TFPT scientific source excerpt Original repository path: experiments/theory-contracts/clock-jordan-response-reconstruction-20261015/PROOF.md Original file SHA-256: dd062d2017de70d3b858452a41a7b9089d3b8cb8dd3984af9cec89a6524cdbc0 Inclusive original lines: 1-127 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. 000001 | # Joint clock probes close neutral quadratic response reconstruction 000002 | 000003 | 15 October 2026. Research result, not verification-ledger promotion. 000004 | Overall verdict **PARTIAL**. Algebraic verdict: complete reconstruction in the 000005 | specified real symmetric quadratic current sector. No complete TFPT solution. 000006 | 000007 | ## Question and reused source 000008 | 000009 | The original `neutral-response-tomography-20261013/PROOF.txt` solves the 000010 | three-block scalar problem, but explicitly leaves arbitrary off-diagonal 000011 | quadratic matrices open. Its self-correlators cannot distinguish isospectral 000012 | matrices with different carrier alignment. Separately, 000013 | `native-mark-memory-20261014/clock_probe.py` proves a scalar joint commutant for 000014 | four actual clock transports of the same carrier projector. We combine these 000015 | two results through the native free-current algebra, without changing a 000016 | Hamiltonian or fitting a new source. 000017 | 000018 | Use `tfpt_explorer/neutral_source_response.py`, its original vacuum contraction 000019 | `h_i(z) h_j(w) ~ delta_ij/(z-w)^2`, and 000020 | 000021 | P = diag(1,1,1,1,1,0,0,0), 000022 | Q0=P, Q1=C P C^T, Q2=J P J^T, Q3=C^2 P (C^2)^T. 000023 | 000024 | C and J are the actual `clock_matrices.*.vector` in 000025 | `compiler-integral-triality-20260918/certificate.json`. The checker verifies 000026 | the source fingerprint, imports the native P, and checks that C and J preserve 000027 | the same 240 E8 roots in this basis. Eight indices are not eight physical 000028 | quantum states; these clocks are not assumed to be time symmetries. 000029 | 000030 | ## 1. Actual current products supply Jordan multiplication 000031 | 000032 | For real symmetric A, define the existing local field 000033 | 000034 | T_A(z) = (1/2) sum_ij A_ij :h_i(z) h_j(z):. 000035 | 000036 | The standard vacuum Wick rule gives 000037 | 000038 | T_A(z) T_B(w) = (Tr(AB)/2)/(z-w)^4 000039 | + 2 T_(A o B)(w)/(z-w)^2 + O((z-w)^(-1)), 000040 | A o B = (AB+BA)/2. 000041 | 000042 | Proof: two complete pairings give Tr(AB)/2. The four single contractions 000043 | cancel the prefactor 1/4, leaving h(z)^T AB h(w). At the double pole the fields 000044 | are evaluated at w and their symmetric normal product kills the antisymmetric 000045 | part of AB. Thus h^T(AB+BA)h/2 = 2 T_(A o B). This holds for arbitrary A,B, 000046 | not only for the checked projectors. The simple pole is intentionally not 000047 | identified with just a derivative: for noncommuting matrices it can contain 000048 | additional antisymmetric weight-three terms. 000049 | 000050 | This uses standard free-boson Wick algebra, not a new general theorem. A 000051 | primary reference for the normalizations is Paul Ginsparg, *Applied Conformal 000052 | Field Theory*, section 2.3, equations (2.16)-(2.17): 000053 | https://arxiv.org/pdf/hep-th/9108028. 000054 | 000055 | Consequently repeated extraction of the weight-two double-pole coefficient 000056 | generates exactly the corresponding iterated Jordan products as local fields 000057 | in this stipulated source algebra. This is a field/OPE identity, not the claim 000058 | that multiplying scalar measurement outputs implements an OPE. 000059 | 000060 | ## 2. Four source fields generate all 36 symmetric directions 000061 | 000062 | Start only with Q0,Q1,Q2,Q3. Iteratively adjoin B o Qj when linearly independent. 000063 | The explicit rational certificate contains 36 matrices, with degree histogram 000064 | 000065 | degree 1: 4; degree 2: 6; degree 3: 16; degree 4: 10. 000066 | 000067 | Every matrix is symmetric. Their coordinate matrix is nonsingular over Q: 000068 | the checker gives an exact rational two-sided inverse. Full rank modulo 101 000069 | is a second algebraic certificate, not a floating-point singular-value test. 000070 | An independent reconstruction without the identity seed also attained rank 000071 | 36 modulo 103; the traceless projections attained rank 35 modulo 107. 000072 | 000073 | The vector space of real symmetric 8 by 8 matrices has dimension 36. These 000074 | 36 independent Jordan words therefore span the entire space. Identity is 000075 | already in their span and is not required as a fifth generator. No theorem 000076 | that every irreducible associative algebra has full Jordan closure is assumed; 000077 | the explicit certificate resolves this question for the actual four Q here. 000078 | 000079 | The displayed degree-four construction is sufficient. We make no universal 000080 | minimality claim about probe count, correlator order, or experimental cost. 000081 | 000082 | ## 3. Explicit inverse from mixed responses 000083 | 000084 | Let B_l be the certificate basis. The native mixed two-point coefficient is 000085 | 000086 | m_l = coefficient of = Tr(B_l A)/2. 000087 | 000088 | Writing a_(ij)=A_ij for i<=j, we obtain m=W a, where 000089 | 000090 | W_(l,ii)=B_(l,ii)/2; W_(l,ij)=B_(l,ij) for i