TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-CONNECTION-AND-WHOLE-CURRENT-SOURCE.txt
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000001 | TORSION20261104D. Actual connection fiber and whole auxiliary/current source.
000002 | 4 November 2026. Anonymous original-source continuation; no replacement model.
000003 | 
000004 | 1. QUESTION, ORIGINAL DATA AND THE PRECISE NEW TEST
000005 | The original operative geometric caller is K_sigma=f D_geo f+B_rel,
000006 | f=exp(-sigma/2), D_geo=D. It already exists and is transported together
000007 | with its bound reference, full state, original time, domain and source legs.
000008 | TORSION20261104C excluded ordinary inner SM/scale and fixed single-Higgs
000009 | phase identifications. It did NOT exclude independent metric spin
000010 | connections or all original B_rel components. This note tests that remaining
000011 | connection class and the whole regulated historical P25 source response.
000012 | 
000013 | Original geometric statements: source45 metrology001:15-128. Its reconstruction
000014 | of g and the Levi-Civita connection from the metric does not imply that
000015 | every lower-order term of the actual D is the Levi-Civita spin connection.
000016 | The primitive integration variables are not selected by that inference.
000017 | 
000018 | 2. THE ENTIRE METRIC SPIN-CONNECTION FIBER, NOT SELECTED EXAMPLES
000019 | Use exactly the original actual_product_pairing.py Clifford assignments,
000020 | Hermitian gamma1..4, gamma5=gamma1gamma2gamma3gamma4, normal n=4.
000021 | For real K_muab=-K_muba, set omega_mu=K_muab gamma^a gamma^b/4.
000022 | There are 24 real connection coefficients at a point, and
000023 |  V(K)=-i gamma^mu omega_mu
000024 | has an eight-dimensional image: four vector and four axial components.
000025 | The formal Hermiticity conditions on V impose four independent linear
000026 | constraints. The symmetric connection fiber has dimension20, its Dirac
000027 | image dimension4, and the invisible kernel dimension16. Its whole visible
000028 | image is precisely
000029 |  W[S]=(3i/4)S_mu gamma^mu gamma5.
000030 | The actual representative K_muab=epsilon_muabnu S_nu/2 reproduces this
000031 | formula exactly. It is a representative modulo the invisible kernel,
000032 | not a new physical choice or an integration measure over all24 coefficients.
000033 | 
000034 | The original principal symbol and [D,f]=-i gamma.df do not change.
000035 | W acts as identity on the finite original algebra, so [W,pi(a)]=0 and
000036 | the original first-order relation [[D,pi(a)],Jpi(b)J^-1]=0 is preserved.
000037 | This is NOT a statement that arbitrary zero-order additions preserve the
000038 | finite first-order relation. J reality and total oddness are retained.
000039 | The visible axial coefficient is recovered without fitting by
000040 |  S_nu=Tr_spin(gamma5 gamma^nu V)/(3i)
000041 | in this scalar internal channel (divide additionally by dF if tracing F).
000042 | 
000043 | Compatibility must be typed. A Clifford connection relative to FIXED
000044 | Levi-Civita tangent differentiation has differences commuting with every
000045 | gamma^nu. On irreducible complex Spin4 their full commutant is scalars;
000046 | with the finite factor it is I_spin tensor End(F). Its contracted image
000047 | is a Clifford vector, not W. A general METRIC tangent connection with
000048 | torsion changes the tangent differentiation in the compatibility relation;
000049 | its spin lift need not commute with gamma. In fact
000050 |  [W,gamma^nu]=-(3i/2)S_nu gamma5.
000051 | Do not equate the represented internal algebra with the Clifford algebra,
000052 | or put gamma5 into the commutant of the full Clifford action.
000053 | 
000054 | This classification is standard geometry applied to the actual original
000055 | consumer, with the whole coefficient map checked exactly. Primary source:
000056 | https://arxiv.org/pdf/1101.1424, section4, especially Corollary4.6/Lemma4.7.
000057 | It supports the geometric class, not selection of the primitive TFPT field.
000058 | 
000059 | 3. ACTUAL COLLAR MAP: THE TANGENTIAL TORSION IS A NORMAL CONNECTION
000060 | On the given flat product collar write D=-i gamma4(partial_r+A_r+B_r).
000061 | Keeping the ENTIRE original Phi, the exact decomposition is
000062 |  B0=gamma4 gamma^i partial_i+i gamma4 gamma5 Phi,
000063 |  B_r=B0-(3/4)S4 gamma5,
000064 |  A_r=-(3/4)sum_i S_i gamma4 gamma^i gamma5.
000065 | Here B_r is formally selfadjoint, {B_r,gamma4}=0,
000066 | A_r^dagger=-A_r, [A_r,gamma4]=0. The naive complete shifted boundary
000067 | generator is B_r+A_r, whose adjoint defect is 2A_r. It is selfadjoint
000068 | iff the three tangential S_i vanish. Thus substituting it blindly into
000069 | the existing selfadjoint C(h) reader changes that reader's assumptions.
000070 | 
000071 | The actual operator determines A_r even though a spin connection is only
000072 | determined modulo its invisible kernel. For the axial representative,
000073 | A_r=3 omega_4; the two are not the same normal connection coefficient.
000074 | The unitary frame U satisfies partial_r U=-A_r U, U(0)=I. Exactly
000075 |  U^dagger D U=-i gamma4(partial_r+U^dagger B_r U).
000076 | B_r is a tangential differential operator: retain every partial_i U term.
000077 | The framed operator is independent of r precisely when
000078 |  partial_r B_r+[A_r,B_r]=0 on the common core.
000079 | This is an operator condition, not an identity for arbitrary S fields.
000080 | Keep BOTH boundary frames, J, the deck, reference, all fields and state/time
000081 | under this transport. It does not remove physical torsion or generate a
000082 | new bosonic action. A moving frame on a dynamical field space also carries
000083 | the corresponding momentum/current terms already established earlier.
000084 | 
000085 | For a variable collar the full Calderon projector depends on propagation
000086 | through the whole collar. It is NOT the instantaneous positive spectral
000087 | projection of B_r. No fixed-domain or eta cancellation is inferred.
000088 | Primary comparison: https://arxiv.org/pdf/1307.3021, sections3.2 and3.3;
000089 | an adapted selfadjoint boundary operator is distinct from the full bulk
000090 | operator/domain. The exact displayed split is computed directly here.
000091 | 
000092 | 4. WHOLE HISTORICAL P25 SOURCE, BOTH INTEGRATION ORDERS