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"2f46e1bc4e998eaf6b50a7ca97976f811f779e5e9799eb5cda2ccedd7cbadf85", "matrix_8x8": [ [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "14d2796e967f2c98bff28b4c79c5953c06e483dee41548ffae5cf1a8c29f121f", "source_records": [ { "equation_record_id": "eq_8605b7c5ccdbcf5f", "object_id": "rrc_eq_b650dd7bdbcd9bcb", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_dbd7ee74be207cc7", "object_id": "rrc_eq_b97583b24c3b2936", "name": "arXiv:2604.21919v1", "equation": "Z = \u27e8\u03c8|\u03c8\u27e9 to 1/ poly(N ) multiplicative\nerror\n(ii) local observables \u27e8OA \u27e9 = \u27e8\u03c8|OA |\u03c8\u27e9/\u27e8\u03c8|\u03c8\u27e9 to\n1/ poly(N ) multiplicative error, given \u27e8OA \u27e9 \u0338= 0\n(iii) correlation functions \u27e8OA OB \u27e9 \u2212 \u27e8OA \u27e9 \u27e8OB \u27e9 to\nO\u0303(1/ poly(N )) additive error\nwhere A, B \u2282 V are disjoint local regions", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "f3d1bc1ba85cb45421849ca7059a865a32e717f62727516bfed2aa6489ebd31d", "matrix_8x8": [ [ 4, 1, 0, 4, 1, 0, 0, 1 ], [ 0, 3, 3, 1, 0, 0, 0, 2 ], [ 3, 0, 1, 0, 5, 1, 0, 0 ], [ 2, 4, 0, 0, 0, 0, 0, 0 ], [ 2, 0, 3, 0, 1, 0, 2, 0 ], [ 0, 0, 2, 1, 0, 5, 2, 0 ], [ 0, 1, 1, 0, 0, 2, 0, 0 ], [ 0, 1, 0, 0, 1, 1, 0, 0 ] ], "global_vocab_hash": "0d42821df7ed04a9555307622e00a0b35eb61ee259da332db748a4434d609f01", "source_records": [ { "equation_record_id": "eq_dbd7ee74be207cc7", "object_id": "rrc_eq_b97583b24c3b2936", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_9138cc21d6e4da27", "object_id": "rrc_eq_d521b2282dde0c38", "name": "arXiv:2604.21919v1", "equation": "e = (v, n) with d(v, A) =\nr, we have\n\u0010\n\u0011\n\u2225\u00b5\u2032\u22c6,\u20d7e \u2212 \u00b5\u22c6,\u20d7e \u22251 = O e\u2212r/\u03be\u2217\n(10)\n\n\f5\nWhere 1/\u03be\u2217 = O(log \u03b5\u2217 /\u03b5)", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "8dfff2da8e227669630596f3bf0f0ce3b624d88c5c07c13d3d26ce83ca65268c", "matrix_8x8": [ [ 1, 0, 1, 1, 1, 0, 0, 2 ], [ 0, 1, 1, 2, 0, 1, 3, 0 ], [ 2, 2, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 2, 0, 0, 0, 1, 0 ], [ 0, 2, 0, 0, 0, 0, 0, 0 ], [ 1, 2, 0, 0, 0, 0, 0, 0 ], [ 2, 0, 1, 0, 0, 2, 1, 0 ], [ 0, 1, 0, 0, 1, 0, 0, 1 ] ], "global_vocab_hash": "6bd3c9b83cf006b07630f8007b026887696953d53fc841119264ff7db0e2c33c", "source_records": [ { "equation_record_id": "eq_9138cc21d6e4da27", "object_id": "rrc_eq_d521b2282dde0c38", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_5be87eaf6cffbaae", "object_id": "rrc_eq_282beac067cdee85", "name": "arXiv:2604.21919v1", "equation": "W = Wnear \u222a Wfar , where clusters in\nWnear are distance at most a cutoff Rth away from B", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "27683e9321b321027ca4f0c54d4ea61b39364f5274f87030abef1d1507f58c76", "matrix_8x8": [ [ 1, 0, 1, 1, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 1 ], [ 2, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0, 1, 0 ] ], "global_vocab_hash": "e9f5e1def8df2bc620eec270a002e5ef30a488587dd12f603135f206f5520b19", "source_records": [ { "equation_record_id": "eq_5be87eaf6cffbaae", "object_id": "rrc_eq_282beac067cdee85", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_927277d74e6531cc", "object_id": "rrc_eq_07b0a4a6f75d86b4", "name": "arXiv:2604.21919v1", "equation": "t < r and\nstarts increasing thereafter, owing to the strict lightcone\nin the message-passing dynamics [see SM for a proof]", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "7f278a691ac6b5ef3a634fd90e9bd25e008a9c0dd48b22da12fe0caf8ddc6614", "matrix_8x8": [ [ 0, 0, 0, 0, 1, 1, 1, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 2 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 2, 0 ], [ 0, 1, 2, 0, 0, 1, 0, 0 ], [ 0, 1, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 2, 0, 0, 0 ], [ 1, 0, 0, 0, 1, 0, 0, 0 ] ], "global_vocab_hash": "7f6673a68565fa944d6fa7974060772995d2a56cb3b3da4e5f682676bd515c0e", "source_records": [ { "equation_record_id": "eq_927277d74e6531cc", "object_id": "rrc_eq_07b0a4a6f75d86b4", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_353043888999f4ac", "object_id": "rrc_eq_295fb47f2cd6ddfb", "name": "arXiv:2604.21919v1", "equation": "t = r\u22121, leading\nto\n\u0010\n\u0011\n\u2225\u00b5\u22c6,\u20d7e \u2212 \u00b5\u2032\u22c6,e \u22251 = O e\u2212r/\u03be\u2217\n(13)\nestablishing locality of the BP fixed-points", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "cb5b2bdff8aba75d095d5f506ee0540735af116fa5c6c7a7b26301e82cf3e9e6", "matrix_8x8": [ [ 2, 0, 0, 1, 2, 1, 0, 0 ], [ 0, 1, 0, 0, 1, 0, 1, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 2 ], [ 2, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 2, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 1, 0, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "924d24d7cfcde3068d9c27feba1f6435fc678ab39edd13204a6f1ccae49b91fc", "source_records": [ { "equation_record_id": "eq_353043888999f4ac", "object_id": "rrc_eq_295fb47f2cd6ddfb", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_532f389442891ed2", "object_id": "rrc_eq_1d0a752af1946515", "name": "arXiv:2604.21919v1", "equation": "d = c \u2212 c0 = O(1)", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "5d231737874debdb8c296f158d2285a50c26bfd693f22452b0208df1a375e3e6", "matrix_8x8": [ [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "4728ec970cc9dbdfd254bb9eecb4bc1fbe53ddd6e91fd9ea1903c876e6725903", "source_records": [ { "equation_record_id": "eq_532f389442891ed2", "object_id": "rrc_eq_1d0a752af1946515", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_fdfae5689dd4a9a5", "object_id": "rrc_eq_3a60fca57229c15b", "name": "arXiv:2604.21919v1", "equation": "R = \u0398(1)", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "83fbe2ced5073a22733e9d83456c8189ac773627f5a79c12c4696e910326155c", "matrix_8x8": [ [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "bacf8be9edbc1522bdec945f75b424ccdacb9187e42bddb542d94e415abee56c", "source_records": [ { "equation_record_id": "eq_fdfae5689dd4a9a5", "object_id": "rrc_eq_3a60fca57229c15b", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_9138aeb942929155", "object_id": "rrc_eq_e25b46d22b7ca0f1", "name": "arXiv:2604.21919v1", "equation": "X = 1}", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "52723f809473a93ec9ec29868c4df83821817a581125514599cf9646cee324fc", "matrix_8x8": [ [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "bd05c25d91fd0de854409183652f4fc879f96e02f9c37a194af1277b7ed211c1", "source_records": [ { "equation_record_id": "eq_9138aeb942929155", "object_id": "rrc_eq_e25b46d22b7ca0f1", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_0cd087779e7532a9", "object_id": "rrc_eq_14fd73dcf1af02b8", "name": "arXiv:2604.21919v1", "equation": "i=1\n\nFor p = \u221e we have,\n\u2225x\u2225\u221e := max |xi |\n1\u2264i\u2264m\n\n(S3)\n\nFor an operator A \u2208 B(H), we define the operator Schatten norm in terms of its singular values \u03c3(A) as,\n\u2225A\u2225p := \u2225\u03c3(A)\u2225p\n\n(S4)\n\nfor any p \u2208 [1, \u221e]", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "a1d74431febc3c67322ec495ac03c78cba4df97dbaccb0deecbea58a623ceb49", "matrix_8x8": [ [ 1, 2, 0, 1, 1, 1, 2, 1 ], [ 0, 2, 1, 1, 2, 0, 1, 2 ], [ 2, 0, 0, 0, 2, 1, 0, 1 ], [ 1, 1, 1, 1, 0, 1, 1, 1 ], [ 2, 0, 1, 0, 0, 1, 0, 2 ], [ 0, 2, 0, 0, 0, 0, 1, 0 ], [ 2, 2, 2, 0, 0, 0, 1, 1 ], [ 1, 0, 1, 4, 1, 0, 2, 0 ] ], "global_vocab_hash": "84baf95e7686554df02fe0d0839795132936a6044cdc17024e39d4f0fd5f86dd", "source_records": [ { "equation_record_id": "eq_0cd087779e7532a9", "object_id": "rrc_eq_14fd73dcf1af02b8", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_5593e036fe6f7dd7", "object_id": "rrc_eq_e6a6d86ede20e5ad", "name": "arXiv:2604.21919v1", "equation": "p \u2264 q \u2264 \u221e and any operator A, we have,\n\u2225A\u2225p \u2265 \u2225A\u2225q\n\n(S5)\n\n\u2225A\u22251 \u2265 \u2225A\u22252 \u2265 \u2225A\u2225\u221e\n\n(S6)\n\nIn particular,\n\n\f10\n2", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "1be9af05f06618e0b11a5ada39278cf2f9168da217a6c334d6dd6699f5cf7e2f", "matrix_8x8": [ [ 1, 0, 0, 0, 0, 1, 1, 1 ], [ 0, 1, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 2, 0, 0 ], [ 0, 0, 0, 0, 1, 2, 0, 0 ], [ 0, 0, 0, 1, 0, 1, 0, 0 ], [ 3, 0, 1, 1, 1, 1, 1, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 1 ], [ 0, 1, 1, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "75ae09195fafbfbcc249bfc6d9f958fdc288efb1f9375b109b680be306a5d68e", "source_records": [ { "equation_record_id": "eq_5593e036fe6f7dd7", "object_id": "rrc_eq_e6a6d86ede20e5ad", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_ea9b6d38a8a0c1b0", "object_id": "rrc_eq_a5fbdc2a7f09759e", "name": "arXiv:2604.21919v1", "equation": "p \u2264 q \u2264 \u221e and any non-zero operator A,\n1\n\n1\n\n\u2225A\u2225p \u2264 rank(A) p \u2212 q \u2225A\u2225q\n\n(S7)\n\np\nrank(A)\u2225A\u2225\u221e\n\n(S8)\n\n\u2225Ax\u22252\n= \u03c3max (A)\nx\u0338=0 \u2225x\u22252\n\n(S9)\n\nIn particular,\n\u2225A\u22252 \u2264\nWe also note that, the \u221e\u2212norm obeys,\n\u2225A\u2225\u221e = sup\n\nwe will denote \u2225 \u00b7 \u2225\u221e by just \u2225 \u00b7 \u2225 for convenience", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "30b05371e315403a1c127ed3b2f1f21922a8bc191e8e3affb7751e602d64a508", "matrix_8x8": [ [ 0, 2, 2, 2, 1, 2, 1, 3 ], [ 3, 0, 1, 1, 2, 0, 0, 3 ], [ 3, 1, 0, 2, 0, 1, 0, 1 ], [ 2, 0, 1, 0, 1, 1, 1, 0 ], [ 0, 2, 0, 0, 1, 1, 2, 1 ], [ 2, 1, 1, 0, 0, 0, 1, 1 ], [ 0, 1, 1, 1, 1, 0, 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0, 0, 1, 0 ], [ 2, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "1b9f1553d70c1dd2facad18e6a70081a37db8539ac12a64d1493013640e0e465", "source_records": [ { "equation_record_id": "eq_e0537400fa963370", "object_id": "rrc_eq_b2fc5edc90c4c538", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_23fc8ebaf9c8ae4a", "object_id": "rrc_eq_05370b4783a8bd6a", "name": "arXiv:2604.21919v1", "equation": "b = (ib1 ,", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "52723f809473a93ec9ec29868c4df83821817a581125514599cf9646cee324fc", "matrix_8x8": [ [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": 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0, 0, 0, 1 ] ], "global_vocab_hash": "0732ed9011ed3cdcb284a1c12d1bc650113b6ebb9e452e39463001c6242aec63", "source_records": [ { "equation_record_id": "eq_5034975bc936419d", "object_id": "rrc_eq_74c2bb3396094843", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_ab6c35762bc86081", "object_id": "rrc_eq_5281edbc9d70f191", "name": "arXiv:2604.21919v1", "equation": "li=1 be finite-dimensional Hilbert spaces (the l virtual legs), and\nlet Hphys be the physical Hilbert space", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "3cd7703f306042bb8a15bf124df454e9d23e10ce286fee4dd685950d30a94732", "matrix_8x8": [ [ 0, 0, 2, 0, 0, 0, 0, 0 ], [ 2, 1, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 1, 2 ], [ 0, 0, 0, 1, 1, 0, 1, 0 ], [ 0, 1, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 1, 0, 0 ], [ 1, 0, 1, 0, 1, 0, 0, 0 ], [ 0, 2, 0, 0, 0, 0, 0, 0 ] 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1, 3, 0, 0, 1 ], [ 2, 2, 1, 0, 0, 0, 0, 0 ], [ 2, 1, 1, 2, 3, 1, 1, 0 ], [ 2, 3, 1, 0, 0, 0, 1, 1 ], [ 1, 1, 0, 0, 3, 1, 3, 0 ], [ 1, 1, 1, 1, 0, 0, 0, 0 ] ], "global_vocab_hash": "c879689de8c2f8176c1883bc703c4876e52f98ed0a2d258894d06236137f6a2e", "source_records": [ { "equation_record_id": "eq_7d5beabf91af593e", "object_id": "rrc_eq_04ac9264dcd7c6c1", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_8daf48607920ba7b", "object_id": "rrc_eq_4795802d1dfe8dc9", "name": "arXiv:2604.21919v1", "equation": "e = \u03b4 \u2208 (0, 1]", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "a5d6909fa28a5cddec9b40200bb1a8ba7edca6f9be00b72adccaa89fb31b392a", "matrix_8x8": [ [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "4f3b19ffc061168e3228382e379b6ed616e26360278cbf529c8f4d4d450da268", "source_records": [ { "equation_record_id": "eq_8daf48607920ba7b", "object_id": "rrc_eq_4795802d1dfe8dc9", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_3b521e5a6156cbc1", "object_id": "rrc_eq_3823a73f30463f6e", "name": "arXiv:2604.21919v1", "equation": "e = 1,\ne\n\ne\n\n(S15)\n\nWe call \u03b4 the injectivity parameter", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "feb917e37e12f62fc4e1874d4e9442dc06db81718240a73ae14a3ba2f02f8159", "matrix_8x8": [ [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, 0, 0 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runs over all connected clusters W", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "f1e847b678586528b599e6f09dc64634c54ba877a6b85d55bb25d8862e5c6062", "matrix_8x8": [ [ 0, 0, 2, 0, 1, 1, 0, 0 ], [ 2, 1, 0, 0, 0, 0, 1, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 1, 2, 0, 0 ], [ 1, 0, 0, 1, 0, 0, 0, 0 ] ], "global_vocab_hash": "64acfc4504309d0899e5710eb83b664f88d8d37dc6b265399aab7ecc11a82295", "source_records": [ { "equation_record_id": "eq_d14e04ca5ba01499", "object_id": "rrc_eq_4d8a407792ae463c", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_405936ddfb9bc98c", "object_id": "rrc_eq_68bc5f8d951cef23", "name": "arXiv:2604.21919v1", "equation": "c > c0 := log(2e\u2206) + 12 such that\n|Zl | \u2264 e\u2212c|l|\n\n(S27)\n\nthen, the series for log Z converges absolutely", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "1eede741ed376299ffc635c953ed0c13b7e436f37c8fb7893ccc9398742f2fbb", "matrix_8x8": [ [ 0, 1, 0, 0, 1, 1, 1, 0 ], [ 0, 0, 1, 1, 0, 1, 1, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 2 ], [ 1, 0, 1, 0, 0, 1, 0, 0 ], [ 1, 2, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 1, 0, 1, 1, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, 1, 0, 0, 0 ] ], "global_vocab_hash": "578dc0e0d67ec50bde4c921a650c453b8fadb2aa77081ac3fe3dbaf5f2180801", "source_records": [ { "equation_record_id": "eq_405936ddfb9bc98c", "object_id": "rrc_eq_68bc5f8d951cef23", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_6a72df8385b3a8fb", "object_id": "rrc_eq_181e73a390899053", "name": "arXiv:2604.21919v1", "equation": "Fm = log ZBP +\n\nX\n\n\u03d5(W)ZW ,\n\n(S28)\n\nconnected W\n|W|\u2264m\n\nis 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W\u2190LA\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8fe\n\uf8f3supp(W)\u2229A\u0338=\u2205\n|W|\u2264m\n\nleads to a relative error \u03b4m = | \u27e8OA \u27e9 \u2212 \u27e8OA \u27e9m |/| \u27e8OA \u27e9 | bounded by\n\u0010\n\u0011\n\u03b4m \u2264 O |A|e\u2212(c\u2212c0 )(m+1)\n\n(S33)\n\nwhere d = c \u2212 c0 = O(1)", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "59c7165e67d5003b3f733535d61f7665aa1cf59c1593940423f8e753d5c44bab", "matrix_8x8": [ [ 2, 0, 2, 1, 2, 0, 0, 1 ], [ 3, 0, 0, 2, 0, 0, 2, 0 ], [ 0, 3, 0, 3, 1, 0, 0, 0 ], [ 0, 1, 2, 1, 1, 5, 0, 0 ], [ 1, 2, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 2, 1, 1, 4, 0, 1 ], [ 1, 0, 0, 0, 0, 1, 2, 1 ], [ 0, 2, 0, 1, 0, 0, 0, 0 ] ], "global_vocab_hash": "e140dddd69e77cc396df2becf5122439910ace59620d404b764aaff81a059af8", "source_records": [ { "equation_record_id": 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"equation": "A = (A1 ,", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "52723f809473a93ec9ec29868c4df83821817a581125514599cf9646cee324fc", "matrix_8x8": [ [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "c8a60a1546f76e608784ca63a2229034624960f92684af2c6d0a7d2b80d2116d", "source_records": [ { "equation_record_id": "eq_2edfb68a37ab3d1f", "object_id": "rrc_eq_068487b9141c4fb6", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_7957e0f2cb86a72c", "object_id": "rrc_eq_8ffeff0c4877aa25", "name": "arXiv:2604.21919v1", "equation": "degree \u2265 2 everywhere except in all regions Ai", "schema": "rrc_pist_predictions_250_v1", 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"rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "e3faf033984d2d8301ff20ce7680f56db6999d0475a8ad5b18d28d1d351b8d40", "matrix_8x8": [ [ 2, 1, 1, 1, 3, 0, 0, 0 ], [ 0, 3, 2, 0, 0, 1, 0, 1 ], [ 2, 1, 0, 0, 0, 1, 0, 1 ], [ 3, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 3, 0 ], [ 0, 0, 0, 2, 0, 0, 0, 0 ], [ 0, 2, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 1, 0, 0, 0 ] ], "global_vocab_hash": "77c24db651b680523f6a7aec4bdf99b8893ba0974df0046f1f0ab2fe3885d243", "source_records": [ { "equation_record_id": "eq_10a508b9d741a6f6", "object_id": "rrc_eq_2c1d4c645f0025fb", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_fb10b15989eae47c", "object_id": "rrc_eq_05407e13b79c062b", "name": "arXiv:2604.21919v1", "equation": "Z = \u27e8\u03c8|\u03c8\u27e9 can be computed to 1/ poly(N ) multiplicative error in poly(N ) time\n(ii) local observables \u27e8OA \u27e9 = \u27e8\u03c8|OA |\u03c8\u27e9/\u27e8\u03c8|\u03c8\u27e9 with \u27e8OA \u27e9 \u0338= 0 can be computed to multiplicative error \u03f5 in poly(1/\u03f5)\ntime\n(iii) 2-point correlation functions can be computed to additive error O\u0303(1/ poly(N )) in poly(N ) time\n\n\f18\nProof", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "47239fb0f2abdf6c1857c48dd161018ebbbac45518968a4052b81149e4df16c4", "matrix_8x8": [ [ 1, 2, 0, 1, 1, 6, 0, 2 ], [ 0, 2, 1, 1, 4, 1, 1, 1 ], [ 5, 1, 3, 0, 3, 0, 0, 0 ], [ 1, 1, 5, 0, 0, 0, 1, 0 ], [ 2, 0, 0, 1, 0, 3, 3, 0 ], [ 0, 5, 0, 3, 1, 1, 0, 1 ], [ 1, 0, 2, 1, 0, 0, 1, 1 ], [ 3, 0, 1, 2, 0, 0, 0, 0 ] ], "global_vocab_hash": "9f28b96214964bb553594328225cbabbd24572df47f2627ad8aaa2127220e8b3", "source_records": [ { "equation_record_id": "eq_fb10b15989eae47c", "object_id": "rrc_eq_05407e13b79c062b", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_984644a6b0dbab3e", "object_id": "rrc_eq_d4e90c1cd69ef948", "name": "arXiv:2604.21919v1", "equation": "m \u2264 N e\u2212d(m+1)\n\n(S37)\n\nHence, to ensure log Z \u2212 F\u0303m < O(1/ poly(N )) we require m = \u2126(log N )", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "a97dc8ace8e12bbd8352be4051cf76591510446de937e495882ff826df7a0bb7", "matrix_8x8": [ [ 0, 1, 0, 2, 3, 1, 0, 0 ], [ 1, 2, 1, 1, 1, 0, 0, 0 ], [ 1, 0, 1, 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 1, 1, 0, 1 ], [ 0, 2, 1, 1, 0, 2, 1, 0 ], [ 1, 2, 0, 0, 0, 0, 0, 1 ], [ 2, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 1, 0 ] ], "global_vocab_hash": "cbfcaaee1eac92abd23143081f8cb1e6c52acbd079d47e42da842a7998822ab4", "source_records": [ { "equation_record_id": "eq_984644a6b0dbab3e", "object_id": "rrc_eq_d4e90c1cd69ef948", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_837e24a6c715af63", "object_id": "rrc_eq_131b32c8ed70796c", "name": "arXiv:2604.21919v1", "equation": "m = \u2126(log N )", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "7a0b3e189372ab528f1b4630bd795a23feb30795f0ada8ba1f3ecc19362cf43d", "matrix_8x8": [ [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "09e5ffa1bc4db6a690c22f3d8beecee8629f449a5d884f3d36abf2e359395f9e", "source_records": [ { "equation_record_id": "eq_837e24a6c715af63", "object_id": "rrc_eq_131b32c8ed70796c", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_e54ad45a68cb6a93", "object_id": "rrc_eq_e6b902735f906c33", "name": "arXiv:2604.21919v1", "equation": "m\n\u2264 O e\u2212d(m+1) \u2264 \u03f5\n\u27e8OB \u27e9\n\n(S38)\n\nwhich can be ensured given m = O(log 1/\u03f5), again computable in poly 1/\u03f5 time", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "05564282cba287a5374274cb2df3042f88cd9f25cead52bbfb3b507d9b53839a", "matrix_8x8": [ [ 1, 1, 1, 0, 1, 1, 0, 2 ], [ 1, 0, 1, 1, 0, 0, 3, 0 ], [ 2, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 1, 1, 0, 1, 1, 0, 0 ], [ 0, 2, 0, 2, 0, 1, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 1, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 2, 0, 0, 0 ] ], "global_vocab_hash": "34a5318efe0e53a7383cab6d3d115f8b673e5c06a8723eb522a406cbcbe54fc5", "source_records": [ { "equation_record_id": "eq_e54ad45a68cb6a93", "object_id": "rrc_eq_e6b902735f906c33", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_21f71c6c28c71003", "object_id": "rrc_eq_c5417aae2492b416", "name": "arXiv:2604.21919v1", "equation": "m = d(A, B) + O(log 1/\u03f5), leading to an error O m2 (|A| + |B|)e\u2212dm , which is\n(note that (|A| + |B|) = O(1),\n!\n\u0012\n\u00132\n\u0012\n\u0013\n1\n1\n2\n\u2212d[d(A,B)+log 1/\u03f5]\nO d(A, B) log\n\u00b7e\n= O log N \u00b7\n= O\u0303(1/ poly(N ))\n(S39)\n\u03f5\npoly(N )\nsince we already have d(A, B) = O(log N ) and we choose \u03f5 = 1/ poly(N )", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "b8278fd18b87c4775fe986ce4f6bb56eeeb326e6a905a788b88b2dbce8a5ae7f", "matrix_8x8": [ [ 1, 9, 2, 2, 2, 1, 2, 0 ], [ 2, 6, 5, 0, 2, 6, 7, 0 ], [ 0, 0, 0, 3, 1, 2, 1, 1 ], [ 6, 1, 1, 0, 2, 0, 1, 2 ], [ 4, 2, 0, 4, 1, 1, 0, 0 ], [ 1, 7, 0, 0, 1, 0, 0, 1 ], [ 0, 4, 0, 4, 3, 0, 1, 1 ], [ 5, 0, 0, 0, 0, 0, 1, 2 ] ], "global_vocab_hash": "331ef8bce896b8d0a5ccd624f89de97ded368d083561f46458613a9e0537a6b4", "source_records": [ { "equation_record_id": "eq_21f71c6c28c71003", "object_id": "rrc_eq_c5417aae2492b416", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_717ace3c9565c03b", "object_id": "rrc_eq_c3dde450e7ce933a", "name": "arXiv:2604.21919v1", "equation": "Ka = D1L\u0338=i", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "3e2c4d683c5f8c2cefc0ef593e8db78aba11b877fd068302b56a273aab710994", "matrix_8x8": [ [ 0, 1, 1, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "a1008cfd88698fc750cdffdb5e9ccd7e4cac0f12eb699350b168e54c5d31467d", "source_records": [ { "equation_record_id": "eq_717ace3c9565c03b", "object_id": "rrc_eq_c3dde450e7ce933a", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_ccca83f09f701b34", "object_id": "rrc_eq_d75b1352c39fe13e", "name": "arXiv:2604.21919v1", "equation": "a \u2265 \u03b4 2 > 0 and\n\n\u2020\na Ka Ka = D\n\nP\n\n(S43)\n\na\n\n1,\nX\n\n\u03bb2a Ka\u2020 Ka \u2ab0 \u03b4 2\n\nX\n\na\n\nKa\u2020 Ka = \u03b4 2 D 1,\n\na\n\nhence\nTr f (X) \u2265 \u03b4 2 D Tr(X) = \u03b4 2 D", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "ace99b82eca49b82c4d500c325787f729fbb2e23d7a61966dc7c7224b8ac3676", "matrix_8x8": [ [ 2, 2, 0, 0, 0, 1, 1, 2 ], [ 0, 0, 0, 1, 1, 1, 1, 2 ], [ 2, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 2 ], [ 1, 3, 1, 0, 4, 0, 2, 0 ], [ 1, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 2, 0, 0, 2, 0, 0, 0 ], [ 1, 0, 1, 2, 3, 0, 0, 0 ] ], "global_vocab_hash": "6410f75f5f8582857ad2a045f89acf2898355df0e39ded66b850bfa00d98791e", "source_records": [ { "equation_record_id": "eq_ccca83f09f701b34", "object_id": "rrc_eq_d75b1352c39fe13e", "name": "arXiv:2604.21919v1" } ] }, { "equation_id": "eq_bc5bda52965ac0f8", "object_id": "rrc_eq_80edc83f7fb3d80e", "name": "arXiv:2604.21919v1", "equation": "e = (v, n) \u2208 E", "schema": "rrc_pist_predictions_250_v1", "claim_boundary": "matrix-only;no-classifier;no-lean-spectral", "proxy_pred": null, "exact_pred": null, "matrix_schema": "token_strand_adjacency_8x8_v1", "matrix_hash": "670b072a4227ed9f367c7427ada3d0d16ec42b2b39f2cb3e30d3a35cb6bd8ee1", "matrix_8x8": [ [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ] ], "global_vocab_hash": "a8a3d1f9205a1aff6dbd3172fbf1cfe0aa820be5a3d9162861adc9fef4e3809a", 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