TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/P1-INSERTION-TANGENT.txt Original file SHA-256: b49aa4ec77bc22029545c4f8f2584e80e63fd8bb6b8a0ce8536baf4137d3ea55 Inclusive original lines: 34-132 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. 000034 | 2. What can be differentiated before assuming a coupling 000035 | 000036 | For a differentiable source curve with fixed reflection R, fixed f and chi, 000037 | V=dot D gives V_rel=(V-RVR)/2 and V_ref=(V+RVR)/2. In a finite regulator, 000038 | dot S_prim = chi^-1 Tr[f'(D_rel/chi) V_rel] 000039 | -chi^-1 Tr[f'(D_ref/chi) V_ref] + i*pi*dot Delta_eta/2. 000040 | This uses trace cyclicity, not commutativity of D and V. If the original 000041 | chi=lambda_1^+(|B|)^-1 varies, its derivative also contributes. If R varies, 000042 | dot(RDR)=dot R D R+R dot D R+R D dot R. These terms cannot be dropped while 000043 | claiming an unrestricted source derivation. Spectral crossings, eta jumps 000044 | and changing domains require separate treatment. 000045 | 000046 | The unknown V is precisely the insertion to be derived. Substituting a desired 000047 | matrix into V would beg the question. The one-form mechanism likewise needs 000048 | the ACTUAL represented a,b in A_+: A=sum a[D,b]. Its usual real completion 000049 | requires order one; without it retain the generalized quadratic term, rather 000050 | than importing an SM calculation. Chamseddine/Connes/van Suijlekom, 000051 | arXiv:1304.7583, equations(2),(3) and section III. 000052 | 000053 | 3. Follow the existing modular/covariance route without replacing it 000054 | 000055 | verification/v258_dirac_covariance_induction.py:65-76 supplies the actual reader 000056 | h(C)=log(I-C)-log C, 0 H_Sigma, 000070 | not a silent basis identification. Then 000071 | C_F = Q* C_Sigma Q, 000072 | dot C_F = Q* dot C_Sigma Q + dot Q* C_Sigma Q + Q* C_Sigma dot Q. 000073 | Even a fixed ambient covariance can have a changing compressed covariance 000074 | when Q moves. Source charge/field maps and their motion must be derived. 000075 | 000076 | For finite faithful C with eigenvalues c_m, diagonalize C, not dot C. The exact 000077 | Frechet derivative is 000078 | [dot h(C)]_mn = h^[1](c_m,c_n) [dot C]_mn, 000079 | where h^[1](x,y)=(h(x)-h(y))/(x-y) for x!=y and 000080 | h^[1](x,x)=-1/[x(1-x)]. 000081 | Proof: differentiate both matrix logarithms using 000082 | D log(X)[E]=integral_0^infty (X+t)^-1 E (X+t)^-1 dt. 000083 | For the first logarithm delta X=-E; for the second it is E with a minus sign. 000084 | Every coefficient is strictly negative. Thus the derivative is an invertible 000085 | linear map on finite Hermitian matrices. This is not the commuting shortcut 000086 | -E/[C(I-C)] when E and C do not commute. 000087 | 000088 | If the stated sector projectors commute with the background C, the same 000089 | spectral decomposition can be taken within their sectors. The Frechet map 000090 | restricts invertibly to every sector block. Consequently 000091 | P_a dot h(C) P_b = 0 iff P_a dot C P_b = 0. 000092 | It neither creates a missing sector block nor chooses one. Do not apply this 000093 | block claim to arbitrary projectors not commuting with C. Chirality-odd 000094 | projection, a varying chirality, a varying scale and moving readouts are extra 000095 | explicit operations; their source provenance must be retained. In particular 000096 | the result does not exclude preexisting background mixing or nonlinear effects. 000097 | An infinite-dimensional/unbounded extension needs its own domains and bounds. 000098 | 000099 | 4. Resolve the projection-versus-occupation distinction on the retained source 000100 | 000101 | The original Calderon Cauchy operator is a PROJECTION. h(P_C) is not a finite 000102 | faithful-covariance logarithm. For finite isometric Q, 000103 | Q* P_C Q has eigenvalue1 iff Ran Q meets Ran P_C nontrivially, 000104 | Q* P_C Q has eigenvalue0 iff Ran Q meets ker P_C nontrivially. 000105 | Proof: norm equalities for an orthogonal projection. In particular, restriction 000106 | to its own range is I, not a mixed occupation. Therefore one cannot identify 000107 | the same positive admissibility subspace with a faithful occupation simply by 000108 | using a compression symbol. This is a conditional consistency test, not a 000109 | claim that every compression of a projection is pure. 000110 | 000111 | The previously constructed TWO-FACE collar supplies the correct mixed object. 000112 | For its given selfadjoint B and positive length ell, put R_c=exp(-ell B). Then 000113 | A=(I+R_c^2)^-1, 000114 | P_graph=[[A,A R_c],[R_c A,R_c^2 A]], 000115 | n=I-A=(I+exp(2ell B))^-1. 000116 | The full P_graph is a projection; n is the mixed one-face occupation. Hence 000117 | h(n)=2ell B, 000118 | D h(n)[dot n]=2ell dot B+2 dot ell B. 000119 | The last identity holds also for a noncommuting bounded variation on a common 000120 | appropriate finite/regulated domain; it follows by differentiating the exact 000121 | functional inverse, without a freely chosen vertex. It recovers the insertion 000122 | already in B(Phi), and cannot supply a previously unspecified dot B. 000123 | 000124 | Reuse the retained lowest charged particle/hole transfer pair R_c=diag(1/4,1/64). 000125 | The exact graph projection has A=diag(16/17,4096/4097), 000126 | n=diag(1/17,1/4097), h(n)=diag(log16,log4096). 000127 | At ell=8log2 this returns B=diag(1/4,3/4), with the already specified transfer 000128 | and quarter source in its POSITIVE particle/hole excitation chart. It is not 000129 | the signed raw one-particle B before the sea/particle-hole transformation. 000130 | Its own-range covariance is I, exactly as above. 000131 | This preserves the prior positive construction; it is not a new source choice. 000132 | Numerical eigenvalue clipping in v258 is a regulator, not a proof of h(I).