TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/pfaff_source_review/PFAFF-INDEPENDENT-REVIEW.txt
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000096 |   delta log Pf(A)=(1/2)Tr(A^-1 delta A).
000097 | 
000098 | For FIXED Jloc, this trace is(1/2)Tr(D^-1 delta D). Further variations of
000099 | Jloc, the measure, the domain or the reference require their own terms.
000100 | These statements need an actual invertible regulator or a specified
000101 | Pfaffian line; the mere skewness of a differential expression does not
000102 | choose its global boundary domain or Pfaffian orientation.
000103 | 
000104 | The result in[F] specifically does not identify D with h or with the graph:
000105 |   D is old-complex-linear but new-complex-antilinear;
000106 |   beta_D(Ev,Ew)=0 on one positive total-grade chart;
000107 |   beta_h(v,w)=<c v,h w> is instead symmetric.
000108 | The nonzero beta_D couples BOTH total grades through the actual normal
000109 | operators partial_r+h and partial_r-h. Its full off-diagonal structure is
000110 | precisely relevant to the counting in the next section.
000111 | 
000112 | 3. THE HALF FACTOR: CORRECT ONLY WITH THE COUNTED SPACE
000113 | 
000114 | For any invertible N-by-N K,
000115 |   A_K=[[0,K],[-K^t,0]],
000116 |   Pf(A_K)=(-1)^(N(N-1)/2) det K.                         (1)
000117 | This follows directly by pairing the first N Grassmann variables with the
000118 | last N in the top coefficient. For fixed orientation the displayed sign is
000119 | independent of K and cancels in a relative ratio. Thus a full skew doubled
000120 | form is NOT automatically sqrt(det K) on the UNDUPLICATED K space.
000121 | Its full trace formula still has1/2 because
000122 |   (1/2)Tr_(2N)(A_K^-1 delta A_K)=Tr_N(K^-1 delta K).
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000135 | 
000136 | This agrees with(1): representing the same block by its one-dimensional
000137 | off-diagonal kinetic K=i lambda uses Pf=det K and a single logarithm.
000138 | Counting only that single K eigenvalue gives x f'=-1. Counting the full
000139 | signed Dirac pair gives(2). The change is not a new dynamical mechanism.
000140 | The trace space in the actual proposed action must be stated before its
000141 | coefficient is meaningful. A further physical Majorana or chiral reduction
000142 | cannot be inferred from the formal doubled chart.
000143 | 
000144 | 4. EXACT PROFILE TEST IN THE ACTUAL BERNSTEIN CLASS
000145 | 
000146 | Equation(2), on any open interval of variable x>0 in this direct comparison
000147 | class, forces f(x)=-(1/2)log x+C. This does not equal a regular original
000148 | positive heat profile with finite moments. Such f is nonnegative and, when
000149 | the zeroth moment is finite, bounded by mu([0,infinity)); the logarithmic
000150 | profile diverges at0 and becomes negative at sufficiently large x for
000151 | every finite C.
000152 | 
000153 | This conclusion applies even if equality is proposed only on an open
000154 | positive spectral interval. A positive Laplace heat profile finite for
000155 | all x>0 is real analytic there. Identity on an interval extends to the
000156 | positive axis, where the boundedness/positivity conflict remains.
000157 | It is not a claim about equality at one isolated spectral value.
000158 | 
000159 | The smallest allowed-profile discriminator needs no spectrum or simulation:
000160 | take the original allowed heat measure delta_1, so f(x)=exp(-x^2). On the
000161 | full paired block the variation residual against minus log Pf is
000162 |   2 x f'(x)+1=1-4x^2 exp(-x^2),
000163 | which at x=1 is1-4/e!=0. References constant in x cancel, and eta phases
000164 | cannot change this real derivative. A different positive finite moment
000165 | profile changes this residual but does not become the exact logarithm on
000166 | an open interval.
000167 | 
000168 | The LATER version[C] permits other declared profiles, so it does not ban
000169 | declaring a suitably renormalized relative logarithm. It also does not
000170 | derive that declaration. This is the exact scope of the negative test:
000171 | the proposed universal direct identification with the original fixed
000172 | regular profile fails; a theory containing a separate bosonic spectral
000173 | action and a fermionic Pfaffian is not excluded. A specially constrained
000174 | original family which does not vary x at fixed chi/reference would need
000175 | its actual joint derivative checked; it is not supplied by(2).
000176 | 
000177 | 5. CONSTRUCTIVE RELATIVE PROPER-TIME FORMULA, ALREADY PARTLY IN THE CORPUS
000178 | 
000179 | Assume a fixed Jloc, invertible finite regulators, equal counted ranks,
000180 | compatible reference and positive scale. Then ordinary scalar Frullani
000181 | integration, applied by spectral calculus and trace, gives
000182 | 
000183 |   -log|Pf_rel(A)|=-(1/4)Tr log(D^2/D_ref^2)
000184 |     =(1/4)integral_0^infinity
000185 |        Tr[exp(-tD^2)-exp(-tD_ref^2)] dt/t.              (3)
000186 | 
000187 | The quotient inside the logarithmic trace is shorthand for the difference
000188 | of the two log traces; no noncommuting matrix-log identity is asserted.
000189 | The D_ref in(3) is a bona fide fermionic kinetic reference for the same
000190 | skew form. It is NOT automatically the primitive deck-even D_ref from[C].
000191 | 
000192 | At finite0<epsilon<R, f_epsilon,R(x)=(1/4)integral_epsilon^R exp(-tx^2)dt/t
000193 | is a positive finite-moment heat profile. Its exact derivative is
000194 |   x f_epsilon,R'(x)=-(1/2)[exp(-epsilon x^2)-exp(-R x^2)].
000195 | Removing the cutoffs yields(2) for x>0. The individual profiles diverge
000196 | as epsilon tends to0. The relative subtraction may cancel that divergence
000197 | in the finite equal-rank problem, while the limiting positive measure
000198 | dt/(4t) is outside the original finite-moment measure class.
000199 | For differential operators, actual UV heat coefficients, zero modes,
000200 | boundary terms, reference and local counterterms determine whether the
000201 | relative integral exists or needs renormalization. Formula(3) does not
000202 | select those counterterms or their finite parts.
000203 | 
000204 | This is NOT a missing mathematical idea in the originals. Paragraph[G]
000205 | already writes exactly
000206 |   -ln a=integral_0^infinity(e^-ta-e^-t)dt/t
000207 |        =2 integral_0^infinity(e^(-a/chi^2)-e^(-1/chi^2))dchi/chi.