TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/GEOMETRIC-ARC-TRANSFER.txt Original file SHA-256: eacdbe333028eab285bf703a182d4004b65a763918ccfd368c1d512e39c06517 Inclusive original lines: 1-74 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 | GEOMETRIC EXTERIOR RESPONSE AND THE ACTUAL QUARTER ARC-TRANSFER TEST 000002 | 26October2026. Same source and diagnostic geometry as GEOMETRIC-SOURCE-RETURN. 000003 | 000004 | 1. Why this is the next necessary source test 000005 | The current native-arc-charge-source-interface-20261026/PROOF.txt proves 000006 | that every nonzero native quartic changes under independent phases on the 000007 | proper eight original preimage arcs. Full CAR already contains the required 000008 | bilocal transfers; their actual P1 insertion remains open. Preserve that 000009 | result. Do NOT call a scalar additive density/stress insertion the quartic. 000010 | The present test asks whether the SAME derived bulk response has any such 000011 | transfer, rather than fitting a new native Cartan matrix. 000012 | 000013 | 2. Actual exterior Schur calculation 000014 | Write h=a0+sum_m>=1 h_m cos(pi*m*w). For one original longitudinal momentum 000015 | k, the paired exterior modes(+m,-m), a=pi*m, have the original Dirac block 000016 | D_m=b[[0,k+i*a],[k-i*a,0]], 000017 | and the first geometric coupling column from the normal zero mode is 000018 | B_m=-(b*h_m/2)(k+i*a/2,k-i*a/2)^t. 000019 | Hence the exact second-order Schur response at spectral parameter z is 000020 | B_m* (z-D_m)^-1 B_m 000021 | =h_m^2 [z(2k^2+a^2/2)+b(2k^3+3k*a^2/2)]/[4(z^2-k^2-a^2)]. 000022 | This is the coefficient of epsilon^2 in exterior elimination, not the full 000023 | finite-epsilon response, and not a local physical Hamiltonian assignment. 000024 | At z=0 it splits into 000025 | -b*h_m^2*k/2 + b*h_m^2*a^2*k/[8(k^2+a^2)]. 000026 | The first term is local. The nonlocal remainder for b=+1 is therefore 000027 | R_h(k)=(k/8)sum_m>=1 h_m^2 (pi*m)^2/[k^2+(pi*m)^2]. 000028 | Every coefficient is fixed by the SAME chosen geometry. No target native 000029 | embedding, relative rate, extra field or replacement state has been fitted. 000030 | 000031 | 3. Exact proper-arc witness for the original q=i circle 000032 | Use the proper preimages of the base interval(1/4,3/4): 000033 | I=(1/32,3/32), J=(17/32,19/32), both length L=1/16. 000034 | Their closures are disjoint. Use normalized real indicator smears; the 000035 | bounded remainder R_h is defined on all L2. Smooth compactly supported 000036 | approximations preserve its nonzero matrix element by norm continuity. 000037 | For a>0 the circle kernel of B/(B^2+a^2), B=-i partial_x with twist q=i, 000038 | follows by differentiating the original twisted image Green sum. For0 000046 | =(1-i) sum_m>=1 h_m^2 exp(-a/2)(1+exp(-a))sinh^2(aL/2) 000047 | /[4L(1+exp(-2a))]. 000048 | Every summand is nonnegative and is positive for a nonzero h_m. Thus every 000049 | nonconstant smooth real even profile gives an actual relative-arc transfer. 000050 | The phase is inherited from the GIVEN quarter convention, not predicted as 000051 | a new physical phase. q=-i gives the correspondingly conjugate statement. 000052 | For the SAME collar-preserving probe used in the source machine, the value 000053 | is approximately.001720549637597(1-i), as an epsilon^2 coefficient. 000054 | The independent original Fourier calculation at k_n=2pi(n+1/4), with 000055 | longitudinal bounds64,128,256,512,1024, approaches this value; final error 000056 | is7.09e-12. Numerical sums use128 normal cosine coefficients. The strict 000057 | all-profile/noncancellation statement is the analytic image formula above. 000058 | 000059 | 4. Necessary ingredient is not the missing full dictionary 000060 | For smooth h, sum_m h_m^2(pi*m)^2 is finite, so 000061 | |R_h(k)|<=sum_m h_m^2(pi*m)^2/(8|k|) ->0. 000062 | The remainder is compact on the original circle. A nonzero local folded 000063 | multiplication current M_f tensor A on its infinite base space is not compact. 000064 | Thus R_h cannot itself equal such a native insertion under a unitary Fold. 000065 | This is a small decisive exclusion of that literal identification, not a 000066 | claim that source products, normal-ordered response derivatives or full 000067 | histories cannot recover native fields. The original quartic is a composite, 000068 | not a one-particle matrix. Neither the native five nor3+2 has been derived 000069 | from this probe. The common-geometry source map and the primitive action's 000070 | profile/history selection remain the next necessary original tasks. 000071 | 000072 | Reproduction: geometric_source_return/check_geometric_arc_transfer.py 000073 | checks the exact2x2 Schur identity and the independent images/Fourier 000074 | witness. Normal and-OO result bytes coincide. No ledger gate is closed.