{ "all_configs_exact_integral": true, "all_configs_in_regime_M_ge_d2": true, "all_configs_ordered_floor_le_flash_le_naive": true, "all_ok": true, "approx_attention": { "all_ok": true, "citation": "arXiv:2605.23751 (2026) \u2014 Approaching I/O-optimality for Approximate Attention", "compliance_oracle": { "foil_below_scan_floor_flagged": true, "valid_kernel_safe": true }, "exact_lower_bound": "I/O >= 3\u00b7N\u00b7d \u2014 every entry of Q,K,V loaded at least once (verbatim, Case I); holds for ANY algorithm, exact or approximate, any hardware", "honest_scope": "Only Case I carries an EXACT constant (3Nd, quoted); Cases II-IV are asymptotic Omega-bounds with no explicit constant \u2014 disclosed, NOT pinned. Approximation escapes the QUADRATIC exact floor down to the LINEAR input-scan floor; the oracle now spans both regimes. MODEL, NOT OTA.", "input_scan_floor_matches_pin": true, "input_scan_floor_words_flagship": 6291456, "regime": "Case I of arXiv:2605.23751 (2026): fast memory M = Omega(d\u00b7r); optimal I/O = Theta(n\u00b7d)", "savings_demo": { "M": 262144, "N": 131072, "achievable_below_exact_floor": true, "approx_scan_floor_words": 50331648, "approximation_saves_io": true, "case_I_regime_M_ge_dr": true, "d": 128, "exact_over_approx": 10.666666666666666, "exact_pebble_floor_words": 536870912, "in_savings_regime_Nd_ge_6M": true, "r": 256 }, "z3_ordering_lemma": { "name": "approx_attention_io_floor_ordering", "proven": true } }, "citations": { "approximate_regime": "arXiv:2605.23751 (2026) \u2014 approximate/sparse attention (v99b)", "flashattention": "Dao et al. 2022 (the O(N\u00b2d\u00b2/M) achievability)", "i_o_lower_bound": "Saha & Ye, arXiv:2402.07443 (2024) \u2014 The I/O Complexity of Attention", "pebble_game": "Hong & Kung, 'I/O complexity: the red-blue pebble game', 1981" }, "compliance_oracle": { "foil": { "claimed_io_words": 1048576, "verdict": "FLAG" }, "foil_below_floor_flagged": true, "note": "the converse-gated oracle: a kernel claiming sub-floor I/O is FLAGged (impossible for exact attention even with fast matmul; must disclose approximation/sparsity)", "valid": { "claimed_io_words": 4194304, "verdict": "SAFE" }, "valid_kernel_safe": true }, "declared_model": { "attention": "exact softmax(Q K\u1d40) V, Q,K,V in R^{N x d}", "cost": "red-blue pebble-game I/O: accesses to slow memory (HBM) from a fast cache of M words", "per_part_entry_bound": "each M-partition part computes <= 2M\u00b2/d\u00b2 entries of QK\u1d40 (Saha-Ye, cited premise)", "regime": "M >= d\u00b2 (the large-cache regime where FlashAttention is I/O-optimal)" }, "executed": true, "explicit_non_claim": "NOT a claim about a specific chip's measured I/O (the red-blue pebble game is a two-level-memory MODEL); the floor is for EXACT attention in the M >= d\u00b2 regime \u2014 APPROXIMATION/SPARSITY is a SEPARATE regime (arXiv:2605.23751, the v99b leg) that the oracle force-discloses, and fast matrix multiplication is COVERED (the floor holds even with FMM, via BCH codes). Lean certifies the ORDER only (the pebble per-part entry bound is a cited imported premise, never re-proven; the I/O magnitudes are the producer's exact arithmetic). A law of math is not patentable \u2014 the contribution is the machine-checked certifier + the converse-gated mechanism, not the bound itself. Booked EVIDENCE-ONLY; base byte-identical.", "flagship": { "M": 1048576, "N": 16384, "d": 128, "flash_over_floor": 2, "flashattention_io_words": 4194304, "floor_matches_pinned": true, "headline": "at N=16384 context, head-dim d=128, M=1048576 words (~2 MB fp16 SRAM), NO exact-attention kernel on any hardware can move less than 2,097,152 words \u2014 FlashAttention (2x the floor) is provably I/O-optimal; naive materialization is 259x worse; even fast matrix multiplication cannot beat the floor", "io_floor_bytes_fp16": 4194304, "io_floor_words": 2097152, "naive_io_words": 543162368, "naive_over_floor": 259 }, "flashattention_optimal_within_factor_2_all_configs": true, "grid": [ { "M": 262144, "N": 4096, "d": 64, "flash_over_floor": 2, "flashattention_io_words": 262144, "io_floor_words": 131072, "naive_io_words": 34340864, "naive_over_floor": 262, "ordered": true, "regime_M_ge_d2": true }, { "M": 262144, "N": 4096, "d": 128, "flash_over_floor": 2, "flashattention_io_words": 1048576, "io_floor_words": 524288, "naive_io_words": 35127296, "naive_over_floor": 67, "ordered": true, "regime_M_ge_d2": true }, { "M": 1048576, "N": 16384, "d": 64, "flash_over_floor": 2, "flashattention_io_words": 1048576, "io_floor_words": 524288, "naive_io_words": 540016640, "naive_over_floor": 1030, "ordered": true, "regime_M_ge_d2": true }, { "M": 1048576, "N": 16384, "d": 128, "flash_over_floor": 2, "flashattention_io_words": 4194304, "io_floor_words": 2097152, "naive_io_words": 543162368, "naive_over_floor": 259, "ordered": true, "regime_M_ge_d2": true }, { "M": 4194304, "N": 65536, "d": 128, "flash_over_floor": 2, "flashattention_io_words": 16777216, "io_floor_words": 8388608, "naive_io_words": 8615100416, "naive_over_floor": 1027, "ordered": true, "regime_M_ge_d2": true } ], "honest_scope": "An EXACT machine-checked lower bound in the declared red-blue pebble-game two-level-memory model for EXACT attention with M >= d\u00b2: the floor N\u00b2d\u00b2/(2M) is computed in exact rational arithmetic, the ordering is z3-certified AND kernel-checked in Lean (6 theorems), FlashAttention is optimal within factor 2, and the compliance oracle FLAGs any sub-floor claim. We own the certifier + the gated oracle, NOT the theorem (Saha-Ye's / Hong-Kung's) \u2014 Alice/Mayo. MODEL, NOT OTA.", "lean_rung": { "build_rc": 0, "credited": true, "n_v99_theorems": 16, "n_v99_verified": 16 }, "schema": "attention-io-floor-v99", "skipped": false, "wall_banner_untouched": { "checked": true, "unchanged": true }, "z3_ordering_lemma": { "name": "io_floor_ordering", "proven": true, "tier": "z3-lra" } }