TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/WHOLE-CLUTCH-ENERGY-INDEPENDENT.txt Original file SHA-256: 4d07407c68388a2c84bdd555c7459afe9ef98ee67d6d38a2602d93975055a41c Inclusive original lines: 1-158 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 | WHOLE CLUTCH: INDEPENDENT ENERGY AND ADJOINT-DOMAIN CHECK 000002 | 2026-10-29. Bounded analytic review of the given constant-coefficient case. 000003 | No numerical census, new model, Atlas mutation or old-source changes. 000004 | 000005 | VERDICT 000006 | The claimed mass-difference energy identities and kernel/cokernel 000007 | intersections are correct on the actual unitary H1 clutch domain under 000008 | the precise hypotheses below. Noncommutation of A with U leaves a real 000009 | boundary term in the MASSLESS energy. It does not invalidate the claimed 000010 | difference between massive and massless energies. Whole balanced finite 000011 | grading gives index zero. Invertible Phi removes all kernels and cokernels 000012 | for this closed realization. No normal Majorana-mode statement follows. 000013 | 000014 | 1. EXACT OPERATORS AND DOMAIN 000015 | 000016 | Let F be the whole finite coefficient Hilbert space, gamma=gamma*=gamma^-1, 000017 | Phi=Phi* bounded on F, {gamma,Phi}=0. In this note Phi is independent of 000018 | BOTH r and theta. Let ell>0, 000019 | 000020 | A=-i partial_theta+alpha, alpha real scalar, Dom(A)=H1_per(S1,F), 000021 | U=M_exp(i theta), U e_n=e_(n+1), 000022 | V=-i gamma Phi=V*, V^2=Phi^2, 000023 | L0=partial_r+A gamma, LPhi=L0+V. 000024 | 000025 | The shared closed-domain candidate is exactly 000026 | 000027 | D_U={psi in H1([0,ell] x S1,F): psi(ell,theta)=U psi(0,theta)}. 000028 | 000029 | The equality is an equality of H1/2 traces. It is not a boundary condition 000030 | imposed independently on a Fourier mode, and U does not preserve individual 000031 | Fourier modes. The Hilbert space for the codomain is L2 on the cylinder, 000032 | equivalently L2 sections of the bundle obtained by this unitary clutch. 000033 | 000034 | More general smooth even A and bounded V are possible only if the actual 000035 | commutator, domain and clutch conditions used below are retained. They are 000036 | not being inferred from the scalar calculation. 000037 | 000038 | 2. CLOSEDNESS, FREDHOLM PROPERTY AND THE ACTUAL ADJOINT 000039 | 000040 | U*AU=A+I, U gamma=gamma U. The comparison expression 000041 | 000042 | Lb=partial_r+(A-r/ell)gamma 000043 | 000044 | defines a smooth elliptic first-order operator on the compact clutched 000045 | torus bundle: at the seam (A-1)U=UA. Its principal symbol is 000046 | i xi_r+gamma xi_theta, invertible away from the zero covector. Consequently 000047 | its closed elliptic domain and its adjoint domain are H1 sections, which 000048 | are precisely D_U in the cut chart. This is a comparison for the domain 000049 | proof, not a replacement of the given L0 or its spectrum. 000050 | 000051 | In the cut chart L0=Lb+(r/ell)gamma. The latter multiplication is bounded 000052 | on L2 (it is allowed to have a bounded jump at the seam), so closedness, 000053 | the H1 adjoint domain and Fredholmness remain valid. Adding the bounded V 000054 | has the same properties. The formal Green boundary expression is 000055 | 000056 | - 000057 | =-, 000058 | 000059 | up to conjugation according to which Hermitian argument is linear. Since 000060 | U is unitary, it vanishes for both traces in D_U. Conversely the pairing 000061 | with every clutched trace forces chi(ell)=U chi(0). Elliptic regularity 000062 | supplies H1 for the maximal adjoint domain as above. Thus 000063 | 000064 | Dom(LPhi*)=D_U, 000065 | LPhi*=-partial_r+A gamma+V. 000066 | 000067 | Neither U^-1 on the outgoing trace nor an extra independent domain for the 000068 | conjugate coefficient slot is inserted here. 000069 | 000070 | 3. THE TWO EXACT ENERGY IDENTITIES 000071 | 000072 | The finite algebra is 000073 | 000074 | {A gamma,V}=0, V^2=Phi^2, partial_r V=0, U*VU=V. 000075 | 000076 | The anticommutator uses [A,Phi]=0 and [A,gamma]=0. For smooth clutched 000077 | fields, integration in theta gives 000078 | 000079 | 2 Re =0. 000080 | 000081 | Integration in r gives 000082 | 000083 | 2 Re 000084 | =- 000085 | ==0. 000086 | 000087 | Density and boundedness of V extend the result to every psi in D_U: 000088 | 000089 | ||LPhi psi||^2=||L0 psi||^2+||Phi psi||^2. 000090 | 000091 | Replacing partial_r by -partial_r changes the sign of the boundary 000092 | term, still zero, and gives the corresponding adjoint identity 000093 | 000094 | ||LPhi* psi||^2=||L0* psi||^2+||Phi psi||^2. 000095 | 000096 | No commutation of U with A was used or is true. In fact the massless 000097 | identity, initially on smooth clutched fields and then in form sense, is 000098 | 000099 | ||L0 psi||^2=||partial_r psi||^2+||A psi||^2 000100 | +. 000101 | 000102 | For L0* the last term has a minus sign. This follows from U*AU=A+1. 000103 | Discarding that boundary term would be an error. Keeping the complete 000104 | ||L0 psi||^2 in the mass-difference identity is exactly what avoids it. 000105 | 000106 | 4. KERNELS, COKERNELS AND THE BALANCED INDEX 000107 | 000108 | Since both right-hand summands in the mass-difference identity are 000109 | nonnegative, 000110 | 000111 | ker LPhi=ker L0 intersect ker Phi, 000112 | ker LPhi*=ker L0* intersect ker Phi. 000113 | 000114 | The ranges are closed by Fredholmness, so coker LPhi is identified with 000115 | ker LPhi*. These are identities for the actual common clutch domains. 000116 | For invertible finite Phi they give zero kernel and zero cokernel. 000117 | 000118 | The massless scalar Fourier recurrence independently checks the index. 000119 | In a gamma eigenspace sigma=+1 or -1, homogeneous solutions have 000120 | 000121 | psi_n(r)=exp(-r sigma(n+alpha))a_n, 000122 | a_n=exp(ell sigma(n+alpha))a_(n-1). 000123 | 000124 | Hence a_n=C exp(ell sigma[n(n+1)/2+alpha n]). Square summability for 000125 | all integer n forces C=0 for sigma=+1, and gives one smooth Gaussian 000126 | Fourier solution for sigma=-1. For the adjoint the roles are reversed. 000127 | Thus, writing dplus/dminus for the full grading dimensions, 000128 | 000129 | dim ker L0=dminus, dim ker L0*=dplus, 000130 | index LPhi=index L0=dminus-dplus. 000131 | 000132 | The full v252 grading under review has dplus=dminus=48, so its index is 000133 | zero. The finite mass homotopy is bounded and preserves Fredholmness. 000134 | For constant odd Phi the surviving kernels are respectively the negative 000135 | and positive grading parts of ker Phi; no bare right-channel reduction 000136 | has been made. A claim that a particular supplied numerical Phi is 000137 | invertible still requires that actual coefficient's rank, rather than 000138 | an assumption that all Yukawa inputs are nonzero or physical. 000139 | 000140 | 5. THIS CONSTANT CASE WAS ALREADY DECLARED; THE EXTENSION WAS NOT 000141 | 000142 | graded_boundary_functor/FINITE-COUPLED.txt:11-14 allows smooth bounded 000143 | Phi(theta) and an even connection. Its exact square at43 is 000144 | 000145 | hPhi^2=A^2+Phi^2-i[A,Phi]. 000146 | 000147 | Lines45-48 EXPLICITLY single out constant Phi and internally scalar A, 000148 | where hPhi^2=A^2+Phi^2. Thus the present commutative constant case is an 000149 | existing declared subcase of the given full coefficient, not a new fitted 000150 | matrix class. It is not the assertion for every permitted variable Phi. 000151 | 000152 | For variable Phi(theta), the extra interior term -i[A,Phi] remains; for 000153 | r-dependent V the energy cross term also contains -. 000154 | If U*VU differs from V there is an extra boundary form. The exact positive 000155 | intersection theorem above may not be transported to those cases without 000156 | controlling these terms. Nor does the older open Cauchy graph by itself 000157 | choose this closed clutch realization: that realization is an explicit 000158 | premise of the present test, and has been retained unchanged throughout.