TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-mark-memory-20261014/PROOF.md Original file SHA-256: 10492f2d34e93ffb8db2593b120ef120401e26bd5acc34db0cd9c7e8f15ec364 Inclusive original lines: 31-151 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. 000031 | ## 3. The carry is an internal source coordinate 000032 | 000033 | [OMITTED: non-scientific conversation or execution metadata] 000034 | 000035 | j(q)=4 sum_(i=6)^8 q_i-2 sum_(i=1)^5 q_i, 000036 | j(s)=1, s=(1/2)^8, j mod4=k. 000037 | 000038 | With K=ker j, every q has a unique representation 000039 | 000040 | q=K u+(4m+r)s, u in Z7, m in Z, r=0,1,2,3. 000041 | 000042 | Thus ell2(L) is explicitly unitarily equivalent to ell2(Z7) tensor ell2(Z) tensor C4, with the original cocycle phases retained. The complement coordinates are a recording convention, not a physically preferred subsystem. Integer carry m is not an appended spectator. 000043 | 000044 | The same construction applies to the neutral charge configuration lattice C0 at N sites. Fix adjacent departure x and arrival y. The primitive neutral vector v=(-s at x,+s at y) has j_y(v)=1, so 000045 | 000046 | C0=ker j_y direct_sum Z v. 000047 | 000048 | Every configuration is z+(4m+r)v with z in ker j_y. This directly gives a four-grade factor and the entire original remaining charge memory, plus the unchanged scalar Hilbert space. A grade-preserving complement choice only changes this coordinate representation. 000049 | 000050 | On every bilateral orbit of the native W=W_xy,s, recursively determined unit phases remove its one-dimensional hopping sign. On the orbit starting at zero the original W phase is -1 at every j; d_j=(-1)^j makes W raise j with coefficient+1. This rephasing does not delete the cocycle of other directions. 000051 | 000052 | ## 4. Derive an internal M4 and keep its winding 000053 | 000054 | Let P_r be the four grade projectors on that orbit and Z=W^4. Define 000055 | 000056 | R=W(I-P3)+W^(-3)P3. 000057 | 000058 | In the original (m,r) coordinates, R increments r modulo4 while holding m fixed. Therefore R is unitary, R^4=I, [R,Z]=0, and 000059 | 000060 | W=R[(I-P3)+ZP3]. 000061 | 000062 | The exact native matrix units are 000063 | 000064 | E_ab=R^a P0 R^(-b), a,b=0..3, 000065 | E_ab E_cd=delta_(b,c) E_ad, E_ab*=E_ba. 000066 | 000067 | Consequently a constant M4(C) factor is generated inside the same source-orbit algebra. It is not a quotient that discards winding. The full algebra is 000068 | 000069 | C*(W,P_r) = M4(C(S1)), 000070 | 000071 | and its von Neumann closure in the bilateral representation is M4(L_infinity(S1)). Indeed Fourier transformation of the original carry makes Z multiplication by z on the unit circle; W is the four-step cyclic matrix with wrap entry z. The E_ab generate all constant matrix units, and Z has the full unit-circle spectrum. This proves both inclusions and faithfulness. 000072 | 000073 | No selected Bloch phase or finite periodic carry is introduced. Replacing Z by a fixed phase is a representation of this smaller algebra, not an established reduction of the original time evolution. The R construction uses a cut after grade3; shifting that cut changes the coordinate frame. Nor is this abstract M4 already identified with the old Clifford CPTP channel. 000074 | 000075 | ## 5. The same source couples grade and carry 000076 | 000077 | For the original adjacent v, D(v)=2/N+14nu and, at n=jv, 000078 | 000079 | V(jv,phi)=j^2[D(v)+u(phi_x^2+phi_y^2)]+V_scalar(phi). 000080 | 000081 | The14 consists of4 from the common bond and10 from the other incident bonds. Thus 000082 | 000083 | V((j+4)v,phi)-V(jv,phi) 000084 | =(8j+16)[D(v)+u(phi_x^2+phi_y^2)]. 000085 | 000086 | Every original hopping commutes with W^4: its cocycle commutator with a4s shift has even pairing. K also commutes with W^4. On the common core, 000087 | 000088 | [H,W^4]=W^4[V(n+4v)-V(n)], 000089 | 000090 | which is nonzero, already at n=0. The center of the grade/transport subalgebra is not central in the full dynamical source. A fixed-z fiber therefore does not carry the full H as an invariant direct-integral fiber. Fourfold wrap information affects subsequent source responses. 000091 | 000092 | More strongly, use the same normalized scalar Schwartz vector and z=0 in the full factorization from section3. The product-basis diagonal energies for j=0,1,4,5 obey 000093 | 000094 | E5+E0-E4-E1=8[D(v)+u]>0. 000095 | 000096 | The scalar baseline and48JN cancel; every nonconstant hopping word has zero diagonal charge matrix element. Any tensor-sum H_grade tensor I + I tensor H_memory would have zero mixed difference. Hence this derived grade/memory factorization is dynamically coupled. 000097 | 000098 | The faithful full Gibbs state cannot be rho_grade tensor rho_memory in this factorization. If it were, spectral logarithms of the positive injective densities would give 000099 | 000100 | H=-beta^(-1)(log rho_grade tensor I + I tensor log rho_memory) 000101 | -beta^(-1)log Z_beta I, 000102 | 000103 | a tensor-sum, contradicting the diagonal witness. This proves that the actual source equilibrium already correlates marks and their memory. It does not forbid every other mathematical factorization or derive a particular measurement apparatus. 000104 | 000105 | ## 6. Exact memory for the four observed grades 000106 | 000107 | Let K_rho be the completion of bounded operators in _rho=Tr(rho X*Y). It is isometric to Hilbert-Schmidt space through X->X rho^(1/2). Source Heisenberg evolution alpha_t(X)=e^(itH)Xe^(-itH) extends to a strongly continuous unitary group e^(itL), with self-adjoint Liouvillian L and the commutator action L X=[H,X] on its domain. 000108 | 000109 | At site y put S=span{M0,M1,M2,M3}. All four are nonzero even in the total-neutral sector. Faithfulness gives weights w_a=Tr(rho M_a)>0, so S has dimension4. Its orthogonal GNS projection is 000110 | 000111 | P X=sum_a M_a Tr(rho M_a X)/w_a, Q=I-P. 000112 | 000113 | The unbounded diagonal and scalar terms commute with M_a. The only commutator is [Hhop,M_a], which is bounded. Thus S is in Dom L; the finite-rank maps A=PLP and B=QLP are bounded. The opposite off-diagonal block is B*. Removing B+B* from L is a bounded self-adjoint perturbation, leaving a self-adjoint block diagonal operator. In particular L_Q=QLQ on the Q subspace is self-adjoint with its induced domain. 000114 | 000115 | Duhamel's formula for these bounded off-diagonal couplings gives an exact closed Volterra equation for the uninserted compressed response C(t)=P e^(itL)P: 000116 | 000117 | C'(t)=iA C(t)-integral_0^t K(t-s)C(s)ds, 000118 | K(t)=B* e^(it L_Q)B, C(0)=I_S. 000119 | 000120 | All ingredients are determined by the same source H, rho and original grade; there is no independently chosen bath state or forgetting rate. This is the source-specific projection-memory construction, not new general Mori mathematics. 000121 | 000122 | ## 7. All three nonconstant grade modes need memory 000123 | 000124 | The exact native e8=s neutral hop sends all-zero charges to (-s,+s). Its original W sign is-1; the -J W hopping term therefore has matrix coefficient+J. At the arrival site it changes grade0 to1, and 000125 | 000126 | <(-s,+s)|[H,M0(y)]|(0,0)>=+J. 000127 | 000128 | For any diagonal f(k), [H,f(k)] is off-diagonal in charges. If B f=0 then Lf lies in S and is diagonal, so every off-diagonal entry vanishes. The same e8 hop connects all four grades, forcing f(a+1)=f(a). Thus, for J>0, 000129 | 000130 | ker B=span{I}, rank B=3, 000131 | K(0)=B*B is positive semidefinite, nonzero, rank3. 000132 | 000133 | It is NOT strictly positive on all four dimensions: its constant direction is zero. The finite compressed matrix has the second-derivative identity 000134 | 000135 | C''(0)=-A^2-B*B. 000136 | 000137 | Only S subset Dom L is required: matrix-element differentiation uses rather than assuming every vector is in Dom L^2. An autonomous group e^(itA) would instead have second derivative -A^2. Therefore the original four-grade compression is not that memoryless group. This statement does not rule out every time-dependent or nonunitary phenomenological fit. 000138 | 000139 | The minimal unitary realization of this two-time memory kernel is the closed subspace generated by e^(itL_Q) B S for all real t. It is internal to the source GNS space. No claim that its dimension is finite, or that this two-time subspace suffices for arbitrary interventions, is made. 000140 | 000141 | ## 8. Preserve complete inserted histories 000142 | 000143 | For a complete grade-history response, use the larger source-selected GNS closure generated from I by e^(itL), all its inverses, and left multiplication by the M_a. Left multiplication is bounded and has adjoint left multiplication by M_a itself. The resulting closed subspace is reducing for these operations. It contains every prescribed ordered history vector 000144 | 000145 | C_h=M_(a_n)(t_n)...M_(a_1)(t_1). 000146 | 000147 | Its Gram matrix reproduces the full source decoherence functional 000148 | 000149 | D(h,h')=Tr(C_h rho C_h'*)=_rho. 000150 | 000151 | This is the minimal closed representation retaining these history vectors and actions; the usual Gram/GNS uniqueness is up to an intertwining unitary. It retains interference and any information required by the stated probe family. The unitary kernel in section6 describes only its uninserted two-time compression. Classical event probabilities additionally require a specified measuring instrument or decoherence condition. Their physical occurrence is not selected by forming a Gram matrix.