TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/CONFORMAL-TRACE-AND-WICK.txt
Original file SHA-256: 77a6bffa2192b809f0264406afe7dd57d4718576f7f6632c107a7036de6ff5d1
Inclusive original lines: 1-183
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000001 | UR.NATIVE.ARCCHARGE.CONFORMAL-TRACE.01
000002 | 26 October 2026. Conformal Cauchy data and normalized boundary words.
000003 | 
000004 | 1. Question, original inputs, and decisive test
000005 | The preceding continuation retained the free chiral CAR quantization of the
000006 | actual same-E midpoint annulus. Does averaging its conformal metric factor
000007 | necessarily produce a non-Wick four-point function? Test the actual trace
000008 | map BEFORE assuming that a fluctuating metric means a fluctuating covariance.
000009 | Success: derive the boundary Hilbert-space identification from the metric
000010 | and the original massless Dirac equation, including both labelled faces.
000011 | Abort equality if the transported Cauchy subspace or its orthogonal projector
000012 | changes. No new geometry, mass, connection, state rule or time is selected.
000013 | 
000014 | Original inputs retained:
000015 |  - BOUNDARY-SOURCE.txt, elliptic-moebius-source-bridge-20261026, Sections1/4:
000016 |    the same square conformal curve, annulus0<=t<=1/2, fixed spin and marks,
000017 |    L=partial_t-i partial_s, and its closed Cauchy graph on both boundary faces.
000018 |  - ELLIPTIC-GREEN-RP.txt in that folder: free chiral CAR functor is EXPLICIT.
000019 |  - ELLIPTIC-CURRENT-SOURCE.txt in this folder: quarter covariance and current
000020 |    functional in that state, without replacing it by the old Hardy vacuum.
000021 |  - The original v45 constructive geometric measure integrates geometric
000022 |    data; source_extracts--05_metrology_source/sections/005-section-absolute-
000023 |    dimensionless-metrology-from-the-bounda.tex, lines136-178. Its general
000024 |    support is NOT proved to be this conformal family here.
000025 |  - Original v45 relative generating functional divides by Z_ref[0],
000026 |    source_extracts--04_qft_source/sections/008-section-relative-reflection-
000027 |    positivity-with-fermions.tex, lines139-169. No primitive Gaussian measure
000028 |    is inferred from this source-independent denominator or from RP alone.
000029 | 
000030 | 2. General exact conformal trace theorem
000031 | Let M be a smooth compact spin manifold of bulk dimension d with smooth
000032 | boundary, fixed spin structure, and g_sigma=exp(2sigma)g for smooth real
000033 | sigma. Use the canonical conformal spin-bundle identification. With this
000034 | identification the physical massless spin Dirac operator satisfies
000035 |  D_sigma=exp(-(d+1)sigma/2) D exp((d-1)sigma/2).             (1)
000036 | This is the usual conformal Dirac law. A fixed twisted connection is allowed
000037 | when it obeys this law; varying a connection or adding a mass is NOT covered.
000038 | Smooth harmonic spinors therefore correspond bijectively by
000039 |  psi_sigma=exp(-(d-1)sigma/2) psi_0.                       (2)
000040 | The induced boundary density is
000041 |  dSigma_sigma=exp((d-1)sigma_boundary)dSigma_0.
000042 | Consequently the geometrically specified boundary L2 unitary is
000043 |  V_sigma: H_boundary,sigma -> H_boundary,0,
000044 |  V_sigma f=exp((d-1)sigma_boundary/2) f.                  (3)
000045 | It is the positive scalar density correction for the canonical spin lift;
000046 | no additional phase or fitted field redefinition is introduced. In particular
000047 |  V_sigma Tr psi_sigma=Tr psi_0.                          (4)
000048 | Thus V_sigma sends the smooth Cauchy-data spaces EXACTLY onto each other.
000049 | Since it is unitary, the same holds for their L2 closures. Uniqueness of
000050 | orthogonal projection onto a closed subspace proves
000051 |  V_sigma P_sigma V_sigma^*=P_0.                          (5)
000052 | Here P means the ORTHOGONAL Calderon projector. This proof is about its
000053 | range, not an arbitrarily chosen nonorthogonal Seeley projection.
000054 | Positive smooth multiplication and its inverse preserve the relevant
000055 | Sobolev spaces on compact M. The smooth harmonic-space argument and the
000056 | unitary closure avoid extrapolating a finite Fourier regulator to(5).
000057 | No product collar is required after the conformal change.
000058 | Chirality is preserved, so the same argument applies to the original
000059 | chiral Cauchy space. No APS or other self-adjoint boundary condition is
000060 | imposed before computing that space.
000061 | 
000062 | 3. Correct trace in the common bulk Hilbert space
000063 | The bulk L2 unitary is U_sigma=exp(d sigma/2). Accordingly
000064 |  D_tilde_sigma=U_sigma D_sigma U_sigma^*
000065 |               =exp(-sigma/2) D exp(-sigma/2).             (6)
000066 | For u_sigma=U_sigma psi_sigma, equation(2) becomes
000067 |  u_sigma=exp(sigma/2)u_0.
000068 | The actual fixed-boundary-space trace is NOT the unweighted raw trace:
000069 |  T_sigma=V_sigma Tr U_sigma^*=exp(-sigma_boundary/2)Tr.   (7)
000070 | Therefore T_sigma u_sigma=Tr u_0 on both faces. This is the missing
000071 | normalization that an unweighted common-bulk-space restriction would lose.
000072 | In particular, different positive scale factors on the two faces would
000073 | change the raw graph slope and its raw orthogonal compression. Equation(7)
000074 | compensates those factors exactly. This is not evidence for a new coupling.
000075 | For arbitrary sigma(t,s), the original NS mode
000076 |  u_n=exp(-lambda_n t)exp(i lambda_n s), lambda_n=2pi(n+1/2),
000077 | satisfies L u_n=0 and
000078 |  exp(-sigma/2)L[exp(-sigma/2)exp(sigma/2)u_n]=0.
000079 | Its two weighted traces are the original two traces, without any assumed
000080 | normal homogeneity or tangential mode decoupling of sigma.
000081 | 
000082 | 4. Actual labelled marginal and its quarter/current consequences
000083 | V_sigma is componentwise multiplication on the same labelled faces. It
000084 | therefore intertwines each face projection, as well as each regional support
000085 | projection. Compress(5) to the ORIGINAL second face:
000086 |  V_sigma,1 C_sigma V_sigma,1^*=C_0.                      (8)
000087 | The fixed flat-representative covariance is still
000088 |  C_0=(1+exp(2W B_Sigma))^-1, W=1/2,
000089 |      =(1+exp(2pi B_cov))^-1,
000090 | and still c_r=(1+exp(4pi(r+alpha)))^-1 for alpha=1/4,3/4
000091 | after the existing NS-cover/quarter map. This is equality under the derived
000092 | boundary unitary, not equality of coordinate kernels with different measures.
000093 | The old quarter/fold map is applied AFTER that transport. V_sigma need not
000094 | commute with old Fourier or quarter projectors; their physical images must
000095 | also be transported. No new quarter sectors of a variable intrinsic boundary
000096 | Dirac operator are silently substituted for the original ones.
000097 | The covariance equality transports the bounded CAR algebra, its star
000098 | operation, supports and the entire given quasifree state. All bounded CAR
000099 | words, and the current Weyl functional already constructed, agree exactly.
000100 | Native point-split composites agree when defined as images of the OLD
000101 | operator distributions with the OLD subtraction, domains and smearing.
000102 | The canonical local stress tensor defined anew from each metric can carry
000103 | a conformal anomaly; it is not equated to that fixed transported composite.
000104 | For an unchanged original reflection or clock symmetry, restrict sigma to
000105 | profiles invariant under it; otherwise transport the geometric symmetry
000106 | too. Neither convention proves that original P1 admits those variations.
000107 | 
000108 | 5. Averaging conformal geometries: exact all-order factorization
000109 | Retain the existing explicit FREE conditional CAR source from Sections1/4.
000110 | For every sigma, transport sources/test spinors by the dual of(3), so they
000111 | represent the same bounded CAR insertions. The normalized conditional
000112 | Grassmann functional is the SAME formal expression
000113 |  Z_psi,sigma[eta]/Z_psi,sigma[0]=exp(bar_eta C_0 eta).     (9)
000114 | Equivalently, every conditional word has the same expectation by(8).
000115 | Let w(sigma) include all geometric weights, the source-free fermion
000116 | determinant, reference effects, Jacobians and anomalies. If the integrals
000117 | exist and their denominator is nonzero, source independence gives
000118 |  [integral w(sigma)exp(bar_eta C_0 eta)]/[integral w(sigma)]
000119 |         =exp(bar_eta C_0 eta).                          (10)
000120 | No specific probability law for sigma, saddle approximation or neglect of
000121 | determinant factors is needed. Even a complex weight factors algebraically
000122 | under these existence conditions. Reflection positivity is not being used
000123 | as a replacement for them.
000124 | In particular, for the grouped four-field word with the previous convention,
000125 |  <a_i^* a_k^* a_l a_j>=C_ij C_kl-C_il C_kj,
000126 | so its connected fermionic four-point defect stays ZERO. All higher Wick
000127 | words also remain fixed. Connected correlations of bilinear currents need
000128 | not vanish; those are different observables from connected fermion four-point.
000129 | For the ORIGINAL relative functional use
000130 |  Zhat_rel[J]=Z_rel[J]/Z_rel[0]. A source-independent Z_ref[0] cancels
000131 | in THIS normalization. It can still enter the effective geometry weight
000132 | and its metric derivative; neither is asserted to be constant.
000133 | If other geometric data xi vary as well, the same argument conditionally
000134 | integrates out sigma at each fixed xi. The remaining effective weight may
000135 | change the distribution of C_xi, and hence its non-Wick correlations.
000136 | This corollary does not assert such a disintegration exists for full P1.
000137 | 
000138 | 6. What is simplified, and what is not identified
000139 | The boundary Cauchy covariance only retains the conformal class in this
000140 | massless fixed-spin/fixed-connection setting. It cannot reconstruct the
000141 | Weyl factor. On the fixed complex curve E, every Hermitian Riemannian metric
000142 | is conformal to its fixed representative: at each point a J-compatible
000143 | positive form on a complex line has only one positive scalar coefficient.
000144 | Thus this covers all such compatible metrics on the same marked curve,
000145 | not just a specially chosen normal profile. Arbitrary metrics that change
000146 | its complex structure, moving cuts or varying spin/connection are distinct.
000147 | Bulk Dirac spectra, bulk resolvents, spectral actions, geometric
000148 | forces and proper times can change along the same family. Existing nonzero
000149 | conformal Schur/sea responses concern those other objects and are preserved.
000150 | The constant-sigma identity D_tilde_sigma=exp(-sigma)D already shows why
000151 | spectrum invariance would be false while(5) is true.
000152 | The current result applies to the already given smooth midpoint annulus.
000153 | It does not construct the unresolved closed real trace on the full ramified
000154 | marked equator graph, identify its carrier with original P1, or select its
000155 | spin/domain, clock, physical time, source measure or admissible algebra.
000156 | It does not derive conditional free fermion quantization from S_prim.
000157 | It excludes an automatic interacting four-point correction caused SOLELY
000158 | by averaging this conformal factor with the correct trace and free premise.
000159 | The next four-point comparison must isolate an original variation that
000160 | actually changes the transported covariance, or an original higher fermion
000161 | vertex. Geometry variance alone is not that criterion.
000162 | 
000163 | 7. Validation and literature
000164 | The checker proves the density exponents and differential product rules
000165 | symbolically, uses the actual NS harmonic mode, and verifies a two-mode
000166 | closed-graph projector exactly with independent face scale factors. The
000167 | wrong raw-trace compression is a strict nonzero countercontrol for all
000168 | positive mode slopes when the face factors differ. This finite check is
000169 | diagnostic; Sections2-4 prove the full closed-space statement.
000170 | It also checks the general symbolic four-field and normalized source-weight
000171 | factorizations, and detects a nonzero defect for an explicitly changing
000172 | covariance. That control is an algebraic check, not a proposed physical model.
000173 | Primary source: Guillarmou/Moroianu/Park, Bergman and Calderon projectors
000174 | for Dirac operators, arxiv:1009.3179, Section3.1 (conformal Dirac law,
000175 | orthogonal closed Cauchy space and Theorem11). Proposition7 and the paragraph
000176 | after(12) independently give the boundary conformal conjugation law.
000177 |  https://arxiv.org/abs/1009.3179
000178 |  https://www.imar.ro/~sergium/fisiere/calderon.pdf
000179 | The averaging conclusion(10) is the present elementary consequence under
000180 | the retained free conditional premise, not a claimed quotation of that paper.
000181 | Verdict: EXACT_SCOPED_CONFORMAL_CAUCHY_INVARIANCE;
000182 | CONFORMAL_AVERAGING_PRESERVES_GIVEN_FREE_BOUNDARY_WICK_WORDS;
000183 | ORIGINAL_P1_SOURCE_AND_INTERACTION_IDENTIFICATION_OPEN. No ledger promotion.