TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/LATER-SOURCE.txt
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000001 | LATER ORIGINAL EXISTENCE RESULTS: THE ACTUAL SHARED SOURCE
000002 | L20261102H. joint_origin, 2 November2026. Entry Atlas
000003 | e99ef15ca0af3a2090099835, shared plan87. Canonical unchanged.
000004 | 
000005 | RESULT
000006 | 
000007 | There already IS a proved shared-source existence route in the originals:
000008 | v1013's actual theorem constructs thermodynamic-limit dynamics and at least
000009 | one ground state, and a beta-KMS state for each beta>0, on ONE given
000010 | quasilocal algebra and its Gauss-invariant subalgebra. This is not just its
000011 | arithmetic/numeric checker. The complete proof memorandum was read.
000012 | 
000013 | Its local restrictions are exactly compatible complete quantum states.
000014 | The original proof does NOT require the smaller free-boundary Gibbs weight
000015 | to equal a larger box's integrated marginal. Treating that stronger finite
000016 | equality as a prerequisite for existence would miss this existing route.
000017 | 
000018 | The source constructed here is the stated assembled finite-range lattice
000019 | Hamiltonian class, including all of that model's gauge, hopping, mirror and
000020 | clock terms. The equations below give its complete bounded multitime
000021 | functional, not merely one covariance or a selected moment battery.
000022 | Applying it to another original full S caller uses the actual supplied
000023 | field/interaction identification. This report creates no new source model
000024 | and imposes no additional universal CAR or clock prerequisite.
000025 | 
000026 | 1. THE EXACT ORIGINAL THEOREM AND ITS INPUTS
000027 | 
000028 | The theorem memorandum is documents/articles--2026-08-30--
000029 | tfpt_thermodynamic_dynamics_en/sections/001-section-the-concrete-
000030 | uniformly-local-class.tex, paper-source:4cc6102a923009e972617edc:1-363,
000031 | SHA256 9f53a07911c9e2e8f6b2f594b7c154f5a8dc0faeb2e9a429cc04a229cc6f4294.
000032 | Read in full1-363. Its equations1.1-1.5 specify
000033 | 
000034 |  H_Lambda=SUM_(X subset Lambda) Phi(X),
000035 |  A_Lambda=B(H_Lambda_local),
000036 |  iota_(Lambda,Lambda')(A)=A tensor I,
000037 |  A=norm-closure UNION_Lambda A_Lambda,
000038 |  Phi(X)=Phi(X)*, diam(X)<=R,
000039 |  kappa=sup_x SUM_(X contains x)|X| norm(Phi(X))<infinity.       (1)
000040 | 
000041 | Here H_Lambda in the first line is the Hamiltonian; H_Lambda_local in
000042 | the second denotes its ambient finite tensor-product Hilbert space.
000043 | The interaction Phi(X) is ONE fixed common interaction, not fitted anew
000044 | for each volume. The hypotheses are finite local dimension with a uniform
000045 | bound, finite range and finite interaction moment. The displayed concrete
000046 | specialization is Gamma=Z2, z4, R2, J12/5, kappa16043/450.
000047 | 
000048 | The actual cell factors are four representative edge fermion modes,
000049 | a mirror Fock factor with n_mir4 or16, two outgoing finite-group gauge
000050 | links, and a four-state seam clock. The full DET16 cell dimension is
000051 | 16,777,216. Its complete interaction table includes electric, plaquette,
000052 | gauge-dressed hopping, DET16 projector, seam clock/charge and staggered
000053 | mass. Those factors, not an unspecified classical geometry measure,
000054 | are what the explicit frozen lattice theorem uses.
000055 | 
000056 | The theorem proves a volume-independent Lieb-Robinson bound, then the
000057 | norm limit tau_t(A)=lim_Lambda tau_t^Lambda(A) for every local A, uniform
000058 | on compact REAL time intervals. It proves gauge-fixed-point consistency
000059 | on A^G, rather than a false tensor embedding of projected physical Hilbert
000060 | spaces. Original source001:225-363, equations3.1-4.3.
000061 | 
000062 | The equilibrium-state proof is source002:1-61,
000063 | paper-source:6ffa2ad8a0ae00e8cc167c8e,
000064 | SHA256 32dea10dc2686956bf07afc299b9ebd65c7dd25b39783a1bf87579a87d9056c0.
000065 | It takes the ACTUAL complete finite Gibbs states
000066 | 
000067 |  omega_(n,beta)(A)=Tr_Lambda_n(e^(-beta H_Lambda_n) A)
000068 |                          /Tr_Lambda_n(e^(-beta H_Lambda_n)),    (2)
000069 | 
000070 | extends them to the common A, and uses weak-* compactness for a subnet.
000071 | Gaussian time smearing and the already proved norm dynamics pass the
000072 | finite KMS identities to that subnet limit. Ground-state inequalities
000073 | pass on the local core in the analogous construction. Thus existence of
000074 | a common state is proved, not assumed as an exactly projective finite
000075 | Gibbs family. No uniqueness claim is made.
000076 | 
000077 | Combined theorem and boundary: paper-source:6465508a38e869a2df13859c:
000078 | 107-156. This original explicitly separates its mandatory-dynamics result
000079 | from phase uniqueness, thermodynamic gap and controlled IR universality.
000080 | Ledger source:92ee380e06bf7f64c0b2:760 says a->0 OPTIONAL; |Lambda|->infinity
000081 | and controlled IR universality MANDATORY. This review retains that choice.
000082 | No unconditional continuum gate is inferred from the older v336 route.
000083 | 
000084 | 2. SMALLEST DECISIVE ACTUAL-PATH TEST: FULL MARGINAL COMPATIBILITY
000085 | 
000086 | Take ANY subnet state omega_beta produced by the original proof. For every
000087 | finite ambient region F its restriction is represented by a density rho_F,
000088 | since A_F is its actual full finite matrix algebra. For F subset G,
000089 | 
000090 |  Tr_(G minus F) rho_G = rho_F.                              (3)
000091 | 
000092 | Proof: for EVERY A in A_F,
000093 |  Tr_F[(Tr_(G minus F)rho_G)A]
000094 |   =omega_beta(A tensor I)=omega_beta(A)=Tr_F(rho_F A).
000095 | Testing against a complete matrix-unit basis implies (3). This proves
000096 | the entire density, all mixed coefficients and all local moments at once;
000097 | it is not an inference from a finite chosen word battery. Positivity and
000098 | normalization are inherited from the original full Gibbs/ground states.
000099 | 
000100 | The same identity follows directly from the construction: restrict the
000101 | SAME omega_(n,beta) first to G and then F, then pass to its common subnet.
000102 | Partial trace on these finite algebras is continuous. Therefore the limit
000103 | restrictions commute exactly, although the original free-boundary density
000104 | e^(-beta H_F)/Z_F need not equal rho_F.
000105 | 
000106 | For the gauge-invariant subalgebra, use restriction of the state through
000107 | the actual algebra inclusions. There is no canonical density matrix for
000108 | an abstract quotient unless that representation is supplied; the state
000109 | identity itself is the invariant statement. This keeps v1013's Gauss
000110 | warning, rather than restoring a projected-Hilbert tensor shortcut.
000111 | 
000112 | This analytic all-word test is the decisive check performed here. It
000113 | identifies precisely WHICH weighted marginals are compatible: restrictions
000114 | of the SAME actual thermodynamic state. It does not replace them by newly
000115 | postulated small-box bare spectral weights. No new simulation was needed.
000116 | 
000117 | 3. THE SAME EXISTENCE PROOF GIVES THE COMPLETE BOUNDED MULTITIME SOURCE
000118 | 
000119 | Let A1,...,Ar be actual bounded local observables of this same class and
000120 | t1,...,tr fixed real times. Write W_n=PRODUCT_j tau_tj^Lambda_n(Aj)
000121 | in the original order and W=PRODUCT_j tau_tj(Aj). The proved compact-time
000122 | norm convergence and the telescoping product identity give norm(W_n-W)->0.
000123 | Consequently
000124 | 
000125 |  omega_(n,beta)(W_n) -> omega_beta(W)                       (4)
000126 | 
000127 | along the SAME subnet: the error is bounded by norm(W_n-W) plus the
000128 | weak-* error on the fixed element W. Equation(4) is every finite joint
000129 | time word, with its order and all mixed factors retained.
000130 | 
000131 | More generally, for compactly time-supported bounded local source
000132 | V_n(t)=SUM_j J_j(t)tau_t^Lambda_n(Aj), with integrable J_j, the ordered
000133 | Dyson operator U_n[J]=T exp(i int V_n(t)dt) converges in norm to U[J].
000134 | The standard iterated-integral estimate follows immediately by telescoping
000135 | the products in its series and their common bound exp(int norm(V_n)).
000136 | Thus the actual normalized full source of (2) has a limit
000137 | 
000138 |  Z_beta[J]=omega_beta(U[J]), Z_beta[0]=1.                  (5)
000139 | 
000140 | For each finite set of these source directions its series is entire,
000141 | with a uniform bounded-operator exponential majorant. All its finite
000142 | ordered source derivatives are the joint moments in(4). No additional
000143 | unbounded-field moment or exact-bare-projectivity hypothesis is needed
000144 | for THIS finite-local-dimension bounded source class.
000145 | 
000146 | This is a concrete shared quantum source, not a classical probability
000147 | measure for jointly noncommuting fields. It also does not assign a finite
000148 | global thermodynamic partition scalar where the original uses states or
000149 | free-energy densities. Identity offsets may be removed for dynamics but
000150 | are retained whenever an actual unnormalized finite Z is evaluated.
000151 | The original absolute relative-source caller and the established G frame
000152 | transport keep their own scalar factors; no free global scalar is added.
000153 | 
000154 | 4. LATER ORIGINALS REPLACE THE EARLY v336 REST WORDING IN THEIR SCOPE
000155 | 
000156 | v336 is fully read at source:c96a0a2d8911631128a8:1-120, SHA256
000157 | 6fdcde19f3931065e0f1d0cad230a45209eed987cb50ff6b50c4e5d273455046.
000158 | It is explicitly a literature/application reduction, not a construction
000159 | of a full interacting4D field measure. Its conditional scaling-limit and
000160 | conformal OS route concerns a particular seam net. Its historical residual
000161 | must not be repeated without the later extension/invariant reductions.
000162 | 
000163 | v469, source:c42f470fbaf0adcb57fe:1-208, read in full, SHA256
000164 | 180a65f5e56dce6cca8c519875ab31e3e0834fb8818175f2a63c896133e72089,
000165 | re-founds the spinor extension as a local simple-current crossed product
000166 | and reduces model-realization R1 to its stated invariant package R1'.
000167 | The later architecture original, paper-source:9ea51c45ef95aa398750f1c3:
000168 | 65-113, explicitly retains that improvement, the v489 ground-state witness,
000169 | and the v779/v802 coherent/martingale strand. It preserves conditional
000170 | MMST/OS typing and leaves the unconditional parent separate. The abstract
000171 | extension is not a literal same-weight geometric configuration marginal;
000172 | that is a different source object. No extra global demand is inferred.
000173 | 
000174 | v779, source:f5be3467a933af09a604:1-200, and v802,
000175 | source:912447bbcf6bd577a79a:1-199, were read at their actual source-class,
000176 | state and convergence definitions (not all1883/683 lines of the modules).
000177 | They give a specific Haar CAR scaling tower and a declared chiral-sea
000178 | state; v802 explicitly uses approximate state defects, not an exact
000179 | martingale. Its finite measured envelope is evidence; an all-depth
000180 | summable state-norm defect bound suffices analytically for its unique
000181 | tower limit. This is an existing alternate limit mechanism, not a new
000182 | universal requirement on v1013 or the original full source.
000183 | 
000184 | The primary Osborne-Stottmeister paper confirms the same distinction:
000185 | section3, equations86-90 define the inductive algebra/RG pullback states;
000186 | an existing weak-* scaling limit is automatically projectively consistent.
000187 | It does not require the unrenormalized finite states to be projective.
000188 | Primary source: https://link.springer.com/article/10.1007/s00220-022-04521-8
000189 | and https://arxiv.org/abs/2107.13834 . This is a1+1 dimensional lattice
000190 | fermion scaling theorem; it is not imported as a full4D interaction theorem.
000191 | 
000192 | 5. v369 IS THE ALTERNATIVE REDUNDANCY ROUTE, NOT ANOTHER MEASURE PRODUCER
000193 | 
000194 | source:d9df14dce43de122925d:1-154, read in full, SHA256
000195 | 89e640abf05b9c86212b5e30c896e7f6458e7ad43660a507f62068452fc6b195,
000196 | explicitly says it does not construct the ambient measure. It proposes
000197 | equivalence of bulk realizations with the same boundary state, conditional
000198 | on the named seam/BW reconstruction package. Its code checks arithmetic
000199 | discriminators and records the conditional assembly. There is no competing
000200 | choice of a new weighted finite field family in that script.