t p²) 000094 | implies (log Z)'(1)=-1/4, hence Var(p)=1/(4pi). 000095 | The two quarter variances coincide by r->-q and are independent; therefore 000096 | Var_a(Q)=1/(8pi), b Var_a(Q)=1/2 for BOTH original quarters. 000097 | Their means remain plus/minus0.04134328958148..., not individually zero. 000098 | There is a concrete connection to the ALREADY given elliptic Quillen norm: 000099 | with theta(z|i)=sum_p exp(-pi p²+2pi i p z), the total-charge characteristic 000100 | function is theta(t/(2pi)|i)/theta(0|i). Hence on the real unit-charge probe 000101 | N(v,0)=|omega(exp(2pi i v(Q_1/4+Q_3/4)))|². 000102 | The derived variance gives partial_v² log N(0,0)=-2pi, exactly the 000103 | independently known relative spectral determinant curvature. This identifies 000104 | these specific response functions and arguments. It does NOT identify a 000105 | positive state at arbitrary complex temporal twist, the physical EM counting 000106 | insertion q(alpha), or its exponent5/4. All reference and line types stay. 000107 | This exact half-factor uses BOTH original quarter blocks and the given square 000108 | geometry; it is not inferred from a decimal coincidence or a fitted beta. 000109 | 000110 | Duhamel differentiation of this SAME normalized Gibbs source now gives 000111 | d_epsilon omega_epsilon(J(g))|0 000112 | =-B(f,g)-b Var_a(Q) f0 g0 000113 | =-mean(fg)+(1/2) f0 g0. 000114 | For disjoint supports it is +(1/2) f0 g0, in each quarter. 000115 | This is a nonzero ORIGINAL exterior response to changing the equilibrium 000116 | operator, when the insertion changes its circle holonomy f0. 000117 | It is NOT a signal or the action of a local trace-preserving operation on 000118 | the unchanged state. Such an operation fixes every exterior observable. 000119 | In particular no differentiable family of such local operations can equal 000120 | these normalized equilibrium source states near epsilon0 when f0!=0: 000121 | choose exterior g with g0!=0. The bounded-Weyl derivative supplies the same 000122 | test; no reliance on manipulating only an unbounded observable is needed. 000123 | This is an infinitesimal statement, not a global finite-epsilon no-go. 000124 | Integer holonomy changes and charged/winding implementers are not excluded. 000125 | For common coupling to both quarters the total exterior coefficient is1, 000126 | since the two independent coefficients add; do not call the whole pair1/2. 000127 | Holding total charge fixed, or changing subtraction, changes the question. 000128 | 000129 | 5. Same original source gives exact causal evolution, not a Gibbs reset 000130 | Use the GIVEN conformal H_a time, J_n(t)=exp(-int)J_n. 000131 | For a specified smooth real time-dependent source f_t(x), in the interaction 000132 | picture v_t(x)=f_t(x+t); then J(f_t,t)=J(v_t). 000133 | Put F=integral dt v_t, B_path=-1/2 integral_(t>s)dt ds c(v_t,v_s). 000134 | All higher Magnus commutators vanish because c is central. Consequently 000135 | U[f]=time_order exp[-i integral dt J(v_t)]=exp(B_path) W(-F). 000136 | It is a unitary, not a measurement instrument or a choice of event schedule. 000137 | For two paths define Z_SK[f+,f-]=omega_a(U[f-]*U[f+]). Exactly 000138 | Z_SK=exp(B_+-B_-+c(F_-,F_+)/2) omega_a(W(F_--F_+)). 000139 | The last factor is the original full Theta times coth expression, not a 000140 | replacement noise distribution. Therefore Z_SK[f,f]=1 identically for 000141 | ALL smooth source paths; cross phases and original charge fluctuations stay. 000142 | All finite ordered current/neutral words can be inserted into the same Weyl 000143 | product. This construction carries the original normalization and correlations. 000144 | 000145 | For a pulse epsilon J(f) at time0, observed at t>=0, 000146 | delta omega(J(g,t))=i epsilon c(f,g_t) 000147 | =-epsilon mean(f' g_t), g_t(x)=g(x+t). 000148 | This is exact in epsilon for the current mean, since the commutator is central. 000149 | It vanishes whenever the transported supports do not meet. The Heaviside 000150 | retarded prescription is zero for times before the source. At equal time, 000151 | every exterior bounded observable is fixed by the supported unitary. 000152 | State-dependent coth fluctuations and state-independent causal commutator 000153 | thus come from one and the same original kernel/current/time dictionary. 000154 | The Euclidean half-factor in section4 is not an instantaneous response. 000155 | The geometric propagation here is the GIVEN chiral circle propagation, 000156 | not a derivation of physical3+1 dimensions, a physical light speed or spin2. 000157 | 000158 | 6. Consequence for F1 and next decisive original source test 000159 | This closes the source insertion/normalization/local execution distinction 000160 | in the whole given smooth scalar-current class, retaining both quarters and 000161 | the whole two-face source chart. It produces a concrete local implementation 000162 | from a given forward operator change when holonomy is fixed. It also excludes 000163 | equating its holonomy-changing equilibrium variation with local dynamics. 000164 | It does not equate the free annulus source with primitive exp(-S_TFPT), 000165 | nor supply its interacting boson/Higgs measure or sixty-event selection. 000166 | R054/R063 still forbid replacing a nontrivial integrated Higgs source by the 000167 | unchanged free source in their precise scope. R043 remains the full original 000168 | operator/measure response with contact terms and reference transport. 000169 | 000170 | Next: evaluate the actual original marked nonabelian/seam insertion in the 000171 | same forward manner, carrying its holonomy, full source, bound reference, 000172 | normalization and given time. Test whether it is pure transported connection 000173 | or has a genuine field-strength/mixed source term. The latter term must be 000174 | read from the original action/producer, not supplied as a fitted perturbation. 000175 | Do not infer a four-dimensional spacetime or gravity from the current theorem. 000176 | For gravity require the original metric/normal variation and its common 000177 | source Ward response with the already solved matter, not a new background. 000178 | 000179 | Counterproposal audit: half-sided modular inclusion needs the original net, 000180 | state and its actual inclusion relations; canonical split alone is not one. 000181 | Relative Cauchy evolution presupposes its spacetime/functor and does not by 000182 | itself derive spacetime or quantum spin2. R043 resolvent variation is not 000183 | literally the same type as an observable commutator. Strict0