TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-BOUND-TORSION-REFERENCE-AND-FIELD-IMAGE.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | TORSION20261104C. Bound torsion reference, actual field image and source stages.
000002 | 4 November 2026. Anonymous source-reviewed continuation, not a new model.
000003 | 
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000011 | 
000012 | 1. ORIGINAL SOURCE TYPES AND THE OPERATIVE GEOMETRIC ROUTE
000013 | Original orientation source002:73-110 prints the organizing seed package
000014 |  S_TFPT=Tr f((D_rel+A_seed)/chi_seed)-Tr f(D_ref/chi_seed)+i*pi*DeltaEta/2,
000015 |  A_seed=B_rel directsum nabla_chi_seed.
000016 | The later original metrology001:15-17,30-128 explicitly realizes its
000017 | geometric package using
000018 |  D_geo=D_ref+D_rel=D,
000019 |  K_sigma=exp(-sigma/2)D_geo exp(-sigma/2)+B_rel,
000020 |  E_D=Res_(s=1) Zeta_rel(s), Gamma4=FP_(s=0) Zeta_rel(s),
000021 |  Gamma_grav=-6 chi_geo^2 E_D+Gamma4.
000022 | The existing CURRENT-ARGUMENT-MAP.txt and full source transport are valid.
000023 | They already supply the geometric realization. Do not reopen that map as
000024 | globally missing, classify all A_seed as bounded, or demand a universal
000025 | literal D_rel+A_seed=K_sigma identity across different original versions.
000026 | 
000027 | The question now concerns the actual torsion jet of this whole realization:
000028 | whether the original field image contains independent S, how B_rel varies
000029 | with that same variable, and what the actual full projected source does.
000030 | Naming the superconnection package alone does not evaluate those jets.
000031 | 
000032 | 2. EXACT DECK SPLIT OF AN INDEPENDENT ONE-FORM
000033 | Use the same actual Spin4 matrices and full96 Phi as HIST20261104A.
000034 | W[S]=(3i/4)gamma^mu gamma5 S_mu, with real S and linear deck tau.
000035 | For the normal reflection R, including its pullback on one-form indices,
000036 |  tau W[S] tau^-1=-W[R^*S].
000037 | This follows directly because the spin deck fixes tangential gamma,
000038 | reverses normal gamma and gamma5, and does not conjugate i.
000039 | The internal gamma_F in the second existing lift commutes with W[S].
000040 | Thus BOTH retained original lift charts give
000041 |  S_plus=(S+R^*S)/2, S_minus=(S-R^*S)/2,
000042 |  delta D_ref=W[S_minus], delta D_rel=W[S_plus].
000043 | The labels describe pullback parity, not an imposed physical pseudoscalar
000044 | parity. The numerator D_geo contains both components.
000045 | 
000046 | On a reflection-symmetric product domain their integrated bilinear cross
000047 | terms vanish, as do the cross terms of their exterior derivatives.
000048 | For seam-even H, the original reference masses remain
000049 |  Phi_ref,S=0 and Phi_ref,T=Phi.
000050 | The actual reference, rather than an independently frozen covariance, is used.
000051 | 
000052 | 3. BOUND RELATIVE BULK RESIDUES ON THE EXISTING ORDINARY PRODUCT
000053 | Let d=96, T=Tr_F Phi^2, Tref=Tr_F Phi_ref^2 and q=exp(2 sigma0)>0.
000054 | Write I_plus/minus=int |S_plus/minus|^2 and
000055 | C_plus/minus=int |dS_plus/minus|_form^2, summing mu<nu in the form norm.
000056 | The previously derived complete numerator heat response gives
000057 |  alpha=9d/(32pi^2), beta=3d/(64pi^2), c=9/(32pi^2).
000058 | For the ordinary product anchor with constant sigma0 and full constant Phi,
000059 |  Delta a2 = alpha [q I_plus+(q-1) I_minus],
000060 |  Delta a4 = -beta C_plus-c[T I_plus+(T-Tref) I_minus].
000061 | Here the reference is UNSCALED, as in the actual original geometric pair.
000062 | The numerator a2 scales by q and its a4 is unchanged. Rescaling the
000063 | reference along with the numerator would change the original source.
000064 | 
000065 | The Mellin factor is essential:
000066 |  zeta(s)=Gamma(s)^-1 int_0^infinity t^(s-1) DeltaHeat(t) dt.
000067 | On its usual regular elliptic realization, E_D=Delta a2 and
000068 | FP_0 zeta=Delta a4-Delta n0, plus the separately retained eta phase.
000069 | In a fixed kernel-rank neighborhood Delta n0 has zero field variation.
000070 | This reuses the established FP0 versus zeta-derivative distinction;
000071 | it is not a new general determinant theorem.
000072 | 
000073 | Consequently the quadratic ordinary product contribution is
000074 |  Gamma_grav^(2)=-6 chi_geo^2 alpha[q I_plus+(q-1) I_minus]
000075 |                -beta C_plus-c[T I_plus+(T-Tref) I_minus].
000076 | In BOTH lifts the reference cancels the C_minus coefficient and leaves
000077 | the C_plus coefficient -3d/(64pi^2), or -9/(2pi^2) for full96.
000078 | At sigma0=0 its S_plus algebraic coefficient is
000079 |  -162 chi_geo^2/pi^2-9 T/(32pi^2).
000080 | This negative Hessian coefficient alone is not an unboundedness theorem
000081 | for the complete action, nor a physical ghost or tachyon diagnosis.
000082 | 
000083 | 4. PRECISE LIMIT OF THE CANCELLATION STATEMENT
000084 | The displayed complete quadratic polynomial is for the supplied ordinary
000085 | Dirac product contribution, not an automatic evaluation of arbitrary B_rel.
000086 | The coefficient with two derivatives on S is unchanged by a FIXED,
000087 | S-independent zero-order perturbation with the same Dirac principal symbol;
000088 | other lower-derivative coefficients can change. If the original B_rel has
000089 | its own first-order symbol or S-dependent vertices, compute them instead
000090 | of applying that restricted assertion. No claim that all B_rel is
000091 | boundary-supported or zero order is made.
000092 | 
000093 | For a fixed elliptic APS boundary realization and S supported away from
000094 | the boundary, boundary heat jets are unchanged. A real eta phase cannot
000095 | cancel a real quadratic coefficient in the action. Do not infer that eta
000096 | itself is constant on every original APS domain merely from bulk grading.
000097 | Do not infer that the full Calderon projector is fixed by the boundary
000098 | symbol. Its moving-domain response has NOT been evaluated here.
000099 | The original downstream Padm is not an insertion into the ordinary local
000100 | geometric anchor. A genuinely projected or nonlocal relative trace needs
000101 | its own original map and contacts; no universal cancellation theorem or
000102 | universal impossibility theorem for that trace is claimed.
000103 | 
000104 | 5. WHAT THIS DECIDES ABOUT P25
000105 | The ACTUAL historical parent before fermion integration contains
000106 |  L_aux,E=A S^2-S.(iB da+c0 J5), A=3/(4 kappa_phys^2)>0, c0=3/4.
000107 | Its S Hessian is the multiplication operator 2A, with no curl term.
000108 | The ordinary product relative residue functional above has nonzero curl
000109 | on S_plus. A real compact interior one-form S_2=h(r)f(x1), with h even
000110 | and supported away from the seam and f nonconstant, supplies that direction.
000111 | Therefore these two OFF-SHELL FUNCTIONALS are not identical on that
000112 | declared family. Neither an overall normalization nor a field rescaling
000113 | that preserves the original minimal spin/current coupling removes the
000114 | nonzero derivative order. No fitted counterterm is introduced.
000115 | 
000116 | This does NOT exclude equality of appropriate COMPLETE quantum sources
000117 | after an actually proved integration map. Integrating fermions in P25
000118 | can generate a nonlocal S response; integrating S first encodes the same
000119 | effect in its four-fermion interaction. Bare and effective stages must be
000120 | matched before comparing coefficients. The full B_rel, field-dependent
000121 | domains, allowed field image and projected source remain necessary.
000122 |