Established in scope
Whole physical ground source has Euclidean limit and the same GNS vacuum
In the question: Whole original source and canonical vacuum attachment
New analytic whole-source application on H already constructed original class. Half-word side and actual adjoints explicit. Gibbs selects whole ground trace,not arbitrary degenerate ground vectors. Beta_n is proof regularization,not a physical temperature fit. Positive time spectrum does not imply Lorentz or phase uniqueness. Original Gauss constraints and absolute energy/reference/Z0 factors retained; no new continuum gate or total closure.
Read the exact result, assumptions and evidence
Result
For the same original physical Hamiltonian family all bounded ordered Euclidean ground words converge on the same weak-* subnet to actual H0 GNS semigroup words. Contractivity,fixed Gaussian smoothing and uniform unsmoothing give product-norm control and compact-simplex uniformity. Whole local source converges with exp(M) majorant; its RP Gram completion is unitarily the same GNS space,[1] maps to Omega0,H0>=0. No thermodynamic gap is needed. For the whole finite ground projector,2(d-d0)/d0 exp(-(beta-T)delta) controls every word; explicit diagonal Gibbs regularization gives the same source without a uniform finite gap.
Required assumptions
- Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
- Original local Hamiltonian class and common limiting caller
One fixed original selfadjoint interaction Phi(X) on the existing finite local tensor factors, finite range and finite uniform interaction moment. Original quasilocal algebra, inclusions and free-boundary H_Lambda=sum_X Phi(X). Use the scoped v1013 real-time norm dynamics and common weak-* ground/KMS subnet construction. Graph, frozen model couplings and actual field factors are supplied class data, not newly derived from P1.
- Original regular finite gauge links and complete interior Gauss stars
Actual regular finite-G links with original matter representation. Impose the original Gauss=1 projectors only for complete contained stars preserved by actual H. Open boundary with a path from every projected link component to an unprojected boundary vertex. Local gauge actions commute and preserve each interaction term. Admissible observables belong to A^G; do not add broken seam/boundary stars. No periodic neutrality or alternate sector choice.
- Existing physical ground subnet and bounded original source legs
H original finite-range Hamiltonian class and whole interior free-boundary Gauss projectors. Actual bounded local admissible insertions preserve projectors. Finite densities supported on original physical ground spaces converge on same weak-* subnet to existing ground construction;retain original norm real-time dynamics/convention. Gibbs comparison explicitly uses normalized WHOLE ground projection,not arbitrary degenerate vector. No uniqueness,uniform gap or Lorentz symmetry assumed.
Original evidence
- GROUND-SOURCE.txt lines 1–200
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-local-source-variable-background-20261102/GROUND-SOURCE.txt
bca2dc3b50d4697286c5f1a19cec48f8416c4125bee2c8ec69d75fc9956c3465
- GROUND-SOURCE.txt lines 201–274
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-local-source-variable-background-20261102/GROUND-SOURCE.txt
bca2dc3b50d4697286c5f1a19cec48f8416c4125bee2c8ec69d75fc9956c3465
- J-FINAL-REVIEW.txt lines 1–192
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/original-local-source-variable-background-20261102/J-FINAL-REVIEW.txt
0c066984d5ba922a61fcdc3b283a6c2136746ea2e55005ceee5e192256307272
- 002-section-existence-of-equilibrium-states.tex lines 1–62
Source checked at this snapshot
Original location and fingerprint
documents/articles--2026-08-30--tfpt_thermodynamic_dynamics_en/sections/002-section-existence-of-equilibrium-states.tex
32dea10dc2686956bf07afc299b9ebd65c7dd25b39783a1bf87579a87d9056c0
finding:original-physical-ground-source-has-complete-euclidean-limit-and-same-gns-vacuum