TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/GROUND-SOURCE.txt Original file SHA-256: bca2dc3b50d4697286c5f1a19cec48f8416c4125bee2c8ec69d75fc9956c3465 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 | THE COMPLETE ORIGINAL HAMILTONIAN GROUND SOURCE 000002 | joint_origin, 3 November 2026. J continuation of accepted G/H/I. 000003 | Read-only canonical work. No new Hamiltonian, coupling, temperature fit, 000004 | numerical model or continuum requirement. 000005 | 000006 | RESULT 000007 | 000008 | The actual original finite-range Hamiltonian class already supplies the 000009 | ground source, not merely a selected two-point covariance. Along the SAME 000010 | physical finite-ground-state subnet defining omega_0, ALL bounded ordered 000011 | Euclidean words converge to the ground-state GNS semigroup words. Their 000012 | entire bounded source and reflected-word positivity follow. No uniform 000013 | spectral gap, quasi-free ansatz or additional vacuum choice is required. 000014 | 000015 | There is also a zero-temperature trace realization if the finite ground 000016 | states are the normalized projections onto their WHOLE physical ground 000017 | spaces. A diagonal beta_n -> infinity is an auxiliary proof limit. It 000018 | changes no original interaction and fits no physical temperature. It does 000019 | not approximate an arbitrarily selected vector inside a degenerate ground 000020 | space. The trace route and an arbitrary selected-ground-state route are 000021 | therefore distinguished below. 000022 | 000023 | 1. RETAIN EXACTLY THE ORIGINAL DATA 000024 | 000025 | Use the original quasilocal algebra, the fixed interaction Phi(X), its 000026 | free-boundary H_Lambda, and the norm-convergent real dynamics 000027 | tau_t^Lambda(A)=exp(it H_Lambda) A exp(-it H_Lambda). 000028 | On the actual invariant/admissible observable algebra use the full-star 000029 | physical projections P_Lambda of H, satisfying 000030 | [P_Lambda,H_Lambda]=[P_Lambda,A]=0, P_Lambda !=0. 000031 | Regular finite-group links and the original free boundary supply the 000032 | already proved nonemptiness. No projected-Hilbert-space tensor embedding 000033 | is used. All state limits and operator comparisons are on the original 000034 | common ambient algebra, followed by its actual physical quotient. 000035 | 000036 | Let E_0,Lambda be the least eigenvalue of H_Lambda RESTRICTED to P_Lambda. 000037 | For the estimates only write 000038 | L_Lambda=(H_Lambda-E_0,Lambda)|Ran(P_Lambda) >=0. 000039 | Let omega_Lambda be any normalized density supported on its ground space. 000040 | Extend its finite state to the common algebra and take one weak-* subnet 000041 | omega_Lambda -> omega_0 on the actual invariant algebra. 000042 | 000043 | The original variational inequality passes to the limit because local 000044 | commutators stabilize. Thus omega_0 is a ground state of the SAME tau. 000045 | Its GNS implementation has 000046 | U_0(t)=exp(it H_0), H_0>=0, U_0(t)Omega_0=Omega_0, 000047 | pi_0(tau_t A)=U_0(t)pi_0(A)U_0(t)*. 000048 | The ground-state condition is only asserted on the original allowed 000049 | algebra. Constraint-changing isolated fields do not inherit it merely 000050 | from extending a physical state to the ambient algebra. 000051 | 000052 | Originals: v1013 equilibrium theorem, source S1 below; accepted physical 000053 | Gauss source, S2. The ground state already exists in these results. 000054 | 000055 | 2. THE ACTUAL LATEST-LEFT EUCLIDEAN WORD 000056 | 000057 | For 0<=s_1<=...<=s_k, use precisely H's time convention: 000058 | 000059 | S_Lambda(A_k,s_k;...;A_1,s_1) 000060 | =omega_Lambda(A_k exp(-(s_k-s_(k-1))L_Lambda) ... A_1). (1) 000061 | 000062 | Here omega_Lambda on a product containing semigroups means its trace on 000063 | the physical finite block. Its ground density commutes with L_Lambda and 000064 | is annihilated by L_Lambda. For a ground vector it is the corresponding 000065 | matrix element. Common shifts of all s_j have no effect. On the ground 000066 | GNS space the candidate limit is 000067 | 000068 | S_0(A_k,s_k;...;A_1,s_1) 000069 | =. (2) 000071 | 000072 | For entire analytic legs these equal respectively 000073 | omega_Lambda(tau_(-i s_k)^Lambda(A_k)... 000074 | tau_(-i s_1)^Lambda(A_1)), 000075 | omega_0(tau_(-i s_k)(A_k)...tau_(-i s_1)(A_1)). (3) 000076 | The sign is MINUS i for latest-left order, as in the accepted H proof. 000077 | The equivalent earliest-left arrangement has PLUS i. It is obtained by 000078 | relabeling the whole ordered word; the convention is never mixed midword. 000079 | 000080 | Since every intervening semigroup is a contraction, 000081 | |S_Lambda(A_k,...,A_1)|, |S_0(A_k,...,A_1)| 000082 | <=PRODUCT_j ||A_j||. (4) 000083 | Replacing the jth leg and telescoping gives the uniform multilinear 000084 | Lipschitz estimate SUM_j ||A_j-B_j|| times the products of the other leg 000085 | norm maxima. Neither bound depends on volume, energy norm or a gap. 000086 | 000087 | 3. SAME-SUBNET CONVERGENCE OF ALL UNSMEARED WORDS 000088 | 000089 | For each original bounded LOCAL allowed A, the fixed finite-range 000090 | interaction gives C_A=sup_Lambda||delta_Lambda(A)||0, 000098 | 000099 | tau_z^Lambda(A_(e,Lambda)) 000100 | =int g_e(t-z)tau_t^Lambda(A)dt (6) 000101 | 000102 | converges in norm to tau_z(A_e), uniformly for z in any fixed compact 000103 | complex set. This follows from the original compact-uniform real-time 000104 | norm convergence, isometry, and Gaussian tails. The L1 norm of the 000105 | complex Gaussian is bounded on that compact set for fixed e. No bound 000106 | uniform as e->0 is used or needed. 000107 | 000108 | Consequently the entire-leg products in (3) converge in norm. The SAME 000109 | weak-* convergent state subnet gives convergence of their expectations. 000110 | In the limiting ground GNS representation, the analytic implementation 000111 | identity reduces that expectation exactly to (2) with the smeared legs. 000112 | Apply (4)-(5) to remove the smearing, uniformly in Lambda and in all 000113 | ordered times in each fixed compact simplex. This proves 000114 | 000115 | S_Lambda(A_k,s_k;...;A_1,s_1) -> S_0(A_k,s_k;...;A_1,s_1). (7) 000116 | 000117 | For fixed e the limiting analytic word's time image is norm compact. 000118 | Weak-* convergence of uniformly bounded states is uniform on such a 000119 | compact set. The unsmearing estimate is time independent. Thus (7) is 000120 | uniform on every fixed time simplex, including coincident times with 000121 | their given word order. No new word-dependent subnet is selected. 000122 | This is the ground analogue of H's already proved thermal smearing 000123 | argument, applied to the actual same original H, not to another source. 000124 | 000125 | 4. WHOLE SOURCE, REFLECTION, AND ACTUAL VACUUM VECTOR 000126 | 000127 | For a finite set of the original bounded source directions A_j and 000128 | integrable time profiles on a finite interval, form the ordered source 000129 | series with coefficients (1). Its degree-k term is bounded by M^k/k!, 000130 | M=int SUM_j |J_j(s)| ||A_j|| ds. 000131 | Equation (7) and this common exponential majorant yield the complete 000132 | normalized ground source q_0[J], locally uniformly in all finite complex 000133 | source parameters. Every ordered derivative has the actual coefficient 000134 | (2). This is not a finite-moment fit and does not require a Wick rule. 000135 | 000136 | For explicit reflection conventions, regard a ket-preparing half-word 000137 | F=(A_1,s_1;...;A_k,s_k), 0<=s_1<=...<=s_k, 000138 | as the original source word at times -s_1,...,-s_k. Put 000139 | Psi_F=exp(-s_1 H_0) pi_0(A_1) exp(-(s_2-s_1)H_0)... 000140 | pi_0(A_k)Omega_0. 000141 | For the empty word Psi_1=Omega_0. Reflect that negative-time half-word 000142 | to positive times with the actual adjoints and reflected order. Its 000143 | complete source kernel with a second such half-word G is exactly 000144 | 000145 | K(F,G)=. (8) 000146 | 000147 | Indeed (8) is the latest-left coefficient (2) with positive-time left 000148 | block A_k*,...,A_1*, central gap s_1+t_1, and negative-time right block 000149 | B_1,...,B_m. The whole list of times and adjoints is fixed; this is not a 000150 | two-point inference. Therefore SUM conj(c_i)c_j K(F_i,F_j) is a norm 000151 | square. The OS quotient [F]->Psi_F is an isometry onto the original 000152 | ground GNS space: zero-time local A Omega_0 are dense, or are obtained 000153 | as limits from the open half-line. It sends [1] to Omega_0 and intertwines 000154 | the corresponding positive preparation semigroup exp(-t H_0). 000155 | 000156 | Using the opposite half-line with the same latest-left convention gives 000157 | the corresponding conjugate/opposite preparation convention. One must 000158 | transport the actual operation, adjoint and side map when comparing a 000159 | separate P1 seam Theta. No identity between that Theta and this canonical 000160 | ground time reflection is inferred from positivity alone. On the actual 000161 | even physical algebra (8) is immediate. Charged or odd field words must 000162 | retain their already supplied carriers, grading and reflected adjoints. 000163 | 000164 | Thus the ground vacuum vector, positive TIME spectrum and complete 000165 | ground word source are already reconstructed from the existing family. 000166 | Full Poincare invariance and the JOINT forward light cone additionally 000167 | use the actual full Euclidean covariance/regularity theorem. Groundness 000168 | alone does not supply Euclidean rotations. No such claim is made here. 000169 | 000170 | 5. A WHOLE-PROJECTOR GIBBS REALIZATION WITH A SHARP ERROR BOUND 000171 | 000172 | Let d_Lambda=dim Ran(P_Lambda), P_0,Lambda the whole ground projection, 000173 | d_0,Lambda=rank(P_0,Lambda), 000174 | delta_Lambda=min(spec L_Lambda without zero)>0 000175 | when an excited space exists. This delta is only a finite regulator gap. 000176 | Put omega_(Lambda,0)(A)=Tr(P_0,Lambda A)/d_0,Lambda. 000177 | It is the finite zero-temperature trace state, including all degeneracy. 000178 | 000179 | For a word spanning T=s_k-s_1 and beta>=T, cyclicity rewrites the EXACT 000180 | finite thermal word as 000181 | 000182 | S_(Lambda,beta)=Tr[exp(-(beta-T)L_Lambda) W_Lambda]/Ztilde, 000183 | W_Lambda=A_k exp(-(s_k-s_(k-1))L_Lambda)...A_1, 000184 | Ztilde=Tr exp(-beta L_Lambda). (9) 000185 | 000186 | Let r_u=Tr[(I-P_0,Lambda)exp(-uL_Lambda)]. Then 000187 | ||W_Lambda||<=PRODUCT||A_j||, Ztilde=d_0+r_beta, 000188 | S_(Lambda,0)=Tr(P_0,Lambda W_Lambda)/d_0. 000189 | Separating the numerator and denominator errors gives 000190 | 000191 | |S_(Lambda,beta)-S_(Lambda,0)| 000192 | <= (r_(beta-T)+r_beta)/d_0 PRODUCT||A_j|| 000193 | <= 2(d_Lambda-d_0)/d_0 exp(-(beta-T)delta_Lambda) 000194 | PRODUCT||A_j||. (10) 000195 | 000196 | The beta-T is essential. A bound with exp(-beta delta) alone would not 000197 | control a word that occupies a long fraction of the thermal circle. 000198 | If the full physical space is ground, r_u=0 and equality is exact. 000199 | 000200 | Take the weak-* subnet of the ACTUAL omega_(Lambda,0) family. Section3