TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/VARIABLE-REVIEW.txt Original file SHA-256: 6c8dd202a698c018ed3ecddfb38a0bdd1676101b9cf15b110547e08914095837 Inclusive original lines: 1-200 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. 000001 | INDEPENDENT REVIEW: VARIABLE ORIGINAL HIGGS BACKGROUND AND FORCE MOMENTS 000002 | 2 November 2026. Original-path continuation. Canonical repository read only. 000003 | 000004 | Verdict 000005 | 000006 | The proposed anticommutator/commutator proof is correct, with one domain 000007 | clarification: define the time integral initially in the strong/weak operator 000008 | sense, not as an unproved norm-Bochner integral in B(H). The two weak 000009 | identities then prove the required H1 range and the bounded hA identity. 000010 | The log-MGF inequality is also correct without [C,A]=0. It yields all 000011 | single-force exponential moments and uniform integrability. All ORDERED 000012 | mixed force moments follow from the existing determinant source by a short 000013 | locally uniform complex-analytic argument, not from a noncommutative 000014 | Holder inference using only the separate scalar MGFs. 000015 | 000016 | 1. Given original data and actual changed premise 000017 | 000018 | Reuse the SAME given circle, finite original internal space, gamma=gamma*, 000019 | gamma^2=I, D0=-i partial_theta+alpha, alpha=1/4 or3/4, fixed Yukawa/MR 000020 | coefficients and original positive ell. On periodic H1 sections put 000021 | h=gamma D0+V(theta), V=V*, ||V||<=M, 000022 | C=expit(2ell h). 000023 | The original V=-i gamma Phi(H(theta)) is included. H now varies smoothly 000024 | in space instead of the constant background of R080. It is time independent 000025 | in the h used to define exp(ith). This is not a theorem for arbitrary 000026 | time-dependent background propagators without the corresponding proof. 000027 | 000028 | For the actual Higgs-force insertion let F(t,theta)=F(t,theta)* be a smooth 000029 | compactly time-supported multiplication operator with {gamma,F}=0. It 000030 | is a real linear combination of the original Ta force matrices. Its 000031 | spatial and time derivatives are bounded and integrable in time. Circle 000032 | domains/twists and fiber endomorphisms are the original compatible ones. 000033 | 000034 | V is a bounded selfadjoint perturbation, so h is selfadjoint on Dom(D0)=H1. 000035 | No derivative bound on V is needed for the estimate below. Smooth variable 000036 | H belongs to this larger bounded class. The covariance is the original 000037 | conditional quasifree covariance of THIS h, not an average-covariance 000038 | Gaussian replacement for an integrated interacting Higgs source. 000039 | 000040 | Source basis freshly reread: 000041 | research-source:8f6a57abe438221d13e6e0d7, 000042 | LOCAL-SPACETIME-FORCE-SOURCE-PROOF.txt:1-161, hash 000043 | 2bee4ef1df14985eaceab4b83d00108ce420f31e46bed1d15d29dcc51eb66c8a. 000044 | R080 already supplies the constant-background force field, centered det2 000045 | source, Stone realization, higher ordered unitary words and actual circle 000046 | locality. Those results are extended in their given source class, not 000047 | reconstructed as newly missing objects. 000048 | 000049 | 2. Anticommutator identity, time-commutator sign and domains 000050 | 000051 | For each t, multiplication by F(t) preserves H1. On H1, 000052 | {h,F(t)}=B(t), 000053 | B(t)=-i gamma partial_theta F(t)+{V,F(t)}. (1) 000054 | The derivative terms cancel exactly because {gamma,F}=0. The scalar twist 000055 | alpha also cancels in that anticommutator. B(t) is bounded; for real force 000056 | tests it is selfadjoint, since gamma partial_theta F is antihermitian. 000057 | 000058 | Define 000059 | A=int exp(ith) F(t) exp(-ith) dt, 000060 | B_int=int exp(ith) B(t) exp(-ith) dt, 000061 | D_int=int exp(ith) partial_t F(t) exp(-ith) dt. (2) 000062 | These exist as strong operator integrals by strong continuity and compact 000063 | support, with ||A||<=int ||F(t)||dt. Strong integrability is sufficient. 000064 | For example an odd constant matrix conjugated by an unbounded Dirac 000065 | generator need not vary continuously in operator norm, so norm-Bochner 000066 | integrability should not be assumed without argument. A=A*. 000067 | 000068 | Test between psi,phi in H1. Moving h onto the test vectors justifies 000069 | integration of (1) as a form identity: 000070 | {h,A}=B_int. (3) 000071 | The time derivative of the conjugated insertion, again as an H1 form, is 000072 | d_t[exp(ith)F(t)exp(-ith)] 000073 | =i[h,exp(ith)F(t)exp(-ith)] 000074 | +exp(ith)partial_t F(t)exp(-ith). 000075 | Its time integral vanishes at both endpoints. Therefore the correct sign is 000076 | [h,A]=i D_int. (4) 000077 | Adding the two forms gives, for phi in H1, 000078 | 2 =. 000079 | The right side is bounded in ||psi||. Selfadjointness of h therefore 000080 | implies A phi in Dom(h) and 000081 | 2hA phi=(B_int+i D_int)phi. 000082 | Now approximate arbitrary phi by H1 vectors. Boundedness of A and the 000083 | displayed right side, together with closedness of h, proves 000084 | A(H) subset Dom(h), 2hA=B_int+i D_int in B(H). (5) 000085 | In particular this conclusion was derived, not presupposed when using 000086 | the weak identities. 000087 | 000088 | 3. Uniform Hilbert-Schmidt estimate, no spectral diagonalization required 000089 | 000090 | Write f0=int||F||dt, ftheta=int||partial_theta F||dt, 000091 | ft=int||partial_t F||dt. From (1)-(5), 000092 | ||hA|| <= (ftheta+ft)/2+M f0, 000093 | ||D0 A||=||gamma(hA-VA)|| <= (ftheta+ft)/2+2M f0. 000094 | Since 000095 | A=(D0+i)^-1 (D0 A+iA), 000096 | and the resolvent on this original one-dimensional compact circle obeys 000097 | ||(D0+i)^-1||_2^2 000098 | =d_int sum_(n in Z) [1+(n+alpha)^2]^-1 A strongly. The SAME finite-dimensional identities give a 000121 | uniform bound on ||(D0+i)A_N||. Hence 000122 | ||(1-P_K)A_N||_2 000123 | <=||(1-P_K)(D0+i)^-1||_2 sup_N||(D0+i)A_N|| ->0 000124 | as K->infinity, uniformly in N. Selfadjointness of A_N supplies the 000125 | analogous right tail. Strong convergence on each fixed finite corner 000126 | becomes norm/HS convergence there; combining both tails proves 000127 | A_N -> A in S2. (7) 000128 | This is a consistency proof for the actual recomputed cutoff, not a new 000129 | choice of background. A thermal C_N recomputed from h_N converges strongly 000130 | to C after its finite embedding and stays in [0,I]. Products with S2/S1 000131 | convergent operators therefore have the needed S2/S1 limits. This does not 000132 | identify different finite thermal representations by a guessed unitary: 000133 | the old Stone construction may use the SAME actual infinite C throughout. 000134 | 000135 | 5. Exact log-MGF proof, including noncommuting C and A 000136 | 000137 | In a finite CAR cutoff let 0<=C_N<=I be the actual covariance, A_N=A_N*, 000138 | and W_N=dGamma(A_N)-Tr(C_N A_N) I. The finite quasifree trace formula gives 000139 | log omega_N(exp(s W_N)) 000140 | =-s Tr(C_N A_N)+log det[I+C_N(exp(sA_N)-I)], s real. 000141 | Use Sylvester's determinant identity to write the determinant with 000142 | Y_N=I+C_N^(1/2)(exp(sA_N)-I)C_N^(1/2)>0. 000143 | Indeed Y_N>=exp(-|s| ||A_N||)I, so its real logarithm is legitimate. 000144 | The elementary positive-matrix inequality logdetY<=Tr(Y-I) yields 000145 | log omega_N(exp(s W_N)) 000146 | <=Tr[C_N(exp(sA_N)-I-sA_N)] 000147 | <=(s^2/2) exp(|s| ||A_N||) Tr(C_N A_N^2) 000148 | <=(s^2/2) exp(|s| ||A_N||) ||A_N||_2^2. (8) 000149 | The middle inequality follows by scalar Taylor remainder on the spectrum 000150 | of selfadjoint A_N and then positive functional calculus. [C_N,A_N]=0 000151 | was never used. The same proof applies to either cutoff construction 000152 | above. All constants are uniform on ||V||<=M by(6). 000153 | 000154 | For every t>0, (8) on both signs implies 000155 | sup_N omega_N(exp(t|W_N|)) 000156 | <=2 exp[(t^2/2)exp(t a)b^2], (9) 000157 | where a bounds ||A_N|| and b bounds ||A_N||_2. For each p>0, 000158 | sup_N omega_N(|W_N|^p 1_(|W_N|>R)) 000159 | <=the RHS(9) sup_(x>R) x^p exp(-t x) ->0. 000160 | Thus single-force spectral distributions have all moments and uniformly 000161 | integrable powers. Weak convergence from the existing characteristic 000162 | source plus (9) gives convergence of all their moments; a stronger 000163 | exponential bound gives convergence of each fixed real MGF as well. 000164 | 000165 | 6. All ordered mixed moments: use the existing complete source 000166 | 000167 | Separate scalar MGFs alone do not prove arbitrary noncommuting ordered 000168 | moment convergence. For actual forces A1,...,Ar use the existing R080 000169 | centered source, now with complex z_j: 000170 | U(z)=product_j exp(z_j A_j), L(z)=sum_j z_j A_j, 000171 | chi(z)=det2[I+C(U(z)-I)] exp Tr C[U(z)-I-L(z)]. (10) 000172 | For imaginary z it is the old ordered unitary word, with original order. 000173 | For real z in a finite cutoff it is the ordered centered exponential word. 000174 | No independent infinite linear trace or uncentered determinant is used. 000175 | 000176 | Put S=sum_j |z_j| ||A_j||, B=sum_j |z_j| ||A_j||_2. 000177 | Telescoping the product gives ||U-I||_2<=exp(S)B. 000178 | For U(t)=product_j exp(t z_j A_j), U(0)=I, U'(0)=L. 000179 | Every term of U''(t) contains two S2 factors, so 000180 | ||U''(t)||_1<=exp(t S) B^2. 000181 | Integrating its second-order remainder yields 000182 | ||U-I-L||_1<=(exp(S)/2) B^2. (11) 000183 | The estimates hold on every complex compact set, uniformly in the actual 000184 | cutoffs and bounded background class. S2 continuity of the exponential 000185 | product and S1 continuity of the remainder, with det2 continuity, give 000186 | locally uniform convergence of(10). All functions involved are entire in 000187 | the finitely many z_j. Cauchy's formula therefore passes every ordered 000188 | mixed derivative to the limit. This supplies the full ordered moment 000189 | functional, not just Gaussian cumulants or symmetrized responses. 000190 | The positive real unitary-word kernel and the existing Stone realization 000191 | are retained. An additional essential-selfadjointness core claim is not 000192 | needed or inferred from this calculation. 000193 | 000194 | 7. First assumptions, scope and novelty 000195 | 000196 | The changed premise is the SAME original force source at bounded variable 000197 | spatial H(theta). Within the given circle/source class, the smoothing 000198 | proof is analytic and uniform; it avoids mode-by-mode diagonalization. 000199 | No new geometry, coupling, state rule, thermal calibration, probability 000200 | law for H, Gaussianization or physical active-clock assumption is added.