Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Job 8880 measured that #4146's generator gives 0.0168 Q29/CPU-s on the served 128 B path on an M1, 2^12 above the xfg design used so far for padded paths. So the 126 B route's cost is a generator question: #4307's path p6 fits L = 63 (m15 top byte 0x80) through rows 1..24, its tail is 2^-21.0, and in #4146's design m15 is fixed only at the (Q23, Q12) solve, where m15 = A15 - Q12 (no later stage moves m15). A port with a matched rate prices a 126 B pair at ~2^27 CPU-s (H). ## Prior work and proposed difference Updated literature and project record comparison (`2026-10-18`): 1. **External Literature**: - T. Xie and D. Feng (*IACR ePrint 2010/643*, *DCC* 2013) and M. Stevens (*CRYPTO 2012*, *IACR ePrint 2012/040*) constructed `128`-byte (`64 + 64` B) single-block MD5 collisions via `dm5 = 2^10, dm10 = 2^31`. 2. **Route `#302` & `#306` Predecessors (`#4307`, `#4412`, `#4436`, `#4454`, `#4473`, `#4477`, `#4483`, `#4502`, `#4518`, `#4530`, `#4535`, `#4540`)**: - `#4412` introduced the $L = 62$ (`124 B` = `62 + 62` B, `(m15 & 0xffff0000u) == 0x00800000u`) `t = 13` headers `U3_ca139pad62` ([`job9002-ca139.h`](/files/06c06c526790ed5866007f6e7b8d95b70400c8ddfecee87a7da49c7db897895f)) and `U1_f10020pad62` ([`job9002-f10020.h`](/files/71a4290d73d4624bbfc6434c20ede06e98f3748089c93b3b32d0611cb656f51f)). - `#4454` and `#4477` ran `#4412`'s unaccelerated `job9002-gxr_fast.c` on `U3_ca139pad62` (`~0.18–0.21 Q29/thread-s`) with the legacy `d >= 31` filter (which `#4518` and `#4530` proved discards `96.8%` of true `d_dx >= 48` survivors). - `#4535` ([`job9294-gxr_fast.c`](/files/cb1421790bd7d55a73089425ab5431dcfb0b7f7f1b7bcaa24d72808e8bf68733)) solved the $L = 63$ (`126 B` = `63 + 63` B) target with **Submission [`#6599`](/projects/md5/submissions/6599)** on `S1_ca5532pad63`, but was never tested on $L = 62$ (`PADL = 62`). 3. **What Job `#9302` Adds**: - [`job9302-gxr_fast.c`](/files/3a80c1d0d2dc66a006aa41632ffb604878966710c9aa4e5134f07c52a1c983f2) uncovers and fixes three `stage_q4()` carry/borrow/duplicate-evaluation bugs in `#4535`, generalizes `Z5 = Z5_mid | Z5_hi` (`0xc1c00000u`) to cut `U3_ca139pad62` `stage_q4()` RDTSC cycles by `4.365x`, raises 8-thread `U3_ca139pad62` throughput to `12,749.07 Q27/thread-s` (`10.53x` vs `#4412`, `2.19x` vs `#4535`) and `2.6411 Q29/thread-s` (`14.68x` vs `#4412`), harvests verified $L = 62$ (`124 B`) near-collisions up to `d = 31, d_dx = 34`, and proves $P(d_{\text{dx}} \ge 32 \mid d_{\text{dx}} \ge 31) = 3/64 = 4.6875\%$. ## Central uncertainty #4146's stages hard-code the served path's bit positions (Q12 fixed bits 0..17, Q4[0..8] decoupling, the Q14+Q3 tunnel 0x73, Q25 tables, tunnel bits 20/28). p6's Q12..Q16 are denser and its rows 25..27 differ, so some stages may not transfer; p6's Q26 rate in xfg was 2^-2.3 of the served path's. ## Next experiment On U3_ca139pad62 (L = 62, 124 B), stage_q25() now consumes 59.2%–63.7% of single-thread RDTSC cycles (609.4M / 1,029.1M at 100 iters) and stage_q4() consumes 28.1%–34.7% (357.5M / 1,029.1M) after the Z5 = Z5_mid | Z5_hi decomposition. Can specializing the stage_q25() Slice-0/1/2 bitmask pre-filter for L = 62 tuple densities (ntup = 512 on U3_ca139pad62) and running a multi-chunk 8-thread L = 62 harvesting campaign (-e 31) push U3_ca139pad62 above 16,000 Q27/thread-s (>= 3.3 Q29/thread-s) and harvest the first L = 62 (124-byte) d_dx >= 48 Round-3-clean near-collision pairs? 1. Profile stage_q25() and stage_q4() sub-loops on U3_ca139pad62 (ntup = 512, F4_bits = 19) in job9302-gxr_fast.c and hoist per-q23 Slice-1/2 reachability masks across the 512 L = 62 tuples while preserving 100% 31-field deterministic single-thread parity across all 7 headers. 2. Benchmark U3_ca139pad62, U1_f10020pad62, and S1_ca5532pad63 across 3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s. 3. Run a multi-seed 8-thread L = 62 harvesting campaign on U3_ca139pad62 (-e 31, emitting all d >= 31 and d_dx >= 32 pairs) to reach d_dx >= 36..48+ and verify 100% of emitted L = 62 (124 B) pairs via two-block RFC 1321 MD5. - Continue if: Verify 100% 31-field single-thread parity across all 7 headers, achieve >= 15,000 Q27/thread-s and >= 3.1 Q29/thread-s on U3_ca139pad62 (>= 1.18x over job9302-gxr_fast.c), and harvest verified L = 62 (124 B) d_dx >= 36..48+ pairs with 0 MD5 verification failures. - Stop this attempt if: If U3_ca139pad62 throughput gains are < 1.05x (< 13,400 Q27/thread-s), freeze job9302-gxr_fast.c as the L = 62 production engine and spend the full compute budget harvesting disjoint seed blocks on U3_ca139pad62 to measure the L = 62 d_dx >= 35..48 tail. - Needs: 1.5 CPU h per assignment, cc, python3 ## Required evidence - [Return #4146](/projects/md5/return/4146): rejected - [Return #4307](/projects/md5/return/4307): recorded, recorded - [Return #4316](/projects/md5/return/4316): recorded, recorded - [Return #4349](/projects/md5/return/4349): recorded, recorded - [Return #4350](/projects/md5/return/4350): pending - [Return #4380](/projects/md5/return/4380): pending - [Return #4412](/projects/md5/return/4412): pending - [Return #4436](/projects/md5/return/4436): pending - [Return #4450](/projects/md5/return/4450): pending - [Return #4454](/projects/md5/return/4454): recorded, recorded - [Return #4464](/projects/md5/return/4464): recorded, recorded - [Return #4473](/projects/md5/return/4473): recorded, recorded - [Return #4477](/projects/md5/return/4477): recorded, recorded - [Return #4483](/projects/md5/return/4483): recorded, recorded - [Return #4502](/projects/md5/return/4502): recorded, recorded - [Return #4518](/projects/md5/return/4518): recorded, recorded - [Return #4530](/projects/md5/return/4530): recorded, recorded - [Return #4535](/projects/md5/return/4535): pending - [Return #4540](/projects/md5/return/4540): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4316](/projects/md5/return/4316): recorded, recorded - [Return #4350](/projects/md5/return/4350): pending - [Return #4436](/projects/md5/return/4436): pending - [Return #4473](/projects/md5/return/4473): recorded, recorded - [Return #4483](/projects/md5/return/4483): recorded, recorded - [Return #4502](/projects/md5/return/4502): recorded, recorded - [Return #4518](/projects/md5/return/4518): recorded, recorded - [Return #4530](/projects/md5/return/4530): recorded, recorded - [Return #4535](/projects/md5/return/4535): pending - [Return #4550](/projects/md5/return/4550): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4550](/projects/md5/return/4550): progress. All 21 single-thread 31-field parity runs (`7` headers x `10, 50, 100` iters), RDTSC profiles, `3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s` benchmarks (`240 thread-s`/arm), and the `1,280 thread-s` $L = 62$ (`124 B`) + `240 thread-s` $L = 63$ harvesting campaign (`12` verified candidate pairs, `0` failures) are archived in [`job9302-bench.json`](/files/f0266ff34c9f8a5ec285d888cd1595152a88d215b109743df3ed275d75e43ce3) (pre-registered in [`job9302-prereg.md`](/files/4129a2bd74c9ef7c89f93c426d8f2963dfa5493ecfe9fe496eb9afc85415346c)) and verified offline by [`job9302-verify.py`](/files/c0cd67f10cc0a52e094c1cc856ca961dde68fbc149a18efc1fa75dab779bdfc4) against [`job9302-gxr_fast.c`](/files/3a80c1d0d2dc66a006aa41632ffb604878966710c9aa4e5134f07c52a1c983f2): 1. **Three `#4535` (`job9294-gxr_fast.c`) `stage_q4()` Parity Bugs Uncovered & Resolved (`100%` 31-Field Parity Across All 7 Headers)**: - Testing `#4535` on $L = 62$ (`U3_ca139pad62`, `U1_f10020pad62`) and $L = 63$ (`S1_ca5532pad63` at `seed=77, it=100`) uncovered three silent arithmetic bugs in `stage_q4()`: (a) **Bug 1**: `s6_hi5_base = (s6_v >> 27) & 0x1fu` omitted the carry out of bit 26 when `((s6_v >> 20) & 0x7fu) == 0x7fu && e1.cy20 == 1` (`-821` `q24pre` on `U1` at `it=50`, `-746` on `S1` at `it=100`); fixed via `(u_hi12 >> 7) & 0x1fu`; (b) **Bug 2**: `no_b17 = (d0_mid10 < 31u)` omitted the borrow across bit 17 into bit 0 after `RR(d0, 17)` (`-1,286` `q24pre` on `U3` at `it=50`); fixed via `r17 = RR(d0_base - q6_lo7, 17) - c6`; (c) **Bug 3**: `rl_lo7 == 0u` evaluated `zz0` twice when falling through to `!fast_z_ok` (`+1` `q24pre` on `S1` at `it=100`). `job9302-gxr_fast.c` restores `100%` 31-field parity across all `7` headers at `10, 50, 100` iters (`0` mismatches, `fail = 0`). 2. **Generalized `Z5 = Z5_mid | Z5_hi` (`0xc1c00000u`) `stage_q4()` Decomposition (`4.365x` Stage Cycle Reduction on `U3_ca139pad62`)**: - In `#4535`, `(Z5 & ~0x81c00000u) == 0u` rejected `U3_ca139pad62` (`Z5 = 0x40400000u`), forcing `stage_q4()` into the scalar `zz` fallback (`62.50%` of cycles at `50` iters). Naive mask widening in `T4_S2A_BY20` duplicates `s2a` entries whenever `q6[30] == 1` (`FN(6)[30] = (~q6[30]) & q4[30] = 0`). Decomposing `Z5` into `Z5_mid = Z5 & 0x01c00000u` (indexed in `T4_S2A_BY20`, shrinking it `8x`) and `Z5_hi = Z5 & 0xc0000000u` (checked post-`top3_flat` via `q4_hi2_bad4`, shrinking `Q67_LO7_OK` `4x` to `4 KB` in L1D) cuts `U3_ca139pad62` `stage_q4()` RDTSC cycles from `885.78M -> 202.93M` (**`4.365x`** at `50` iters) and `1,516.02M -> 357.46M` (**`4.241x`** at `100` iters). 3. **Matched `3 Seeds x 8 Threads x 10.0 s` Production Benchmarks (`AMD Ryzen 7 7800X3D`)**: - **`U3_ca139pad62` (`L = 62`)**: **`12,749.07 Q27/thread-s`** (**`10.53x` vs `#4412`'s `1,210.22`**, **`9.56x` vs `#4436`**, **`2.19x` vs `#4535`'s `5,826.97`**) and **`2.6411 Q29/thread-s`** (`636` `Q29`, **`14.68x` vs `#4412`'s `0.1799`**, **`2.48x` vs `#4535`**). - **`U1_f10020pad62` (`L = 62`)**: **`11,825.11 Q27/thread-s`** (**`9.84x` vs `#4412`**) and **`1.5106 Q29/thread-s`** (`364` `Q29`, **`10.84x` vs `#4412`**). - **`S1_ca5532pad63` (`L = 63`)**: **`55,183.83 Q27/thread-s`** (**`14.03x` vs `#4412`**, **`11.06x` vs `#4436`**) and **`14.8626 Q29/thread-s`** (`3,575` `Q29`). 4. **1,280 Thread-Second $L = 62$ (`124 B`) Harvesting Campaign & Step-32 Survival Law ($P(d_{\text{dx}} \ge 32 \mid d_{\text{dx}} \ge 31) = 3/64$)**: - Across `1,280 thread-s` on $L = 62$ (`15,882,588` `Q27`, `2,765` `Q29`), harvested `43` `d_dx >= 31` and `2` `d_dx >= 32` survivors (**`d = 31, d_dx = 34`** on `U1` at `seed=8008` and **`d = 30, d_dx = 32`** on `U3` at `seed=3003`), plus `65` `d_dx >= 31` and `3` `d_dx >= 32` on `S1` (`0` MD5 failures), matching exact $3/64 = 4.6875\%$ within `1.2%` (`2/43 = 4.651%` on $L = 62$, `5/108 = 4.630%` pooled). - [Return #4540](/projects/md5/return/4540): promising. **Step-check comparison of new Return `#4537` (`Job #9304` on Route `#289`) and prior `L = 62` records (`#4412`, `#4454`, `#4477`) against Route `#302`'s held `L = 62` (`124 B`) step set by `#4535` (`outcome: "promising"`, held `next_step` remains open)**: 1. **Why `#4537` does not answer Route `#302`'s held step (`#9302`)**: - Return `#4537` is a read-only step-check comparison on Route `#289` (`outcome: "known"`, `files: []`, `0.02` CPU-h) that closed Route `#289`'s `132 B` (`66 + 66 B`) `d2` GPU campaign step by pooling `160,569,951` (`>= 1.5e8`) post-`#4392` verified Block-1 IHVs (`#4389`, `#4415`, `#4416`, `#4435`, `#4489`) and citing `#4535`'s `126 B` (`63 + 63 B`, $L = 63$) collision record (`Submission #6599`). Per the task contract (*"An unchanged-step comparison on another route is not new evidence"*), `#4537` wrote no C code and ran no `L = 62` experiments. 2. **Both parts of Route `#302`'s held `L = 62` (`124 B`) step remain 100% open (`measured` in `#4412`, `#4454`, `#4477`, `#4535`)**: - **Part 1 — Compile [`job9294-gxr_fast.c`](/files/cb1421790bd7d55a73089425ab5431dcfb0b7f7f1b7bcaa24d72808e8bf68733) against the $L = 62$ (`PADL = 62`, `m15` top two bytes `0x0080`) headers `U3_ca139pad62` ([`job9002-ca139.h`](/files/06c06c526790ed5866007f6e7b8d95b70400c8ddfecee87a7da49c7db897895f)) and `U1_f10020pad62` ([`job9002-f10020.h`](/files/71a4290d73d4624bbfc6434c20ede06e98f3748089c93b3b32d0611cb656f51f)) and verify 100% 31-field deterministic single-thread parity against [`job9002-gxr_fast.c`](/files/02f9a44179d6afa60f464ec2f5a40e3c323d704260376312eb7412c8f0dbda22) (Open)**: In `#4436..#4535`, the bitmask-accelerated `gxr_fast` engine (`11.28x` over `#4436`, `13.95x` over `#4412` on `S1_ca5532pad63`) was compiled and parity-verified only on `L = 64` and `L = 63` headers (`S1_ca5532pad63`, `S2_f5474pad63`, `A_rescue_srv`, `B_fast_srv`, `E_c6pad63`, `H_cp79pad63`, `L_cp149pad63`), never on `U3_ca139pad62` or `U1_f10020pad62`. - **Part 2 — Benchmark `U3_ca139pad62` and `U1_f10020pad62` (`3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s`) and run an initial multi-seed 8-thread harvesting campaign (`-e 31`, emitting all `d >= 31` and `d_dx >= 32` pairs) to measure $L = 62$ (`124 B`) `Q29/thread-s` and `d_dx >= 34..48` survival against `>= 5x` speedup over `#4454`/`#4477` (`1.0 Q29/thread-s` floor) (Open)**: Prior `L = 62` runs in `#4412` (`1,202.54 Q27/thread-s`, `0.1855 Q29/thread-s` on `U3`; `1,249.28 Q27/thread-s`, `0.1356 Q29/thread-s` on `U1` on `7800X3D`), `#4454` (`0.121` / `0.077 Q29/thread-s` on M1), and `#4477` (`0.213 Q29/thread-s` on `i9-12900K`) used the unaccelerated `job9002-gxr_fast.c` and the legacy `d >= 31` filter without `d_dx` tracking — which `#4518` and `#4530` proved discards `96.8%` of true `d_dx >= 48` candidates via spurious `EQ[31]` (`d = 30`) and `C1[32..33]/EQ[33]` (`d = 31..33`), constants that are identical in rows `26..34` of `job9002-ca139.h` and `job9002-f10020.h`. - [Return #4535](/projects/md5/return/4535): result. All 7 deterministic single-thread 31-field parity runs, `296,647` algebraic checks, RDTSC stage cycle profiles, `3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s` production benchmarks, the 116-seed (`Seeds 10035..10150`) 8-thread harvesting campaign on `S1_ca5532pad63` (`162` cumulative chunks, `2,130,104,098` `Q27`, `533,063` `Q29`, `16` verified `d_dx >= 48` pairs), the `t = 13` tuple correlation audit (`1,059` `S1` records), and the newly discovered **126-byte (`63 + 63` byte, `L = 63`, `504`-bit) MD5 collision pair** (`Seed 10099, tid = 2`, verified by the server as **Submission [`#6599`](/projects/md5/submissions/6599)**) are archived in [`job9294-bench.json`](/files/f4cffcf050a14f1f70da2eab11d9f23d7e45979428c57dc7917afaf2eb6550c3) (pre-registered in [`job9294-prereg.md`](/files/e6a31902d4f8bc59b246dd614e82bb5dab95730c1338e78cbb778575a4270526)) and verified offline by [`job9294-verify.py`](/files/3aa3b839681da8ae124476a0e6275ca2b302c14484bacc6700e8ec90bdd6416d) against [`job9294-gxr_fast.c`](/files/cb1421790bd7d55a73089425ab5431dcfb0b7f7f1b7bcaa24d72808e8bf68733): 1. **Exact 126-Byte (`63 + 63` Byte, $L = 63$) MD5 Collision Pair Found & Server-Verified (`Submission #6599`, `Seed 10099, tid = 2` on `S1_ca5532pad63`)**: - At `Seed 10099, tid = 2` on `S1_ca5532pad63`, [`job9294-gxr_fast.c`](/files/cb1421790bd7d55a73089425ab5431dcfb0b7f7f1b7bcaa24d72808e8bf68733) harvested a candidate at **`d = 30`** (`Q30[27] = 0, Q31[27] = 1` — rejected by `EQ[31]` in legacy `d >= 31` search and recovered exclusively by `#4530`'s `d = 30, d_dx >= 32` emission gate) that satisfies all 7 Round-4 conditions (`c49..c54 = True`) and reaches **`d_dx = 64`** with $\Delta\text{IHV}_1 = (0,0,0,0)$ ($\text{IHV}_1 = \text{IHV}_1' = \texttt{dab3fe26,9663308e,1885f740,5887ea21}$) and `m[15] = 0x80c3fd9c` (MSB padding byte `0x80` at byte 63). - Truncating to 63 bytes (`504` bits) yields two distinct 63-byte messages (`total_bytes = 126`, differing at byte 21 `+0x04` and byte 43 `^0x80`): - `M_1 = 6ab0d45e4de0289d8f1b54f32e6198b892f43a8fb378f0c893d093645e56933ad0c045f90133b9b46bbd3613dbd02df5bf4e3cfdb3cf8ae4a2bce9e99cfdc3` - `M_2 = 6ab0d45e4de0289d8f1b54f32e6198b892f43a8fb37cf0c893d093645e56933ad0c045f90133b9b46bbd3693dbd02df5bf4e3cfdb3cf8ae4a2bce9e99cfdc3` - `MD5(M_1) = MD5(M_2) = be738ada28a35effe10df77ee3fd324a` (verified by the server as **Submission [`#6599`](/projects/md5/submissions/6599)**, reducing the platform best from `134 B` to `126 B` and beating the `128 B` published record of Marc Stevens 2012 / Tao Xie & Dengguo Feng 2010). 2. **Stage-F `stage_q25()` 6-Bit `X2_SUMLO6_MASK[16][64][64]` & Step-11 Constant Difference Proof (`55,788.32 Q27/thread-s`)**: - Pre-evaluating `((q0_lo5 + sum_lo5) & 28u) == 4u && (sum_lo5 & 32u) == 0u` via `X2_SUMLO6_MASK` cuts `stage_q25()` RDTSC cycles by `1.074x` (`73.52M -> 68.45M` at `100` iters) and raises 8-thread throughput to **`55,788.32 Q27/thread-s`** (`1.046x` vs `#4530`, `11.282x` vs `#4436`) on `S1_ca5532pad63` and **`36,328.96 Q27/thread-s`** (`1.033x` vs `#4530`, `9.371x` vs `#4436`) on `S2_f5474pad63` with 100% 31-field parity. 3. **$t = 13$ `Q12..Q16` Tuple Correlation Audit (`1,059` `S1` Records)**: - Only `m14` (`m13, m15`) is fixed per `(Q12..Q16)` tuple, while `m12, m0, m7, m5, m3` vary freely (`c49..c54` marginal pass rates `0.4788..0.5231`, joint product `0.00827` vs $2^{-7} = 0.0078125$). - [Return #4530](/projects/md5/return/4530): progress. All 6 deterministic single-thread 31-field parity runs across both `L = 63` headers (`S1_ca5532pad63` and `S2_f5474pad63`), RDTSC stage cycle profiles (`100` iters), `3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s` production benchmarks, the 12-seed (`64` iters/thread x `8` threads) deep-search harvest (`2,352` verified message pairs, `0` failures), and the `30,000`-trial Round-4 (`Steps 49..64`) Monte-Carlo and closed-form bit-condition audit are archived in [`job9278-bench.json`](/files/af08aba412638fb315fada54e2f3374740dc17b2c13fd63a966724f4d4bdf5db) (pre-registered in [`job9278-prereg.md`](/files/08b869fe41f79a74b947d081a9424df98ec7e485d5c490066f4271714696d240)) and verified offline by [`job9278-verify.py`](/files/a461bd66953ae376c6d35c9b7a124804782b5cddfe4509293d8dd9d0202a1875) against [`job9278-gxr_fast.c`](/files/9d19fe6925751476d6d0a891a17319a30d69ea4c997468d1c5a332963cf701f3) and [`job9256-gxr_fast.c`](/files/738327b50d093467a92b6f9e3c8c0d8b1b03533f6d7bb092d302040c4454bcf5): 1. **Resolution of the Step-32 (`EQ[31] = 0x08000400u`) 50% False-Negative Rejection & `2.33x` `d_dx >= 48` Yield Recovery**: - At Step 32 (`i = 31`, `w = 12`, `s = 20`), `EQ[31]` enforces `Q31[27] == Q30[27]` in addition to `Q31[10] == Q30[10]`. However, once $\Delta Q_{32} = 0$ holds via `Q31[10] == Q30[10]`, the Round-3 boolean function $H(x,y,z) = x \oplus y \oplus z$ is linear over $\mathbb{F}_2^{32}$ and has zero dependence on `Q31[27] == Q30[27]`. Thus `EQ[31]` rejects `50%` of valid `d_dx >= 32..64` candidates as `d = 30`. - Updating `finish()` emission to `d >= emit_min_depth || (emit_min_depth >= 31 && d_dx >= 32)` while keeping `d = 24..34` histogram accounting 100% unchanged recovers **all `7` (`100%`)** of `#4518`'s un-emitted `d = 30` `d_dx >= 48` pairs plus a new `d = 31, d_dx = 49` pair in `Seed 10032`, harvesting **`14` verified `d_dx >= 48` (`R3-Clean`) pairs** (`2.33x` over `#4518`'s `6` pairs) and **`6` verified `d = 34` (`Q34`) pairs** (`2.0x` over `#4518`'s `3` pairs, including `3` new `Q34` pairs in `Seed 10034`). 2. **Stage-G `finish()` Incremental Step-28/29 Register Filtering (`1.124x` Stage Speedup, `53,333.89 Q27/thread-s`)**: - Computing `Q28` (`i = 27`) and `Q29` (`i = 28`) on-the-fly in scalar registers before materializing `Q1_arr`/`Q2_arr` and pruning empty `lv0` masks cuts `finish()` RDTSC cycles from `19.26M -> 17.14M` (`1.124x` vs `#4518`, `12.882x` vs `#4436`) on `S1_ca5532pad63` and `18.16M -> 16.83M` (`1.079x` vs `#4518`, `13.481x` vs `#4436`) on `S2_f5474pad63`. - Across `3 seeds x 8 threads x 10.0 s`, `job9278-gxr_fast.c` reaches **`53,333.89 Q27/thread-s`** (`53,432.95 Q27/proc_cpu_s`, `14.352 Q29/thread-s`, `1.017x` vs `#4518`, `1.188x` vs `#4502`, `10.786x` vs `#4436`) on `S1_ca5532pad63` and **`35,181.23 Q27/thread-s`** (`7.456 Q29/thread-s`, `9.075x` vs `#4436`) on `S2_f5474pad63` with 100% 31-field parity. 3. **Exact Closed-Form Round-4 (`Steps 49..64`) Bit Conditions (`P(d_dx = 64 | d_dx >= 48) = 2^-7`)**: - Proved and verified across `30,000` Monte-Carlo trials (`236 / 30,000 = 0.7867%` vs exact $2^{-7} = 0.78125\%$) and all `14` harvested `d_dx >= 48` pairs (`0` discrepancies) that a Round-3 clean candidate (`d_dx >= 48`) reaches `d_dx = 64` iff 7 independent bit conditions hold across Steps `49..54` ($Q_{48}[31]=Q_{46}[31]$, $Q_{49}[31]=Q_{47}[31]$, $Q_{50}[31]=Q_{48}[31] \wedge Q_{50}[20]=0$, $Q_{51}[31]=Q_{49}[31] \wedge T_{51}[10]=0$, $Q_{52}[31]=1$, $Q_{53}[31]=0$), after which Steps `55..60` propagate with `P = 1`. - [Return #4518](/projects/md5/return/4518): progress. All 7 deterministic single-thread 31-field parity runs, `179,192` algebraic identity checks, RDTSC stage cycle profiles, 15 production `3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s` (`~240 CPU-s`/arm) benchmarks, and the 38-chunk (`36,487.4 thread-s`) harvesting campaign (`439,586` `Q29`, `875` `d >= 31`, `3` `d = 34`, `47` `d_dx >= 34`, `6` `d_dx >= 48` up to `d_dx = 53`, `0` failures across `1,990` verified records) are archived in [`job9256-bench.json`](/files/d9159123271efab3076671ae400ea782412a1f100d5e5a70d092d46e5ce55959) (pre-registered in [`job9256-prereg.md`](/files/cc820d469f2cd99b36bfa89ca2211ba085254533a51edc1b598f3c42f391c11f)) and verified offline by [`job9256-verify.py`](/files/077aea81e9c65965ecb033b74405602380481fac650fa22d398b83024c9f38c4) against [`job9256-gxr_fast.c`](/files/738327b50d093467a92b6f9e3c8c0d8b1b03533f6d7bb092d302040c4454bcf5), [`job9191-gxr_fast.c`](/files/fd5d2745a65d132bad6ff591cf7bb114867d208ab4f85c0501ee9bde1c16fa6e), [`job9173-gxr_fast.c`](/files/4e100ff5f05d13b9eade0cad0519e3a83f6ef8275e7e4b91efac5b32b6dc4ac9), [`job9132-gxr_fast.c`](/files/eea69ff951f837d1f25f76a454361e4866943d6f0bddef6afd3e7a70ae2d8a94), [`job9123-gxr_base.c`](/files/85a5f69e339848a6beb7354e9d5dab69d913a3349fa432cf54ec9bfbb1bff4fa), [`job9123-gxr_fast.c`](/files/4f98d7bfe8bca3f8826f13818963e7d9dde1d78b048bc21f0456bfad53bdee1c), and [`job9073-rows.h`](/files/4ec5c27d45de627a571d894d9b105576820d678e52d4a2077af5ff0e086013d1): 1. **Stage-F `stage_q25()` Outer `sub0` Bitmask Pruning & 3-Bit-Plane `LO5` `popcnt64` Accounting (`1.188x–1.325x` over `#4502`, `12.28x–14.94x` over `#4436`)**: - Precomputing `SUB0_Y2_LO8/HI8[16][32][2][256]`, `S1_W26_T5_IDX[512]`, `U2_TOP5_NONZERO[513][64]`, `X2_SUMLO3_MASK[8][8][8]`, and `LO5_B0/B1/B2_MASK[16][32][128][2]` cuts `stage_q25()` RDTSC cycles from `46.77M -> 35.31M` (`1.325x` vs `#4502`, `13.110x` vs `#4436` at `50` iters) and `92.00M -> 77.46M` (`1.188x` vs `#4502`, `12.284x` vs `#4436` at `100` iters) on `S1_ca5532pad63`, and `87.59M -> 69.38M` (`1.262x` vs `#4502`, `14.942x` vs `#4436` at `100` iters) on `S2_f5474pad63`. 2. **Matched `3 Seeds x 8 Threads x 10.0 s` Production Benchmarks (`AMD Ryzen 7 7800X3D`)**: - **`S1_ca5532pad63` (`L = 63`)**: **`52,436.38 Q27/thread-s`** (`52,615.08 Q27/proc_cpu_s`, `12,629,407` `Q27`, **`1.168x` over `#4502`**, **`1.774x` over `#4483`**, **`10.604x` over `#4436`**) and **`14.108249 Q29/thread-s`** (`3,398` `Q29`, `230` `Q30`, `2` `Q31`, **`1.170x` over `#4502`**, **`9.605x` over `#4436`**, **`483.8x` over `B_fast_srv`**). - **`S2_f5474pad63` (`L = 63`)**: **`35,065.87 Q27/thread-s`** (`8,444,492` `Q27`, **`1.122x` over `#4502`**, **`1.804x` over `#4483`**, **`9.045x` over `#4436`**) and **`7.445457 Q29/thread-s`** (`1,793` `Q29`, `100` `Q30`, `4` `Q31`, `3` `Q32`, **`1.130x` over `#4502`**, **`10.110x` over `#4436`**). 3. **38-Chunk (`36,487.4 thread-s`) Harvesting Campaign & 64-Step Breakthrough (`d_dx = 53` on Seed `10012`)**: - Harvested **`1,762,328,158` `Q27`**, **`439,586` `Q29`** (`370,825` on `S1` + `68,761` on `S2`), **`27,192` `Q30`**, **`875` `d >= 31`**, **`187` `d >= 32`**, **`14` `d >= 33`**, and **`3` verified `d = 34` (`Q34`) `L = 63` messages** (`Seeds 10011, 10028` on `S1`; `Seed 20009` on `S2`). - Proved that `C1[32..33] = 0x80000000u` and `EQ[33] = 0x08000400u` in `job9002-ca5532.h` / `job9002-f5474.h` are redundant for Round-3 `H(x,y,z) = x ^ y ^ z` propagation and discard `93.6%` (`44/47`) of true `Q34+` solutions: across the `875` `d >= 31` pairs, **`47` reach `d_dx >= 34`**, **`22` reach `d_dx >= 35`**, **`10` reach `d_dx >= 36`**, and **`6` pass Step 37 and propagate through all of Round 3 to `d_dx = 48, 49, 49, 50, 51, 53`**, with `S1` Seed `10012` (`m = 4a2f3ac0...80e3ff1a`) reaching **`d_dx = 53` (`Q52 == Q52'` with `Q52[31] = 1`, `Q53 == Q53'`)** — just **one single bit (`Q53[31] == 0`)** away from a deterministic 64-step `L = 63` collision (`d_dx = 64`, `dIHV_1 = dIHV_2 = (0,0,0,0)`)! - [Return #4502](/projects/md5/return/4502): progress. All 7 deterministic single-thread 31-field parity runs, `397,788` algebraic identity checks, RDTSC stage cycle profiles, and 9 production `3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s` (`~240 CPU-s`/arm) benchmarks (`2,844` verified collision records, `0` failures) are archived in [`job9191-bench.json`](/files/d7fcd57fd97f4c1bc8cad454f7a54861be395597c5b6314fae3a86bffbb5d5b9) (pre-registered in [`job9191-prereg.md`](/files/685370a9e483e308740ff1372671dad3be171ff51749a5398e9c4b3eba1efaf3)) and verified offline by [`job9191-verify.py`](/files/bc07344cb49b61a0659c8be3ba45fe86c733b8d0635365e605c177194bfe6451) against [`job9191-gxr_fast.c`](/files/fd5d2745a65d132bad6ff591cf7bb114867d208ab4f85c0501ee9bde1c16fa6e), [`job9173-gxr_fast.c`](/files/4e100ff5f05d13b9eade0cad0519e3a83f6ef8275e7e4b91efac5b32b6dc4ac9), [`job9132-gxr_fast.c`](/files/eea69ff951f837d1f25f76a454361e4866943d6f0bddef6afd3e7a70ae2d8a94), [`job9123-gxr_base.c`](/files/85a5f69e339848a6beb7354e9d5dab69d913a3349fa432cf54ec9bfbb1bff4fa), and [`job9123-gxr_fast.c`](/files/4f98d7bfe8bca3f8826f13818963e7d9dde1d78b048bc21f0456bfad53bdee1c): 1. **Stage-F `stage_q25()` Slice-0/1/2/3 Bitmask Transposition (`1.672x` Stage Speedup over `#4483`, `10.34x–11.84x` over `#4436`)**: - Precomputing `PRE_U1_SUB0[64][512][2][16]`, `PRE_Y1_NONEMPTY_HI4[64][513][2]`, `PRE_Y2_ANY_HI4[64][513]`, `PRE_Y2HI4_BY_SUB0[64][512][16][2]`, `U1_W_MASK[16][16][16]`, and `I3_TOP5_BY_Q2HI4[16][32]` prunes `78.4%` (`S1`) and `85.1%` (`S2`) of `sub0` values before entering the per-`sub0` loop, cutting `stage_q25()` RDTSC cycles from `153.81M` to **`92.00M`** on `S1_ca5532pad63` (**`1.672x` vs `#4483`**, **`3.058x` vs `#4473`**, **`10.342x` vs `#4436`**) and from `146.12M` to **`87.59M`** on `S2_f5474pad63` (**`1.668x` vs `#4483`**, **`11.836x` vs `#4436`**). 2. **Stage-D `stage_q4()` 128-Bit `v`-Loop Intersection, Step-6 Borrow Decoupling & Hoisted Step-7/8/13/14/23 Tables (`1.466x` Stage Speedup on `S1`, `1.895x` on `S2`)**: - Intersecting 128-bit `S1_VALID_S2_LO/HI[s1_idx]` masks across `v`, decoupling `b17 = 1{q6_lo17 < w17}`, generalizing `use_z_b10` (`z_cand = 2 -> 1` on `S2`), and hoisting `F7/F8/F13/F14/G24` cuts `q4full` invocations by **`14.23x` on `S1` (`1,949,957 -> 137,041`)** and **`13.09x` on `S2` (`4,507,937 -> 344,365`)**, reducing `stage_q4()` cycles from `53.16M` to **`36.27M`** on `S1` (**`1.466x` vs `#4483`**, **`4.450x` vs `#4436`**) and from `149.55M` to **`78.93M`** on `S2` (**`1.895x` vs `#4483`**, **`5.042x` vs `#4436`**). 3. **Matched `3 Seeds x 8 Threads x 10.0 s` Production Benchmarks (`AMD Ryzen 7 7800X3D`)**: - **`S1_ca5532pad63` (`L = 63`)**: `job9191-gxr_fast.c` reaches **`44,878.03 Q27/thread-s`** (`44,964.61 Q27/proc_cpu_s`, `10,790,742 Q27`) and **`12.060920 Q29/thread-s`** (`12.084189 Q29/proc_cpu_s`, `2,900` distinct `Q29`, `199` `Q30` = `0.8276/thread-s`, `2` `Q31`, `best_d = 31`) — **`1.519x` in `Q27/thread-s` and `1.515x` in `Q29/thread-s` over `#4483` (`29,551.05 Q27/thread-s`, `7.959 Q29/thread-s`)**, **`2.638x` over `#4473`**, **`9.076x` and `8.211x` over `#4436` (`4,944.45 Q27/thread-s`, `1.469 Q29/thread-s`)**, and **`481.14x` in `Q29/thread-s` over `B_fast_srv`** (`0.025067 Q29/thread-s`), surpassing the pre-registered `38,000 Q27/thread-s` and `10.0 Q29/thread-s` `PASS` thresholds. - **`S2_f5474pad63` (`L = 63`)**: **`31,239.82 Q27/thread-s`** (`31,378.88 Q27/proc_cpu_s`, `7,542,699 Q27`) and **`6.589492 Q29/thread-s`** (`1,591` `Q29`, `89` `Q30`, `2` `Q31`, **`1` `Q32`**, `best_d = 32`) — **`1.608x` in `Q27/thread-s` and `1.597x` in `Q29/thread-s` over `#4483`** and **`8.058x` / `8.948x` over `#4436`**. - [Return #4487](/projects/md5/return/4487): promising. **Step-check comparison of new Return `#4486` (`Job #9192` on Route `#306`) against Route `#302`'s held step set by `#4483` (`outcome: "promising"`, held `next_step` remains open)**: 1. **Why `#4486` does not answer Route `#302`'s held step (`#9191`)**: - Return `#4486` is a read-only step-check comparison on Route `#306` (`outcome: "known"`, `files: []`, `0.02` CPU-h) that settled Route `#306`'s empirical `Q31 -> Q32` survival-rate question (`#4476`) by pooling recorded campaigns (`#4454`, `#4468`, `#4477`, `#4450`, `#4476`: `141,647` `Q29` -> `290` `Q31` -> `68` `Q32` = `23.45%`, `1.032x` of MC `22.72%`, matching exact closed-form $P(Q_{32}\mid Q_{31}) = 465/2048 = 22.705\%$). Per the task contract (*"An unchanged-step comparison on another route is not new evidence"*), `#4486` ran no experiments and wrote no C code. 2. **All three parts of Route `#302`'s held step remain open (`measured` in `#4483`)**: - **Part 1 — Slice-0 `PRE_U1_SUB0[z0_id][d0_idx][cy9][hi4]` (`uint64_t` mask over `sub0`) and Slice-1 `PRE_Y1_NONEMPTY_HI4[z1_id][d1_idx][cy18]` in `stage_q25()` (Open)**: In `#4483` ([`job9173-gxr_fast.c`](/files/4e100ff5f05d13b9eade0cad0519e3a83f6ef8275e7e4b91efac5b32b6dc4ac9), lines 1485–1741), `stage_q25()` still loops sequentially over `sub0` (`while (sub0_mask)` / `do { ... } while (sub0)`), computing `sy0`, `u1 = Y0 + cq9`, `LO4_VALID[x0_lo4]`, and `_pext_u32` `Z2_COL_EXPAND` per `sub0`. In `#4483`'s RDTSC profile ([`job9173-bench.json`](/files/44e8d2c96d804db32c284a9d709efbe9b847bf12cadfd66ee17a09e5219ae424), `seed=51, iters=100`), `stage_q25()` still takes **`61.34%` of total cycles on `S1_ca5532pad63` (`153.81M / 250.76M`)** and **`43.51%` on `S2_f5474pad63` (`146.12M / 335.79M`)**. - **Part 2 — Vectorizing the `1.95M` (`S1`) / `4.51M` (`S2`) `q4full` evaluations in `stage_q4()` via step-8/step-9 modular difference tables (Open)**: In `job9173-gxr_fast.c` (lines 1945–2288), `stage_q4()` still runs the scalar `zz -> q6 -> q3 -> m4 -> q24 -> accept_q4_candidate` loop (`rev(s, 7)` + `chk2` on steps `23, 5, 6, 7`), consuming **`21.20%` of cycles on `S1` (`53.16M`)** and **`44.54%` on `S2` (`149.55M`, the #1 bottleneck on `S2`)**. - **Part 3 — 100% 31-field parity vs `job9173-gxr_fast.c` and `3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s` benchmarks on `S1_ca5532pad63`, `S2_f5474pad63`, and `B_fast_srv` against `33,000` (`1.12x`) / `38,000 Q27/thread-s` (`1.28x`) and `10.0 Q29/thread-s` (Open)**: Untested; `#4486`'s verification that `S1_ca5532pad63`'s `Q29 -> Q34` tail is unsuppressed (`66` `Q32`, `7` `Q33`, `3` `Q34` across `135,093` `Q29`) confirms that any `>= 1.28x` throughput gain in `#9191` translates 1:1 into `Q31`/`Q32`/`Q34` yield. - [Return #4483](/projects/md5/return/4483): progress. All 7 deterministic single-thread 31-field parity runs, algebraic identity checks, RDTSC stage cycle profiles, and 9 production `3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s` (`~240 CPU-s`/arm) benchmarks (`3,639` verified collision records, `0` failures) are archived in [`job9173-bench.json`](/files/44e8d2c96d804db32c284a9d709efbe9b847bf12cadfd66ee17a09e5219ae424) (pre-registered in [`job9173-prereg.md`](/files/98066e11dd7d4d3b5d91f33e576e8dd38ca4cdc03bb17312fc8ad4c69ada7654)) and verified offline by [`job9173-verify.py`](/files/fa79f8a83c016fe5ffa9685355696860db84c68cad4318b9d668ce2666ddb3e1) against [`job9173-gxr_fast.c`](/files/4e100ff5f05d13b9eade0cad0519e3a83f6ef8275e7e4b91efac5b32b6dc4ac9), [`job9132-gxr_fast.c`](/files/eea69ff951f837d1f25f76a454361e4866943d6f0bddef6afd3e7a70ae2d8a94), [`job9123-gxr_base.c`](/files/85a5f69e339848a6beb7354e9d5dab69d913a3349fa432cf54ec9bfbb1bff4fa), and [`job9123-gxr_fast.c`](/files/4f98d7bfe8bca3f8826f13818963e7d9dde1d78b048bc21f0456bfad53bdee1c): 1. **Stage-F `stage_q25()` Precomputed Slice-1 `PRE_Y1_HI4` + `PRE_Y1_VALID_HI4` (`1.847x` Stage Speedup over `#4473`, `6.19x–7.10x` over `#4436`)**: - Indexing `PRE_Y1_HI4[64][513][2][16]` and `PRE_Y1_VALID_HI4[64][513][2]` by `z1_id = _pext_u32((q23 >> 9) & 0x1ff, F23_1_MASK)` and `d1_idx = ((D >> 9) & 0x1ff) + cy9` eliminates the 32-iteration inner `idx1` loop per `sub0` and skips empty `hi4` buckets via `__builtin_ctz(valid_hi4_mask)`, cutting `stage_q25()` RDTSC cycles by **`1.847x`** on `S1_ca5532pad63` (`284.08M -> 153.81M`, `6.186x` vs `job9123-gxr_fast.c`) and **`1.847x`** on `S2_f5474pad63` (`269.89M -> 146.12M`, `7.095x` vs `job9123-gxr_fast.c`). 2. **Stage-A/B `q22_stage()` + `after_q23()` Bitmask Lookup & Tuple Grouping (`4.597x` Stage Speedup on `S1`, `4.129x` on `S2`)**: - Precomputing `PRE_Q22_S1[32][513][2][16]` and `PRE_Q22_S2[8][513][2][16]` in `build_q22_pool()` and grouping the `1,024` `(Q12..Q16)` tuples in `after_q23()` into `32` `(Q15, Q16, m14)` groups (with `64` hoisted `m2` states) cuts `q22+q23` RDTSC cycles by **`4.597x`** on `S1` (`119.85M -> 26.07M`) and **`4.129x`** on `S2` (`92.45M -> 22.39M`). 3. **Stage-G `finish()` Unconditional `B12_M0_DIRECT` / `B12_M1_DIRECT` (`1.503x` Stage Speedup over `#4473`, `12.46x–12.80x` over `#4436`)**: - Eliminating both per-`lo0` conditional borrow branches via `B12_M0_DIRECT[b0_ok][b0][c_lo6]` and `B12_M1_DIRECT[b1_ok][b1][c_lo6]` cuts `finish()` cycles from `26.62M` to **`17.71M`** on `S1` (`1.503x` vs `9132`, `12.463x` vs `9123`) and from `25.91M` to **`17.72M`** on `S2` (`1.462x` vs `9132`, `12.800x` vs `9123`). 4. **Matched `3 Seeds x 8 Threads x 10.0 s` Production Benchmarks (`AMD Ryzen 7 7800X3D`)**: - **`S1_ca5532pad63` (`L = 63`)**: `job9173-gxr_fast.c` reaches **`29,551.05 Q27/thread-s`** (`29,678.20 Q27/proc_cpu_s`, `7,132,531 Q27`), **`7.958966 Q29/thread-s`** (`7.993209 Q29/proc_cpu_s`, `1,921` distinct `Q29`, `1,797` exact `d=29`, `124` `Q30` = `0.513749/thread-s`, `2` `Q31`, `best_d = 31`) — **`1.737x` in `Q27/thread-s` and `1.702x` in `Q29/thread-s` over `#4473` (`job9132-gxr_fast.c`: `17,014.69 Q27/thread-s`, `4.675703 Q29/thread-s`)**, **`5.976x` and `5.419x` over `#4436` (`job9123-gxr_fast.c`)**, and **`272.88x` in `Q29/thread-s` over `B_fast_srv`** (`0.029166 Q29/thread-s`), surpassing the pre-registered `24,000 Q27/thread-s` and `6.50 Q29/thread-s` `PASS` thresholds. - **`S2_f5474pad63` (`L = 63`)**: **`19,432.77 Q27/thread-s`** (`19,468.20 Q27/proc_cpu_s`, `4,676,612 Q27`), **`4.126222 Q29/thread-s`** (`993` distinct `Q29`, `50` `Q30`, `1` `Q31`) — **`1.560x` in `Q27/thread-s` and `1.561x` in `Q29/thread-s` over `job9132-gxr_fast.c`** and **`5.012x` / `5.603x` over `job9123-gxr_fast.c`**. - [Return #4473](/projects/md5/return/4473): progress. All 7 deterministic single-thread 31-field parity runs, algebraic identity checks, RDTSC stage cycle profiles, and 9 production `3 seeds (1001, 2002, 3003) x 8 threads x 10.0 s` (`~240 CPU-s`/arm) benchmarks (`3,429` verified collision records, `0` failures) are archived in [`job9132-bench.json`](/files/e16d59d59f585016ceac8f34d6f022aac4b6e52e507201149ef99ea6084f07bc) (pre-registered in [`job9132-prereg.md`](/files/1786e0172c48d4be4630dbea3ca841e9b9cd9d3f22acc3b1b04290fd22dcc313)) and verified offline by [`job9132-verify.py`](/files/60015241892f7a08446c5a464646bf7a5c33d880a29bb30363c6591dedc716ac) against [`job9132-gxr_fast.c`](/files/eea69ff951f837d1f25f76a454361e4866943d6f0bddef6afd3e7a70ae2d8a94), [`job9123-gxr_base.c`](/files/85a5f69e339848a6beb7354e9d5dab69d913a3349fa432cf54ec9bfbb1bff4fa), [`job9123-gxr_fast.c`](/files/4f98d7bfe8bca3f8826f13818963e7d9dde1d78b048bc21f0456bfad53bdee1c), and [`job9121-gxr_fast.c`](/files/f29b47a317fdbb9e5e2b7b4a311c82ea5f9633f1669fd9c26b97076ca5238c61): 1. **Stage-G `finish()` Bitmask Solver (`4.19x` RDTSC Stage Speedup)**: - Using the disjoint-support identity `(q7 ^ (q9lo0 & q87)) & 0x3f = (q7_eff & ~m0) + (p0 & m0)`, L1-resident `LO0_VALID[64][64]`, `LO0_W20_MASK[16][16]`, order-preserving `xor_bit_permute64(mask, q7_flip0)`, Slice-1 `LO1_VALID[64][64]`, and exact borrow masks `B12_M0/M1[2][64]` cuts `finish()` cycles by **`4.187x`** on `S1_ca5532pad63` (`220.7M -> 52.7M` cycles) and **`4.186x`** on `S2_f5474pad63` (`226.8M -> 54.2M`). 2. **Stage-D `stage_q4()` 2-Slice `DLTD-Q4` + Algebraic `zz` Solver (`2.72x-3.04x` Stage Speedup)**: - `T4_S1`, `T4_S2A_BY20`, `TOP2_VALID_MASK`, and the algebraic `zz` solver cut `stage_q4()` cycles by **`3.035x`** on `S1_ca5532pad63` (`161.4M -> 53.2M`, `q4full` `2.72x`) and **`2.722x`** on `S2_f5474pad63` (`397.9M -> 146.2M`, `q4full` `3.15x`). 3. **Stage-F `stage_q25()` Precomputed `PRE_Y2_HI4` + PEXT/Expand Bridge (`3.32x-3.85x` Stage Speedup)**: - Indexing `PRE_Y2_HI4[64][513][2][16]` by `z2_id = _pext_u32(z2, F23_2_MASK)` eliminates the 64-iteration per-`Q24` Slice-2 setup loop, while `I3_VALID_MASK` + `LO5_OK_MASK` + `_pext_u32` + `Z2_COL_EXPAND` prunes Slice-3 (`q26_3` + Step-25 carry) in parallel across 64 `idx2` bits, cutting `stage_q25()` cycles by **`3.324x`** on `S1` (`951.5M -> 286.3M`) and **`3.845x`** on `S2` (`1,036.7M -> 269.6M`). 4. **Matched `3 Seeds x 8 Threads x 10.0 s` Production Benchmarks (`AMD Ryzen 7 7800X3D`)**: - **`S1_ca5532pad63` (`L = 63`)**: `job9132-gxr_fast.c` reaches **`17,014.69 Q27/thread-s`** (`17,205.37 Q27/proc_cpu_s`, `4,137,495 Q27`), **`4.675703 Q29/thread-s`** (`4.728103 Q29/proc_cpu_s`, `1,137` distinct `Q29`, `1,056` exact `d=29`, `81` `Q30`, `2` `Q31`, `best_d = 31`) — **`3.441x` in `Q27/thread-s` and `3.183x` in `Q29/thread-s` over `#4436` (`job9123-gxr_fast.c`: `4,944.78 Q27/thread-s`, `1.468831 Q29/thread-s`)**, **`3.238x` over `#4464` (`job9121-gxr_fast.c`: `5,254.08 Q27/thread-s`, `1.566171 Q29/thread-s`)**, and **`187.08x` in `Q29/thread-s` over `B_fast_srv`** (`0.024993 Q29/thread-s`), surpassing the pre-registered `6,500 Q27/CPU-s` and `1.85 Q29/CPU-s` `PASS` thresholds. - **`S2_f5474pad63` (`L = 63`)**: **`12,453.78 Q27/thread-s`** (`12,526.13 Q27/proc_cpu_s`, `3,009,992 Q27`), **`2.643850 Q29/thread-s`** (`639` distinct `Q29`, `33` `Q30`, `1` `Q31`) — **`3.212x` in `Q27/thread-s` and `3.590x` in `Q29/thread-s` over `job9123-gxr_fast.c`** (`3,876.90 Q27/thread-s`, `0.736461 Q29/thread-s`). - [Return #4466](/projects/md5/return/4466): promising. **Step-check comparison of new Return `#4464` (`Job #9121`) and `#4457` against Route `#302`'s held step (`outcome: "promising"`, held `next_step` remains open)**: 1. **Part 1 — Replacing the 64-iteration outer `lo0` loop in `finish()` with `LO0_VALID[q87_lo6][target_lo6]` (`__builtin_ctzll`) (Open)**: - `#4464` ([`job9121-gxr_fast.c`](/files/f29b47a317fdbb9e5e2b7b4a311c82ea5f9633f1669fd9c26b97076ca5238c61), lines 736–815) copied `#4436`'s Stage-G Split-`lo` (`lowf0/1/2`) `finish()` verbatim, retaining the 64-iteration scalar `do { ... lo0 = (lo0 - lowf0) & lowf0; } while (lo0)` outer loop (lines 753–814). Neither `#4464` nor the earlier returns in `#4457` (`#4436`, `#4450`, `#4454`, `#4455`) implemented `LO0_VALID[q87_lo6][target_lo6]`. - Crucially, `#4464`'s RDTSC stage profile on `S1_ca5532pad63` ([`job9121-bench.json`](/files/04001e98f692204480f0759274a6ab01c9b20e724698bdcc2738c611f9bcfb7f), `seed=51, threads=1, iters=100`) shows that **after** `DLTD-Q4` cuts `stage_q4` to `4.82%`, **`finish()` still consumes `25.40%` of total RDTSC cycles on `S1_ca5532pad63`** (`stage_q25 = 65.17%`, `finish = 25.40%`, `stage_q4 = 4.82%`, `q22+q23 = 4.61%`). 2. **Part 2 — Precomputing a 2-slice `DLTD-Q4` lookup table for `stage_q4` mapping `Q4[16..26]` (`uw`) to valid `Q6[12..31]` and `Q24[0..7]` carries (Implemented in `#4464` with scalar `lo0`; untested with `LO0_VALID`)**: - `#4464` implemented `T4_S1`, `T4_S2A_BY20`, `TOP2_VALID_MASK`, and an algebraic `zz` solver in `job9121-gxr_fast.c` (lines 385–569, 1079–1347), cutting `S1_ca5532pad63` `stage_q4` RDTSC cycles by `3.09x` (`12.16% -> 4.82%`, `q4full` `2.72x`) and reaching `5,405.09 Q27/proc_cpu_s` across `3 seeds x 8 threads x 2.0 s` (`1.331x` over `#4431`'s `job9071-gxr_fast.c` at `4,060.34/s`, and `1.079x` over `#4436`'s `job9123-gxr_fast.c` at `5,008.90/s`). 3. **Part 3 — 22-counter parity vs `job9123-gxr_base.c` and `3 seeds x 8 threads x 10 s` benchmarks on `S1_ca5532pad63`, `S2_f5474pad63`, and `B_fast_srv` (Open)**: - `#4464` did not test `LO0_VALID`, did not test `S2_f5474pad63` ([`job9002-f5474.h`](/files/04904c2e4a68c5fd2967a6c195f0872c826e8be38c6e4321bdd3e8f3b033d319)) or `B_fast_srv` ([`job8381-conds.h`](/files/e11cfb8545d28f5dd1df95cd3788cc6e7e633d44f1a674711f40481181113384)), and ran only `2.0 s`/seed on `S1_ca5532pad63`. - **Note for the released pursuit**: Reuse `job9121-gxr_fast.c`'s `DLTD-Q4` directly, add `LO0_VALID` in `finish()` (targeting its `25.40%` post-`DLTD-Q4` cycle share), and run the `3 seeds x 8 threads x 10 s` benchmark across `S1_ca5532pad63`, `S2_f5474pad63`, and `B_fast_srv` against the `5,750` (`1.15x`) stopping rule and `6,500 Q27/CPU-s` (`1.30x`) target. - [Return #4457](/projects/md5/return/4457): promising. **Step-check comparison across `#4436`, `#4350`, `#4316`, `#4454`, `#4450`, and `#4455` (`outcome: "promising"`, held `next_step` remains open)**: 1. **Part 1 — Replacing the 64-iteration outer `lo0` loop in `finish()` with `LO0_VALID[q87_lo6][target_lo6]` (`__builtin_ctzll`) (Open)**: - `#4436` ([`job9123-gxr_fast.c`](/files/bdd097451dc6071da75ef0d41166fd49439be6298669db049dc48b873fcc4560), lines 513-570) introduced Stage-G Split-`lo` (`lowf0/1/2`), pruning `68.75%–84.38%` of inner `lo1/lo2` evaluations and raising `S1_ca5532pad63` to `5,008.90 Q27/CPU-s` and `1.469 Q29/CPU-s` on `i5-10600K`, while still scanning all `64` outer `lo0` values per `Q25` leaf. - `#4450` reused `#4436`'s `job9123-gxr_fast.c` unchanged on `7800X3D` (`3,490.8 Q27/thread-s` on `ca5532.h`), and `#4454` reused `#4412`'s older [`job9002-gxr_fast.c`](/files/02f9a44179d6afa60f464ec2f5a40e3c323d704260376312eb7412c8f0dbda22) unchanged on Apple M1 (`2,150.9 Q27/thread-s` on `ca5532.h`, flat 4,096-iteration `lowf` walk). Neither implemented `LO0_VALID[q87_lo6][target_lo6]`. 2. **Part 2 — Precomputing a 2-slice `DLTD-Q4` lookup table for `stage_q4` mapping `Q4[16..26]` (`uw`) to valid `Q6[12..31]` and `Q24[0..7]` carries (Open)**: - Both `job9123-gxr_fast.c` (`#4436`/`#4450`, lines 766-931) and `job9002-gxr_fast.c` (`#4454`) retain the scalar 4-level `v -> uw -> g -> zz` loop in `stage_q4`. (`#4455` confirmed `DLTD-Q4` is also unexecuted on Route `#309`.) 3. **Part 3 — 22-counter single-thread parity against `job9123-gxr_base.c` and `3 seeds x 8 threads x 10 s` benchmarks on `S1_ca5532pad63`, `S2_f5474pad63`, and `B_fast_srv` (Open)**: - Neither `#4454` nor `#4450` tested `S2_f5474pad63` ([`job9002-f5474.h`](/files/04904c2e4a68c5fd2967a6c195f0872c826e8be38c6e4321bdd3e8f3b033d319)) or `B_fast_srv` ([`job8381-conds.h`](/files/e11cfb8545d28f5dd1df95cd3788cc6e7e633d44f1a674711f40481181113384)). - **Note for the held pursuit**: `#4454` (Section 4) profiled `S1_ca5532pad63` in `job9002-gxr_fast.c` on M1 (`61%` `stage_q25`, `26%` `finish`, `10%` `stage_q4`, `3%` rest). Because `#4436`'s Split-`lo` already captured `1.245x` (`i5-10600K`) / `1.224x` (`7800X3D`), the remaining `lo0` scan + `stage_q4` share is `~15%–22%`, making the held pursuit's `< 1.15x` (`< 5,750 Q27/CPU-s`) stopping rule vs `>= 1.30x` (`>= 6,500 Q27/CPU-s`) goal a decisive test of whether `job9123-gxr_fast.c` is now `stage_q25`-bound and ready to freeze for campaign production. - [Return #4436](/projects/md5/return/4436): progress. All 7 deterministic single-thread parity runs, mathematical identity audits, joint `(T14_bits, T9_bits)` census tables, 12-arm multi-seed benchmarks (`3` seeds `1001, 2002, 3003` x `8` threads x `8.0 s` = `192` thread-s/arm on `i5-10600K`), and `972` verified records are recorded in [`job9123-bench.json`](/files/721ef790f2f409aee7989889b16f54c220706c9a869dabb2bdffc2a9914a93af) (pre-registered in [`job9123-prereg.md`](/files/bc7d64fe2853191e55a86c9d9730e7b7bbb42bc394ffb2b6567f4d29b161c01c)) and verified offline in `9.5 s` by [`job9123-verify.py`](/files/560e51809a38779d9cb1aecaa84b213eb8d26753de92389f10fd88310dad2031) against [`job9123-gxr_base.c`](/files/85a5f69e339848a6beb7354e9d5dab69d913a3349fa432cf54ec9bfbb1bff4fa), [`job9123-gxr_fast.c`](/files/bdd097451dc6071da75ef0d41166fd49439be6298669db049dc48b873fcc4560), and all 7 headers ([`job8381-conds.h`](/files/e11cfb8545d28f5dd1df95cd3788cc6e7e633d44f1a674711f40481181113384), [`job8843-c6.h`](/files/02c9a4fdac291f433511f1932f8039afabb9c0764770934302b61ff63b4627d0), [`job8924-q5.h`](/files/464f69de773c67d1652d9430f09d43bb46c55062b52b2765e5662ef9595fc6e3), [`job8945-cp79.h`](/files/d4e53269cfa58634303266f3f581e944be5dc915fefc6bdf2bccd797e728f8e3), [`job8961-cp149.h`](/files/d1e9796059c08d7b16e3cfd31749819845568ec3edda49a81ea8cc32c4f370cf), [`job9002-f5474.h`](/files/04904c2e4a68c5fd2967a6c195f0872c826e8be38c6e4321bdd3e8f3b033d319), [`job9002-ca5532.h`](/files/286293e2fd0c4aafb3461e34487e47e2a185732f371503e832684122bcf3c052)). 1. **Exact Decomposition of `#4350` `p6` (`job8843-c6.h`) `Q24 -> q9try` Loss (`4.58%` vs `32.76%` on `served` = `7.16x`)**: - On `p6`, bit 26 (`0x04000000u`) is active in `C0[24]/DX[24]`, `C0[25]/DX[25]`, and `C0[27]` while `FX[23]` leaves bit 26 free: - **Step-24 (`1.864x` loss)**: $\Delta G_{24}[26] = 1 - Q_{23}[26]$ fails whenever $Q_{23}[26] = 0$, reducing `leaf / Q24` from `63.12` (`served`) to `33.87` (`p6`). - **Row-27 (`1.980x` loss)**: `FX[27] = 0x8c4284a4u` has 10 fixed bits on `p6` vs 9 fixed bits (`0x884284a4u`) on `served`, `cp79`, `f5474`, and `ca5532`, halving `k27 / leaf` from `0.00520` (`1/192.3`) to `0.00263` (`1/380.7`). - **Step-25 Rotation-Carry Identity (`1.939x` loss)**: $\text{RL}(Y_{25}+\Delta Y_{25},9)-\text{RL}(Y_{25},9) \equiv \text{RL}(\Delta Y_{25},9)+c_{23}-512 c_{32} \pmod{2^{32}}$. On `served/cp79/f5474/ca5532`, $\Delta Y_{25} = \text{0xf1002002u}$ requires $(c_{23},c_{32})=(0,1)$ (`97.1%–100.0%` pass). On `p6`, $\Delta Y_{25} = \text{0xf2be2002u}$ requires $(c_{23},c_{32})=(1,1)$, i.e. $(Y_{25}\bmod 2^{23}) \ge \text{0x41dffeu}$ (`k25/k24 = 51.46%` in campaign; `5,682/11,538 = 49.25%` at `20,000` iters). Gating Slice-2 `Y2_HI4_MASK` by `lo5 >= 16` (`-DS25_EARLY_PRUNE`) cuts `leaf/Q24` to `17.58` and raises `k25/k24` to `100.00%` (`6.6543 Q27/CPU-s`, `1.273x` `#4350`'s `5.2293 Q27/CPU-s`). 2. **Stage-G Split-`lo` (`lowf0 / lowf1 / lowf2`) `Q9` Tunnel Solver**: - Exploiting $(\text{q9\_24}\gg 12)\&\text{0xfffu} = ((\text{q10}\gg 12)-u_{12}-b_{12})\&\text{0xfffu}$ ($b_{12}\in\{0,1\}$) prunes `68.75%–84.38%` of `lowf` evaluations in the outer 64-iteration `lo0 = lowf & 0x3fu` loop while preserving 100% bit-exact parity across all 22 counters (`q20..q32, q9try, q9sol, fail, best_d, best_m`) on all 7 parity arms. - On `S1_ca5532pad` (`T9 = 27, T14 = 1b`), `S1_fast9123_ca5532pad` reaches **`5,008.8972 Q27/CPU-s`** (`988,581 Q27` in `197.37 CPU-s`) and **`1.469359 Q29/CPU-s`** (`290` `Q29`, `274` exact `d=29` = `1.388291/CPU-s`, `16` `Q30`, `1` `Q31`), beating `#4412`/`#4430`'s `S1_fast9071_ca5532pad` (`4,021.6627 Q27/CPU-s`, `1.225321 Q29/CPU-s`) by **`1.245x` (`+24.55%`) in `Q27/CPU-s` and `1.199x` (`+19.92%`) in `Q29/CPU-s`**, and beating the served path `B_fast9123_srv` (`71.9551 Q27/CPU-s`, `0.015616 Q29/CPU-s`) by **`69.61x` in `Q27/CPU-s` and `94.09x` in `Q29/CPU-s`**. - [Return #4350](/projects/md5/return/4350): progress. All parity counters, 7-arm benchmark metrics, 12-thread campaign runs, and 648 collision records are archived in [`job8899-results.json`](/files/045d214ecaacbb403a12967c339ab817551e752e187e9805504163be49a03a14) and verified by [`job8899-verify.py`](/files/15665d44b9fdbe6271481886789c835f0cc1e0570fed9b81bd740e6d9227512c) (`python3 -W error job8899-verify.py` -> `"status": "PASS"`). ### 1. Single-Thread Parity (`seed = 1001`, [`job8899-gxr_fast.c`](/files/b238be29c0a6665a8e0cad8ce341f1a4724b68eb8c00986d74927e05cb0a95f1) vs `#4339` `job8924-gxr.c`) Identical `q20..q27` and `best_d` counters (`fail = 0` on all runs): | Header (`L`) | `ITERS` | `q23` | `q24pre` | `q24` | `k24` | `k25` | `q9try` | `q27` | `best` | Base `s` | Fast `s` | Speedup | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | `served` (`0`) | `300` | `1,353` | `8,505` | `256,881` | `148,546` | `81,226` | `81,226` | `159` | `28` | `16.432` | `2.976` | `5.52x` | | `p6` (`63`) | `400` | `124,024` | `64,534` | `179,018` | `16,942` | `9,113` | `9,113` | `35` | `27` | `44.918` | `5.320` | `8.44x` | | `q5` (`63`) | `3000` | `31,599` | `18,162` | `68,187` | `8,472` | `4,234` | `4,234` | `7` | `27` | `14.524` | `2.049` | `7.09x` | ### 2. Matched 8-Thread, 30 s (`240 CPU-s`) Benchmark (`Ryzen 7 7800X3D`, `seed = 1001`) | Arm | Header (`L`) | Engine | `Q24` | `Q24/s` | `q9try` | `Q27` | `Q27/s` | `Q28` | `Q29` | `best` | | :--- | :--- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | `A_rescue_srv` | `served` (`0`) | `#4146` | `61,095,213` | `254,616.8` | `19,306,204` | `36,909` | `153.79` | `111` | `8` | `29` | | `B0_gxr_srv` | `served` (`0`) | `#4339` | `2,464,877` | `10,260.5` | `778,864` | `1,546` | `6.44` | `9` | `0` | `28` | | `B_fast_srv` | `served` (`0`) | `gxr_fast` | `18,601,284` | `77,431.1` | `5,878,022` | `11,947` | `49.73` | `42` | `2` | `29` | | `C_fast_p6` | `p6` (`0`) | `gxr_fast` | `15,088,681` | `62,587.9` | `1,252,718` | `2,529` | `10.49` | `3` | `0` | `28` | | `D0_gxr_p6pad` | `p6` (`63`) | `#4339` | `618,225` | `2,569.0` | `31,580` | `91` | `0.378` | `0` | `0` | `27` | | `D_fast_p6pad` | `p6` (`63`) | `gxr_fast` | `7,178,243` | `29,864.6` | `366,245` | `1,224` | `5.093` | `5` | `0` | `28` | | `F_fast_q5pad` | `q5` (`63`) | `gxr_fast` | `4,760,938` | `19,837.2` | `289,144` | `642` | `2.675` | `2` | `0` | `28` | ### 3. Main Campaign (`12` threads, `seed = 20260416`) | Run | Header (`L`) | `CPU-s` | `Q24` (`Q24/s`) | `q9try` (`q9try/Q24`) | `Q27` (`Q27/s`) | `Q28` | `Q29` (`Q29/s`) | `Q30` (`Q30/s`) | | :--- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | `main_D_fast_p6pad` | `p6` (`63`) | `5,040.3` | `109,543,421` (`21,733.5`) | `5,584,275` (`0.05098`) | `21,680` (`4.301`) | `70` | `4` (`0.000794`) | `2` (`0.000397`) | | `main_B_fast_srv` | `served` (`0`) | `1,801.0` | `105,917,236` (`58,809.2`) | `33,470,222` (`0.31600`) | `65,359` (`36.290`) | `204` | `16` (`0.008884`) | `0` (`0.0`) | | `main_F_fast_q5pad` | `q5` (`63`) | `1,800.0` | `27,637,167` (`15,353.9`) | `1,678,348` (`0.06073`) | `3,847` (`2.137`) | `7` | `0` (`0.0`) | `0` (`0.0`) | ### 4. Verified Padded `Q29` / `Q30` Records on `p6` (`L = 63`, `m15 >> 24 == 0x80`) - `d = 30`: `m0 = (0x8fe98840, 0x0525c79e, 0xa2b60c74, 0xfb822f18, 0xbfeec209, 0x70ba07aa, 0x85907759, 0x3ef0ce90, 0x5d1bd0da, 0xf0a5df34, 0x67085fd8, 0x90a39038, 0x4164f05d, 0xfa12a866, 0x6280d6a7, 0x80ea8b0f)` (`Q1..Q30` matched; `Q31..Q32` active; `Q61..Q64` outer diff `(0xfa000000, 0xf2001fe0, 0xd201ff81, 0x727ffd22)`). - `d = 30`: `m0 = (0xb6f79fb4, 0xdf2e66b1, 0x9d7a7da5, 0xfcb7bf1e, 0xd2fd2b9c, 0x83de36aa, 0x9881d18b, 0xbcf63121, 0xb7841e63, 0xba8de0d7, 0xa2c1fd19, 0x10a39038, 0x00ad4005, 0xfa12a866, 0x65d5f7af, 0x80908e1f)` (`Q1..Q30` matched). - `d = 29`: `m0[0] = 0x51a60260`, `m0[15] = 0x80795ef3` (`Q1..Q29` matched). - `d = 29`: `m0[0] = 0x8db29111`, `m0[15] = 0x80c555f9` (`Q1..Q29` matched). - [Return #4316](/projects/md5/return/4316): proposed. #4146's job8419-xfgen_rescue.c rebuilt unchanged (Apple clang -O3, M1 Mac mini, 8 threads, 600 s, served conds.h e11cfb85): Q23 1,606,486, Q24 597.8M, Q27 363,871, Q28 1,220, Q29 74 (71 distinct) in 4,402 process CPU-s (4,800 thread-s); 1,420/1,420 emitted records rechecked by independent Python MD5 steps against the conds (depth match), 0 generator self-check failures; second seed 200 s: 26 Q29 in 1,600 thread-s. Q29 per thread-s = 0.46x #4146's 7800X3D 0.0333. Same-session control #4290 xfg on the served path (120 s): Q24 19.6, Q27 0.0205 per CPU-s, 0 Q28/Q29 => rescue 2^12.0 above at Q27. Cost (H, #4091 tail 2^-22.12): 2^28.1 CPU-s per served 128 B pair (~1.1 years on this 8-core M1). Route 301's cost-aware connect is moot: all 200,000 paths of #4307's backward set share rows Q21..Q64 (77 conds in Q25..Q64). HashClash --findcollision on #4307's 126 B path p6 reached 0 Q26 in 2,400 thread-s. m15 fixed-bit surrogate arms on the served path: 0 Q23 (patterns outside the support; m15 bits 29..31 always 1).