TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/COMOVING-CLOCK-CONTROL.txt
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000001 | CONSTANT HIGGS PAIR: TRANSPORTED DIAGNOSTIC CLOCK AND ITS SOURCE TEST
000002 | 
000003 | Scope. R081's fixed constant Higgs quark singular block, R084's explicitly
000004 | diagnostic selective shift, both quarter phases. This is NOT an identification
000005 | of the original P1 seam clock, its real reflection, or its common source.
000006 | The preexisting whole 36-dimensional quark covariance and its coupled P/Q
000007 | fields are retained. Neither the full dynamical Higgs integral nor a physical
000008 | 4D theory follows from this check.
000009 | 
000010 | Let p=n+alpha, alpha=1/4 or 3/4, n integer, h_p=p sigma_z+m sigma_x,
000011 | E_p=sqrt(p^2+m^2), ell>0 and C_p=(I+tanh(ell E_p)h_p/E_p)/2.
000012 | For m real after the given singular-block change of basis define
000013 |  theta_p=arctan(m/p) in (-pi/2,pi/2),
000014 |  W_p=exp(-i theta_p sigma_y/2),
000015 |  U0=S P_+ + P_-, with S e_n=e_(n+1), P_+=(I+sigma_z)/2.
000016 | Then W_p* h_p W_p=sign(p) E_p sigma_z. W is the direct sum of W_p
000017 | and the proposed new conditional clock is Um=W U0 W*.
000018 | 
000019 | 1. Exact transported result. W* C W is diagonal with upper occupation
000020 |  c(p)=(1+sign(p)tanh(ell E_p))/2. As n increases, c is nondecreasing,
000021 | with limits zero and one. Thus U0 W*CW U0* - W*CW is diagonal with entries
000022 | c(n-1+alpha)-c(n+alpha) on the first internal component and zero on the
000023 | second. The telescoping sum is absolutely convergent and has total absolute
000024 | value 1. Therefore Um C Um* - C is trace class with exact trace norm 1.
000025 | For 0<z<1 its ordinary relative determinant equals (1-z)^(-1). No product
000026 | of separately compressed interacting covariances is used.
000027 | 
000028 | The same argument holds for all fixed finite constant V with Gamma^2=I,
000029 | V=V*, {Gamma,V}=0 and equal finite Gamma sectors, by SVD decomposition. If
000030 | each of r paired Gamma+ directions is shifted, direct-sum multiplicity gives
000031 | (1-z)^(-r); in the complete R081 quark Gamma+ block r=18. This does not
000032 | settle the relationship between the full source and an original single
000033 | P1 winding. The two quarter phases are not physically identified by this.
000034 | 
000035 | 2. The new clock differs from the previously tested fixed one. For nonzero
000036 | m, W-I is Hilbert Schmidt since W_p-I=O(1/|p|); its off-diagonal diagonal
000037 | matrix elements behave like m/(2p), so W-I is not trace class. Um-U0 is
000038 | Hilbert Schmidt from the product identity. Its mode-diagonal block equals
000039 | W_p P_- W_p*, with mixed element -sign(p)m/(2E_p). The absolute sum of
000040 | these same-index matrix elements diverges. Hence Um-U0 is NOT trace class.
000041 | The comparison is between two stipulated operators, neither automatically
000042 | the original physical clock. This is compatible with Um C Um* - C being
000043 | trace class: the latter is a different operator.
000044 | 
000045 | 3. A concrete source/locality discriminator. A pointwise multiplication
000046 | operator on the circle has the same n,n Fourier block for every n. Here
000047 | (Um)_{nn}=W_{n+alpha} P_- W_{n+alpha}* depends on n if m != 0. Therefore
000048 | Um cannot be pointwise multiplication by an internal matrix-valued function.
000049 | This does not rule it out as an operator in a larger OS algebra, as a dressed
000050 | operation, or as a permitted originally sourced nonlocal transformation.
000051 | Its construction alone does not establish any of these. It makes an explicit
000052 | test possible: derive the actual original source map and compare its mode
000053 | blocks to these, with both paired source sectors, domains, time, reference,
000054 | eta, Z0 and all mixed words intact.
000055 | 
000056 | 4. Conditional quarter reflection illustration. At the purely 2x2 algebra
000057 | level, p -> -p maps quarter alpha to 1-alpha with n -> -n-1 and conjugation
000058 | by sigma_x sends W_p to W_-p and h_p to h_-p. The same conjugation exchanges
000059 | P_+ and P_- and reverses S. Thus it sends U0 to S* P_- + P_+, not U0*.
000060 | This check uses a stated model antiunitary/reflection convention; it does not
000061 | claim the actual P1 reflection has this action. The actual charge, orientation
000062 | and reflection dictionary must be derived before drawing a P1 conclusion.
000063 | 
000064 | Reproduction: python3 check_comoving_clock.py. The 32 finite and explicit
000065 | algebra controls include the seed252 singular value, both quarter phases,
000066 | exact diagonalization, endpoint product, and growing lower bounds on the
000067 | new-vs-fixed clock. The analytic estimates above, not finite truncations,
000068 | establish the Schatten classes. No Lean, original-source proof, physical
000069 | validation, absolute determinant, eta or ledger promotion is claimed.