TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/primitive-seam-operator-reconstruction-20261025/TOEPLITZ-SEAM-RECONSTRUCTION.txt
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000001 | TOEPLITZ SEAM RECONSTRUCTION FROM THE ORIGINAL OPERATORS
000002 | Date: 2026-10-25
000003 | Verdict: conditional positive embedded circle reconstruction, but the required Toeplitz and time-covariance lemmas are not proved by the original source
000004 | 
000005 | Question
000006 | ========
000007 | 
000008 | Can the previously free primitive seam vector be removed by forming
000009 | 
000010 |     H_C = Ran(P_C),             S = P_C u_Sigma P_C | H_C,
000011 |     Pi_Sigma = P_C - S S*,
000012 | 
000013 | and deriving a multiplicity one Hardy shift directly from the original
000014 | Calderon projector P_C and seam unitary u_Sigma?
000015 | 
000016 | Exact conditional theorem
000017 | =========================
000018 | 
000019 | Let P be an orthogonal projection on H, let u be unitary, and put
000020 | S = P u P on PH. Then
000021 | 
000022 |     S* S = P - P u* (1-P) u P.                              (1)
000023 | 
000024 | Consequently S is an isometry exactly when
000025 | 
000026 |     (1-P) u P = 0, equivalently u(PH) is contained in PH.   (2)
000027 | 
000028 | Assume (2) and assume S is Fredholm with ind(S) = -1.  Then Pi=P-SS*
000029 | is the orthogonal projector onto ker(S*) and
000030 | 
000031 |     rank(Pi) = dim ker(S*) = 1.                              (3)
000032 | 
000033 | No purity assumption is needed for this conclusion.  The Wold decomposition
000034 | gives a canonical orthogonal sum
000035 | 
000036 |     PH = H_shift direct_sum H_unitary,
000037 |     H_shift = direct sum over n >= 0 of S^n Ran(Pi).         (4)
000038 | 
000039 | The line Ran(Pi) and the unilateral multiplicity one shift H_shift are
000040 | canonical functions of the supplied pair (P,u).  A normalized vector in that
000041 | line is still defined only up to phase, but every such vector gives the same
000042 | projector and the same subspace.
000043 | 
000044 | Because u itself is already unitary on the original H, no external unitary
000045 | dilation is required.  For any unit psi in Ran(Pi), put
000046 | 
000047 |     e_n = u^n psi, n in Z.                                  (5)
000048 | 
000049 | The vectors e_n are orthonormal.  For n>=0 they lie in PH, while for n<0
000050 | they lie in (1-P)H.  Their bilateral span
000051 | 
000052 |     H_circle = closure span{e_n : n in Z}                    (6)
000053 | 
000054 | reduces both u and P.  On H_circle, u is the bilateral shift and P is exactly
000055 | the Hardy cut onto n>=0.  Changing psi by a phase changes no operator or
000056 | subspace.  This is a canonical embedded rank-one circle sector in the
000057 | original Hilbert space.
000058 | 
000059 | The additional purity condition
000060 | 
000061 |     intersection over n >= 0 of S^n(PH) = {0}                (7)
000062 | 
000063 | is needed only for the stronger assertion H_shift=PH, namely that the whole
000064 | Calderon range is this multiplicity one Hardy space.  It is not needed to
000065 | remove the free psi_Sigma direction or to obtain H_circle.
000066 | 
000067 | The sign convention is harmless but must be stated.  In the standard Hardy
000068 | orientation, wind(u)=+1 gives ind(PuP)=-1.  If the source spectral-flow
000069 | convention has the opposite sign, replace u by u*.  This orientation change
000070 | does not supply (2) or (3).
000071 | 
000072 | What the original source actually proves
000073 | ========================================
000074 | 
000075 | 1. paper-source:2c00b99902dffffff710a671, corresponding to
000076 |    documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000077 |    sections/001-section-primitive-core-and-carrier-interface-handoff.tex,
000078 |    lines 105-166, constructs P_C as the Calderon projector and identifies
000079 |    Ran(P_C) with the even boundary Cauchy data.  It gives no displayed
000080 |    relation between P_C and u_Sigma.
000081 | 
000082 | 2. paper-source:c439790c9d6f51b9bcaa037f, corresponding to
000083 |    documents/archive--tfpt-45--source_extracts--03_em_flavor_source/
000084 |    sections/001-section-kernel-numerics-and-electromagnetic-closure.tex,
000085 |    lines 9-45, proves only
000086 | 
000087 |        [u_Sigma]=1 in K^1(S^1),  SF(U_Sigma)=1,  Delta_Sigma=2 pi.
000088 | 
000089 |    The wider original datum already contains a unitary u_Sigma, so no new
000090 |    unitary or dilation has to be chosen.  The displayed primitive theorem,
000091 |    however, records only its K^1 class and spectral flow.  It does not prove
000092 |    the decisive relation (1-P_C)u_Sigma P_C=0.
000093 | 
000094 | 3. paper-source:575dd43392a0abd5fa5c93ab, corresponding to
000095 |    documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000096 |    sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex,
000097 |    lines 166-182 and 281-305, defines
000098 | 
000099 |        d0 = |SF(U_Sigma)|,
000100 |        d1 = essential rank of the finite primitive carrier,
000101 |        d3 = h_Sigma^red, the residual boundary nullity after Calderon normal form.
000102 | 
000103 |    These are different typed quantities from dim ker(S*), dim ker(S), and the
000104 |    Wold unitary summand.  Lexicographic minimality of d1 and h_Sigma^red=0
000105 |    therefore supplies neither the isometry condition (2), the defect rank,
000106 |    nor purity (7).  No source theorem identifies these quantities.
000107 | 
000108 | 4. paper-source:c439790c9d6f51b9bcaa037f, lines 94-122, subsequently says
000109 |    that the primitive seam theorem has singled out a normalized psi_Sigma and
000110 |    defines |psi_Sigma><psi_Sigma|.  The cited theorem in lines 9-45 does not
000111 |    construct that vector.  Hence this is exactly the gap that the conditional
000112 |    Wold construction would repair, but its isometry and index hypotheses are
000113 |    absent.
000114 | 
000115 | No occurrence of a Toeplitz, Hardy, or Wold condition was found in these
000116 | original boundary and electromagnetic source sections.  In particular, the
000117 | collar identity D_+=gamma_n(partial_n+B_Sigma) gives an adapted tangential
000118 | operator.  It does not give a covariance relation between B_Sigma and
000119 | u_Sigma.
000120 | 
000121 | Why index one is insufficient
000122 | =============================
000123 | 
000124 | The Fredholm formula is
000125 | 
000126 |     ind(S) = dim ker(S) - dim ker(S*).
000127 | 
000128 | It yields rank ker(S*)=1 only after injectivity, which follows here from the
000129 | separate isometry hypothesis.  For example
000130 | 
000131 |     S0 = unilateral_shift direct_sum 0_C
000132 | 
000133 | has ind(S0)=-1, but it is not an isometry and its cokernel has dimension two.
000134 | 
000135 | Even isometry plus one dimensional defect does not prove that the whole
000136 | Calderon range is pure:
000137 | 
000138 |     S1 = unilateral_shift direct_sum identity_C
000139 | 
000140 | is an isometry, has ind(S1)=-1 and rank(1-S1 S1*)=1, but its one dimensional
000141 | unitary Wold summand survives in every S1^n H.  Its canonical rank-one shift
000142 | part still exists, exactly as in (4).
000143 | 
000144 | The smallest index counterprobe can be made with one fixed bilateral unitary.
000145 | Let H=l2(Z), let u e_n=e_(n+1), and let P_A project onto
000146 | 
000147 |     span{e_0,e_2,e_3,e_4,...}.
000148 | 
000149 | For S_A=P_A u P_A on P_A H,
000150 | 
000151 |     ker(S_A)=span{e_0},
000152 |     coker(S_A)=span{e_0,e_2},
000153 |     ind(S_A)=1-2=-1.
000154 | 
000155 | Yet S_A e_0=0, so S_A is not an isometry.  The incoming defect has dimension
000156 | one while the outgoing defect has dimension two.  This uses the same full
000157 | unitary u and shows directly that index -1 does not imply leakage zero.  It
000158 | is an operator counterprobe only, not a claim that P_A is the physical TFPT
000159 | Calderon projector.
000160 | 
000161 | There is also a counterprobe inside the exact scalar circle setting.  Take
000162 | the standard Hardy projector C and, for 0<a<1/3,
000163 | 
000164 |     b_a(z) = (z-a)/(1-a z),        u_a(z) = z^2/b_a(z).
000165 | 
000166 | The symbol u_a is smooth, unimodular, strictly orientation preserving, and
000167 | has winding one.  Nevertheless
000168 | 
000169 |     ||(1-C) u_a e_0||^2 = a^4(1-a^2),
000170 | 
000171 | which is 15/4096 at a=1/4.  Thus C M_{u_a} C is not an isometry although all
000172 | index and orientation data agree.  This exact control is recorded in
000173 | research-source:10843ef87f7062e60e223846, lines 69-128.  It is the smallest
000174 | decisive counterprobe to the inference
000175 | 
000176 |     positive generator plus Calderon projection implies isometric shift.
000177 | 
000178 | The smallest decisive original-source test is therefore the leakage operator
000179 | 
000180 |     L_Sigma = P_C u_Sigma* (1-P_C) u_Sigma P_C.             (12)
000181 | 
000182 | Equation (1) says S is isometric exactly when L_Sigma=0.  If (12) is nonzero,
000183 | the proposed repair stops immediately.  If it vanishes, the next independent
000184 | test is strong convergence S*^n -> 0, equivalently (7), if one wants to
000185 | identify all of P_C H with the Hardy sector.  Neither test can be replaced by
000186 | spectral flow.
000187 | 
000188 | Clock dictionary and the missing quarter offset
000189 | ================================================
000190 | 
000191 | Suppose leakage zero and index -1 are proved for the original pair.  To
000192 | reconstruct the claimed first order clock from the supplied tangential
000193 | operator B_Sigma, the clean sufficient condition is the strong covariance
000194 | 
000195 |     exp(it B_Sigma) u_Sigma exp(-it B_Sigma)
000196 |       = exp(it) u_Sigma for every real t,                    (8)
000197 | 
000198 | together with strong commutation [B_Sigma,P_C]=0.  The infinitesimal form is
000199 | [B_Sigma,u_Sigma]=u_Sigma on a common invariant core, but the domain statement
000200 | must be supplied rather than inferred formally.