TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DOUBLE-WALL-REVIEW.txt Original file SHA-256: f8635d1b544e80626b79ca7a3a23120d568b8f3878ace11acacfadaa98ab759c Inclusive original lines: 1-162 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 | INDEPENDENT REVIEW: CLOSED DOUBLE, ANTIPERIODIC WALLS AND NORMAL MODES 000002 | 29 October 2026 000003 | 000004 | Verdict 000005 | 000006 | The homogeneous antiperiodic Green-function formula is correct once that 000007 | closed source and its spin/domain data are GIVEN. The local face reality 000008 | operator C_faces=(C0,-C0) does not by itself select this global closure or 000009 | identify those Green functions with the original Schwinger functional. 000010 | 000011 | [OMITTED: non-scientific conversation or execution metadata] 000012 | [OMITTED: non-scientific conversation or execution metadata] 000013 | [OMITTED: non-scientific conversation or execution metadata] 000014 | [OMITTED: non-scientific conversation or execution metadata] 000015 | [OMITTED: non-scientific conversation or execution metadata] 000016 | [OMITTED: non-scientific conversation or execution metadata] 000017 | 000018 | No original source or registered file is changed. No numerical run or new 000019 | wall model was used for this review. 000020 | 000021 | 1. Sources actually compared 000022 | 000023 | [OMITTED: non-scientific conversation or execution metadata] 000024 | 000025 | [OMITTED: non-scientific conversation or execution metadata] 000026 | [OMITTED: non-scientific conversation or execution metadata] 000027 | 000028 | Relevant accessibility-text items are 642-645 (new source assumption and 000029 | asserted antiperiodicity), 648-656 (homogeneous D1), 659-670 (wall D2 and 000030 | claimed exact modes), and 676-680/726 (claimed physical consequences). 000031 | The attached numerical script was not independently inspected or run. 000032 | 000033 | Existing local source calculations reused here: 000034 | 000035 | graded_boundary_functor/normal_dependent_transport/PROOF.txt, sections1-3: 000036 | the SAME constant Clifford chart transports the full variable-coefficient 000037 | Cauchy graph, with C0 K C0=(K*)^-1, not generally K^-1. 000038 | 000039 | graded_boundary_functor/EVEN-REFLECTION-POSITIVITY.txt, geometric setup: 000040 | C_faces=(C0,-C0), F=S gamma_F and the actual product reflection. 000041 | 000042 | original_insertions/PROOF.txt, section7, reusing 000043 | ../original-admissible-source-closure-20261025/ 000044 | NORMAL-INDEX-TRANSPORT.txt, sections1-2: 000045 | the genuine spin Dirac on S2 twisted by L2=O(1) has a one-dimensional 000046 | positive kernel and zero negative kernel under the stated operator choice. 000047 | 000048 | The last normal operator is NOT the finite Yukawa matrix Phi. Its original 000049 | operator/family identification remains subject to that existing contract. 000050 | 000051 | 2. What D1 proves, and its first additional input 000052 | 000053 | Let h be the actual given selfadjoint charged boundary generator. Impose 000054 | the homogeneous closed operator L=partial_r+h with antiperiodic condition 000055 | on a circle of length beta=2ell. In each spectral component of h its Green 000056 | kernel for -beta0)-n1_(tau<0)], 000059 | n=(1+exp(beta h))^-1. 000060 | 000061 | The jump is I and G(tau+beta)=-G(tau). Functional calculus therefore gives 000062 | 000063 | G(ell)=exp(-ell h)/(1+exp(-2ell h)) 000064 | =(2 cosh(ell h))^-1. 000065 | 000066 | [OMITTED: non-scientific conversation or execution metadata] 000067 | [OMITTED: non-scientific conversation or execution metadata] 000068 | [OMITTED: non-scientific conversation or execution metadata] 000069 | [OMITTED: non-scientific conversation or execution metadata] 000070 | [OMITTED: non-scientific conversation or execution metadata] 000071 | 000072 | Local face orientation/reality and global spin holonomy are distinct data. 000073 | For example, any linear clutch U satisfying the local compatibility 000074 | condition C1 U=U C0 has -U satisfying the SAME condition. This homogeneous 000075 | compatibility equation cannot select the gluing sign. A theorem with a 000076 | specified global spin bundle, lift and clutch can select a closure, but 000077 | those data and their induced domain must be tracked. They are not encoded 000078 | in the displayed formula C_faces alone. 000079 | 000080 | An ordinary fermionic trace has antiperiodic coherent-state closure; the 000081 | corresponding supertrace has periodic closure. Using that familiar rule 000082 | presupposes the relevant trace/source realization. It is not a derivation 000083 | of that realization from local face orientation. In particular an arbitrary 000084 | periodic spin structure on a geometric double is not ruled out merely by 000085 | naming the supertrace of a different chosen quantization. 000086 | 000087 | 3. The smallest exact finite-wall test 000088 | 000089 | Use precisely the stated normal matrices 000090 | 000091 | N=[[0,-1],[1,0]], Z=diag(1,-1), S=NZ=[[0,1],[1,0]]. 000092 | 000093 | Thus N^2=-I, Z^2=S^2=I, NZ=S, ZN=-S, and S anticommutes with N and Z. 000094 | At one actual tangential Fourier value p, take finite-dimensional constant 000095 | Phi=Phi*, a smooth real periodic scalar sigma(r), and the scalar 000096 | antiperiodic domain on the circle of length 2ell. Set 000097 | 000098 | Q=N partial_r+Z sigma(r)Phi, 000099 | D=Q+pS. 000100 | 000101 | Internal tensor identities are implicit; Phi commutes with N,Z,S. Q and D 000102 | are selfadjoint on H1_antiperiodic by bounded perturbation of N partial_r. 000103 | They have compact resolvent. On H2_antiperiodic, with the corresponding 000104 | first-derivative boundary trace, direct multiplication gives 000105 | 000106 | {S,Q}=0, 000107 | Q^2=-partial_r^2+sigma(r)^2 Phi^2+S sigma'(r)Phi, 000108 | D^2=Q^2+p^2 I. (1) 000109 | 000110 | In particular the derivative term has the PLUS sign. To test an exact 000111 | lambda_D=+/-p it suffices to solve Q psi=0. Multiplying this equation by N 000112 | gives 000113 | 000114 | partial_r psi=sigma(r)(S Phi)psi, 000115 | psi(r)=exp(S Phi F(r))psi(0), 000116 | F(r)=integral_0^r sigma(u)du. (2) 000117 | 000118 | For the claimed odd periodic wall profile, including a profile of the 000119 | displayed tanh(sin) type, the integral around the circle is zero. Its normal 000120 | transfer in (2) is consequently I. The antiperiodic boundary condition 000121 | requires that transfer to act as -I on the initial value. Hence psi(0)=0, 000122 | and there is NO nonzero normal zero mode. 000123 | 000124 | In fact the conclusion is stronger within this restricted commuting class: 000125 | for any real sigma and constant Hermitian Phi, exp(S Phi integral sigma) 000126 | is positive definite, so it has no eigenvalue -1. No scalar antiperiodic 000127 | normal zero mode exists even without the zero-mean premise. This statement 000128 | does not cover noncommuting background transport or a different clutch. 000129 | 000130 | Compact resolvent now gives mu0=min |spec Q|>0 for each fixed such source. 000131 | By (1), every spectral value of D satisfies 000132 | 000133 | |lambda_D| >= sqrt(p^2+mu0^2) > |p|. (3) 000134 | 000135 | The anticommutation pairs Q-eigenvalues mu and -mu, and the corresponding 000136 | two-dimensional block has D-eigenvalues +/-sqrt(p^2+mu^2). Thus exact 000137 | eigenvalues +/-p are absent. mu0 can become very small in a separation or 000138 | large-mass limit; (3) is not a uniform lower bound over such limits. 000139 | 000140 | The statement that the modes are exact at |p| and the later allowance for 000141 | exponentially small coupling are different statements. Only the latter can 000142 | describe approximate isolated-wall modes on this finite circle. The number 000143 | 0.2501 reported for p=0.25 already fails equality at the printed precision; 000144 | the analytic argument, rather than that rounding, decides the exact claim. 000145 | 000146 | There is also an explicit inverse in this SAME zero-mean, constant-Phi, 000147 | scalar-antiperiodic class. Write L=2ell, K=S Phi and 000148 | Fmat(r)=exp(K integral_0^r sigma), so Fmat(L)=I. Since 000149 | Q=N(partial_r-K sigma), variation of constants and the anti-I closure give 000150 | 000151 | Q^-1(r,s)=Fmat(r)[1_(r>s)-1/2]Fmat(s)^-1(-N). (4) 000152 | 000153 | The derivative of the bracket is delta(r-s); the endpoint values are 000154 | opposites. Thus (4) maps L2 to the actual H1 antiperiodic domain and is the 000155 | unique inverse, not just an exclusion of one trial zero mode. It requires 000156 | exactly the fixed commuting coefficient and zero-mean premises displayed 000157 | here; it is not an arbitrary normally varying background Green formula. 000158 | 000159 | 4. What the infinite single-wall calculation does prove 000160 | 000161 | Retain constant Hermitian Phi but put r on R. Suppose sigma approaches -1 000162 | at -infinity and +1 at +infinity. Then F(r) in (2) grows as |r| at both ends.