TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/admissible_pair_20261028/SIXTY-VERSUS-HODGE-20261102.txt Original file SHA-256: 7b82b79efb6081ec1e5fdac93d8a60a566210973f5f4b775ddb380df8614701f Inclusive original lines: 1-86 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 | SIXTY SOURCE CHANNEL AND FIVE HODGE AMPLITUDES: DIRECT COMPARISON 000002 | ================================================================= 000003 | 2 November 2026. Finite channel statements, not a derivation or exclusion 000004 | of a shared physical seam process with different instruments or records. 000005 | 000006 | Use the actual 60 Gaussian E8 source reflections U_l on Sym^2(C^4), and the 000007 | five original epsilon matrices C_m on Lambda^2(C^5), with the six already 000008 | constructed mark-dependent Bell/duad charts W_q. The unnormalized Choi 000009 | operators and normalized maximally entangled input outputs are 000010 | 000011 | J60=(1/60) sum_l |U_l>><><=24/25 in these six charts. 000023 | This is not the earlier marked normalized twenty-jump process, where the 000024 | same P5 probability was 1/5. P5 is a valid internal operator; its realization 000025 | as an allowed physical preparation and measurement remains a separate source 000026 | question, as does the actual chart's operator and state correspondence. 000027 | 000028 | The conclusion also does not depend on choosing these charts for the weaker 000029 | bound: the normalized Choi eigenvalues of the 60 channel are 4/25 once, 000030 | 2/75 nine times, 1/75 45 times, and 0 otherwise. The original 60x60 Gram 000031 | matrix is integral. The checker verifies its annihilating polynomial 000032 | x(x-8)(x-16)(x-96) and trace moments 600, 14400, 944640, which uniquely 000033 | fix multiplicities 0^5, 8^45, 16^9, 96^1. Dividing by 600 yields the 000034 | Choi eigenvalues. 000035 | 000036 | Under ANY unitary identification of the ten-dimensional input and output 000037 | spaces, rho60 is unitarily conjugated on the Choi tensor product and retains 000038 | these eigenvalues. By the Ky Fan variational inequality, for any orthogonal 000039 | rank-five projector P, 000040 | 000041 | Tr(P rho60) <= 4/25+4*(2/75)=4/15. 000042 | 000043 | Since rhoH is supported on P5, the maximally entangled input followed by 000044 | P5 yields half diamond distance >= 1-4/15=11/15, uniformly for all such 000045 | unitary input and output identifications. This is a lower bound, not a claim 000046 | of the exact optimal channel distance. The stronger 24/25 holds in the six 000047 | explicit source charts; it is not asserted for arbitrary identifications. 000048 | 000049 | A pre-existing stronger representation-specific obstacle is documented in 000050 | compiler-correlated-event-clock-20260919/PROOF.txt section 6: on the native 000051 | quartic five carrier g=diag(1,1,-1,-1) acts trivially, whereas its Sym^2(C4) 000052 | action has trace 2, so no global linear intertwiner with the native wedge 000053 | representation exists for that action. The Bell/duad ray bijection survives. 000054 | Neither obstruction excludes a common finer source followed by different 000055 | instruments, memory updates, selections or coarse grainings. Producing that 000056 | source map and its ordered words remains the relevant original question. 000057 | 000058 | Reproduce from repository root: 000059 | python3 -B -OO experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/admissible_pair_20261028/check_60_hodge_separation_20261102.py 000060 | 000061 | No Lean run, TFPT physical measurement, reconstructed seam instrument, or 000062 | claim promotion is asserted. 000063 | 000064 | CLASSICAL RECORD SELECTION DOES NOT GENERATE THE HODGE EVENT 000065 | ----------------------------------------------------------- 000066 | Let a nonzero classical selection of the original recorded unitary events 000067 | have CP map E(X)=sum_l w_l U_l X U_l*, w_l>=0. If E=p Phi_H for p>0, 000068 | its Choi support is span{|C_m>>}. A positive sum of rank-one projectors has 000069 | support equal to the span of all its positively weighted vectors, so every 000070 | U_l with w_l>0 must lie in span{C_m}. The original R088 builder proves every 000071 | nonzero combination sum z_m C_m has rank exactly 6, whereas each U_l is 000072 | unitary of rank 10. Contradiction. Fixed invertible/unitary input and output 000073 | identifications preserve ranks and do not change this argument. 000074 | 000075 | The same argument applies to positive classical selection of finite record 000076 | histories when every selected history is a product of the original unitaries: 000077 | each such product is still rank 10. It does not apply if the source supports 000078 | an additional nonunitary instrument, coherent manipulation, a modified 000079 | record/environment, or an intervention before or between the original 000080 | unitaries. A refinement of an individual unitary channel that averages back 000081 | to exactly that SAME channel can only use Kraus operators proportional to the 000082 | unitary, by rank-one Choi support. The existing marked stay/jump instrument 000083 | is consistent with this: its refinement preserves the unmonitored channel 000084 | after averaging the source labels, not for each individually fixed label. 000085 | The actual original seam must decide whether and how such a refinement and 000086 | recording are realized. No independent event rate or measurement is assumed.