TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-SPIN-CURRENT-INSERTION-PROOF.txt Original file SHA-256: ab74be0de99bc0d417e7e14ba88eae8e1e21985f72fbf5b17a7c31501fc54fa0 Inclusive original lines: 1-179 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 | R072. ACTUAL ARC AND RESIDUE SOURCE INSERTION 000002 | 1 November2026. Scoped source test, not completion of physical F1. 000003 | 000004 | 1. First identification, before another endpoint calculation 000005 | R071 already gives the entire conditional mixed spinor/CAR word and NS/R 000006 | sewing. We do not recompute it. The test here is whether the ORIGINAL scalar 000007 | cover-current operation produces the smooth Cartan transport in the actual 000008 | q direction. This is a necessary test of that specific proposed insertion, 000009 | not a universal requirement on the original neutral source route. 000010 | 000011 | Use the actual source fold, from ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM- 000012 | GRADING.txt:108-160: 000013 | eta_r(x)=psi((x+2pi r)/8)/sqrt8, 000014 | U_alpha(r,k)=exp(2pi i(k+alpha)r/8)/sqrt8, chi=U_alpha^* eta. 000015 | The chi_k are the residue channels, not the eight geometric preimage arcs. 000016 | The original scalar Weyl operation acts in the ARC chart as 000017 | G_phi(x)=diag_r exp(-i phi((x+2pi r)/8)). (1) 000018 | One smooth scalar cover profile can have different values on the eight 000019 | arcs. We do NOT assume those eight values coincide. The decisive property 000020 | is that(1) is DIAGONAL in the arc chart. 000021 | 000022 | For the actual four missing color roots beta=+/-b,+/-(a+b), a=e1-e2, 000023 | b=e2-e3, keep the original M from fold-results.json and put q=M^T beta. 000024 | For beta=b this gives 000025 | q=(1,-1,-1,-1,-1,-1,1,-1)/2. (2) 000026 | Every one of these four vectors has two signs of one type and six of the 000027 | other, q.q=2, sum(q)=+/-2. These are original fixed data, not a fitted frame. 000028 | 000029 | 2. Complete scalar-profile class decided 000030 | A smooth closed Cartan rotation in this q direction acts on chi as 000031 | D_q(b)=diag_k exp(-i q_k b(x)), 000032 | so its REQUIRED arc action is 000033 | V_q(b)=U_alpha D_q(b) U_alpha^*. (3) 000034 | Here b is a real smooth periodic test function. It may vanish outside an 000035 | interval and have plateau2pi. It need not wind around the circle. 000036 | 000037 | Write U_alpha=P_alpha F_8, where P_alpha is diagonal and F_8 is the DFT. 000038 | F_8 diag(d_k) F_8^* is diagonal iff every nonzero discrete Fourier 000039 | coefficient of d vanishes, equivalently d_k is constant. Thus the arc and 000040 | residue diagonal algebras intersect in C I. This is an all-profile proof. 000041 | For the actual signs(2), all exp(-iq_k b) coincide iff b in2pi Z. 000042 | On a connected circle, a smooth b with that property is constant. If b 000043 | vanishes outside a proper interval it is identically zero. 000044 | 000045 | There is therefore NO scalar original cover profile phi implementing a 000046 | nontrivial smooth closed q rotation(3). On plateau b=2pi the matrix is -I, 000047 | which a scalar phase can implement there; its smooth transitions require 000048 | the offdiagonal arc matrix. No scalar interpolation fixes this obstruction. 000049 | 000050 | The defect is explicit and independent of alpha: 000051 | A_q=U_alpha diag(q) U_alpha^*, (A_q)_rr=sum(q)/8=+/-1/4, 000052 | ||offdiag A_q||_HS^2=2-8(1/4)^2=3/2, 000053 | V_q(b)=cos(b/2)I-2i sin(b/2) A_q, 000054 | ||offdiag V_q(b)||_HS^2=6 sin^2(b/2). (4) 000055 | At b=pi the missing offdiagonal norm squared is6. A diagonal cover action 000056 | has zero in all these entries. Both original quarters obey exactly(4). 000057 | The independent symbolic Fourier equations have rank7 and constant kernel. 000058 | No scan over millions of scalar profiles can change this class decision. 000059 | 000060 | 3. What the actual available composite insertion is 000061 | The existing folded current prescription already supplies 000062 | J_q(x)=:chi(x)^* diag(q) chi(x): 000063 | =:eta(x)^* A_q eta(x):, (5) 000064 | with its ORIGINAL point splitting, sea and reference subtraction. Formula 000065 | (5) is a fixed combination of the original residue Cartan currents. In 000066 | the original cover it contains psi(theta_r)^* psi(theta_s), r!=s, at the 000067 | same quotient point x. It couples preimages separated by2pi(r-s)/8. 000068 | It is not a scalar density psi(theta)^*psi(theta) at one cover point. 000069 | For a test profile supported in quotient K, its algebraic support stays 000070 | in the actual union of eight preimages F^-1(K), not one newly declared 000071 | cover interval. The full Hilbert space/source remains the original one. 000072 | 000073 | Smooth exponentiation is available on this GIVEN folded CAR representation: 000074 | the normal-ordered modes J_q,n obey 000075 | [J_q,n,J_q,m]=(q.q)n delta_(n+m,0)=2n delta_(n+m,0). 000076 | For real smooth b, sum_(n>0)n|b_n|^20)[b_n J_q,-n+bar(b_n)J_q,n], 000084 | [J_q(b),J_q(c)]=2sum_(n>0)n[bar(b_n)c_n-b_n bar(c_n)] I, 000085 | W_q(b)W_q(c)=exp(-[J_q(b),J_q(c)]/2) W_q(b+c). 000086 | This is the original current convention with its forced level2; zero-mode 000087 | q.p and its actual source expectation are retained. The FOCK exponential 000088 | W_q and its ONE-PARTICLE action V_q are different operators. The latter 000089 | does not by itself supply the phase or expectation of the former. 000090 | These statements concern the existing CAR representation and its algebra 000091 | with the actual multi-arc support. They do not assert that the primitive 000092 | physical action already selects(5) or identify a new physical local net. 000093 | 000094 | There is a second exact distinction. Write U=union_r K_r=F^-1(K). The 000095 | even part of the full CAR algebra on U contains cross-arc bilinears. The 000096 | ADDITIVE algebra generated by the individual even CAR(K_r) does not 000097 | automatically contain them. Choose an original smooth scalar phase equal 000098 | to pi on K_r and zero on the other seven arcs, with transitions outside 000099 | U. Its Weyl implementer commutes with each individual even CAR(K_s), 000100 | hence with their generated von Neumann algebra. On U its one-particle 000101 | action is R_r=I-2|r>