TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PHYSICAL-THERMAL-SOURCE.txt
Original file SHA-256: 04b9873c44060b77cd9d5d246602f8188bddee50743de294455f452ae826e52a
Inclusive original lines: 1-198
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 | PHYSICAL GAUSS SECTOR AND COMPLETE THERMAL WORD SOURCE
000002 | L20261102H. 2 November 2026. Constructive application to the original class.
000003 | 
000004 | QUESTION AND OUTCOME
000005 | 
000006 | Does the existing thermodynamic Hamiltonian construction supply a full
000007 | physical local source, rather than only selected covariances, and can its
000008 | Gauss support and thermal reflected words be carried to the same limit?
000009 | 
000010 | Yes in the original finite-range, finite-local-dimension Hamiltonian class.
000011 | There is a physical-sector existence construction, a complete bounded real
000012 | and Euclidean time-word limit, and a thermal reflection-positive word kernel.
000013 | This applies the actual original H and gauge projectors. It chooses no new
000014 | coupling, field measure or oscillator. The physical projected state can differ
000015 | from the earlier ambient Gibbs state: those callers are explicitly separated.
000016 | It does not identify this class with every original S_adm or derive its IR
000017 | universality. Original frozen model parameters remain supplied inputs.
000018 | 
000019 | 1. RETAIN THE ACTUAL ORIGINAL DATA AND THEIR EXISTENCE THEOREM
000020 | 
000021 | Use the single original interaction Phi(X)=Phi(X)* with finite range R,
000022 | uniform finite local dimension, and finite interaction moment kappa. Its
000023 | H_Lambda=sum_(X subset Lambda)Phi(X) is the existing free-boundary family.
000024 | v1013 proves norm convergence of tau_t^Lambda(A)=exp(itH_Lambda)Aexp(-itH_Lambda)
000025 | on local A, uniformly on compact real time intervals, and weak-* subnet
000026 | existence of ground and beta-KMS states on the quasilocal algebra A.
000027 | The gauge action commutes with this dynamics and fixes each interaction.
000028 | The physical observable algebra below is the actual invariant subalgebra
000029 | A^G, with the original admissible even fields/words when parity is required.
000030 | Neither a tensor embedding of projected Hilbert spaces nor projectivity of
000031 | newly restarted small-box bare Gibbs states is assumed.
000032 | 
000033 | The original Ledger QFT4D.LATTICE.FUNDAMENTAL.01 requires the thermodynamic
000034 | limit and controlled IR universality. An exact lattice-spacing limit a->0
000035 | is optional. This report does not restore it as a universal physical gate.
000036 | 
000037 | 2. GAUGE INVARIANCE AND THE PHYSICAL GAUSS SECTOR ARE DISTINCT
000038 | 
000039 | For every finite Lambda take the actual original full-star Gauss projectors
000040 | P_x whose entire support lies in Lambda and whose transformations are
000041 | respected by that original Hamiltonian. Put P_Lambda=product_x P_x.
000042 | These commute. Then [P_Lambda,H_Lambda]=0 and [P_Lambda,A]=0 for the actual
000043 | admissible gauge-invariant A. No broken boundary/seam constraint is added.
000044 | 
000045 | The original ambient Gibbs rho=exp(-beta H)/Tr(exp(-beta H)) has full rank.
000046 | When 0<P<I it therefore has Tr(rho(I-P))>0. Averaging a state over gauge
000047 | transformations, or restricting it to A^G, does not turn this number into0:
000048 | P itself is gauge invariant. The existing A^G dynamics theorem remains
000049 | correct; the stronger assertion of Gauss=1 support needs its actual sector.
000050 | 
000051 | Nonemptiness is constructive for the original regular finite gauge links
000052 | with open boundary. Every connected link component has an unprojected
000053 | boundary vertex. If a gauge transformation on the projected interior fixes
000054 | a link configuration u, its value is forced to1 at an adjacent boundary
000055 | edge and then along every path to the interior. The interior group action
000056 | on link configurations is therefore free. For any matter vector v!=0,
000057 | 
000058 |  ||P_Lambda(|u> tensor v)||^2=||v||^2/|G|^(number of projected vertices)>0.
000059 | 
000060 | This is an existence witness, not a selected new boundary state. The
000061 | assumptions are actual regular link representations and the stated open
000062 | graph. It is not asserted for an arbitrary closed topology or gauge sector.
000063 | 
000064 | 3. CONSTRUCT THE PHYSICAL STATE WITHOUT A FALSE HILBERT EMBEDDING
000065 | 
000066 | Use the SAME original H and its actual physical projector:
000067 | 
000068 |  rho_Lambda^phys=P_Lambda exp(-beta H_Lambda)/Z_Lambda^phys,
000069 |  Z_Lambda^phys=Tr(P_Lambda exp(-beta H_Lambda))>0.
000070 | 
000071 | Regard this as an ambient finite state and extend it to A. Weak-* compactness
000072 | gives a subnet limit omega_beta^phys. On the admissible A^G, finite KMS
000073 | identities hold because H and A preserve P_Lambda. The original v1013
000074 | Gaussian-smearing proof then passes KMS to the SAME limiting dynamics.
000075 | Choosing finite ground vectors in P_Lambda gives a ground-state limit on
000076 | A^G by the corresponding variational inequality. Neither assertion is
000077 | made for isolated constraint-changing fields on the full ambient algebra.
000078 | 
000079 | For each fixed interior Gauss projector P_F, eventually P_Lambda P_F=P_Lambda.
000080 | Thus omega_beta^phys(P_F)=1. The P_F are central relative to A^G, so its GNS
000081 | representation annihilates the ideal generated by I-P_F. Dynamics preserves
000082 | that ideal. This is actual physical Gauss support in the limit, obtained
000083 | without embedding P_Lambda H_Lambda as independent tensor factors.
000084 | 
000085 | The earlier ambient omega_beta, its restriction, and this stronger projected
000086 | state are all legitimate named constructions. They need not coincide. No
000087 | P1 seam state is silently replaced by this physical-sector Gibbs state.
000088 | 
000089 | 4. COMPLETE REAL-TIME WORDS AND DYSON SOURCE
000090 | 
000091 | For actual bounded admissible local A_j and fixed real times t_j,
000092 | W_Lambda=product_j tau_tj^Lambda(A_j) converges in norm to W in the same order.
000093 | Along the chosen state subnet,
000094 | 
000095 |  |omega_Lambda^phys(W_Lambda)-omega_beta^phys(W)|
000096 |  <=||W_Lambda-W||+|omega_Lambda^phys(W)-omega_beta^phys(W)| ->0.
000097 | 
000098 | This retains every finite mixed clock, matter and gauge-invariant word in
000099 | this source algebra. It is not a reconstruction from a selected finite
000100 | moment battery. For V_Lambda(t)=sum_j J_j(t)tau_t^Lambda(A_j), integrable
000101 | compactly supported J_j, the ordered Dyson series converges in norm and
000102 | its expectation converges. Its majorant is exp(sum_j||J_j||_1||A_j||).
000103 | The resulting normalized local source is entire in every finite collection
000104 | of these bounded source directions. Admissible charged composite legs must
000105 | keep their actual gauge carriers; arbitrary gauge-variant sources are not
000106 | declared physical merely by this construction.
000107 | 
000108 | 5. COMPLETE UNSMEARED EUCLIDEAN WORDS HAVE THE SAME LIMIT
000109 | 
000110 | On the finite physical Hilbert space use the actual thermal trace, for
000111 | 0<=s1<=...<=sk<=beta,
000112 | 
000113 |  S_Lambda(A_k,...,A_1;s)=Tr_P[
000114 |  exp(-(beta-s_k)H) A_k exp(-(s_k-s_(k-1))H) ... A_1 exp(-s1 H)]/Z_P.
000115 | 
000116 | For tau_t(A)=exp(itH)Aexp(-itH), this is the ordered product with
000117 | tau_(-is_k)(A_k)...tau_(-is1)(A_1), not +is with that same written order.
000118 | Schatten Holder gives |S_Lambda|<=product_j||A_j||: the positive thermal
000119 | gaps sum to beta, and ||exp(-uH)||_(beta/u)=Z_P^(u/beta).
000120 | At u=0 use the operator norm. By telescoping in the inserted A_j, the
000121 | same bound gives a volume-independent Lipschitz estimate for each leg.
000122 | 
000123 | Let g_epsilon(t) be a positive normalized Gaussian of width epsilon and
000124 | A_(epsilon,Lambda)=integral g_epsilon(t)tau_t^Lambda(A)dt. Locality gives
000125 | sup_Lambda||delta_Lambda(A)||<=C_A<infinity, hence
000126 | sup_Lambda||A_(epsilon,Lambda)-A||<=C_A integral |t|g_epsilon(t)dt ->0.
000127 | For each FIXED epsilon, the entire continuation
000128 | 
000129 |  tau_z^Lambda(A_(epsilon,Lambda))
000130 |    =integral g_epsilon(t-z)tau_t^Lambda(A)dt
000131 | 
000132 | converges in norm on compact z sets, including |Im z|<=beta. Its Gaussian
000133 | L1 majorant is uniform in Lambda for that fixed epsilon. Products therefore
000134 | converge in norm; the same weak-* state subnet gives convergence of their
000135 | expectations. The Holder Lipschitz estimate then removes the smearing
000136 | uniformly in Lambda. Thus every UNSMEARED bounded ordered Euclidean word
000137 | has a limit. No norm convergence of unsmeared imaginary-time dynamics is
000138 | assumed. Coincident times retain their original word order.
000139 | 
000140 | Expanding the finite thermal source with these ordered insertions has the
000141 | same exp(sum||J_j||_1||A_j||) majorant. Coefficient convergence passes to
000142 | the whole source, locally uniformly in finitely many complex source
000143 | parameters. All its ordered source derivatives are the complete limiting
000144 | words above. This is a quantum word functional, not a positive measure
000145 | assigned to Grassmann or ghost variables.
000146 | 
000147 | 6. THERMAL TIME REFLECTION IS POSITIVE ON WHOLE WORDS
000148 | 
000149 | For a positive-half word F with 0<s1<...<sk<beta/2 put
000150 | 
000151 |  B_Lambda(F)=exp(-(beta/2-s_k)H) A_k exp(-(s_k-s_(k-1))H)
000152 |                   ... A_1 exp(-s1 H).
000153 | 
000154 | The reflected adjoint/reversed word at beta-s has the exact finite kernel
000155 | 
000156 |  K_Lambda(F_i,F_j)=Tr_P(B_Lambda(F_i)^* B_Lambda(F_j))/Z_P.
000157 | 
000158 | For any finite coefficients c_i its quadratic form is the Hilbert-Schmidt
000159 | norm square of sum_i c_i B_Lambda(F_i), divided by Z_P. It is nonnegative.
000160 | The full Euclidean word convergence above passes this positivity to the
000161 | limit. This is the canonical THERMAL time reflection with the actual
000162 | operator adjoint. Identifying it with a separately supplied P1 seam Theta
000163 | still uses the original reflection/field map. A graded field dictionary
000164 | must retain its original signs; no ordinary commuting-field rule is imposed.
000165 | 
000166 | 7. KEEP NORMALIZATION, ENERGY INSERTIONS AND THE PHYSICAL SCOPE
000167 | 
000168 | The local generating functional above is q[J] with q[0]=1. It does not
000169 | redefine the original unnormalized Z_rel. At each actual finite source call
000170 | retain its own Z_rel,Lambda[0] and q_Lambda[J] separately. No finite global
000171 | thermodynamic partition number is assigned by the existence of a state.
000172 | Identity offsets in the original H matter for Z and for the original energy
000173 | insertions even though they cancel from commutators. In the v1013 frozen
000174 | plaquette, deleting its constant 1/g^2=25/36 per plaquette is legitimate in
000175 | a dynamics estimate only. The complete source keeps the actual original
000176 | offset. It is not a new freely adjustable vacuum counterterm.
000177 | 
000178 | The fixed lattice class now has a genuine complete bounded physical source,
000179 | Gauss support and a thermal RP kernel, not merely a finite toy covariance.
000180 | Its actual frozen couplings and graph are still supplied class data. This
000181 | does not prove phase uniqueness, the limit gap, controlled IR universality,
000182 | a common physical light cone, or equality with every primitive/geometric
000183 | S_adm caller. The latter uses the actual original source dictionary. No
000184 | global continuum condition is added and no full TFPT solution is claimed.
000185 | 
000186 | EVIDENCE
000187 | 
000188 | LATER-SOURCE.txt: original v1013 proof, complete multitime construction and
000189 | later v336/v469/v779/v802/v369 scopes; original ledger760.
000190 | GAUSS-SOURCE-PROOF.txt: independent sector construction and nonemptiness.
000191 | EUCLIDEAN-SOURCE-REVIEW.txt: independent Holder/Gaussian/HS-square review.
000192 | GAUSS-NATIVE.txt and its JSON: unchanged original v989 four-site Z4 control,
000193 | dimension1024 and physical rank64. This is a different original example,
000194 | not execution of the16,777,216-dimensional v1013 DET16 cell or a numerical
000195 | measurement of the all-volume class. The whole-class proof is analytic.
000196 | BLOCKING-PATH.txt and CONSTRUCTIVE-CRITERION.txt: original exact blocking,
000197 | compatible state construction, and unnecessary global-density conditions.
000198 | G remains intact. No original paper, verification or ledger status is edited.