TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/recursive-mirrored-source-20261103/MATRIX-KERNEL-PROOF.txt
Original file SHA-256: ff2db6b22553834ea847c7986c19000f2d19b2e43aa6ff00c592db97abd7e988
Inclusive original lines: 191-293
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000191 | Then for fixed t,
000192 |  (L_(t/n)^S)^n -> exp(-tH) in norm.                      (13)
000193 | Proof: finite H gives exp(-epsilon H)=I-epsilon H+O(epsilon^2).
000194 | Telescoping the difference of n products yields
000195 |  ||(L_epsilon^S)^n-exp(-n epsilon H)||
000196 |  <=n exp(C' n epsilon) ||L_epsilon^S-exp(-epsilon H)||,
000197 | which tends to zero for epsilon=t/n. Finite actual insertions follow
000198 | by telescoping products interval by interval; bounded-source Dyson
000199 | functionals follow by their norm exponential majorant. Actual boundary
000200 | and P contractions are continuous, hence carry the same whole source.
000201 | 
000202 | Thus the decisive comparison is the evaluated whole short-time block
000203 |  E_y dot L_0^S E_x = -H_yx for EVERY actual block and internal coefficient,
000204 | plus the identity at zero width and original sewing. It is not enough
000205 | to compare the Fock vacuum block, eigenvalues, normalized graph kernel,
000206 | or quadratic Grassmann coefficient alone. For a supplied coherent CAR
000207 | description compare the complete finite Grassmann symbol, including
000208 | scalar and all higher coefficients, on the original pairing convention.
000209 | 
000210 | An arbitrary rescaling L_epsilon->(1-c epsilon)L_epsilon changes H by
000211 | cI and the absolute source. It fails this test unless c is the actual
000212 | original scalar derivative. No normalized-fiber loss is allowed.
000213 | 
000214 | This is a sufficient finite-regulator identification theorem, not an
000215 | evaluation of an unavailable primitive strip integral. It does not assume
000216 | a spectral trace is automatically local in physical time. If the actual
000217 | original S_adm kernel retains memory/rest legs, keep those legs in the
000218 | common strip space rather than resetting them after each slice. Neither
000219 | the norm o(epsilon) bound nor common-domain convergence at an unbounded
000220 | regulator follows from a formal core commutator; a separate actual
000221 | operator convergence proof would be needed there.
000222 | 
000223 | 5. Actual original finite matrix test and independent review
000224 | ------------------------------------------------------------
000225 | Graph-first code discovery in the indexed owning repository and exact snippets:
000226 |  experiments/tfpt-discovery/tfpt3p1d_coupled_wall_probe.py
000227 |  TreeGauge.canonicalize:334-361, coupled_hamiltonian:421-517.
000228 | Root executed the UNCHANGED coupled_hamiltonian at g=0.8, the first declared
000229 | coupling in the original model. Every other constructor argument retained
000230 | its original default. This is not a derived primitive physical coupling.
000231 | 
000232 | Whole native dimension4096 =32 original Z2 cycle coordinates times128 even
000233 | matter states; H has78336 stored entries. Hermiticity, all12 electric
000234 | involutions, their pairwise commutation and absence of cross-coordinate
000235 | blocks in the retained R are exact in the computed matrix entries.
000236 | H=H_E+R has floating residual8.88e-16. No term or original offset is removed.
000237 | With c=g²/2, the original electric quotient operators X_l give
000238 |  H_E=c sum_l(I-X_l),
000239 |  exp(-s H_E)=exp(-12cs) product_l[cosh(cs)I+sinh(cs)X_l].
000240 | The full R coordinate blocks retain magnetic,mass,matter hopping and DET.
000241 | Thus exact block exponentials and symmetric Strang composition implement
000242 | the full same-source field integration with all its internal matrix marks.
000243 | 
000244 | At diagnostic t0.2, full-vector errors versus independent sparse exponential
000245 | for4,8,16 slices are2.693321492e-4,6.742529507e-5,1.686211705e-5.
000246 | One actual boson-coordinate mark and even matter mark give whole-word vector
000247 | errors1.808698568e-6,4.528993390e-7,1.132702707e-7 for the same slice counts.
000248 | These are two numerical legs,not exhaustive all-source tests. All finite
000249 | bounded source insertions follow analytically from operator-norm Strang
000250 | convergence and the kernel identities. No new convergence claim from counts.
000251 | 
000252 | The q0->q6 electric block has entries-0.32,+0.32 and no internal offdiagonal
000253 | entries. Replacing it by the EMPTY OCCUPATION coordinate entry times identity
000254 | has norm error0.64=g². This is not the physical OS vacuum or ground state.
000255 | Analytically the block is -(g²/2)diag((-1)^popcount(gauge_mask & occupancy)),
000256 | with gauge_mask supplied by the actual original canonicalize. Its path jump
000257 | must retain this whole occupation-dependent phase. No coordinate-independent
000258 | positivity obstruction or new physical interaction is inferred.
000259 | 
000260 | A separate analytical source/code review checked the electric exponential,
000261 | all retained terms,projection/normalization conventions and numerical scope.
000262 | No sign/projection/normalization error was found. Scalar positivity of the
000263 | reference sampling process does not turn these matrix weights into physical
000264 | positive scalar boson probabilities. Existing R113 and physical thermal
000265 | source prerequisites remain. Root g0.8 check/results are retained in
000266 | check_native_matrix_source.py and native-matrix-source-results.json.
000267 | 
000268 | 6. Original-source and mathematical scope
000269 | ----------------------------------------
000270 | R113 PROJECTED-JOINT-SOURCE-GENERATOR-20261103.txt gives the existing
000271 | conditional joint generator and operator-valued H_B body. Sections 1-4
000272 | here apply after that whole generator has ACTUALLY been identified and
000273 | evaluated. They do not calculate the unknown primitive H_B by definition.
000274 | ADMISSIBLE-ACTION-DERIVATION-20261103.txt retains its original physical
000275 | linear-fermion-contact premise; this proof does not silently assert it.
000276 | 
000277 | Existing finite physical Hamiltonian source: PHYSICAL-THERMAL-SOURCE.txt
000278 | supplies actual reducing P, whole bounded Euclidean words and their
000279 | thermodynamic limit. Every bounded kernel-expanded word equals that same
000280 | finite word, so its sum has the same already established limit. Individual
000281 | configuration kernels need not have independent thermodynamic limits.
000282 | No global finite infinite-volume partition number is inferred.
000283 | 
000284 | For the primitive spectral S_adm branch, (12) is still to be evaluated on
000285 | its actual entire original field/domain/reference chart. No result here
000286 | declares that equality established, renames -log omega as the action,
000287 | or promotes the primitive physical spectrum/ledger/F1 to solved.
000288 | 
000289 | Primary context checked: arXiv:2311.08564 is WITHDRAWN and states that its
000290 | technical contributions were superseded by arXiv:2405.12316. It is not used
000291 | as an authority for any claim here. The finite matrix identities above
000292 | are proven directly by ODE uniqueness and the Dyson expansion, with
000293 | independent analytical/code review. No novelty priority is claimed.