TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DECK-REFERENCE-AND-ACTION.txt
Original file SHA-256: d392cfc2c0b3b50d39a041c6c1ee8bd3a1e173f80e767f35e05ba82344eaf0f1
Inclusive original lines: 101-197
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.

000101 | cannot be made by silently replacing S with the RP reflection or vice versa.
000102 | 
000103 | The old example supplies S over r reversal in the FREE local double. It
000104 | does not prove that the original coupled doubled bundle, source J and
000105 | backgrounds use precisely the extension in(1). One must transport those
000106 | same labels and domains. If the actual deck also translates theta by pi,
000107 | the local constant A commutes with that translation, but Phi is evaluated
000108 | at the translated point and the actual spin/quarter transition still enters.
000109 | The square of a half-period pullback on twisted sections can be its full
000110 | holonomy, so an involutive lift is not automatic from the base map alone.
000111 | 
000112 | Nor can an extra J_F be hidden in S: it is antilinear, fixes the given
000113 | J-real Phi and sends A_alpha to+i partial_theta+alpha. The Fourier and
000114 | quarter-domain action then changes, unless the actual base/reflection map
000115 | compensates it. The correct J_F and winding transports are already retained
000116 | in DERIVED-PAIRING.txt. This review does not choose an alternative lift.
000117 | 
000118 | For unbounded operators(1) additionally requires a common tau-stable global
000119 | domain. If the proposed APS projector is changed with B_Phi, its domain
000120 | variation must be included; it is not merely a bounded algebraic perturbation
000121 | on an automatically unchanged APS domain. A supplied closed-double H1 domain
000122 | can avoid that issue only when its actual global bundle/lift has been fixed.
000123 | 
000124 | 4. SMALLEST DECISIVE ACTION TEST: THE FREE LIMIT
000125 | 
000126 | The original formula is, literally,
000127 |   Tr f(D_rel/chi)-Tr f(D_ref/chi)+(i pi/2)Delta eta_Sigma. (3)
000128 | It is not printed as Tr f(D_Phi/chi)-Tr f(D0/chi).
000129 | Assume the conditional identification(1) with even Phi and a nonzero
000130 | infinite-dimensional spatial source Hilbert space. At Phi=0 it yields
000131 |   D_rel=0, D_ref=D0.
000132 | For the expressly allowed heat profile f(x)=exp(-x^2),
000133 |   f(D_rel/chi)=I.
000134 | On a compact elliptic realization with a genuine selfadjoint APS domain,
000135 | exp[-(D0/chi)^2] is trace class. Hence
000136 |   I-exp[-(D0/chi)^2]
000137 | is not compact and is not trace class. Even interpreting(3) as the trace
000138 | of a single operator difference does not cancel its high-frequency tail.
000139 | No nonzero mass, fitting, profile scan or new model is needed for this test.
000140 | 
000141 | For a constant finite coefficient Phi, f(Z Phi/chi) likewise has a fixed
000142 | nonzero finite matrix repeated on infinitely many spatial modes. It is
000143 | not compact for a nonzero positive heat profile. A finite spectral regulator
000144 | makes its trace finite, but removing that regulator requires an actual trace
000145 | or counterterm prescription; the primitive action does not acquire a new
000146 | meaning merely because another determinant has a finite or logarithmic limit.
000147 | 
000148 | 5. WHAT THE ORIGINAL RESTRICTIONS DO AND DO NOT CHANGE
000149 | 
000150 | The displayed action(3) uses an ordinary tr without an H_car trace, normalized
000151 | trace density, supertrace or specified spectral cutoff in that definition.
000152 | The primitive source is not automatically its later finite carrier module.
000153 | The given H+ boundary problem and the usual Cauchy/positive-energy restriction
000154 | still retain infinitely many spatial modes. The displayed primitive gap
000155 | penalty removes the LOW-energy spectral interval, not the UV tail. The later
000156 | color/determinant restrictions are finite internal selection rules; no cited
000157 | formula makes their nonzero physical field sector finite dimensional.
000158 | 
000159 | At Phi=0 any nonzero infinite-dimensional retained space still has the
000160 | identity from f(0). Its ordinary trace is infinite. Consequently a named
000161 | admissibility restriction alone does not repair this particular action test.
000162 | An actually supplied finite-dimensional cutoff or different trace could
000163 | alter the conclusion and must be specified. The older theorem's compact
000164 | geometric M_Lambda has an infinite-dimensional L2 Hilbert space; compactness
000165 | of the base is not finiteness of the number of modes.
000166 | 
000167 | The test is scoped. It excludes the COMBINATION of the literal action(3),
000168 | that ordinary infinite trace, the allowed heat profile and the claimed
000169 | bounded D_rel identification. It does not prove that every admissible
000170 | original realization has bounded D_rel, or that every possible declared
000171 | profile fails. The newer wider profile contract and a genuine different
000172 | trace prescription cannot be replaced by assumptions about this example.
000173 | 
000174 | 6. NO UNIQUE REWRITE OR RENORMALIZATION SCALE FOLLOWS
000175 | 
000176 | Replacing(3) by Tr f(D_Phi/chi)-Tr f(D0/chi) changes its operator arguments.
000177 | Those two pairs have already different values at a finite regulator; they
000178 | are not an algebraic restatement of the original even/odd definition.
000179 | An undefined literal branch can reveal a source/notation inconsistency,
000180 | but cannot prove this particular correction was intended or uniquely follows.
000181 | The original relative-objects table also permits a declared relative PAIR
000182 | as a convention; that must be explicitly reconciled with source001's actual
000183 | even/odd operator definition, not silently substituted into its action.
000184 | 
000185 | The massive-minus-massless sea energy has the known logarithmic UV term:
000186 | sqrt(p^2+m^2)-|p|=m^2/(2|p|)+O(|p|^-3). With the actual signed-mode and
000187 | internal pair counts it leads to the stated log divergence. This is a valid
000188 | conditional source calculation. It is not the high-frequency behavior of
000189 | the literal heat trace of bounded D_rel in(3), nor does it select that rewrite.
000190 | 
000191 | The existing chi_seed=lambda_1^+(|B|)^-1 supplies a possible source scale.
000192 | It does not by itself prescribe mu=1/chi_seed as a renormalization condition,
000193 | its finite counterterm, eta/reference line or normalization. Those data can
000194 | be inherited from an actual full operational Seed, which already contains
000195 | its state and time. They are not generated by renaming the same scale.
000196 | 
000197 | No original, registered review or Atlas record was altered. No suite was run.