TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PAIR-TYPING.txt
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000001 | THE ACTUAL OLD REFERENCE PAIR IN THE EXISTING BOUNDARY CHART
000002 | L20261102G, 2 November2026. Source derivation and exact operator review.
000003 | Live Atlas at entry a30058cba8bd21f1e5fd77b9; shared plan84.
000004 | Canonical sources unchanged. No new source, lift, Y, field or trace chosen.
000005 | 
000006 | RESULT
000007 | 
000008 | The old bound reference can be transported by the SAME existing E chart.
000009 | For the already established R032 S/T collar lifts its charged generators are
000010 | 
000011 |  h_ref,S=A gamma-i gamma L(m),
000012 |  h_ref,T=A gamma-i gamma[L(p)+D_M],
000013 | 
000014 | where p=(H+H_tau)/2 and m=(H-H_tau)/2 retain the actual old base pullback.
000015 | The whole original coefficient remains Phi=L(H)+D_M and
000016 |  h=A gamma-i gamma Phi.
000017 | No reference is selected independently. These formulas apply to the actual
000018 | lift supplied as input; neither lift is newly declared the primitive choice.
000019 | 
000020 | More strongly, the complete OLD bulk pair is unitarily transported to
000021 |  (mathfrakD_h,mathfrakD_href),
000022 |  mathfrakD_h=[[0,-partial_r+h],[partial_r+h,0]],
000023 | with the SAME transported original full domain. This identifies both members
000024 | of the pair, their reference difference and their full functional calculus.
000025 | It does not substitute a tangential h trace for a bulk Dirac trace.
000026 | 
000027 | The original primitive f(D_rel) caller and the later geometric full-pair
000028 | caller are explicitly different in the originals. The later geometric text
000029 | already fixes D_geo=D_ref+D_rel=D as the local anchor and uses its relative
000030 | PAIR. That is a positive existing clarification of geometric typing. It is
000031 | not an identity rewriting the earlier primitive action or the current
000032 | matter/gauge action without its actual A_Sigma and field map.
000033 | 
000034 | 1. SMALLEST DECISIVE SOURCE AND SYMBOL TEST
000035 | 
000036 | Original boundary001 explicitly defines
000037 |  D_ref=(D+tau D tau^-1)/2,
000038 |  D_rel=(D-tau D tau^-1)/2,
000039 | and its Primitive Euclidean action explicitly reads
000040 |  Tr f(D_rel/chi_seed)-Tr f(D_ref/chi_seed)+i*pi*Delta_eta/2.
000041 | The actual profile-on-odd caller is therefore boundary001:302-316, not an
000042 | inference from the word relative. Exact hash-read source:
000043 |  paper-source:2c00b99902dffffff710a671:291-345,
000044 |  documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000045 |  sections/001-section-primitive-core-and-carrier-interface-handoff.tex,
000046 |  SHA256 7d6e673d99c29d950dc601283cd51c1cc0b113567eeb79c8396fec9c82a02c79.
000047 | The split itself is at122-154 in the same original.
000048 | 
000049 | On the actual R032 same-symbol collar, D_ref=D0+Z Phi_ref and
000050 | D=D0+Z Phi. Thus D-D_ref=Z(Phi-Phi_ref) is bounded order zero, whereas
000051 | D and D_ref have the same first-order principal symbol. This is already
000052 | the decisive operator-order test. A fixed unitary frame cannot turn that
000053 | bounded difference into the whole first-order h or a geometric Dirac.
000054 | No new heat-divergence scan is needed or performed.
000055 | 
000056 | The later original metrology001 explicitly states at15-17
000057 |  D_geo:=D_ref+D_rel=D.
000058 | Its reconstruction theorem says D_rel is a comparison operator, not the
000059 | local principal-symbol anchor. Its displayed relative spectral residues
000060 | at105-128 use
000061 |  Tr_rel(|D_sigma,geo+B_rel|^(-2s)-|D_ref|^(-2s))+i*pi*Delta_eta/2.
000062 | That is the actual full geometric pair caller, with its own scale and
000063 | additional B_rel preserved. Exact original:
000064 |  research-source:20271554929f9198008f7754:1-146,
000065 |  documents/archive--tfpt-45--source_extracts--05_metrology_source/
000066 |  sections/001-subsection-from-bridge-to-geometric-completion.tex,
000067 |  SHA256 f03609effe2abf6629a0057fa024c29fba0c59176072588599bfa1646f48939e.
000068 | 
000069 | The original relative-objects table permits D-D_ref OR a specified pair;
000070 | its geometric APS appendix explicitly says to work with (D,D_ref).
000071 |  paper-source:26e6fa4e9f9895ec3416fd94:1-22,190-209,
000072 |  boundary003-subsection-relative-objects.tex,
000073 |  SHA256 d1136b33055198aedd6d1322912672977374f13f30e65dd073312ed272df0858.
000074 | This licenses the explicit later pair contract in its own layer. It does
000075 | not make Tr f(D-D_ref)-Tr f(D_ref) equal Tr f(D)-Tr f(D_ref).
000076 | The two displayed expressions are not universally interchangeable.
000077 | 
000078 | 2. TRANSPORT THE ACTUAL PAIR, INCLUDING ITS NORMAL SYMBOL
000079 | 
000080 | Reuse the existing fixed product collar coefficients
000081 |  N=[[0,-1],[1,0]], Z=diag(1,-1), S=NZ,
000082 |  D=D0+Z Phi, D0=N partial_r+S A,
000083 |  B=Z A-S Phi, so D=N(partial_r+B).
000084 | Here Phi=Phi*, {gamma,Phi}=0, [A,gamma]=0. The given actual gamma is the
000085 | whole v252 finite grading; Phi retains every original Y and D_M block.
000086 | 
000087 | The already supplied E is
000088 |  Pplus=(I+gamma)/2, Pminus=(I-gamma)/2,
000089 |  E v=(Pplus v,i Pminus v).
000090 | It satisfies E*E=I, B E=E h for h=A gamma-i gamma Phi. It is constant in
000091 | r and theta. Put F=N E and U=(E,F), retaining the full second summand.
000092 | 
000093 | Because {N,Gamma_tot}=0 and Ran(E) is the full positive total grade,
000094 |  E*F=0, F*F=I, EE*+FF*=I.
000095 | Hence U is a unitary between the original doubled complex coefficient
000096 | space and two copies of its ENTIRE original coefficient space.
000097 | Also B N=-N B. Direct action on test sections gives
000098 |  D E v=F(partial_r+h)v,
000099 |  D F w=E(-partial_r+h)w.
000100 | Therefore, on the TRANSPORTED original full domain,
000101 | 
000102 |  U* D U=mathfrakD_h=[[0,-partial_r+h],[partial_r+h,0]].       (1)
000103 | 
000104 | No boundary condition is invented: Dom(mathfrakD_h)=U*Dom(D). If the
000105 | original closed elliptic boundary condition couples the two slots, the
000106 | transported domain keeps that coupling. The notation L*= -partial_r+h
000107 | is a formal expression; its actual adjoint domain is the transported one.
000108 | One must not impose independent APS or periodic normal domains afterward.
000109 | 
000110 | For ANY actual reference having the same collar form
000111 |  D_ref=N(partial_r+B_ref), B_ref=Z A-S Phi_ref,
000112 | with the same supplied normal symbol and compatible original domain, the
000113 | same calculation with the same U gives
000114 | 
000115 |  U* D_ref U=mathfrakD_href,
000116 |  h_ref=A gamma-i gamma Phi_ref.                            (2)
000117 | 
000118 | The same-symbol premise is verified by the R032 references below; it is
000119 | not a theorem for arbitrary tau lifts. An arbitrary even half of D can
000120 | have another principal symbol, in which case (2) is not applicable.
000121 | 
000122 | The old difference is now explicit as well:
000123 |  U*(D-D_ref)U=[[0,delta_h],[delta_h,0]],
000124 |  delta_h=h-h_ref=-i gamma(Phi-Phi_ref).                     (3)
000125 | It has no hidden normal or tangential derivative. Equations(1)-(3) hold
000126 | for smooth bounded variable coefficients without [A,Phi]=0. Product-rule
000127 | derivatives are retained on squaring, rather than put to zero by a
000128 | constant-momentum identity.
000129 | 
000130 | This is functorial for the ACTUAL supplied involution, not just for a
000131 | reference potential written afterward. Define tau^U=U*tau U, retaining
000132 | its base pullback and its transported domain. Then identically
000133 |  U*D_ref U=(mathfrakD_h+tau^U mathfrakD_h (tau^U)^-1)/2.
000134 | For the two supplied collar lifts its fiber action is computable:
000135 |  U* S U=[[0,gamma],[gamma,0]],
000136 |  U*(S gamma)U=[[0,I],[I,0]].
000137 | Thus tauS^U=[[0,gamma],[gamma,0]]R_r and
000138 | tauT^U=[[0,I],[I,0]]R_r. Since R_r partial_r R_r=-partial_r,
000139 | their actual transformed half-operators remain of the form(2), with
000140 |  h_ref,S=(h+gamma h_tau gamma)/2,
000141 |  h_ref,T=(h+h_tau)/2,
000142 | where h_tau=A gamma-i gamma Phi_tau in the R032 fixed-A chart.
000143 | These equations derive (5) directly from the OLD operator half-split.
000144 | An actual geometry-dependent A transport is retained inside h_tau
000145 | instead of assumed fixed outside this chart.
000146 | 
000147 | 3. INSERT THE ACTUAL TWO R032 REFERENCES
000148 | 
000149 | The exact existing proof is
000150 |  research-source:039eb8d38b194a51c6147af4:50-108,
000151 |  ORIGINAL-HIGGS-BOUND-REFERENCE-PROOF.txt,
000152 |  SHA256 13e79f2be052ca918fd2be8faf8cf6f1669fc3ad535ccf9a3fdff5b7c0b1b3c5.
000153 | It keeps the full base pullback r->-r and the actual unchanged Y/MR
000154 | transport. Its two supplied lifts give
000155 | 
000156 |  tauS=S R_r:
000157 |  D_ref,S=D0+Z L(m), D_rel,S=Z[L(p)+D_M];
000158 |  tauT=S gamma R_r:
000159 |  D_ref,T=D0+Z[L(p)+D_M], D_rel,T=Z L(m).                    (4)
000160 | 
000161 | Consequently (2) produces, with no new reference prescription,
000162 |  h_ref,S=A gamma-i gamma L(m),
000163 |  h_ref,T=A gamma-i gamma[L(p)+D_M].                        (5)
000164 | The whole reference jets follow from real linearity of L and the actual
000165 | H_tau variation. In the fixed A/tau field chart,
000166 |  delta h_ref,S=-i gamma L(delta m),
000167 |  delta h_ref,T=-i gamma L(delta p).
000168 | There is no extra Majorana variation when its original D_M is fixed.
000169 | Moving geometry, grading, pullback or Y/MR transport contributes its
000170 | actual derivative instead of being suppressed by (5).
000171 | 
000172 | For the existing normal-independent product source H_tau=H, so m=0:
000173 |  h_ref,S=A gamma,
000174 |  h_ref,T=h.
000175 | In particular branch S removes D_M from its reference, exactly as (4)
000176 | says. Branch T retains it and has a vanishing whole unscaled pair in
000177 | this even-H chart. Neither outcome is a new physical selection rule.
000178 | At the seam of a general smooth reflected field m|Sigma=0, but that
000179 | boundary equality must not erase m off the seam in the normal transfer.
000180 | 
000181 | 4. THE WHOLE PAIR FUNCTIONAL, SCALE, TRACE AND CHARGE
000182 | 
000183 | Unitary functional calculus transports the actual pair, not just a
000184 | matching spectrum:
000185 |  U*[f(D/chi)-f(D_ref/chi)]U
000186 |    =f(mathfrakD_h/chi)-f(mathfrakD_href/chi),               (6)
000187 | on its genuine functional-calculus/relative-trace domain. The same
000188 | statement holds for bounded regularized functions and for justified
000189 | relative zeta/heat functions with their actual trace prescription.
000190 | Both full normal slots remain. No factor of two, chirality division or
000191 | extra independent species has been inserted.
000192 | 
000193 | The geometric original has D_sigma=e^(-sigma/2)D e^(-sigma/2) and an
000194 | UNSCALED reference in its displayed subtraction. Since U is internal
000195 | and constant, its exact transported numerator is
000196 |  e^(-sigma/2)mathfrakD_h e^(-sigma/2)+U*B_rel U,
000197 | and its reference remains mathfrakD_href. The original chi_geo, chi_seed,
000198 | and the modular scale below are distinct until their actual source map
000199 | equates them. The original paired trace, eta, zero modes and boundary
000200 | domains are transported. They are not replaced by a determinant of h.