TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CALDERON-OPERATION-SOURCE-PROOF.txt
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000001 | R074. ORIGINAL TWO-FACE OPERATIONS: REDUCTION, IMPLEMENTATION, SOURCE CHANGE
000002 | 1 November2026. Scoped continuation of R073, not completion of F1.
000003 | 
000004 | 1. Decisive question and original source types
000005 | The required result remains the original common physical TFPT source.
000006 | Do not replace it by construction of another compatible free model.
000007 | We test the FIRST proposed identification: does preserving the original
000008 | Cauchy range correctly express physical admissibility of a local operation?
000009 | The original archive defines iota_C=2P_C-I on Cauchy data, but defines the
000010 | physical net AFTER OS by operators A and A* preserving H_adm. Its color
000011 | P_sing is explicitly (I+Z_c+Z_c^2)/3, not full nonabelian SU3 invariance.
000012 | No Gamma(P_C), Q_Theta=CAR parity, Dadm=L0 or Padm=G is supplied.
000013 | 
000014 | Reuse the actual labelled product-collar graph, beta=4pi, alpha=1/4,3/4:
000015 |  p=r+alpha, r in Z; C(p)=1/(1+exp(beta p)),
000016 |  X(p)=1/(2cosh(beta p/2)),
000017 |  G(p)=[[1-C(p),X(p)],[X(p),C(p)]].                    (1)
000018 | The SECOND original face has occupation C. G is an orthogonal projection.
000019 | Swapping both faces conjugates G by a fixed unitary and changes none of
000020 | the norm/commutant statements; it does NOT silently exchange the labels.
000021 | The imported original graph_projection(exp(-beta*p/2)) reproduces(1).
000022 | Both signed spectral tails, the complete sea and cross coherence remain.
000023 | Identification of this geometric G with the full physical covariance is
000024 | still the existing conditional source premise, not proved by naming P_C.
000025 | 
000026 | 2. Complete bounded matrix-multiplication commutant
000027 | Let A(theta) be any bounded measurable2x2 matrix multiplication (smooth
000028 | is sufficient). Its Fourier coefficient A_m obeys [M_A,G]=0 iff
000029 |  G(p+m) A_m=A_m G(p) for every r,m.                    (2)
000030 | Taking r to +infinity in(2) gives [diag(1,0),A_m]=0.
000031 | Thus A_m=diag(a,d), without assuming A Hermitian or unitary. For m!=0,
000032 | strict monotonicity of C gives a=d=0 from the diagonal entries. For m=0,
000033 | the offdiagonal X(p)>0 gives a=d. Conversely a constant cI2 commutes.
000034 | Therefore the whole multiplication commutant is precisely C I2.
000035 | In particular no nonzero properly supported multiplier reduces G.
000036 | For bounded operators, A and A* preserving RanG is equivalent to(2).
000037 | The full bounded-operator commutant is much larger; no claim about all
000038 | nonmultiplication operators or the physical local net follows.
000039 | 
000040 | Use the original eightfold MODE decomposition in a PERIODIC frame:
000041 |  p_(n,k)=8n+k+alpha, n in Z, k=0,...,7,
000042 |  B=8(-i partial_x)+diag(k+alpha).                      (3)
000043 | Removing the original fractional phases from the quasiperiodic residue
000044 | fields produces this frame and its displayed flat connection. Keeping
000045 | quasiperiodic fields instead requires the corresponding frame transport;
000046 | neither the connection nor derivative terms may be dropped.
000047 | For arbitrary bounded16x16 matrix multiplication its2x2 block condition is
000048 |  G(p_(n+m,k)) A_m[k,l]=A_m[k,l] G(p_(n,l)).             (4)
000049 | The same tail and monotonicity argument leaves a block only if
000050 |  delta=8m+k-l=0, hence m=0,k=l, and then A_0[k,k]=c_k I2.
000051 | Thus the entire folded commutant is constant C^8, same coefficient on
000052 | both faces. It is not restricted to scalar profiles, a chosen q, or a
000053 | finite list of Fourier modes. The two original quarters separately obey
000054 | this statement; no new interquarter identification is introduced.
000055 | 
000056 | 3. Entire smooth unitary multiplier implementability class
000057 | For a unitary multiplier V, its CAR automorphism is implementable in the
000058 | G-Fock representation iff [G,V] is Hilbert-Schmidt. This is the charged
000059 | CAR form of the standard criterion (Araki1970,Theorems6/7); it does not
000060 | assert physical availability, locality of an implementer or primitive
000061 | selection. Pure-state equivalence and unchanged state are different.
000062 | 
000063 | In the folded periodic frame, the EXACT smooth multiplier class is
000064 |  V(x)=V_L(x) direct_sum V_R(x), V_L,V_R in U(8).        (5)
000065 | Necessity: for each fixed Fourier coefficient V_m, the matrix entries of
000066 | [G,V] along n -> +infinity tend to [P_face,V_m]. A nonzero off-face block
000067 | therefore gives an infinite sum of nonzero squared entries. HS forces all
000068 | off-face Fourier coefficients to vanish. This proves necessity on the
000069 | whole unitary matrix class, not merely an example.
000070 | Sufficiency: for block-diagonal V the asymptotic projections commute with
000071 | each V_m. Outside O(1+|m|) modes, G_(n+m,k)-its same-tail limit and G_(n,l)
000072 | minus its limit decay exponentially. Uniformly over the finite k,l,
000073 | the summed squared block norm is bounded by const*(1+|m|)||V_m||_HS^2.
000074 | Smooth Fourier decay makes the sum finite. This also supplies a direct
000075 | infinite-dimensional proof, without a finite-CCR construction.
000076 | For the SEPARATE class of cover-pointwise2x2 multipliers the answer is
000077 | U(1)xU(1) multiplication on the two faces. This is not the pullback of
000078 | every folded U(8)xU(8) multiplier: a general folded channel mixing pulls
000079 | back to a multi-arc, bilocal cover operator. Keep R072/R073 locality types.
000080 | A constant bare face swap is NOT implementable: its commutator
000081 | has nondecaying off-face entries at both spectral ends. The already given
000082 | combined reflection with reversed tangential operator is a DIFFERENT
000083 | operator, not excluded. No alternative reflection is searched here.
000084 | 
000085 | Reducing unitaries from section2 are the much smaller constant U(1)^8,
000086 | with the SAME residue phase on both faces. Implementability of a general
000087 | loop does not force charge neutrality, even fermion parity or membership
000088 | in a specified local observable algebra. Those remain original tests.
000089 | 
000090 | 4. Exact all-mode size of an actual common source change
000091 | For scalar multiplication v on BOTH faces (not a one-face lift),
000092 |  ||[M_v,G]||_HS^2=sum_(m in Z) 2m tanh(beta*m/4)|v_m|^2. (6)
000093 | The m=0 coefficient is0. The expression is positive for every m!=0 and
000094 | does not depend on alpha. It is a norm, not a probability or coupling.
000095 | Derive it directly from the original2x2 graph. Put t=exp(beta*p/2)>0,
000096 |  s=exp(beta*m/2)>0. Its blocks are rank-one projectors, and
000097 |  tr((G(p+m)-G(p))^2)
000098 |  =2t^2(s-1)^2/((1+t^2)(1+s^2*t^2))
000099 |  =2(s-1)/(s+1)*(C(p)-C(p+m)).                         (7)
000100 | The two tails of C are1,0. Thus the absolutely convergent whole-sea sum
000101 |  sum_r(C(r+alpha)-C(r+alpha+m))=m                     (8)
000102 | by telescoping, for positive or negative integer m. Combining(7),(8)
000103 | with Parseval proves(6). A finite window is only a numerical control.
000104 | For general beta>0 the weights are comparable to |m| at large |m|;
000105 | smooth v is sufficient. The beta->0 limit must not be interchanged with
000106 | the whole infinite sum without an additional argument.
000107 | 
000108 | For the existing fixed q=M^T b from R072, q.q=2, and real smooth b(x),
000109 | let A=b(x)diag(q) act commonly on both faces in the frame(3). Then
000110 |  ||[A,G]||_HS^2=4 sum_m m tanh(2beta*m)|b_m|^2.        (9)
000111 | Each residue uses step8; its telescoping sum is m, not8m. Summing q_k^2
000112 | gives2. This explains both factors in(9). No new current normalization.
000113 | For any implementable selfadjoint generator A, normal ordered to have
000114 | zero source mean, Var_G(dGamma(A))=Tr(G A(1-G)A)=1/2||[G,A]||_HS^2.
000115 | This covariance response concerns the COMMON two-face lift. The original
000116 | single-face q current and its central level are not replaced by it.
000117 | The old source already distinguishes common and relative lifts and their
000118 | cocycles; that distinction is reused, not counted as a new result.
000119 | 
000120 | 5. Correct covariance transport and original locality