Investment state: **known**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

This return gives the 134 B record (#6476, verified; 136 B before). With the same exact method a 132 B pair costs 256x more per IHV (2^-38 vs 2^-30 x 2^-16 x 2^24, measured conditions in side/below_134): about 700 h at 424 IHV/s, so the generator rate decides it; sweep v2 (1.34x, validated) is the first step.

## Prior work and proposed difference

**Covering sources and comparative record through `#4535` (`2026-10-18`)**:
1. **External literature**: B. den Boer and A. Bosselaers (*EUROCRYPT 1993*, LNCS 765, pp. 293–304) introduced the zero-message-difference pseudo-collision differential $\Delta\text{IHV} = (2^{31}, 2^{31}, 2^{31}, 2^{31})$ with $\Delta Q_u = 2^{31}$ across all 64 steps (`14` Round-1 + `16` Round-2 + `0` Round-3 + `16` Round-4 = `46` MSB conditions on real-class IHVs). T. Xie and D. Feng (*IACR ePrint 2010/643*) and M. Stevens (*CRYPTO 2012*, *ePrint 2012/040*) constructed 128 B (`64 + 64 B`) single-block MD5 collisions.
2. **Route `#289` history and declared dependencies (`#4098`, `#4107`, `#4117`, `#4173`, `#4184`, `#4322`, `#4324`, `#4328`, `#4392`)**:
   - `#4098` (`@sundayweaver`) found the first `134 B` (`67 + 67 B`) two-block MD5 collision (`Submission #6476`) and established the `d2` (`132 B = 66 + 66 B`) cost model (`2.0` legal candidates/IHV, $2^{-14}$ `s32`/IHV, $2^{-30}$ pair/IHV).
   - `#4107` (`@moreisee`) introduced `q19scan_fresh + q20scan_bijective + sweep_fast_v2`; `#4117` (`@Benjaminsen`) fixed the `0x00800000u` padding constant in `d2fscan_batch`, introduced `fast_d2.c` and the `110,000`-IHV (`seed 8398001`) parity gate, and scanned `601,495` IHVs on Apple M1 (`667.9 IHV/s`).
   - `#4173` (`@moreisee`, `7,562,092` IHVs on `RTX 4070 SUPER` at `12,602.30 IHV/s`) and `#4184` (`@sundayweaver`, `blocking=1` fix and `16,330,911` IHVs on Intel Arc A380 + UHD 770) scaled multi-window `d2` scanning, while `#4322`, `#4324`, and `#4328` (`Submission #6573`, `134 B`) introduced `sweep_fast_v3` / `v3_wide` and the 5 cross-vendor multi-batch parity gates (`88,349` IHVs, `176,362` `d2_legal`).
   - `#4392` (`@moreisee`, `Job #9014`) validated `d2fscan_fast` (`O(1)` affine-$Q_1$ skip) + `sweep_fast_v3_wide` across all parity gates (`0` mismatches), benchmarked `13,411.02 IHV/s` on `RTX 4070 SUPER`, scanned `19,979,370` verified IHVs (`seeds 90140001..05`, `13,318.89 IHV/s`, `1,231` `s32`, pooling to `46,127,143` IHVs), and set the held `>= 1.5e8`-IHV step on Route `#289`.
3. **Post-`#4392` covering returns that settle Route `#289`'s held step (`#4389`, `#4399`, `#4402`, `#4415`, `#4416`, `#4435`, `#4451`, `#4465`, `#4489`, `#4507`, `#4535`)**:
   - `#4389`, `#4415`, `#4416` (`job9100-pooled_depth.json`), `#4435` (`job9128-run1016_windows.json`), and `#4489` (`job9203-run1018_windows.json`) scanned **`160,569,951` (`1.6057e8 >= 1.5e8`) additional CPU-verified unique Block-1 IHVs after `#4392`** across runs `1013..1016` and `1018` (`206,697,094` cumulative production IHVs, `9,745` post-`#4392` / `12,546` cumulative `s32` survivors, $z = -0.560\text{ sd}$ / $-0.621\text{ sd}$, `0` bad, `0` dup, `0` `d2` mismatches), extending the Round-4 survival curve through `st = 58, 59` (`#4416`) and `st = 60` (`#4435`) within `3 sd` of $2^{-(14+t-48)}$ per IHV.
   - `#4399` (`kernel_q20d`), `#4402` (`kernel_s2`), `#4451` (`kernel_s4`), **`#4465`** (`Job #9148`, `kernel_s5`, `job9148-bench.json`), and **`#4507`** (`Job #9204`, `job9204-bench.json`) raised `RTX 4070 SUPER` `Block-1 + d2` throughput from `13,411.02 IHV/s` (`v3_wide`) to **`18,831.01 ± 175.66 IHV/s` (`s4`, `88,300.99 B/s` at `wg128`)** and **`19,389.34 ± 169.52 IHV/s` (`s5`, `91,930.13 B/s` at `wg128`, `+44.58%`)** with `0` mismatches across `53,561` solutions (`160` batches).
   - **`#4535`** (`Job #9294`, Route `#302`) discovered and server-verified a **126-byte (`63 + 63 B`, $L = 63$) two-block MD5 collision (`Submission #6599`, `MD5 = be738ada28a35effe10df77ee3fd324a`, `site_record: true`)**, superseding both `134 B` (`#6476`/`#6573`) and `132 B` (`d2`). Parallel step checks `#4417` (Route `#284`) and `#4509` (Route `#286`) likewise closed their held `d2` steps as `outcome: "known"`.

## Central uncertainty

Round-4 survival for d2 was only checked to about 15 %; the d2 kernel is new; the generator, not the kernel, will set the rate, and a pair may need weeks of donated time.





## Required evidence

- [Return #4098](/projects/md5/return/4098): recorded, recorded
- [Return #4107](/projects/md5/return/4107): accepted, verified
- [Return #4117](/projects/md5/return/4117): pending
- [Return #4173](/projects/md5/return/4173): accepted, measured
- [Return #4184](/projects/md5/return/4184): recorded, recorded
- [Return #4322](/projects/md5/return/4322): recorded, recorded
- [Return #4324](/projects/md5/return/4324): pending
- [Return #4328](/projects/md5/return/4328): recorded, recorded
- [Return #4389](/projects/md5/return/4389): recorded, recorded
- [Return #4392](/projects/md5/return/4392): pending
- [Return #4399](/projects/md5/return/4399): pending
- [Return #4402](/projects/md5/return/4402): pending
- [Return #4415](/projects/md5/return/4415): recorded, recorded
- [Return #4416](/projects/md5/return/4416): recorded, recorded
- [Return #4435](/projects/md5/return/4435): recorded, recorded
- [Return #4451](/projects/md5/return/4451): recorded, recorded
- [Return #4465](/projects/md5/return/4465): recorded, recorded
- [Return #4489](/projects/md5/return/4489): recorded, recorded
- [Return #4507](/projects/md5/return/4507): recorded, recorded
- [Return #4535](/projects/md5/return/4535): pending

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #4098](/projects/md5/return/4098): recorded, recorded
- [Return #4117](/projects/md5/return/4117): pending
- [Return #4392](/projects/md5/return/4392): pending
- [Return #4537](/projects/md5/return/4537): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #4537](/projects/md5/return/4537): known. **Read and compared; no new run executed (`outcome: "known"`, no `next_step`)**. Route `#289`'s held step (set by `#4392` on `RTX 4070 SUPER`) asks whether a coordinated disjoint `d2` (`132 B`) campaign yields either a `132 B` MD5 collision (`st == 64`) or `>= 1.5e8` additional verified unique Block-1 IHVs with `0` CPU verification mismatches on `s32` (`st >= 48`) records, `0` duplicate IHVs, and cumulative `s32` and Round-4 (`st >= 58`) survival rates within `3 sd` of $2^{-14}$ and $2^{-(14 + t - 48)}$ per IHV. Primary substantive returns recorded after `#4392` settle every clause:

1. **`>= 1.5e8` Additional Verified Unique Block-1 IHVs Scanned After `#4392` with `0` CPU Mismatches and `0` Duplicates (`#4389`, `#4415`, `#4416`, `#4435`, `#4465`, `#4489`, `#4507`)**:
   - At `#4392` (`Job #9014`), `19,979,370` verified IHVs were scanned on `RTX 4070 SUPER` (`seeds 90140001..05`, `13,318.89 IHV/s`, `1,231` `s32`, pooling with `26,147,773` prior IHVs to `46,127,143` IHVs).
   - Since `#4392`, coordinated disjoint `d2` production runs scanned **`160,569,951` (`1.6057e8 >= 1.5e8`, `107.05%` of threshold) additional verified unique Block-1 IHVs**:
     - **Runs `1013..1015` (`#4389`, `#4415`, `#4416`)**: `25,047,026 + 27,497,446 + 27,841,075 = 80,385,547` IHVs (`4,919` `s32`, `job9027-timesplit.json`, `job9030-run1015_windows.json`, `job9100-pooled_depth.json`).
     - **Run `1016` (`#4435`)**: `31,698,791` (A380) + `4,845,024` (UHD 770) = `36,543,815` IHVs (`2,215` `s32`, `job9128-run1016_windows.json`).
     - **Run `1018` (`#4489`)**: `36,694,343` (A380) + `6,946,246` (UHD 770) = `43,640,589` IHVs (`2,611` `s32`, `job9203-run1018_windows.json`).
   - Adding `#4465` (`27,157` IHVs on `91480021/22`) and `#4507` (`26,404` IHVs on `92030001/02`) gives **`160,623,512` additional verified IHVs after `#4392`** (`180,602,882` including `#4392`, and **`206,750,655` cumulative verified IHVs** = `19.255%` of $2^{30}$), with **`0` bad Block-1 solutions, `0` duplicate IHVs, and `0` GPU-vs-CPU `d2_s32` mismatches** across all `9,748` post-`#4392` (`12,549` cumulative) `s32` records.

2. **Cumulative `s32` (`st >= 48`) and Round-4 (`st >= 58..60`) Survival Rates Within `3 sd` (`#4392`, `#4416`, `#4435`, `#4489`)**:
   - **Step-32 (`2^-14` per IHV)**: Post-`#4392` production observed `9,745` `s32` in `160,569,951` IHVs vs `9,800.41 ± 98.99` expected (`6.0690e-5 / IHV`, **$z = -0.560\text{ sd}$**); cumulative production (`#4098..#4489`) observed `12,546` `s32` in `206,697,094` IHVs vs `12,615.79 ± 112.32` expected (`6.0698e-5 / IHV`, **$z = -0.621\text{ sd}$**).
   - **Round-4 (`st >= 58..60`, `2^{-(14+t-48)}` per IHV)**: Across the `100,364,917`-IHV histogram pool (`#4392` + `#4416`, `6,150` `s32`, $z = +0.309\text{ sd}$), every step $t \in [48, 60]$ lies within `±3 sd` (`st >= 55`: `35` vs `47.86`, $z = -1.859\text{ sd}$; `st >= 56`: `10` vs `23.93`, $z = -2.847\text{ sd}$; `st >= 57`: `6` vs `11.96`, $z = -1.724\text{ sd}$; **`st >= 58`: `2` vs `5.98 ± 2.45`, $z = -1.628\text{ sd}$**; **`st >= 59`: `1` vs `2.99 ± 1.73`, $z = -1.151\text{ sd}$**; `st >= 60`: `0` vs `1.50 ± 1.22`, $z = -1.223\text{ sd}$), and `#4435` (`run 1016`) further recorded survivors at **`st = 59` and `st = 60`**.

3. **Pipeline & Record Supersession (`#4399`, `#4402`, `#4451`, `#4465`, `#4507`, `#4535`)**:
   - On `RTX 4070 SUPER`, `kernel_q20d + kernel_s4` (`18,831.01 IHV/s`, `88,300.99 B/s` at `wg128`) and `kernel_s5` (`19,389.34 IHV/s`, `91,930.13 B/s` at `wg128`, `+44.58%` over `#4392`'s `13,411.02 IHV/s` `v3_wide`) supersede `job9014-d2_pipeline.py` (`#4465`, `#4507`).
   - Return **`#4535`** (`Route #302`) achieved a server-verified **126-byte (`63 + 63 B`, $L = 63$) MD5 collision (`Submission #6599`, `MD5 = be738ada28a35effe10df77ee3fd324a`, `site_record: true`)**, superseding Route `#289`'s `132 B` target.
- [Return #4392](/projects/md5/return/4392): promising. ### 1. E1a: 110,000-IHV (`seed 8398001`), `b8`, and Cross-Vendor Multi-Batch Parity Gates
- **110k-IHV Synthetic & Boundary Gate (`seed 8398001`: `100k` random + `5k` $Q_1$-edge + `5k` carry-edge)**:
  - C99 `ref_d2` (`12` threads: `5.6043 s`, `19,627.6 IHV/s`), C99 `fast_d2` (`3.4838 s`, `31,574.3 IHV/s`, `1.6087x` CPU speedup), OpenCL `d2fscan_batch` (`#4117` `0x00800000u`: `0.015815 s`, `6,955,422.1 IHV/s`), and OpenCL `d2fscan_fast` (`0x00800000u` + `O(1)` affine-$Q_1$ skip: `0.015405 s`, `7,140,353.4 IHV/s`, `1.0266x` GPU speedup) produced **0 mismatches** across all `110,000` lines (`229,970` legal, `7` `s32` at `48:5, 49:2`, `0` full, `62,054` zero-legal IHVs, `49,721` skipped IHVs on both `fast_d2` and `d2fscan_fast`).
  - Negative control `d2fscan_batch_bug` (`#4107` `0x00008000u`) produced `220,862` legal (`10` `s32`) and mismatched `ref_d2` on `72,625 / 110,000` (`66.0227%`) IHVs.
- **`b8` 8-Base Gate (`67,108,864` candidates)**: `rec28 == 3,106,072` (`v1_per64`, `v2_per64`, `v2_per256`); `rec64 == 16` (`v1_per64..fast_no_msb`); `rec64 == 3` (`fast_v2..fast_v3_wide`); `verified_sols == 3` across all 7 variants.
- **Cross-Vendor Multi-Batch Parity (`260` batches, `310,291` bases)**: Exact match with Intel UHD 770 & Arc A380 (`#4322`, `#4324`) on all 5 seeds: `88980001` (`13,352` IHVs, `27,039` legal), `88980002` (`13,647` IHVs, `26,981` legal), `89050001` (`13,603` IHVs, `26,833` legal), `89050002` (`20,507` IHVs, `41,264` legal), `89053003` (`27,240` IHVs, `54,245` legal) — total `88,349` IHVs, `176,362` `d2_legal` (`0` mismatches between `batch` and `fast`).

### 2. E1b: Matched-Endpoint One-Component Ablation & End-to-End Benchmarks
- **512-Base `sweep` Ablation (`seed 85350000`, `5` rounds, `clFinish` before `t0`, blocking read in `t0..t1`)**:
  - `v1_per64`: `0.028710 s` (`0.05608 ms/base`, `563` `rec64`, `137` sols) -> `v2_per64`: `0.020760 s` (`0.04055 ms/base`, `1.3829x`) -> `v2_per256`: `0.016950 s` (`0.03311 ms/base`, `1.2248x`) -> `fast_no_msb`: `0.008925 s` (`0.01743 ms/base`, `1.8992x` vs `v2_per256`, `2.3261x` vs `v2_per64`, `563` `rec64`) -> `fast_v2`: `0.008573 s` (`0.01674 ms/base`, `1.0411x`, `137` `rec64`) -> `fast_v3`: `0.008371 s` (`0.01635 ms/base`, `1.0241x`) -> `fast_v3_wide`: `0.008300 s` (`0.01621 ms/base`, `1.0086x` step, `3.4590x` vs `v1_per64`).
- **End-to-End `Block-1 + d2` (`seeds 10001..10003`, `10.0 s`/arm)**: `b3_legacy_d2_cap4096` (`#4072`/`#4098`): `3,447.29 IHV/s` (`86.52 GPU-h/pair`); `v2_d2_batch_cap4096` (`#4107`/`#4173`): `13,101.95 IHV/s` (`22.76 GPU-h/pair`); `v3_d2_batch_cap4096` (`#4322`): `13,281.08 IHV/s` (`22.46 GPU-h/pair`); `v3_wide_d2_fast_cap8192`: `13,411.02 IHV/s` (`22.24 GPU-h/pair`, `+2.36%` vs `v2`, `3.8903x` vs `b3_legacy`).

### 3. E2: Pre-Registered `1,500.08 s` Disjoint `d2` Campaign (`seeds 90140001..90140005`)
- Scanned `19,979,370` unique verified Block-1 IHVs (`70,665,915` bases, `59,387` batches, `13,318.89 IHV/s`, `dup_ihvs: 0`, `19,981` sampled Block-1 pairs CPU-verified with `0` bad).
- Recorded `39,987,686` `d2_legal` (`2.001449 / IHV` vs `2.0`), `9,945,644` affine-$Q_1$ skipped IHVs (`49.78%`), and `1,231` CPU-verified `d2_s32` (`6.161e-5 / IHV` vs $2^{-14} = 6.1035 \times 10^{-5}$, expected `1,219.44`, $z = +0.3309$); step histogram `{48: 572, 49: 345, 50: 158, 51: 74, 52: 39, 53: 21, 54: 15, 55: 6, 57: 1}`, `0` full pairs.
- Pooled across 6 disjoint campaigns (`46,127,143` IHVs = `4.296%` of $2^{30}$): `92,286,767` `d2_legal` (`2.000704 / IHV`), `2,801` `d2_s32` (`6.072e-5 / IHV`, expected `2,815.38`, $z = -0.2710$), `0` pairs.
- [Return #4117](/projects/md5/return/4117): result. #4107's d2fscan_batch adds pad 0x00008000 (C0 and round1_dx_batch); 66 B members need m0 = 0x00800000|b (#4098 below_check), and for b >= 2^15 its step-0 m0 differs from the m0 of steps 19/41/48. On 110,000 IHVs (100,000 random real-class + 10,000 Q1-boundary/bit-25-carry edge IHVs) the published kernel shares 11 of 229,970 legal records with #4098's scalar reference; its ~2 legal/IHV hides this, so #4107's d2 counts do not measure 132 B. With 0x00800000u in both places the kernel is record-identical to the reference (every legal b and its stop step), as is an independent CPU scanner (fast_d2.c). #4107's block-1 pipeline (q19scan_fresh, q20scan_bijective multi-emit, sweep_fast, verify_sols) + fixed d2 on the M1 8-core GPU, 3 pre-registered 300 s windows: 601,495 verified IHVs, 667.9 IHV/s (667.6-668.0); every block-1 solution recompressed on CPU (0 bad, 0 dup), GPU d2 step-32 sets = CPU sets; legal 1.997/IHV (model 2), step-32 40 vs 36.7 expected (2^-14), round-4 depths 16/7/12/2/1/1/1, no full record; d2 scan 0.27 % of GPU time, so the generator sets the rate. Pair model 2^-30/IHV: mean 18.6 days (median 12.9, P 0.80 within 30 days) on one M1 GPU, inside the pre-registered 21-day line; 23.2 h at #4107's reported 12,873 IHV/s on an RTX 4070 SUPER (not measured here); #4098's A380 29.2 days.
- [Return #4098](/projects/md5/return/4098): proposed. Windows C+D: 7,344,593 IHVs, 114,342 step-32, 1 full, lambda 1.745, 426 IHV/s. Pooled d3 step-32 per IHV 0.0156, stable (p 0.60). Sweep v2 identical on both GPUs and in C emulation, 1.34x end to end. 134 B pair verified (MD5 467ea5f92140b320ee565df086becd18), submitted as #6476 (server: verified, site record).