{ "all_ok": true, "bridge_external": true, "brute_force_validation": { "formula_never_exceeds_true_min": true, "rows": [ { "brute_force_min": 3, "formula_floor": 2.9282032302755088, "formula_is_lower_bound": true, "n_devices": 2, "n_tokens": 4, "slack": 0.07179676972449123 }, { "brute_force_min": 6, "formula_floor": 5.211102550927978, "formula_is_lower_bound": true, "n_devices": 4, "n_tokens": 4, "slack": 0.7888974490720218 }, { "brute_force_min": 5, "formula_floor": 4.392304845413263, "formula_is_lower_bound": true, "n_devices": 2, "n_tokens": 6, "slack": 0.6076951545867368 }, { "brute_force_min": 7, "formula_floor": 6.583005244258362, "formula_is_lower_bound": true, "n_devices": 3, "n_tokens": 6, "slack": 0.416994755741638 }, { "brute_force_min": 6, "formula_floor": 5.8564064605510175, "formula_is_lower_bound": true, "n_devices": 2, "n_tokens": 8, "slack": 0.14359353944898245 }, { "brute_force_min": 11, "formula_floor": 10.422205101855956, "formula_is_lower_bound": true, "n_devices": 4, "n_tokens": 8, "slack": 0.5777948981440435 } ] }, "citations": [ "Saha & Ye (2024), arXiv:2402.07443 \u2014 I/O complexity of exact attention (single device)", "Hong & Kung (1981) \u2014 the red-blue pebble game", "Liu, Zaharia & Abbeel (2023), arXiv:2310.01889 \u2014 ring attention (an in-model schedule)" ], "composition_with_saha_ye": { "binding_axis_switches": true, "compose_by": "max (independent axes; neither implies the other)", "rows": [ { "binding_axis": "network", "composed_floor_max": 383805.4537986715, "dim": 128, "fast_memory_words": 1048576, "fraction_of_saturation": 0.3660254037844386, "n_devices": 2, "n_tokens": 4096, "network_words_floor": 383805.4537986715, "saturation_limit_2dN": 1048576.0, "single_device_io_floor": 131072.0 }, { "binding_axis": "network", "composed_floor_max": 858499.8990998077, "dim": 128, "fast_memory_words": 1048576, "fraction_of_saturation": 0.8187293044088437, "n_devices": 8, "n_tokens": 4096, "network_words_floor": 858499.8990998077, "saturation_limit_2dN": 1048576.0, "single_device_io_floor": 131072.0 }, { "binding_axis": "network", "composed_floor_max": 1024096.7514067716, "dim": 128, "fast_memory_words": 1048576, "fraction_of_saturation": 0.9766547693317142, "n_devices": 64, "n_tokens": 4096, "network_words_floor": 1024096.7514067716, "saturation_limit_2dN": 1048576.0, "single_device_io_floor": 131072.0 }, { "binding_axis": "network", "composed_floor_max": 767610.907597343, "dim": 128, "fast_memory_words": 4194304, "fraction_of_saturation": 0.3660254037844386, "n_devices": 2, "n_tokens": 8192, "network_words_floor": 767610.907597343, "saturation_limit_2dN": 2097152.0, "single_device_io_floor": 131072.0 }, { "binding_axis": "network", "composed_floor_max": 1716999.7981996154, "dim": 128, "fast_memory_words": 4194304, "fraction_of_saturation": 0.8187293044088437, "n_devices": 8, "n_tokens": 8192, "network_words_floor": 1716999.7981996154, "saturation_limit_2dN": 2097152.0, "single_device_io_floor": 131072.0 }, { "binding_axis": "network", "composed_floor_max": 2048193.5028135432, "dim": 128, "fast_memory_words": 4194304, "fraction_of_saturation": 0.9766547693317142, "n_devices": 64, "n_tokens": 8192, "network_words_floor": 2048193.5028135432, "saturation_limit_2dN": 2097152.0, "single_device_io_floor": 131072.0 }, { "binding_axis": "single_device_io", "composed_floor_max": 2097152.0, "dim": 64, "fast_memory_words": 1048576, "fraction_of_saturation": 0.3660254037844386, "n_devices": 2, "n_tokens": 32768, "network_words_floor": 1535221.815194686, "saturation_limit_2dN": 4194304.0, "single_device_io_floor": 2097152.0 }, { "binding_axis": "network", "composed_floor_max": 3433999.596399231, "dim": 64, "fast_memory_words": 1048576, "fraction_of_saturation": 0.8187293044088437, "n_devices": 8, "n_tokens": 32768, "network_words_floor": 3433999.596399231, "saturation_limit_2dN": 4194304.0, "single_device_io_floor": 2097152.0 }, { "binding_axis": "network", "composed_floor_max": 4096387.0056270864, "dim": 64, "fast_memory_words": 1048576, "fraction_of_saturation": 0.9766547693317142, "n_devices": 64, "n_tokens": 32768, "network_words_floor": 4096387.0056270864, "saturation_limit_2dN": 4194304.0, "single_device_io_floor": 2097152.0 } ] }, "declared_abstraction": { "attention": "exact; every score pair must be computed somewhere", "cost_unit": "a query row or key column costs d words to move", "in_model_note": "ring/block schedules that circulate whole K/V blocks ARE covered \u2014 they move exactly the rows/columns this bound counts", "out_of_model": [ "fast-memory capacity M (that is Saha-Ye's axis)", "V movement and the softmax normalisation pass", "PARTIAL-SUM FORWARDING (splitting one inner product across devices)", "network topology, latency, and contention" ], "sharding": "even: device p natively holds N/P query rows and N/P key columns" }, "executed": true, "explicit_non_claim": "MODEL, not a measurement. No novelty is claimed in I/O complexity theory; the underlying single-device theorem is Saha-Ye's and is cited. The contribution is the network axis, its brute-force validation, and the composition. BRIDGE-EXTERNAL: pinned counts and the modeled base untouched.", "floor_curve": { "approaches_saturation": true, "below_saturation": true, "dim": 128, "monotone_in_devices": true, "n_tokens": 4096, "rows": [ { "n_devices": 1, "words_floor": 0.0 }, { "n_devices": 2, "words_floor": 383805.4537986715 }, { "n_devices": 4, "words_floor": 683029.633555232 }, { "n_devices": 8, "words_floor": 858499.8990998077 }, { "n_devices": 16, "words_floor": 951856.312894097 }, { "n_devices": 32, "words_floor": 999814.021551799 }, { "n_devices": 64, "words_floor": 1024096.7514067716 } ], "saturation_limit_2dN": 1048576.0, "single_device_floor_is_zero": true }, "honest_scope": "A converse of the DECLARED abstraction: no in-model schedule moves fewer words. It does not bound latency, does not model topology or contention, ignores fast-memory capacity, and assumes no partial-sum forwarding.", "schema": "distributed-attention-floor-v101", "skipped": false, "summary": { "audit_note": "The bound is a COUNTING argument and is deliberately loose; its value is that it is brute-force validated (never exceeds the true integer minimum on any tested configuration) and that it opens an axis Saha-Ye does not cover. A TIGHT distributed bound is OPEN and is not claimed here.", "headline": "exact attention sharded across P devices must move at least 2dN(sqrt(1-(P-1)/P^2) - 1/P) words; at N=4096, d=128 this rises to 1.024e+06 words at P=64 and SATURATES at 2dN = 1.049e+06 \u2014 scaling out never amortises it away" }, "wall_banner_untouched": { "checked": true, "slope_db_per_decade": 5.673, "unchanged": true }, "z3_ordering_lemma": { "name": "distributed_attention_floor_ordering", "proven": true, "statement": "DISTRIBUTED-ATTENTION NETWORK FLOOR ORDERING (v101): the counting bound on inter-device words for exact sharded attention rests on AM-GM (a fixed pair product is cheapest at x=y), on CONCENTRATION being cheaper than an even split for a sum-of-squares target (so the minimiser loads one device \u2014 an even-split assumption overstates the floor), and on the saturation step s^2<=1 => s<=1, which caps the floor at 2dN for ANY device count. The floor values, the brute-force validation and the saturation point are producer numerics, not z3 facts; the bound is deliberately loose (ignores fast-memory M, assumes no partial-sum forwarding) and a tight distributed bound stays OPEN.", "tier": "z3-lra" } }