TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/B1-NORMAL-REAL-AND-FRAME.txt Original file SHA-256: 4af165a08652c4954210a81625dd84f4b7f60c069135db625bfe0ba990d4b903 Inclusive original lines: 1-111 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 | B1: NORMAL REAL MAPS, QUARTER TRANSPORT AND COORDINATE FRAMES 000002 | 30 October 2026. Source reviewed continuation, not a physical source selection. 000003 | 000004 | 1. Exact given class and notation 000005 | Use the original FINITE-COUPLED.txt product collar and v252 full96 functions. 000006 | N=[[0,-1],[1,0]], Z=diag(1,-1), S=NZ, NS=-Z. 000007 | D=N partial_r+S tensor A+Z tensor Phi, B=Z tensor A-S tensor Phi. 000008 | Gamma=Z tensor gamma. A is gamma-even, selfadjoint on circle H1; 000009 | Phi is bounded, Hermitian, gamma-odd and JF-real. JF^2=1 and JF gamma=-gamma JF. 000010 | In section2 ONLY impose JF A JF^-1=-A. It holds for -i partial_theta plus 000011 | an actual JF-odd real gauge connection. It fails for scalar quarter shift. 000012 | Coefficients are normal independent. R_r reverses the normal coordinate. 000013 | Old complex i and Ical=i_old Gamma must remain distinct. 000014 | 000015 | 2. Complete classification inside the stated SEPARABLE ansatz 000016 | J=(M tensor gamma^j JF)R_r^k, M constant unitary2, j,k in{0,1}. 000017 | This is not a classification of all antiunitaries on the whole96 module. 000018 | On the common H1 core, JD=epsilon DJ gives the coefficient equations 000019 | M N=(-1)^k epsilon N M, 000020 | M S=-epsilon S M, 000021 | M Z=epsilon(-1)^j Z M. 000022 | Coefficient independence is at the level of this product class, not a 000023 | single accidental spectrum. The Pauli product relation leaves one complex 000024 | line of solutions per pair(j,k), then unitarity fixes its modulus: 000025 | j k M epsilon J^2 JGammaJ^-1 JBJ^-1 Ical type 000026 | 0 0 S - + +Gamma +B antilinear 000027 | 0 1 Z + + -Gamma -B linear 000028 | 1 0 N + + +Gamma +B antilinear 000029 | 1 1 I - - -Gamma -B linear. 000030 | All four are OLD antilinear. The Ical column follows from J i_old=-i_old J. 000031 | The exact linear solver confirms each solution line. Formal derivative 000032 | identities and full192 coefficient controls use the actual complex v252 000033 | matrices, with non-diagonal Yukawas and symmetric complex Majorana. 000034 | No physical OS reflection or original archived inv is selected by this table. 000035 | 000036 | 3. Existing deck as a product, with whole source retained 000037 | JS=S tensor JF, JR=(Z tensor JF)R_r. Then 000038 | JS JR=(N tensor I)R_r=T, JR JS=-T, T^2=-I. 000039 | This is precisely the already given canonical normal Clifford deck. 000040 | It reverses D, Gamma and B, is OLD linear and NEW antilinear. 000041 | For JS the same real functional calculus preserves every G block. For JR/T, 000042 | B changes sign; the face exchange restores the whole two-face projection G. 000043 | This relates real-map types to an existing deck, without identifying it 000044 | with the older inv whose original Gamma dictionary remains open. 000045 | C0=i_old N and the charged Fuchs real map are separate original operations. 000046 | 000047 | 4. Actual scalar quarter is an image transport, not a fixed-bundle symmetry 000048 | Let A_alpha=-i partial_theta+a_odd+alpha I on PERIODIC H1, alpha=1/4, 000049 | JF a_odd JF^-1=-a_odd. Then 000050 | JF A_alpha JF^-1=-A_alpha+2alpha I, 000051 | JS D_alpha JS^-1=-D_alpha+2alpha S=-D_{-alpha}, 000052 | JS B_alpha JS^-1=B_alpha-2alpha Z=B_{-alpha}. 000053 | Thus the stronger premise of section2 cannot be silently applied. 000054 | The scalar holonomies i and -i are conjugate, not equal. In the convention 000055 | Hol=exp(-2pi i alpha) the same pair is written -i and+i. 000056 | A twisted domain H1_U is carried by a given fiber antiunitary K kappa into 000057 | H1_{K bar(U) K^-1}. A fixed-domain endomorphism requires 000058 | U=K bar(U) K^-1; a source-to-image map requires U_p=K bar(U_q) K^-1. 000059 | For a periodic gauge-corrected map g(theta)K kappa one must also solve the 000060 | connection transport, Phi and grading intertwiners and its square. Holonomy 000061 | conjugacy alone is not sufficient for all these conditions. 000062 | In the commuting scalar case, the required multiplier exp(-2i alpha theta) 000063 | is not periodic at1/4. The already existing inverse-holonomy image3/4 is a 000064 | valid target with the corresponding source/domain transport. It is not a 000065 | new independent source. This agrees with the prior SPIN2 annulus review. 000066 | No global absence theorem for every possible96 antiunitary is asserted. 000067 | 000068 | 5. Phase freedom survives a fixed normal real map as a coordinate change 000069 | E_theta=[Pplus; exp(i theta)Pminus], Pplus/minus=(I+/-gamma)/2. 000070 | JS E_theta=exp(-i theta)E_theta JF. After the overall rephasing 000071 | Et_theta=exp(-i theta/2)E_theta=E_0 exp(-i theta gamma/2), 000072 | JS Et_theta=Et_theta JF for EVERY theta. 000073 | Antilinearity and JF gamma=-gamma JF imply 000074 | JF exp(-i theta gamma/2) JF^-1=exp(-i theta gamma/2). 000075 | Hence normal reality does not turn this coordinate phase into a physical 000076 | quarter clock. The intrinsic operator and all its fields must transform 000077 | together; changing only one field is a different question. 000078 | 000079 | 6. Exact gauge and real compatible constant frame group 000080 | The already proved original P2 real gauge dictionary decomposes gamma=+ 000081 | into six inequivalent complex irreps Q,L,u^c,d^c,e^c,nu^c, each with family 000082 | multiplicity3. All constant unitary frames preserving THIS full gauge 000083 | representation and gamma are U(3)^6 on gamma=+. JF fixes the action on 000084 | gamma=- by Uminus=JF Uplus JF^-1. Schur's lemma proves necessity. 000085 | This is also the compatible commutant of the represented full left algebra 000086 | and opposite algebra in this v252 bimodule. The exact actual-slot check 000087 | constructs all54 anti-Hermitian generators and verifies gamma, JF and every 000088 | one of the24 real left algebra basis elements; JF then gives the opposite. 000089 | Sufficiency is therefore checked on the actual field slots, not dimensions. 000090 | The count54 is the real dimension of coordinate frames, not54 physical 000091 | parameters. If a concrete connection, Phi, clock or family dictionary is 000092 | also held numerically fixed, intersect with ITS stabilizer; U(3)^6 is not 000093 | claimed to stabilize all those extra data. Transforming them together gives 000094 | coordinate-equivalent descriptions. Their invariant couplings remain to be 000095 | derived; coordinate freedom itself is not a new physical selection gap. 000096 | 000097 | [OMITTED: non-scientific conversation or execution metadata] 000098 | [OMITTED: non-scientific conversation or execution metadata] 000099 | [OMITTED: non-scientific conversation or execution metadata] 000100 | [OMITTED: non-scientific conversation or execution metadata] 000101 | [OMITTED: non-scientific conversation or execution metadata] 000102 | [OMITTED: non-scientific conversation or execution metadata] 000103 | [OMITTED: non-scientific conversation or execution metadata] 000104 | [OMITTED: non-scientific conversation or execution metadata] 000105 | [OMITTED: non-scientific conversation or execution metadata] 000106 | 000107 | [OMITTED: non-scientific conversation or execution metadata] 000108 | [OMITTED: non-scientific conversation or execution metadata] 000109 | [OMITTED: non-scientific conversation or execution metadata] 000110 | [OMITTED: non-scientific conversation or execution metadata] 000111 | [OMITTED: non-scientific conversation or execution metadata]