TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/APS-DOMAIN-AND-STATE.txt Original file SHA-256: 540f4d191ff1a1f0f46397d55bd72b32dc6a9cf3639530ad16a7e6414bf51e06 Inclusive original lines: 1-194 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 | [OMITTED: non-scientific conversation or execution metadata] 000002 | [OMITTED: non-scientific conversation or execution metadata] 000003 | [OMITTED: non-scientific conversation or execution metadata] 000004 | [OMITTED: non-scientific conversation or execution metadata] 000005 | 000006 | Verdict 000007 | The proposed opposite endpoint APS signs, adjoint-domain transport and inverse 000008 | are correct for the GIVEN constant, gapped, full product operator. They give a 000009 | valid closed skew realization of the existing L=-N D pairing. The cited original 000010 | theorem does not select the operational Seed's state or physical radial time. 000011 | It explicitly chooses a provisional geometric realization. Its physical source 000012 | identification is a separate comparison, not permission to choose a new state 000013 | for an already complete Seed. 000014 | 000015 | 1. Actual source scope, including its later qualification 000016 | All source paths in this section are relative to documents/. 000017 | 000018 | archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/ 000019 | 001-section-primitive-core-and-carrier-interface-handoff.tex:353-377: 000020 | finite compact cutoff; formally selfadjoint geometric Dirac anchor; 000021 | adapted boundary operator A_partial; CHOOSE a selfadjoint elliptic modified 000022 | APS boundary condition; conclude selfadjoint Fredholm realization/discrete 000023 | spectrum. The statement contains no CAR field, state omega, Schwinger family 000024 | or identification of a radial coordinate with physical time. 000025 | Registered source paper-source:8027564739689dd02e02f11c:353-377 was read. 000026 | 000027 | The continuation in 002-section-intrinsic-closed-branch-and-canonical-rigid- 000028 | obje.tex:33-42 explicitly calls APSdomprov a provisional analytic starting 000029 | representative and refers to a later Calderon upgrade. The symbol guide 000030 | 003-subsection-relative-objects.tex:121-122 classifies APSdomprov as an analytic 000031 | choice and distinguishes the later closed-branch domain APSdom. 000032 | 000033 | The actual upgrade is in archive--tfpt-45--source_extracts--04_qft_source/ 000034 | sections/002-section-closure-theorems-from-the-admissibility-complex.tex:167-198. 000035 | It asserts a canonical Calderon-compatible domain in the same admissible 000036 | elliptic class. It does not write the full quantum source or identify its state 000037 | with the particular APS vacuum kernel below. No such extra conclusion follows 000038 | just from membership in an elliptic class. 000039 | 000040 | In contrast the original operational Seed definition, 01/sections/002:103-119, 000041 | already includes (A_loc,tau_t,Theta,omega,[u_Sigma],D_coll). Its omega is RP and 000042 | clustering. These existing data must be compared with the constructed source. 000043 | The earlier Atlas finding original-operational-seed-and-given-green-pairing 000044 | and its registered PROOF.txt:449-480 were reread and are reused here. 000045 | 000046 | 2. Endpoint signs and closed adjoint identity, actually established 000047 | Keep the existing normal-independent coupled operator 000048 | D=N(partial_r+B), L=-N D=partial_r+B, 0<=r<=ell, 000049 | B*=B, C0 B C0=-B, C0^2=I, gap |B|>=delta>0. 000050 | All C0 statements use the existing NEW complex structure. At the original 000051 | scalar quarters, bounded constant odd Phi gives B^2=A^2+Phi^2, hence delta>=1/4 000052 | in A=-i partial_theta+alpha units. No new normal datum is introduced. 000053 | 000054 | For inward normals, the product-adapted operators are B at r=0 and -B at r=ell. 000055 | Choosing these actual adapted operators and negative APS boundary spaces gives 000056 | Dom L: x(0) in H_-(B), x(ell) in H_+(B), with H1 regularity. 000057 | The Hilbert adjoint has expression L*=-partial_r+B and 000058 | Dom L*: y(0) in H_+(B), y(ell) in H_-(B). 000059 | C0 flips the spectral signs and preserves the relevant regularity. Therefore 000060 | C0 Dom L=Dom L*, C0 L C0=-L* 000061 | as closed operators. Each endpoint separately has =0 for x,y in the 000062 | same allowed spectral half. Thus the already proved oriented boundary term 000063 | beta_L(x,y)+beta_L(y,x)=[]_0^ell 000064 | vanishes on this realization. This is a constructive global completion of 000065 | the previous interior pairing for this explicit chosen endpoint domain. 000066 | 000067 | The qualification 'choosing these actual adapted operators' matters: an 000068 | adapted boundary operator is fixed principally; admissible zero-order choices 000069 | are not excluded merely by its name. Standard primary reference: Baer and 000070 | Ballmann, Guide to elliptic boundary value problems for Dirac-type operators, 000071 | https://arxiv.org/pdf/1307.3021 , sections3.2-3.3, theorem4.11, example5.12. 000072 | This reference supports the geometric boundary statement, not state selection. 000073 | 000074 | 3. Exact inverse and the smallest state comparison 000075 | For this constant B the claimed inverse is correct: 000076 | G_APS(r,s)=1_(r>s) exp(-(r-s)B)P_+ 000077 | -1_(s>r) exp(-(r-s)B)P_-. 000078 | Away from r=s it solves L G=0; its jump is P_++P_-=I. Its r=0 range is H_- 000079 | and its r=ell range is H_+. A homogeneous solution obeying both endpoint 000080 | conditions is zero. The formula is therefore the inverse, not just a formal 000081 | parametrix. Spectral calculus gives bounded decaying exponentials on each 000082 | indicated half. The kernel's Hilbert operator norm is <=1 pointwise; a safe 000083 | L2 inverse bound is ||L^-1||<=delta^-1. An unqualified inverse norm <=1 does 000084 | not follow from the quarter gap. Generic Fredholmness alone is not invertibility. 000085 | 000086 | Given charged CAR quantization, this is the time-ordered vacuum propagator 000087 | with sea P_-(h), and the usual particle/hole Fock construction has excitation 000088 | Hamiltonian sum |lambda| n_exc >=0. This is a correct source realization for 000089 | the given h. The analytic realization theorem alone does not identify its 000090 | field operators, state or radial translations with the original Seed. 000091 | 000092 | It also does not replace the already established whole finite-collar graph. 000093 | Its labeled face occupation is C(h)=(1+exp(2ell h))^-1, with nonzero cross 000094 | X=(2cosh(ell h))^-1. On any positive eigenmode lambda, the APS vacuum has 000095 | occupation0, whereas C(lambda)>0 at finite ell. These are distinguishable 000096 | normalized source data. To call one the original omega requires the actual 000097 | source map and its correlations, rather than reinterpretation of a boundary 000098 | realization theorem. The given graph, HS source and their proven circle time 000099 | dictionary remain valid; no existing completed source is discarded here. 000100 | 000101 | 4. Original insertion versus a background-dependent endpoint domain 000102 | On the actual constant odd Yukawa insertion, at Phi=0, 000103 | B0=Z A, delta B=-S delta Phi, {B0,delta B}=0, 000104 | delta P_-(B)|_0=-delta B/(2|A|)=S delta Phi/(2|A|). 000105 | For delta Phi nonzero this changes the APS trace subspace. Consequently the 000106 | family APS(B_Phi) is not a single fixed boundary domain for all original 000107 | insertions. If this family is intended, its domain transport and boundary 000108 | variation terms must accompany the derivative of its determinant/source. 000109 | If instead the original fixed APS(B0) domain is retained, the actual bounded 000110 | zero-order Yukawa potential defines a valid same-domain perturbation, but the 000111 | simple G_APS(B_Phi) above is not thereby its Green function. These are two 000112 | specific existing-domain contracts, not an invitation to select a new source. 000113 | 000114 | The broad first-order relative-bound-zero assertion in the original theorem 000115 | was already excluded in Atlas findings generic-first-order-relative-bound- 000116 | zero-counterexample-and-correct-domain-repair and order-one-infinitesimal- 000117 | bound-inference-fails. It is not a new finding here. The actual bounded 000118 | zero-order constant Phi used above satisfies the needed perturbation theorem. 000119 | 000120 | 5. Exact actual odd-block witness, independently checked analytically 000121 | On the charged two-grade odd singular block, keep the quarter p=1/4 and 000122 | insertion m=1/3: h=p sigma_z+m sigma_y, E=5/12. Then 000123 | P_h=P_-(h)=[[1/5,2i/5],[-2i/5,4/5]], P0=diag(0,1). 000124 | The proposed moving-domain Green function at r=0 violates the fixed old 000125 | boundary condition by the strictly nonzero amount 000126 | ||(I-P0)G_h(0,s)||_HS^2=exp(-5s/6)/5, 00, 000147 | P=P_-(h0), R_t=exp(-t h). 000148 | The endpoint conditions remain x(0) in Ran P, P x(ell)=0. Variation of 000149 | constants gives exactly 000150 | M=P R_ell P|RanP, 000151 | G_fixed(r,s)=1_(r>s)R_(r-s)-R_r P M^-1 P R_(ell-s). 000152 | Here 000153 | M=[cosh(ell E)+|p| E^-1 sinh(ell E)]|RanP >=I, 000154 | since E commutes with P, P V P=0 and P h0 P=-|p|P. Thus M is invertible. 000155 | The correction enforces both fixed endpoint conditions, while its derivative 000156 | does not change the jump I. This proves the actual fixed-domain inverse. 000157 | 000158 | There is also a direct all-mode estimate. On this fixed domain, 000159 | ||Lx||^2=||partial_r x||^2+||h x||^2+[]_0^ell 000160 | >=p^2||x||^2, 000161 | because the bracket equals |p|(||x(0)||^2+||x(ell)||^2). The V endpoint 000162 | expectations vanish on each allowed h0 half. For L*=-partial_r+h on the 000163 | complementary endpoint domain the cross term has the opposite sign, and 000164 | its boundary values give the same positive contribution. Thus both closed 000165 | operators have lower bound |p|. Closed range and ker L*=0 prove surjectivity. 000166 | 000167 | Taking the direct sum over ALL p=n+alpha gives 000168 | ||L_fixed^-1||<=1/min_n|n+alpha|=4 000169 | at both original quarters. This is a full circle estimate for the stated 000170 | constant finite internal coefficient, not extrapolation of a finite cutoff. 000171 | The kernel formula is interpreted modewise and then as this bounded inverse. 000172 | Individual exponential terms at unbounded signed h must not be separately 000173 | claimed bounded: their controlled combination has the stated closure. 000174 | 000175 | At a fixed domain a bounded original coefficient variation consequently has 000176 | the genuine resolvent response 000177 | delta G_fixed=-G_fixed (delta h) G_fixed, 000178 | ||delta G_fixed||<=16||delta h||. 000179 | This is a geometric operator/source-response formula on the actual existing 000180 | class. It does not assert that the original quantum source equals a Gaussian 000181 | integral of this inverse, or assign its determinant scalar or physical time. 000182 | 000183 | G_fixed, the moving G_APS(h), and the homogeneous whole Cauchy graph G(h) 000184 | answer different precisely specified boundary questions. The fixed-domain 000185 | formula completes the first of these without replacing the already proven 000186 | graph or its given quantum realization. It is the appropriate direct 000187 | comparison candidate if the original fixed B_sa insertion contract is used. 000188 | 000189 | Next decisive comparison 000190 | Identify which actual original domain is used under the v254 insertion and 000191 | compute its independent original source two-point response on one nonzero 000192 | Yukawa block, retaining the already fixed omega and tau. Compare that result 000193 | with the corresponding fixed-domain Green variation or transported moving- 000194 | domain variation. That tests the first source identification directly.