TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DECISION.txt
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000001 | APS SOURCE REVIEW: FIXED INVERSE CONSTRUCTED; PHYSICAL SELECTION NOT IMPLIED
000002 | 29 October 2026. Continue the same complete original-source comparison.
000003 | 
000004 | 1. The actual original source was read
000005 | source001353-377 CHOOSES modified APS as analytic geometric realization.
000006 | source00233-42 and source003121 classify it as provisional analytic choice;
000007 | source00278-84 places Lorentzian reconstruction after carrier/closure inputs.
000008 | Operational Seed source002103-145 already includes the net, omega and tau.
000009 | The state must therefore be transported and compared, not chosen anew by
000010 | calling a geometric operator selfadjoint. Standard Baer/Ballmann boundary
000011 | theory supports the geometric domain statement, not that physical choice.
000012 | The generic first-order relative-bound-zero inference in the original was
000013 | already excluded by two Atlas findings and is not new here. Bounded odd
000014 | Phi, the actual class being tested, satisfies the needed perturbation result.
000015 | 
000016 | 2. Positive closed-domain result, retained
000017 | For each fixed constant gapped B and inward normals on the two faces,
000018 |   x0 in P_-(B), xell in P_+(B)
000019 | gives C0 Dom L=Dom L*, C0LC0=-L*, and the closed skew pairing. The inverse
000020 |   G_APS(r,s)=1_(r>s)e^(-(r-s)B)P_+
000021 |              -1_(s>r)e^(-(r-s)B)P_-
000022 | is correct, with pointwise kernel norm<=1 and inverse norm<=1/gap.
000023 | Given the specific CAR quantization it is a vacuum propagator. This does
000024 | not identify it with the finite thermal face marginal of the already given
000025 | whole Cauchy graph or with the complete operational Seed omega.
000026 | 
000027 | 3. Original insertion needs its actual boundary-domain contract
000028 | The projector derivative under the actual odd insertion is
000029 |   delta P_-(B)|0=-delta B/(2|A|).
000030 | Thus changing APS(B_Phi) changes the endpoint field domain. A fixed original
000031 | APS(B0) domain and a moving APS(B_Phi) domain are different source families.
000032 | The minimal quarter block p1/4,m1/3 has E5/12 and
000033 |   P_h=[[1/5,2i/5],[-2i/5,4/5]].
000034 | The moving G_APS(h) violates the old fixed initial condition by
000035 |   ||(I-P0)G_APS(h)(0,s)||HS²=e^(-5s/6)/5>0.
000036 | Both realizations are mathematically valid; they cannot be interchanged
000037 | while claiming the same unchanged insertion/reference. The exact existing
000038 | exterior cap amplitude ratio is4/5+(1/5)e^(-5ell/6), not1.
000039 | 
000040 | 4. Constructive solution on the WHOLE fixed old domain
000041 | This source family can be solved directly. At every original p=n+alpha,
000042 |   h0=p gamma_F, h=h0-i gamma_F Phi, E=sqrt(p²+Phi²), P=P_-(h0),
000043 |   R_t=e^-t h, M=P R_ell P|RanP,
000044 |   M=cosh(ellE)+|p|E^-1sinh(ellE)>=I.
000045 | Then the fixed-domain Green kernel is exactly
000046 |   Gfixed(r,s)=1_(r>s)R_(r-s)-R_r P M^-1 P R_(ell-s).
000047 | The energy identities for L and L* give lower bound |p| and hence
000048 |   ||Gfixed||<=4
000049 | on the full direct sum of ALL Fourier modes at either original quarter.
000050 | The terms in the displayed kernel are combined modewise before taking the
000051 | bounded direct-sum closure; signed exponentials alone are not bounded.
000052 | The same fixed-domain operator has actual bounded insertion response
000053 |   deltaGfixed=-Gfixed delta h Gfixed, ||deltaGfixed||<=16||delta h||.
000054 | No new field/state/kernel choice has been fitted. Root independently
000055 | verified the exact ODE, jump and both boundary conditions on the minimal
000056 | nonzero odd block. The full mode theorem is analytic, not a cutoff estimate.
000057 | 
000058 | 5. Finite boundary correction from the SAME exterior transfer
000059 | At each p, the fixed/moving cap quotient is
000060 |   C_p=(I+|p|E^-1)/2+(I-|p|E^-1)e^(-2ellE)/2 on RanP0.
000061 | It obeys C_p>=I/2 and 0<=I-C_p<=Phi²/(4p²). Thus the whole direct sum
000062 | is I plus trace class, and its Fredholm determinant is positive/nonzero.
000063 | The actual96er chart gives Tr(I-C)<=pi² Tr_KPhi²/4 at either quarter;
000064 | conjugate test slots are not counted again. This finite boundary factor
000065 | corrects the pure vacuum-energy amplitude. The remaining bulk vacuum-energy
000066 | factor still needs its actual original renormalization/reference. This is
000067 | a relative boundary ratio, not a new choice of absolute physical scalar.
000068 | 
000069 | 6. Actual deck and action comparison
000070 | With FULL tau=S R_r and even Phi, the conditional identity Dref=D0 and
000071 | Drel=Z Phi is correct. S alone omits the normal-derivative reflection.
000072 | The different existing graded product-reflection S gamma_F R_r preserves
000073 | D_Phi instead, so its Drel is0. Original deck, field transport and OS
000074 | reflection cannot be silently exchanged. This is not a new choice of lift.
000075 | For the claimed mass-only Drel at Phi0, the original ordinary heat profile
000076 | f(x)=e^-x² yields I-e^[-(D0/chi)²], not even compact on an infinite source
000077 | space. Compact base, a low-energy gap removal and internal selectors do
000078 | not turn it into a finite-dimensional trace. This excludes THAT combined
000079 | literal action/deck identification and ordinary-trace prescription.
000080 | The original relative-object table also permits a declared relative PAIR.
000081 | That alternative requires an explicit reconciliation with the even/odd
000082 | operator formula; failure of one reading does not uniquely prove another.
000083 | No replacement of original operator arguments or mu=1/chi renormalization
000084 | condition is inferred by fiat. No all-P1 or all-profile exclusion is made.
000085 | 
000086 | 7. Joint quantum source retained
000087 | The original source007 defines W_k using a formal integral of S_adm and
000088 | propagates its supplied Hessian/higher vertices. Those formulae do not,
000089 | by themselves, evaluate a microscopic Phi insertion. They remain available
000090 | for the actual common source comparison. A common bosonic-plus-fermionic
000091 | functional need not equate its bosonic spectral action with logPf.
000092 | The full given Seed already contains omega and tau; its missing comparison
000093 | must not be replaced with another free Gaussian state or another Fock layer.
000094 | 
000095 | Next: compare the independently specified original omega, field insertion
000096 | and domain/reference transport with the now explicit fixed Green response,
000097 | or with a correctly transported moving-domain response. Use the actual full
000098 | source map before calling radial propagation the original physical time.