TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/pfaff_source_review/QUARTER-COVER-AND-SCALE-DECISION.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000013 | 
000014 | 2. Exact normal-symbol obstruction, in its narrow scope
000015 | They are not thereby the two normal spinor slots or the two radial faces.
000016 | For any old-complex-linear local zero-order N(x),
000017 |   sigma_1(B_cov)=2xi I, sigma_1({B_cov,N})=4xi N(x).
000018 | Anticommutation on all smooth modes forces N=0, incompatible with N²=-I.
000019 | The actual V commutes with B_cov; [B_cov,u_cov]=u_cov. Neither is the
000020 | claimed normal Clifford multiplication on this same coefficient space.
000021 | This is the minimal exact test of the proposed deck-to-normal step.
000022 | The valid antilinear J' does not contradict a statement about LINEAR N.
000023 | A full transported normal bundle or the existing corrected Clifford source
000024 | is not excluded. No complete P1 countermodel or universal cover no-go is made.
000025 | 
000026 | 3. A finite relative heat integral does not select its measure
000027 | Define for the single circle pair
000028 |   Delta(t)=Tr(exp(-t A_1/4²)-exp(-t A_1/2²)).
000029 | Poisson summation gives exponential cancellation as t decreases to zero;
000030 | both operators have a gap, so the large t endpoint also converges. Thus
000031 |   integral Delta(t) dt/t=log2
000032 | is valid. On the paired cover Delta_pair=2Delta; the stated Pfaff weight1/4
000033 | then gives log(sqrt2), as claimed for this DECLARED log prescription.
000034 | 
000035 | However both positive measures dt/t and dt/t+c exp(-t)dt, c>0, give finite
000036 | integrals on the SAME pair. Their single-pair difference is cR with
000037 |   R=Tr((A_1/4²+1)^-1-(A_1/2²+1)^-1)
000038 |    =pi sinh(2pi)/(cosh(2pi)(cosh(2pi)+1))>0.
000039 | The circle resolvent formula used here is
000040 |   sum_n 1/((n+a)^2+1)=pi sinh(2pi)/(cosh(2pi)-cos(2pi a)).
000041 | Tonelli for each positive trace or absolute endpoint estimates justify the
000042 | exponentially damped integral. On the paired Pfaff expression the change
000043 | is cR/2. Therefore convergence ALONE cannot choose the logarithmic measure.
000044 | These alternatives are tests of that implication, not claims of two full
000045 | admissible P1 theories or of compatibility with an extra dilation Ward.
000046 | If an original independent scale law selects dt/t, it must be used as such.
000047 | Original009:125-146 instead explicitly calls the KMS scale measure TFPT input.
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000056 | 
000057 | The source norm, its NS comparison, the actual deck-even D_ref, primitive
000058 | bosonic weight and the charged determinant line must be matched in their
000059 | actual original prescription. A joint bosonic-plus-fermionic source does
000060 | NOT in general require bosonic spectral action to equal a fermion logdet.
000061 | The direct-equality test is only one candidate and cannot become a new P1
000062 | axiom. Reuse the positive L=-ND kinetic bridge and whole original source.