TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-LOCAL-WINDING-AND-FUTURE-RESPONSE-20261102.txt Original file SHA-256: 1e178d1f117418179538e1a08aa16fd6541e750cf6d7989fcbb99db15311b455 Inclusive original lines: 1-163 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 | R099. ORIGINAL LOCAL WINDING AND FUTURE RESPONSE 000002 | ================================================ 000003 | 2 November 2026. Analytic result on the existing quarter CAR source. 000004 | No new source, field measure, clock, metric, or search software is chosen. 000005 | This does not close the primitive physical origin question F1. 000006 | 000007 | 1. Exact original inputs 000008 | 000009 | The source is ONE cover-circle CAR field, with modes exp(i(m+alpha)theta), 000010 | m in Z, alpha=1/4 or 3/4, and the original sea m<0. Retain its given 000011 | K_alpha=(L_cover-ground)/8, including normal ordering and quarter holonomy. 000012 | The fold F(theta)=8theta mod 2pi has eight preimage arcs for a quotient 000013 | interval. Its sheet fields are eta_r(x)=psi((x+2pi r)/8)/sqrt8. 000014 | 000015 | The existing global winding implementer W4 acts on the annihilation field 000016 | by multiplication with v(theta)=exp(4i theta), in the retained convention. 000017 | On the eigenchannels it is B4(x)=exp(ix/2)P, P:k<->k+4, with 000018 | P D_alpha=-D_alpha P. Only the PRODUCT obeys the original twisted domain. 000019 | The original source already proves 000020 | [Q,W4]=4W4, 000021 | [K_alpha,W4]=W4(Q/2+3/4+alpha/2), 000022 | and T(p)=Pp+a, a=(1,1,1,1,0,0,0,0) in D8. 000023 | These are reused, not new results. W4 preserves the raw D8 sector; neither 000024 | the formal split into s=(1/2)^8 and r0 nor its energy match with s selects 000025 | an E8 extension. The global cocycle-compatible factorization in the 000026 | stipulated E8 extension in the original sections20/21 remains closed. 000027 | 000028 | Original smooth Weyls obey 000029 | W(f)psi(theta)W(f)*=exp(-i f(theta))psi(theta). 000030 | They are bounded in the existing representation and preserve Q for real 000031 | smooth periodic f. No extra eight species or product sea is introduced. 000032 | 000033 | 2. A local realization from this same source 000034 | 000035 | Choose nonempty open quotient intervals J,I with closure(J) contained in I 000036 | and closure(I) avoiding the quotient cut. Let b be real smooth, b=1 on J, 000037 | supp b contained in I. On the chosen cover coordinate 0<=theta<2pi put 000038 | phi(theta)=4theta b(8theta mod 2pi), 000039 | U=W(-phi), u(theta)=exp(i phi(theta)). 000040 | Near EVERY theta=2pi r/8 the bump is zero. Thus phi extends as a real 000041 | smooth periodic function across all fold cuts and the cover-circle cut. 000042 | Its real lift has zero net increment; wind(u)=0 and [Q,U]=0. 000043 | On F^{-1}(J), phi(theta)=4theta, hence u=v. Therefore 000044 | Ad U(A)=Ad W4(A) (1) 000045 | for EVERY bounded CAR word supported on F^{-1}(J), including their 000046 | norm-closed algebra. The full adjoint actions agree there, not just a 000047 | finite list of moments. Both adjoints agree on the same algebra too. 000048 | Thus all state evaluations of these adjoint responses agree initially. 000049 | 000050 | U and W4 are different global operators and have charges0 and4. This is 000051 | not equality of source amplitudes with bare U/W4 insertions, of absolute 000052 | partition functions, of histories already probing the complement, or of 000053 | the full original source functional. It is specifically equality of the 000054 | local adjoint responses (1). The bump is a diagnostic comparison, not a 000055 | P1-selected physical field or new reference operator. No norm-C* local 000056 | membership or sharp endpoint theorem is needed for (1). 000057 | 000058 | 3. The original time forces a future distinction 000059 | 000060 | Let tau_t be the original K_alpha time on source observables, with either 000061 | consistent Heisenberg sign. It translates cover supports by +/-t/8. A 000062 | common alpha-dependent field phase cancels in neutral bilinears. 000063 | Set w=u/v=exp(i phi-4i theta). This is a smooth circle-valued function with 000064 | wind(w)=(phi(2pi)-phi(0)-8pi)/(2pi)=-4. (2) 000065 | 000066 | Suppose that (1) also held for ALL tau_t(a*(f)a(g)), where f,g are smooth 000067 | and supported in any one component of F^{-1}(J), for all 0<=t<=2pi. 000068 | The corresponding rank-one one-particle kernels on each translated arc 000069 | would be invariant under multiplication with w. This implies that w is 000070 | constant on that arc. One can see it from the kernels, or from their 000071 | commutators with arbitrary creation operators, which recover |f>