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: 201-274
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.

000201 | already supplies its complete ground source. Along an exhaustion indexed
000202 | by n, for example
000203 | 
000204 |  beta_n=max(n, 2(log d_n+n)/delta_n)                     (11)
000205 | 
000206 | when there are excited states, and beta_n=n otherwise. For any fixed
000207 | time-span bound T_0, eventually beta_n>=2T_0 and (10) is at most
000208 |  2 exp(-n) PRODUCT||A_j||.
000209 | Thus the same diagonal physical finite Gibbs source converges on ALL
000210 | bounded finite Euclidean words and, by the common entire majorant, on
000211 | the complete finite-support source to this same ground source. No
000212 | uniform lower bound on delta_n is assumed. Formula (11) specifies a
000213 | sufficient proof regulator, not a fitted physical thermal parameter.
000214 | 
000215 | A different degenerate-ground-vector selection is not automatically this
000216 | trace limit. If the actual phase is uniquely selected by its original
000217 | boundary/state rule or a separately established uniqueness theorem,
000218 | the distinction disappears. No uniqueness is claimed from this argument.
000219 | 
000220 | 6. ORIGINAL ABSOLUTE FACTORS AND WHAT HAS ACTUALLY CLOSED
000221 | 
000222 | The shift L=H-E_0 was only an estimate and a preparation-semigroup
000223 | normalization. The ORIGINAL finite partition factor is
000224 |  Z_phys,Lambda(beta)=exp(-beta E_0,Lambda) Ztilde_Lambda(beta).
000225 | That factor, every actual Z_ref denominator and eta phase remain in the
000226 | unnormalized original S/Z caller. No energy offset is erased and no
000227 | finite absolute thermodynamic Z number is inferred from a state limit.
000228 | G's full-source coordinate transport carries these original factors.
000229 | 
000230 | Closed here: existence of the whole bounded physical ground source,
000231 | same-subnet finite-ground and suitable whole-ground-projector Gibbs
000232 | realizations, canonical ground reflection, H_0>=0, and Omega_0=[1].
000233 | All use the already constructed original Hamiltonian family. Neither
000234 | clustering nor a mass gap is needed for these claims. Euclidean/Poincare
000235 | covariance, phase uniqueness, original master/P1 source identification
000236 | and controlled IR behavior keep exactly their original scoped hypotheses.
000237 | An a->0 limit is not reinstated as a universal physical gate.
000238 | 
000239 | SOURCES READ AND REUSED
000240 | 
000241 | S1. Original v1013 existence theorem, actual finite ground inequality and
000242 | Gaussian imaginary-time argument, read_source full relevant1-62:
000243 |  paper-source:d5493e3dbebe594aa35c7b65,
000244 |  documents/articles--2026-08-30--tfpt_thermodynamic_dynamics_en/
000245 |  sections/002-section-existence-of-equilibrium-states.tex:1-62,
000246 |  SHA25632dea10dc2686956bf07afc299b9ebd65c7dd25b39783a1bf87579a87d9056c0.
000247 | Original common interaction/dynamics theorem:
000248 |  paper-source:4cc6102a923009e972617edc,
000249 |  sections/001-section-the-concrete-uniformly-local-class.tex:225-362,
000250 |  SHA2569f53a07911c9e2e8f6b2f594b7c154f5a8dc0faeb2e9a429cc04a229cc6f4294.
000251 | Its class, support and dynamics were fully reviewed in accepted H.
000252 | 
000253 | S2. Accepted H physical Gauss proof, reread relevant1-119:
000254 |  research-source:b8b5207c4c32a966c51567a1,1-200, and
000255 |  research-source:95d8288c21f1b8b6fca8e038,201-213,
000256 |  experiments/theory-contracts/original-local-source-variable-background-
000257 |  20261102/GAUSS-SOURCE-PROOF.txt,
000258 |  SHA25698c2394d9e60ca195dd12d89f54d144c6bf4ad83a4d08b4c31da9958db53b76e.
000259 | 
000260 | S3. Accepted H whole thermal source and independent complete review,
000261 |  the latter reread1-199:
000262 |  research-source:1ea97421a9a1bd5dc9b4e46a,1-198,
000263 |  PHYSICAL-THERMAL-SOURCE.txt,
000264 |  SHA25604b9873c44060b77cd9d5d246602f8188bddee50743de294455f452ae826e52a;
000265 |  research-source:3566c0d8399cf600b2541f92,1-199,
000266 |  EUCLIDEAN-SOURCE-REVIEW.txt,
000267 |  SHA256090a2c8e818ee24b053a41182d126940741037ee2ccc9bbfa4e7dc52f3a8ef13.
000268 | Same contract directory as S2. Equations(5)-(7) here apply its existing
000269 | smearing/unsmearing method to ground contractions. No new model is used.
000270 | 
000271 | No numerical test was run. The decisive test is the exact complete word
000272 | identity(3), the uniform unsmearing bound(4)-(6), and the exact numerator
000273 | and denominator comparison(9)-(10). These prove the claimed whole-source
000274 | limit, rather than testing a finite collection of moments.