TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/FULL-FUNCTOR-REVIEW.txt
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000001 | FULL ORIGINAL SOURCE UNDER THE EXISTING U: INDEPENDENT REVIEW
000002 | L20261102G, 2 November2026. Canonical sources unchanged.
000003 | 
000004 | VERDICT
000005 | 
000006 | YES: when the FULL existing U=(E,NE) is used as a change of coordinates
000007 | of the supplied original source, the entire original S/Z, including its
000008 | boson integration and fermion measure, is transported exactly. There is
000009 | no independent measure, reference scalar or interaction to choose again.
000010 | The claim is stronger than invariance of a kinetic matrix and weaker than
000011 | equality of two differently defined historical source functionals.
000012 | 
000013 | U covers the ENTIRE original normal-double coefficient space in the
000014 | given collar realization, not only Ran(E). At every point it is a unitary
000015 | from two copies of the whole original96 coefficient space onto the
000016 | original192 normal-double space. On the circle it acts on every mode;
000017 | it is not a finite-rank cutoff of the field Hilbert space. The bosonic,
000018 | gauge, ghost and geometric variables are still present as the SAME
000019 | original variables. Acting by the identity on those variables extends
000020 | U to a change of coordinates of their full joint integral.
000021 | 
000022 | The concrete limit of this statement is the actual argument of the
000023 | current spectral caller. The transported full Dirac is mathfrakD_h,
000024 | not h and not the bounded internal R_Phi supplied by the odd v258 reader.
000025 | v259 computes scalar profile moments. It does not itself replace any
000026 | whole Dirac operator by a finite matrix or supply a new whole trace.
000027 | The existing4D product consumer and R044/R045 expansions remain their
000028 | own original, explicitly scoped constructions.
000029 | 
000030 | 1. WHAT SPACE IS ACTUALLY RETAINED?
000031 | 
000032 | The source read in graded_boundary_functor/FINITE-COUPLED.txt:11-38 is
000033 |  B=Z A-S Phi, E v=(Pplus v,i Pminus v),
000034 |  gamma=gamma_F, Phi=Phi*, {gamma,Phi}=0, [A,gamma]=0.
000035 | E alone maps the whole96 coefficient onto the positive TOTAL grade.
000036 | It is an isometry with a proper range in the old normal double. The
000037 | full U=(E,NE) keeps both complementary total grades and is unitary.
000038 | PAIR-TYPING.txt and the independent FUNCTOR.txt prove on the original
000039 | transported domain
000040 | 
000041 |  U*DU=mathfrakD_h=[[0,-partial_r+h],[partial_r+h,0]],
000042 |  h=A gamma-i gamma Phi,
000043 |  U*D_ref U=mathfrakD_href.
000044 | 
000045 | The actual tau, J, boundary Green form and constraints are transported
000046 | with these operators. U does not declare both new slots independent
000047 | physical fermion species, discard the original KO/chiral restriction,
000048 | or replace the original skew pairing by <cv,hw> on one E slot.
000049 | 
000050 | This is a full chart of the represented collar source. It is not a
000051 | claim that the entire physical configuration space is just C96, or
000052 | that the circle is four-dimensional spacetime. For an original source
000053 | with further spacetime/bundle patches, fields and their transition data
000054 | outside this collar remain present. A local frame change carries the
000055 | overlap transition maps as well. It does not require deleting the
000056 | remaining fields or asserting a new global Cartesian bundle product.
000057 | 
000058 | The normal-independent classical-background scope in the original
000059 | FINITE-COUPLED.txt:5-8 is explicit. Its h/C formulas describe the given
000060 | conditional fermion source. They do not turn the joint H integration
000061 | into a single quasi-free state. The full transformed joint integral
000062 | retains that original H integration and every induced higher word.
000063 | 
000064 | 2. EXACT WHOLE-SOURCE CHANGE OF COORDINATES
000065 | 
000066 | Start with the ACTUAL original finite regulated tuple
000067 |  (b,psi,barpsi), b=(H,g,A,ghosts,other original fields),
000068 | its actual domain/support, integration operation and complete weights.
000069 | Use psi=U chi, barpsi=barchi U*, leaving b unchanged. The source terms
000070 | bar_eta psi+barpsi eta become
000071 |  bar_eta' chi+barchi eta',
000072 |  bar_eta'=bar_eta U, eta'=U*eta.
000073 | Bosonic sources J_b continue to couple to the same b. All original
000074 | composite operators are reexpressed with these same substitutions.
000075 | 
000076 | For the paired complex Berezin coordinates, the two inverse determinant
000077 | Jacobians multiply to1. Therefore a bilinear matrix K[b] transforms to
000078 | U*K[b]U, its determinant and every sourced minor transform consistently,
000079 | and no phase or vacuum multiplier is created. If the original physical
000080 | caller instead uses a skew Majorana/chiral pairing, its matrix transforms
000081 | by congruence M->U^T M U; Pfaffian det(U) and the inverse Grassmann
000082 | Jacobian compensate, with the ORIGINAL orientation transported. This
000083 | statement uses the original pairing and constraint subspace. It does not
000084 | choose a complex determinant instead of the original Pfaffian.
000085 | 
000086 | The boson measure is the original one because b has not been changed.
000087 | In particular H is NOT replaced by a flat integration over R_Phi.
000088 | Its kinetic term, potential, domain, gauge/ghost factors, determinant
000089 | weights, higher mixed vertices, and all non-Gaussian correlations remain.
000090 | Expressing an A or Phi coefficient in the new fermion basis is a change
000091 | of its represented matrix, not a new integration rule for its original
000092 | bosonic components. A later genuine boson coordinate change must carry
000093 | its induced original measure, rather than choose a fresh flat density.
000094 | 
000095 | Let scriptU denote this full change of coordinates, including all
000096 | original constraint/support data and dual sources. Define the transported
000097 | action by substitution, not by a new spectral or Gaussian ansatz:
000098 |  S_adm^U[b,chi,barchi]=S_adm[b,U chi,barchi U*].
000099 | The exact resulting identity is
000100 | 
000101 |  Z_adm^U[J_b,eta',bar_eta']=Z_adm[J_b,eta,bar_eta],
000102 |  Z_ref^U[0]=Z_ref[0],
000103 |  Z_rel^U[J_b,eta',bar_eta']=Z_rel[J_b,eta,bar_eta].          (1)
000104 | 
000105 | Every original reference field factor is transported by the same rule.
000106 | There is no need to choose a second w_ref or reset Z_ref to1. If the
000107 | original reference caller is supplied as a full functional rather than
000108 | as an evaluated number, (1) transports that functional; it does not
000109 | claim to numerically evaluate its original integration.
000110 | 
000111 | Each original finite regulator P_N, IR R_k, selector P_adm and boundary
000112 | domain is transported as well. In the displayed constant internal U,
000113 | ordinary angular mode cutoffs commute with U; more general regulators
000114 | are conjugated. Exact equality at every member of the SAME original
000115 | limiting family passes to the same existing limit. It needs no separate
000116 | new moment or domination hypothesis for the transformed copy. It does
000117 | not prove existence of a limit that the original source lacks.
000118 | 
000119 | 3. TERMS AND DERIVATIVES THAT NECESSARILY GO WITH U
000120 | 
000121 | The original spectral calls transport by full functional calculus:
000122 |  f(D), f(D_ref), f(D-D_ref), and the actual full geometric pair each
000123 | retain THEIR actual arguments. Their traces, zero modes, eta branches,
000124 | spectral scales and original subtraction are unchanged under the
000125 | simultaneously transported domain and regulator. The symmetric conformal
000126 | sandwich becomes e^(-sigma/2)mathfrakD_h e^(-sigma/2), with its stated
000127 | unscaled or scaled reference exactly as originally declared.
000128 | 
000129 | All original interactions are substituted, not discarded after the
000130 | bilinear term. Thus a source current delta S_adm/dH transports as that
000131 | same current, including reference and contact terms. Since this U is
000132 | H independent, H derivatives commute with its substitution. The same
000133 | holds for all external sources and their mixed higher moments. Full
000134 | Z[0] and its background derivatives remain in the same integral.
000135 | 
000136 | The reflected algebra, Theta, original time translations, state and
000137 | local support are transported with the fields. A constant pointwise
000138 | fiber unitary preserves support and the full reflected quadratic form.
000139 | An existing OS null quotient and reconstruction therefore carry to an
000140 | equivalent representation under their original hypotheses. This does
000141 | not identify geometric normal evolution with physical time anew.
000142 | The original effective-action Legendre and regulator procedure is
000143 | likewise transported; bare S, W=log Z and Gamma are not equated.
000144 | 
000145 | For geometry variations, H independence alone does not assert delta_g U=0.
000146 | The displayed fixed product chart has fixed N and gamma, hence constant
000147 | U there. If an actual moving geometric frame gives U_g, the coordinate
000148 | change is made fiberwise with that determined U_g. Its differentiated
000149 | operator contains the commutator connection
000150 |  delta(U_g*D_gU_g)=U_g*[delta D_g+[D_g,Q_g]]U_g,
000151 |  Q_g=(delta U_g)U_g*.
000152 | The same terms occur for the reference, sources and moving domains.
000153 | They are forced coordinate terms, not independently selectable physical
000154 | counterterms. For a point-dependent frame the differential expression
000155 | also carries U_g*partial U_g. One cannot keep the simple fixed-frame
000156 | h formula while discarding these known frame derivatives.
000157 | 
000158 | 4. SMALLEST DECISIVE TEST AT THE CURRENT SPECTRAL CALL
000159 | 
000160 | The actual v258 original definition24-27 is
000161 |  R_Phi=mu_geo Pi_odd^gamma log((I-C)C^-1)
000162 |       =-2ell mu_geo i gamma Phi.
000163 | Its exact inverse retains the given A, ell and mu_geo. This reconstructs
000164 | h and hence the full old pair, as R003 already proves. It does NOT mean
000165 | that applying a spectral function to R_Phi is the full spectral function
000166 | of mathfrakD_h. Inversion of a reader is not equality of these arguments.
000167 | 
000168 | Use the existing original constant-Phi product source and its actual
000169 | quarter angular mode p=n+alpha. No new normal boundary condition is
000170 | needed: compactly supported normal test functions belong to the common
000171 | interior core of the original closed realizations. On that core,
000172 | 
000173 |  mathfrakD_h^2=diag(-partial_r^2+p^2+Phi^2,
000174 |                    -partial_r^2+p^2+Phi^2),
000175 |  R_Phi^2=(2ell mu_geo)^2 Phi^2.                           (2)
000176 | 
000177 | For a fixed compact normal test function rho and internal vector v, the
000178 | whole operator squared form contains
000179 |  ||rho'||^2||v||^2+p^2||rho||^2||v||^2+||rho||^2||Phi v||^2.
000180 | The internal reader squared form has no p^2 or normal derivative term.
000181 | This actual-mode identity is the smallest decisive type test: the two
000182 | operators are not related by the full U coordinate change. Their domains,
000183 | orders and functional calculi have different roles. No new numerical
000184 | model or old infinite heat obstruction is required.
000185 | 
000186 | v259_modular_cutoff_kappa.py:50-114 evaluates scalar integrals of
000187 | f(u)=exp(-u), specifically f0/f2/f4 and ratios. It does not receive D,
000188 | h, Phi, a normal domain, or an operator trace. The original v255 uses
000189 | finite internal traces as coefficients in its stated expansion. Such
000190 | coefficients can correctly enter the ALREADY GIVEN geometric/product
000191 | consumer; they are not an exact standalone evaluation of the complete
000192 | Dirac trace. R044/R045's argument convention, spacetime momentum
000193 | integration and reference cancellations remain required parts of THAT
000194 | existing calculation, not newly missing physical hypotheses here.
000195 | 
000196 | In particular the given4D constant product heat calculation retains the
000197 | spacetime factor/momentum integral multiplying the internal Phi part.
000198 | Dropping that factor is not the U transformation. Conversely, using the
000199 | existing full4D product consumer with its already stated assumptions is
000200 | a valid scoped call and does not need to be reconstructed from zero.