TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/PRIMITIVE-GEOMETRIC-SELECTION.txt
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000001 | PRIMITIVE GEOMETRIC SELECTION: COUPLED VARIATION AND A SHARP DATA BOUNDARY
000002 | 26 October 2026. Continuation of GEOMETRIC-SOURCE-RETURN.txt, section5.
000003 | Experiment-only mathematical source review. Overall contract stays PARTIAL.
000004 | No new original ledger claim or physical gate is promoted.
000005 | 
000006 | 1. Source contract and the precise question
000007 | 
000008 | The v31 primitive section, lines50-92, and v42 primitive section, lines66-72,
000009 | define on a common declared Hilbert space and reflection domain
000010 |  A=D_ref=(D+RDR)/2, B=D_rel=(D-RDR)/2, R=R*=R^-1.
000011 | The retained v45 boundary-kernel source has the same pair at lines122-129
000012 | and S=Tr F(B/chi)-Tr F(A/chi)+(i*pi/2)DeltaEta at lines304-320.
000013 | It treats a Bernstein representation as an analytic realization of an already
000014 | fixed transfer channel, not as an independent origin of admissibility.
000015 | This last qualification is ESSENTIAL to the conclusion below.
000016 | 
000017 | The previous Pin geometric machine supplies a comparison D_s, not a proved
000018 | identification D_s=B of this pair. Its Pin deck, normal projector, original
000019 | geometric involution R and Cauchy polarization are not equated here.
000020 | Question: what information fixes the geometry response of the original pair,
000021 | and can endpoint transfer values plus positivity alone determine that response?
000022 | 
000023 | 2. Exact coupled conformal derivative on a fixed reflection domain
000024 | 
000025 | Assume common transported domains, fixed R, and a conformal Dirac variation
000026 |  D_s=exp(-s h/2) D exp(-s h/2), h=h*, deltaD=-{h,D}/2.
000027 | This is the standard conformal covariance formula after unitary L2 density
000028 | transport; Ginoux, The Dirac Spectrum, Proposition1.3.10.
000029 | Set h_+=(h+RhR)/2 and h_-=(h-RhR)/2. Expanding the definitions, without
000030 | any assumption that A and B commute, gives
000031 |  deltaA=-({h_+,A}+{h_-,B})/2,
000032 |  deltaB=-({h_+,B}+{h_-,A})/2.                         (1)
000033 | Even profiles change both pieces. Odd profiles mix them. Formula(1) is
000034 | kinematic: odd profiles need not belong to the actual doubled geometry.
000035 | If B=0 and h is even, deltaB=0; an even conformal probe cannot create an
000036 | odd relative component from zero. This is not a conclusion about every P1
000037 | source, whose original D and R must first be identified.
000038 | 
000039 | Let E_X(t)=exp(-t X^2/chi^2), F(x)=integral exp(-t x^2)dmu(t).
000040 | Assume finite dimension or an ACTUAL valid differentiable relative trace
000041 | contract. Heat positivity/finite moments alone do not supply that contract.
000042 | For fixed chi, cyclic trace differentiation gives
000043 |  delta S_real=-(2/chi^2) integral t Tr[B E_B deltaB-A E_A deltaA]dmu.
000044 | Inserting(1) yields
000045 |  delta_+ S_real=(2/chi^2) integral t Tr h_+[B^2 E_B-A^2 E_A]dmu,
000046 |  delta_- S_real=(1/chi^2) integral t Tr h_-[{A,B E_B}-{B,A E_A}]dmu. (2)
000047 | The ordered products in the odd term must be retained.
000048 | For kappa=delta chi/chi add
000049 |  (2*kappa/chi^2) integral t Tr[B^2 E_B-A^2 E_A]dmu.     (3)
000050 | For a varying representation measure add
000051 |  integral Tr[E_B(t)-E_A(t)]delta mu(dt).               (4)
000052 | Retain (i*pi/2)delta DeltaEta wherever differentiable, and actual spectral-flow
000053 | jumps where appropriate. If R itself varies, (1) gains the terms
000054 |  +[deltaR D R+R D deltaR]/2 in deltaA and their negatives in deltaB.
000055 | Moving selfadjoint domains require their actual transported family. No new
000056 | physical metric law follows merely from these identities.
000057 | 
000058 | The original QFT source lines144-169 writes Z_rel[J]=Z_adm[J]/Z_ref[0].
000059 | Its denominator is independent of INSERTION SOURCES. That assertion does not
000060 | make it independent of geometric data. Thus at fixed J
000061 |  delta_g log Z_rel=delta_g log Z_adm[J]-delta_g log Z_ref[0].
000062 | Dropping the reference response solely because its source argument is0 is
000063 | invalid. A genuinely fixed reference along a specified family is a different
000064 | contract. Quantum stationarity still concerns the complete original measure
000065 | or effective action, not automatically the bare spectral seed alone.
000066 | 
000067 | 3. Exact two-measure counterexample in the stated algebraic formula
000068 | 
000069 | Choose on C2
000070 |  R=diag(1,-1), A=2I, B=sigma_x, D=A+B.
000071 | These obey the original even/odd split exactly. Let chi=1 and scale the
000072 | ENTIRE operator D(s)=exp(-s)D, retaining both derived pieces and a fixed eta
000073 | phase. This finite algebraic response test is not a claimed P1 geometry.
000074 | Take q_j=(1/2,2/3,3/4,4/5), t_j=-log q_j, and
000075 |  F_w(x)=sum_j w_j exp(-t_j x^2).
000076 | Every positive w defines an even positive heat profile in the displayed
000077 | positive-measure class; no assertion is made that every such w represents
000078 | the same originally FIXED channel on all geometries.
000079 | 
000080 | The exact rational vector
000081 |  v=(-11809/112368,91611/149824,-1,221875/449472)
000082 | satisfies sum v_j=sum v_j q_j=sum v_j q_j^4=0.
000083 | Define c_j=t_j(q_j-4q_j^4), r=-sum_(j>1)c_j/c_1>0,
000084 |  w0=(r,1,1,1)/(r+3), delta=(1/2)min_j w0_j,
000085 |  w_plus=w0+delta v, w_minus=w0-delta v.
000086 | Since |v_j|<=1, both measures are strictly positive and normalized.
000087 | Their values at x=0,1,2 coincide EXACTLY. Therefore
000088 |  F_plus(A)=F_minus(A), F_plus(B)=F_minus(B), S_plus(0)=S_minus(0).
000089 | The entire given finite pair response agrees, not only its scalar trace.
000090 | 
000091 | But S_w(s)=2[F_w(exp(-s))-F_w(2exp(-s))] and
000092 |  S_w'(0)=4 sum_j w_j c_j.
000093 | The center w0 has zero derivative and
000094 |  c dot v = (285893/149824)log2-(88563/149824)log3
000095 |            -(46505/112368)log5
000096 |          = .0071660522580729957712998046167... >0.     (5)
000097 | The rational independence of prime logarithms proves it is not zero:
000098 | clearing denominators and exponentiating a zero value would contradict
000099 | unique factorization. A separate outward interval evaluation certifies the
000100 | displayed sign. Hence the two derivatives are exactly nonzero opposites.
000101 | 
000102 | Numbers shown only for convenient inspection:
000103 |  S_plus(0)=S_minus(0)=.8675847696814872,
000104 |  S_plus'(0)=+.0027432473040925574,
000105 |  S_minus'(0)=-.0027432473040925574.
000106 | The endpoint operators alone cannot determine a unique geometry force.
000107 | A deterministic recursion using only their identical values and identical
000108 | initial data keeps identical iterates by induction. It cannot reconstruct
000109 | which of the opposite derivatives is the original one. Querying the actual
000110 | deformed source would add information; it would no longer be endpoint-only.
000111 | 
000112 | 4. General finite all-profile stationarity theorem
000113 | 
000114 | Assume a finite selfadjoint pair, fixed chi and a permitted common scale path.
000115 | If the real spectral action is stationary at one scale for EVERY positive
000116 | atomic heat measure, then for every t>0
000117 |  sum_(lambda in spec B) lambda^2 exp(-t lambda^2/chi^2)
000118 |  =sum_(lambda in spec A) lambda^2 exp(-t lambda^2/chi^2).
000119 | Group distinct positive squared eigenvalues x_1<...<x_n. Independence of the
000120 | finite functions exp(-t x_j) forces equal multiplicities at every x_j: divide
000121 | by exp(-t x_1) and send t to infinity, then iterate. Zero multiplicities are
000122 | invisible to this derivative. Thus the action is constant along the ENTIRE
000123 | scale orbit, equal only to the zero-multiplicity difference times F(0).
000124 | There is no isolated selected scale from this universal stationarity demand.
000125 | This theorem does not apply if chi is tied to scale, scale is disallowed,
000126 | the quantum measure supplies other terms, or the actual channel fixes F.
000127 | 
000128 | 5. Minimal extra response data; preserve the already fixed scalar transfer
000129 | 
000130 | For a finite Hermitian X, the Frechet derivative of F(X) in an eigenbasis is
000131 |  (dF_X[V])_ij=F[lambda_i,lambda_j] V_ij.
000132 | Offdiagonal divided differences use existing values for distinct eigenvalues;
000133 | the repeated-eigenvalue value is F'(lambda). Therefore the trace force needs
000134 | F' on the current spectrum and the ACTUAL insertion deltaX. It does not need
000135 | an invented new learning rate or necessarily an entire new spectral measure.
000136 | Endpoint values do not determine these derivative values, as section3 proves.
000137 | Full history requires this response along its actual family.
000138 | 
000139 | In particular the original directed Fredholm/transfer contract already fixes
000140 |  T(q)=(1-q)^(-5/4), delta log T=(5/4)delta q/(1-q).
000141 | Under the retained historical weight q(alpha)=delta_top exp(-2alpha),
000142 |  partial_alpha log T=-(5/2)q/(1-q).
000143 | These derivatives are determined and are NOT reopened by the counterexample.
000144 | The missing geometric bridge is the original delta g -> delta q insertion,
000145 | with retained residual sector ratio, reference/measure terms and eta/domain
000146 | data. A full fixed transfer functional on a neighborhood fixes its derivatives;
000147 | arbitrary independent heat profiles cannot then be declared physical choices.
000148 | The actual local native field/word map remains necessary for that full source.
000149 | 
000150 | 6. Evidence and scope
000151 | 
000152 | geometric_source_return/check_primitive_selection.py contains the exact
000153 | rational nullspace, symbolic parity check, prime-log certificate, outward
000154 | interval sign and90-digit independent matrix-action derivative. The latter
000155 | includes a NONCOMMUTING pair, odd h and moving chi; error3.1e-92.
000156 | Normal and python -OO output bytes agree. Results are in
000157 | geometric_source_return/primitive_selection.json.
000158 | No fitted histories, chosen force law or new autonomous geometry model is
000159 | introduced. The original geometric machine and its derived full memory are
000160 | retained. This excludes an inference from finite endpoint data and heat
000161 | positivity; it neither excludes the full TFPT source nor completes it.
000162 | 
000163 | Original source paths:
000164 |  documents/archive--paper-latex--tfpt-31/sections/002-subsection-primitive-relative-spectral-dynamics.tex
000165 |  documents/archive--paper-latex--tfpt-42/sections/003-subsection-primitive-relative-spectral-dynamics.tex
000166 |  documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/001-section-primitive-core-and-carrier-interface-handoff.tex
000167 |  documents/archive--tfpt-45--source_extracts--04_qft_source/sections/008-section-relative-reflection-positivity-with-fermions.tex
000168 | Existing source-reviewed retained transfer/measure contracts:
000169 |  DIRECT-DEFECT-CLOSURE.txt, REWEIGHTING-REVIEW.txt,
000170 |  EM-MEASURE-AND-QUILLEN.txt.