TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/B1-POSITIVE-SOURCE-HEAT.txt
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Inclusive original lines: 1-101
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | EXISTING AP POSITIVE FACTOR TO HEAT MIXTURE, WITH FULL SOURCE ORIENTATION
000002 | 30 October 2026. Constructive continuation of B1/B3; no new physical model.
000003 | 
000004 | 1. Original factor, not a replacement action
000005 | ROOT-CANONICAL-DOUBLE-AND-ORIGINAL-LIFT.txt:48-76 already computes
000006 |  F_AP=Tr[beta E_mu+2log(I+exp(-beta E_mu))], beta=2ell,
000007 |  E_mu=sqrt(B^2+mu^2), for each finite tangential selection; the full-circle
000008 | Fredholm gluing term is defined separately from the divergent bulk.
000009 | At mu=0, on the given charged gamma sector E=|h|, with
000010 |  h=p gamma-i gamma Phi, h^2=p^2+Phi^2, p=n+1/4.
000011 | The same positive factor Q=exp(-beta E) occurs in that existing determinant.
000012 | No statement S_bos=-logPf, new coupling, or independent source is introduced.
000013 | The physical K_car origin, reference, eta and4D action remain separate.
000014 | 
000015 | 2. Lossless polar source representation
000016 | Let h be the GIVEN selfadjoint tangent source, with its actual domain and a
000017 | gap at0 as in this constant quarter class. Define E=|h| and R=sgn(h), R^2=I.
000018 | The spectral theorem gives Q=exp(-beta E), 0<Q<I, commuting with R.
000019 | The entire Fermi source and crossface block follow exactly:
000020 |  C=(I-R)/2+R Q(I+Q)^-1,
000021 |  X=Q^(1/2)(I+Q)^-1,
000022 |  G=[[I-C,X],[X,C]].
000023 | This is proved separately on the positive and negative h spectral sectors:
000024 | there C=Q/(1+Q) and1/(1+Q), respectively. The crossblock is even in h.
000025 | Conversely C determines k=log((I-C)C^-1)=beta h by its unbounded spectral
000026 | functional calculus, hence determines R and Q=exp(-|k|).
000027 | Thus (R,Q) carries the whole source; Q by itself loses orientation. Indeed
000028 | h and-h give the same Q,X but complementary C. No new sign choice is made:
000029 | R is inherited from the original h, not selected from Q.
000030 | If an original zero mode were present, keep its projection and C=1/2 there;
000031 | the invertible quarter hypothesis avoids this extra bookkeeping here.
000032 | 
000033 | This constructs the bounded positive factor that the signed one-particle
000034 | exp(-k) could not supply. It does not identify either with the many-particle
000035 | Fock density, GNS modular operator, or full bosonic spectral action.
000036 | 
000037 | 3. Exact positive heat representation of the SAME factor
000038 | To test the squared-energy argument set u=(beta E)^2, by functional calculus,
000039 | so Q=F(u) with F(u)=exp(-sqrt(u)). This is an argument conversion on the
000040 | existing factor. It is not a postulate that the physical action uses u.
000041 | For all scalar u>=0 the standard subordination formula gives
000042 |  F(u)=integral_0^infinity exp(-t u)dnu(t),
000043 |  dnu(t)=[1/(2sqrt(pi))]t^(-3/2)exp[-1/(4t)]dt.
000044 | The measure is strictly positive, has total mass1, and is diffuse rather than
000045 | the point mass delta1 associated with exp(-u).
000046 | A scalar Gaussian integral proves the formula; positive bounded spectral
000047 | calculus then gives the corresponding strong operator integral on the
000048 | whole given source. No Fourier or internal dimension cutoff is needed.
000049 | The tangent E here is NOT silently identified with the current whole4D
000050 | D_rel or the full normal-double D; those require the original map.
000051 | 
000052 | Primary mathematical context read: Ferguson, Mei, Simanek,
000053 | H-infinity-calculus for semigroup generators on BMO,
000054 | https://arxiv.org/pdf/1701.06623 , introduction and section1 equation3.
000055 | It distinguishes the Poisson e^(-t sqrt(L)) and heat e^(-t L) semigroups and
000056 | their positive subordination. The present application uses the elementary
000057 | selfadjoint spectral identity, not its additional BMO hypotheses.
000058 | 
000059 | 4. Whole-profile shape differs from the unit-moment exponential
000060 | By u=v^2 and elementary gamma integrals,
000061 |  m0=F(0)=1,
000062 |  m1=integral F(u)du=2 integral v exp(-v)dv=2,
000063 |  m2=integral u F(u)du=2 integral v^3 exp(-v)dv=12.
000064 | The same values follow from integral t^-1dnu=2 and integral t^-2dnu=12.
000065 | Consequently the positive-measure shape invariant is
000066 |  J=m0*m2/m1^2=3,
000067 | whereas exp(-u), with its point heat measure, has J=1.
000068 | For any amplitude a>0 and argument rescaling b>0, F_ab(u)=a F(bu),
000069 |  (m0,m1,m2)->(a m0,a m1/b,a m2/b^2), so J is unchanged.
000070 | No such normalization turns the entire triple(1,2,12) into(1,1,1).
000071 | Equivalently the variance of inverse heat time is12-4=8, not0.
000072 | Thus the positive-source conversion and the previous exact moment theorem
000073 | fit together: one is a diffuse positive measure, the other a unique point.
000074 | This is an exact shape distinction, not a failed numerical fit.
000075 | 
000076 | 5. Precise consequence for the original source/action bridge
000077 | If the proposed identification is literally the existing positive source
000078 | factor expressed in its squared-energy argument, the profile is the stated
000079 | subordinated one and is not the v259 unit-moment exp(-u) profile, even after
000080 | an overall amplitude and scale change. If the action instead uses the
000081 | linear positive energy as argument, the scalar exp(-u) notation has a
000082 | different meaning; its Seeley moment convention and operator map must be
000083 | transported accordingly. The printed first-order D notation alone cannot
000084 | identify these arguments. v255's displayed moment convention is retained
000085 | as printed, not silently corrected or applied to the wrong variable.
000086 | Nothing here proves that the physical bosonic action MUST equal the
000087 | fermionic gluing functional or this single source factor. These are given
000088 | consumers of the source, not an independently derived whole4D interaction
000089 | measure. The original same-source argument and reference transport remains
000090 | the concrete unresolved equation, now with an explicit positive comparison.
000091 | 
000092 | 6. Evidence and next step
000093 | check_b1_positive_source_heat.py and B1-POSITIVE-SOURCE-HEAT.json contain18
000094 | successful new controls: exact normalization, inverse-time moments and
000095 | sample subordination values, all required profile moments, scale invariant,
000096 | and actual96 reconstruction of C, X and whole G from (R,Q). The analytic
000097 | spectral theorem above supplies the all-spectrum result; the matrix probes
000098 | are controls rather than an infinite-dimensional proof.
000099 | Retain this source-to-positive-heat map and its orientation in B3.M5.
000100 | No physical B1/B3 parent or complete TFPT solution is thereby closed.
000101 | The new source comparison is not a new reason to rerun the earlier73 tests.