TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/EUCLIDEAN-SOURCE-REVIEW.txt Original file SHA-256: 090a2c8e818ee24b053a41182d126940741037ee2ccc9bbfa4e7dc52f3a8ef13 Inclusive original lines: 1-199 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 | H: INDEPENDENT REVIEW OF THE WHOLE THERMAL EUCLIDEAN SOURCE 000002 | 2 November2026. relational_logic. Canonical unchanged. 000003 | 000004 | VERDICT 000005 | 000006 | The proposed construction is correct for the ACTUAL v1013 bounded local 000007 | Hamiltonian class, at each given beta>0, with the qualifications below. 000008 | It constructs ALL finite ordered Euclidean words and their entire bounded 000009 | source functional, with a positive reflected half-circle word kernel. 000010 | It uses the same Hamiltonians, dynamics and selected weak-* subnet state; 000011 | there is no new measure, new spacetime or independently chosen interaction. 000012 | 000013 | Three precision corrections are necessary: 000014 | (i) v1013 itself uses ambient finite Gibbs traces, then A^G restriction. 000015 | A Gauss-projected Gibbs construction is also covered if [P,H]=0 and 000016 | the physical insertions commute with P, but can select another state. 000017 | (ii) With tau_t(A)=exp(itH)A exp(-itH), the latest-left Schwinger trace 000018 | below uses tau_(-is), not tau_(+is) in that same order. 000019 | (iii) RP is the CANONICAL thermal reflection s->beta-s with adjoint and 000020 | word reversal. Identifying it with a separate P1 seam Theta is not 000021 | automatic from v1013. The full-source/master identity is not claimed. 000022 | 000023 | 1. THE FINITE COMPLETE WORD AND ITS UNIFORM BOUND 000024 | 000025 | For precisely the original finite H_Lambda, put 000026 | rho_Lambda=exp(-beta H_Lambda)/Tr exp(-beta H_Lambda). 000027 | For 0<=s1<=...<=sk<=beta define 000028 | 000029 | S_Lambda(Ak,sk;...;A1,s1) 000030 | =Tr[ exp(-(beta-sk)H) Ak exp(-(sk-s(k-1))H) ... 000031 | A1 exp(-s1 H)] / Z_Lambda. (1) 000032 | 000033 | All other original finite terms in H are retained. Constant energy shifts 000034 | cancel in this normalized expression; this does not delete their actual 000035 | unnormalized partition factors from another caller. 000036 | 000037 | Let a0=(beta-sk)/beta, a1=(sk-s(k-1))/beta,...,ak=s1/beta. 000038 | They are nonnegative and sum to1. Expression(1) is 000039 | Tr[rho^a0 Ak rho^a1 ... A1 rho^ak]. Schatten Holder with 000040 | ||rho^a||_(1/a)=1 and ||Ai||_infinity=||Ai|| gives exactly 000041 | 000042 | |S_Lambda(Ak,...,A1)| <= PRODUCT_i ||Ai||. (2) 000043 | 000044 | For zero a use the infinity norm of the identity. Thus coincident times 000045 | and the closed endpoints are included; keep their specified word order. 000046 | Multilinearity and telescoping also give 000047 | |S_Lambda(A)-S_Lambda(B)| 000048 | <=SUM_j ||Aj-Bj|| PRODUCT_(i!=j) max(||Ai||,||Bi||). (3) 000049 | These bounds are independent of volume and of the extensive energy norm. 000050 | No spectral gap, unbounded moment estimate or positive classical ghost 000051 | measure is used. 000052 | 000053 | For the projected physical trace, replace the trace space by P_Lambda H 000054 | and H by its restriction. Equations(1)–(3) remain exact if P is an 000055 | orthogonal original projection with [P,H]=[P,Ai]=0 and P H is nonzero. 000056 | Alternatively embed its density P exp(-beta H)/Tr(P exp(-beta H)) in the 000057 | ambient matrix algebra. Do not tensor-embed projected physical spaces. 000058 | The actual original v1013 equations4.1–4.3 use the consistent algebra A^G; 000059 | equation5.2 is its separate ambient Gibbs construction. Both callers must 000060 | retain their own subnet state and normalization. 000061 | 000062 | 2. GAUSSIAN SMEARING CLOSES THE COMPLEX-TIME STEP 000063 | 000064 | For a fixed bounded original local A, finite range and the fixed original 000065 | interaction give 000066 | C_A=sup_Lambda ||delta_Lambda(A)||0 its entire analytic continuation has the exact formula 000078 | 000079 | tau_z^Lambda(A_e,Lambda) 000080 | =int g_e(t-z)tau_t^Lambda(A)dt. (5) 000081 | 000082 | The existing real-time compact-uniform norm convergence from v1013, 000083 | the bound ||tau_t^Lambda(A)||=||A||, and the Gaussian tails imply norm 000084 | convergence in(5), uniformly for z in every compact strip subset. 000085 | For |Im z|<=beta its L1 kernel norm is <=exp(beta^2/(4e)). 000086 | This constant may diverge as e->0; NO bound uniform in e is needed. 000087 | The order of limits is fixed e first, thermodynamic limit second, 000088 | and unsmearing last. 000089 | 000090 | The sign/order identity for(1), using precisely v1013's real-time convention, 000091 | is 000092 | S_Lambda =omega_Lambda( tau_(-isk)^Lambda(Ak) ... 000093 | tau_(-is1)^Lambda(A1)). (6) 000094 | For smeared Ai, all factors in(6) converge in norm by(5). The SAME 000095 | weak-* convergent state subnet therefore gives the limit of their 000096 | complete product. No assumption about norm convergence of UNSMEARED 000097 | imaginary-time operators is made. 000098 | 000099 | Now use(3) and(4) to bound the difference between the original Schwinger 000100 | word and its smeared word uniformly in Lambda by a constant times sqrt(e). 000101 | The limiting smeared words consequently form a Cauchy family as e->0. 000102 | The usual three-term error bound proves convergence of the ORIGINAL 000103 | unsmeared words along the same thermodynamic subnet. 000104 | 000105 | This proves the limit for EVERY fixed ordered word. It also gives uniform 000106 | convergence on the closed ordered-time simplex for each fixed tuple Ai: 000107 | for fixed e the analytic product's time-image is norm compact, and weak-* 000108 | convergence of uniformly bounded states is uniform on such compact sets; 000109 | the unsmearing estimate is already independent of the times. No extra 000110 | word-wise choice of subnet is required once the original state subnet is 000111 | fixed. Nonuniqueness of that thermodynamic state is not erased. 000112 | 000113 | 3. THE REFLECTED WORD GRAM IS AN EXACT SQUARE 000114 | 000115 | For a positive half-circle word F=(A1,s1;...;Ak,sk), 000116 | 0<=s1<=...<=sk<=beta/2, define the finite transfer amplitude 000117 | 000118 | X_Lambda(F)=exp(-(beta/2-sk)H) Ak exp(-(sk-s(k-1))H)... 000119 | A1 exp(-s1 H). 000120 | For the empty word use X_Lambda(1)=exp(-beta H/2). 000121 | Reflect by beta-s, replace each insertion by its adjoint, and keep the 000122 | induced reversed word order. The complete reflected mixed Schwinger word 000123 | is exactly 000124 | 000125 | K_Lambda(F,G)=Tr[X_Lambda(F)^* X_Lambda(G)]/Z_Lambda. (7) 000126 | 000127 | Thus for every finite set of original words and complex coefficients, 000128 | SUM_ij conjugate(c_i)c_j K_Lambda(F_i,F_j) 000129 | =||SUM_j c_j X_Lambda(F_j)||_2^2/Z_Lambda >=0. (8) 000130 | 000131 | Every entry is one of the convergent complete Euclidean words from 000132 | section2. Passing to the SAME subnet limit in this finite sum proves RP 000133 | of the entire limiting half-circle word kernel. Coincident-time order 000134 | and the endpoints remain the original ones in(7). 000135 | 000136 | Equations(7)–(8) need no physical time-reversal symmetry of H. They use 000137 | selfadjointness, the thermal trace and adjoint reflection. They establish 000138 | thermal-circle RP, not zero-temperature vacuum spectral positivity or 000139 | four-dimensional Euclidean covariance. 000140 | 000141 | Use(7) directly on the actual even physical observable algebra. With odd 000142 | fermion fields, their graded Euclidean time ordering and original reflected 000143 | adjoint/twist must be transported consistently before applying the square. 000144 | One may not add fermionic permutation signs to(1) but leave the reflection 000145 | untwisted and then call it the same Gram. The ordinary even physical 000146 | algebra theorem is already complete for that original scope; it does not 000147 | postulate a positive ghost or Grassmann configuration measure. 000148 | 000149 | 4. FULL BOUNDED EUCLIDEAN SOURCE, NOT A FINITE MOMENT BATTERY 000150 | 000151 | For a finite list of these original bounded local source directions and 000152 | integrable time profiles, integrate the ordered coefficients(1) over their 000153 | time simplexes to form the full Euclidean ordered source series. Put 000154 | M=int_0^beta SUM_j |J_j(s)| ||Aj|| ds. 000155 | Equation(2) bounds its degree-k term by M^k/k!. Hence the full series is 000156 | entire in these source parameters with volume-independent majorant e^M. 000157 | Coefficientwise convergence from section2 and this majorant give the 000158 | complete limiting Euclidean source and all its ordered source derivatives. 000159 | The RP Gram also extends to the corresponding positive-half source 000160 | exponentials by this absolute series convergence. 000161 | 000162 | This source uses the original bounded finite-local-dimension Hamiltonian 000163 | class. There is no additional unbounded-field UI premise for these actual 000164 | operators. Original external unnormalized relative factors, if attached 000165 | by another caller, remain attached; the normalized Gibbs construction 000166 | does not determine or reset an independent original Z_rel[0]. 000167 | 000168 | 5. ORIGINAL AND PRIMARY EVIDENCE 000169 | 000170 | Actual original dynamics, Gaussian KMS passage and Gauss consistency: 000171 | documents/articles--2026-08-30--tfpt_thermodynamic_dynamics_en/ 000172 | sections/001-section-the-concrete-uniformly-local-class.tex:225–362, 000173 | paper-source:4cc6102a923009e972617edc, 000174 | SHA2569f53a07911c9e2e8f6b2f594b7c154f5a8dc0faeb2e9a429cc04a229cc6f4294. 000175 | sections/002-section-existence-of-equilibrium-states.tex:1–62, 000176 | paper-source:d5493e3dbebe594aa35c7b65, 000177 | SHA25632dea10dc2686956bf07afc299b9ebd65c7dd25b39783a1bf87579a87d9056c0. 000178 | Both source ranges were independently read through validated read_source. 000179 | 000180 | Primary literature confirmation of the thermal KMS/circle-RP distinction: 000181 | Neeb and Olafsson, Reflection positivity for the circle group, 000182 | section5, Proposition5.2 and Remark5.3: 000183 | https://arxiv.org/pdf/1411.2439 . 000184 | It formulates the strip boundary relation on sesquilinear forms and 000185 | derives circle RP for selfadjoint insertion directions. It does not turn 000186 | this into a higher-dimensional interacting-field theorem. The complete 000187 | finite-word proof(1)–(8) above was checked directly, rather than inferred 000188 | from a two-point statement in that paper. 000189 | 000190 | REVIEW SCOPE 000191 | 000192 | This is a positive complete thermal Euclidean functional and RP kernel 000193 | for the same original Hamiltonian class. The selected ambient v1013 KMS 000194 | state and the separately constructed physical-Gauss state are kept 000195 | distinct. The actual original S_adm/master equality, P1 reflection 000196 | dictionary, phase uniqueness and controlled IR universality are not 000197 | asserted by this theorem. An a->0 continuum limit is not introduced as a 000198 | universal requirement; the original lattice-fundamental decision remains. 000199 | No numerical harness, different source model or canonical edit was used.