TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/COMOVING-WINDING-QUARTER-DYNAMICS-20261102.txt
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000001 | CO-MOVING WINDING, CHARGED QUARTER AND THE DYNAMIC FIELD TERM
000002 | 
000003 | Scope: the given constant singular block and diagnostic shift of R087, not an
000004 | identification of the primitive P1 clock or a new physical source. The given
000005 | alpha is 1/4 or 3/4, p=n+alpha, n in Z. Write P+=(I+sigma_z)/2, P-=I-P+,
000006 | h_p=p sigma_z+m sigma_x, W_p=exp(-i atan(m/p) sigma_y/2),
000007 | U_m=W (S P+ + P-) W*, S|n>=|n+1>. The original free charged quarter and
000008 | its explicitly stated time-loop relation are in seam-collar-multifield-
000009 | 20261016/PROOF.txt, sections 6, 8, particularly lines 166-170, 249-256.
000010 | R087's whole covariance and its hypotheses remain unchanged.
000011 | 
000012 | 1. Exact spectral flow; the direction and index are not guessed.
000013 | Let P0 be the negative-energy projector of p sigma_z, so that it occupies
000014 | |n,+> when n<0 and |n,-> when n>=0. Let P_m=W P0 W* be the actual negative
000015 | spectral projector of h on the stated block. Directly,
000016 | 
000017 |  U_m P_m U_m* - P_m = W |0,+><0,+| W* .                    (1)
000018 | 
000019 | The right side is exactly one rank-one projector, trace +1, for every finite
000020 | constant m and either quarter. This is a unit spectral flow across the chosen
000021 | filled sea. It is not the 18-fold result unless all 18 modes are selected.
000022 | W-I is Hilbert-Schmidt, so its off-diagonal blocks relative to P0 are Hilbert-
000023 | Schmidt; the polarized CAR representation of P_m is unitarily equivalent to
000024 | that of P0. The Bogoliubov transformation U_m is implementable as its
000025 | commutator with P_m has finite rank. A corresponding implementer takes the
000026 | filled reference to the +1 charged sector; by the original particle/hole
000027 | charge convention Q=Np-Nh, it obeys
000028 | 
000029 |  exp(i pi Q_m/2) Uhat_m exp(-i pi Q_m/2) = i Uhat_m.       (2)
000030 | 
000031 | Q_m and its reference are transported WITH W. The phase in (2) is independent
000032 | of the undetermined overall phase of Uhat_m. This is a structural compatibility
000033 | of a charged winding with a co-transported charged quarter action, not a proof
000034 | that Uhat_m is an admissible primitive-source operation, nor a derivation of
000035 | its determinant/eta/absolute partition phase. In particular, the unitarily
000036 | transported Q_m has not been identified with the original interacting Q.
000037 | 
000038 | U_m is not itself an order-four quarter operator: U_m^4 is unitarily equivalent
000039 | to S^4 P+ + P-, which is not I. Its spectrum includes the full unit circle,
000040 | whereas an operator whose fourth power is I has spectrum inside the four roots
000041 | of unity. The original P1 winding, geometric loop and quarter representation
000042 | may be distinct operations: this refutes only their DIRECT identification.
000043 | 
000044 | 2. An explicit antiunitary opposite-quarter diagnostic.
000045 | Define R by R|n,v>_alpha=|-n-1,sigma_x conjugate(v)>_(1-alpha).
000046 | Since sigma_x W_p sigma_x=W_-p, it sends P_m to the corresponding opposite-
000047 | quarter P_m and sends U_m to the co-moving S* P- + P+ shift. The transported
000048 | rank-one defect (1) then sits at |-1,-> in the reflected frame, still with
000049 | trace +1. Antiunitarity conjugates the scalar i in (2). This is a precisely
000050 | given reflection test, NOT an asserted identification of R with the original
000051 | P1 reflection, time reversal, charge conjugation or the physical OS map.
000052 | 
000053 | 3. Actual dynamic-field connection and a decisive source-generator test.
000054 | For differentiable real m(t), retain ALL of the stated time-dependent block.
000055 | With psi=W chi and i dot(psi)=h psi, the transformed equation is
000056 | 
000057 |  i dot(chi) = [ sign(p) E_p sigma_z
000058 |                 - (p dot(m)/(2(p^2+m^2))) sigma_y ] chi,     (3)
000059 |  E_p=sqrt(p^2+m^2).
000060 | 
000061 | The second term is compulsory and follows from -i W* dot(W). It is nonzero for
000062 | p!=0 and dot(m)!=0, couples both eigensectors and must remain in time ordered
000063 | words. Example p=1/4,m=0,dot(m)=1 gives -2 sigma_y. The complete source may
000064 | have further terms and field dependencies; (3) neither selects its measure nor
000065 | replaces all 96 components or the mixed vertices. It proves that replacing a
000066 | time dependent source by only the instantaneous diagonal E_p fails this
000067 | specified exact moving-frame equation.
000068 | 
000069 | The original FREE charged-source model explicitly has Wq=exp(2pi i Hfree).
000070 | If one additionally identifies the same unrescaled 2pi generator, at nonzero m,
000071 | with the positive quasiparticle energy E_n and retains the co-transported
000072 | quarter eigenvalue exp(2pi i alpha), this proposed identification fails for
000073 | any fixed nonzero m. For n>=0,
000074 | 
000075 |  E_n-(n+alpha)=m^2/(sqrt((n+alpha)^2+m^2)+(n+alpha)) >0,
000076 | 
000077 | and it tends to 0; eventually it lies strictly between 0 and 1, so the two
000078 | exponentials cannot agree. This is a conditional failure of THAT SAME-generator
000079 | identification. In particular it does not imply that geometric angular
000080 | translation must equal the massive Hamiltonian, and does not refute a separate
000081 | original quarter operator or any full TFPT theory. The actual originating map,
000082 | its time generator, charge, coupling, field/frame transport and reflection must
000083 | be compared with these identities. Eta and reference phase remain separate.
000084 | 
000085 | Reproduction: python3 check_comoving_winding_reflection.py checks both quarters,
000086 | three masses, interior finite-window rank-one and opposite-quarter reflection,
000087 | U^4 != I, and three finite-difference checks of (3). Infinite identities and
000088 | the nonzero-m mismatch are proved analytically above; truncation is not their
000089 | proof. No new Lean proof, physical experiment or ledger promotion is asserted.