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# Results

## Table 1 — Addition: exact match at ONE forward pass

The whole answer committed in a single pass. Depth scaled as log2(digits); learning rate 1e-4 with 4000 warmup steps, which matters - see Table 4.

| digits | MDLM baseline | full method | seeds >90% (base → ours) |
|---|---|---|---|
| **8** | 100.0 [100,100] | 100.0 [100,100] | 4/4 → 4/4 |
| **12** | 76.7 [30,100] | 100.0 [100,100] | 3/4 → 4/4 |
| **16** | 25.0 [0,75] | 75.0 [25,100] | 1/4 → 3/4 |
| **24** | 0.0 [0,0] | 0.0 [0,0] | 0/4 → 0/4 |
| **32** | 0.0 [0,0] | 0.0 [0,0] | 0/4 → 0/4 |

## Table 2 — Morphological inflection: where the problem does NOT arise

Answers are unique but only ~10 characters with shallow, local dependencies. One pass already suffices for the plain baseline, so there is no gap for any method to close. This is the boundary the account predicts, and it is reported as such.

| language | 1 pass | 32 passes | gap |
|---|---|---|---|
| english | 95.5 | 95.1 | -0.4 |
| english | 95.0 | 95.2 | +0.2 |
| english | 95.3 | 95.2 | -0.1 |
| finnish | 50.9 | 55.8 | +4.9 |
| finnish | 48.6 | 56.6 | +8.0 |
| finnish | 48.3 | 53.5 | +5.3 |
| finnish | 50.6 | 55.6 | +4.9 |
| finnish | 50.9 | 55.8 | +4.9 |
| finnish | 52.9 | 57.9 | +4.9 |
| finnish | 50.3 | 57.0 | +6.7 |
| finnish | 53.6 | 60.9 | +7.3 |
| finnish | 48.6 | 56.6 | +8.0 |
| finnish | 54.3 | 61.7 | +7.4 |
| georgian | 94.3 | 94.1 | -0.2 |
| georgian | 95.4 | 95.3 | -0.1 |
| georgian | 94.8 | 94.5 | -0.3 |
| georgian | 93.9 | 93.8 | -0.1 |
| georgian | 94.3 | 94.1 | -0.2 |
| georgian | 93.8 | 94.0 | +0.1 |
| georgian | 94.4 | 94.9 | +0.5 |
| georgian | 94.1 | 93.8 | -0.3 |
| georgian | 95.4 | 95.3 | -0.1 |
| georgian | 93.9 | 93.9 | +0.0 |
| german | 78.9 | 79.3 | +0.4 |
| german | 75.6 | 79.5 | +3.9 |
| german | 78.6 | 78.9 | +0.3 |
| german | 77.9 | 78.5 | +0.7 |
| german | 78.9 | 79.3 | +0.4 |
| german | 77.9 | 78.0 | +0.1 |
| german | 79.4 | 81.0 | +1.7 |
| german | 76.5 | 77.6 | +1.0 |
| german | 75.6 | 79.5 | +3.9 |
| german | 76.4 | 78.9 | +2.5 |
| german | 77.2 | 78.3 | +1.0 |
| navajo | 29.5 | 33.8 | +4.3 |
| navajo | 31.4 | 40.2 | +8.9 |
| navajo | 33.7 | 39.0 | +5.2 |
| navajo | 24.2 | 30.8 | +6.6 |
| navajo | 29.5 | 33.8 | +4.3 |
| navajo | 24.3 | 29.0 | +4.8 |
| navajo | 31.1 | 39.3 | +8.2 |
| navajo | 26.3 | 34.0 | +7.7 |
| navajo | 31.4 | 40.2 | +8.9 |
| navajo | 24.6 | 31.7 | +7.1 |
| russian | 72.7 | 72.0 | -0.7 |
| russian | 69.1 | 71.7 | +2.6 |
| russian | 73.5 | 73.0 | -0.5 |
| russian | 74.2 | 74.5 | +0.4 |
| russian | 72.7 | 72.0 | -0.7 |
| russian | 71.9 | 72.0 | +0.1 |
| russian | 73.1 | 73.3 | +0.3 |
| russian | 76.3 | 76.2 | -0.1 |
| russian | 69.1 | 71.7 | +2.6 |
| russian | 74.1 | 75.9 | +1.8 |
| spanish | 84.9 | 84.7 | -0.2 |
| spanish | 83.9 | 84.1 | +0.1 |
| turkish | 80.2 | 81.2 | +1.0 |
| turkish | 77.5 | 82.2 | +4.7 |
| turkish | 77.3 | 78.9 | +1.6 |
| turkish | 83.1 | 83.9 | +0.8 |
| turkish | 80.2 | 81.2 | +1.0 |
| turkish | 84.5 | 85.1 | +0.7 |
| turkish | 77.0 | 80.9 | +3.9 |
| turkish | 80.4 | 83.8 | +3.4 |
| turkish | 77.5 | 82.2 | +4.7 |
| turkish | 79.9 | 82.2 | +2.3 |
| turkish | 76.4 | 81.8 | +5.4 |

## Table 3 — Non-unique answers: the limit no training removes

On probes where a group must agree but the value is free, the likelihood-optimal marginal is uniform over the valid values, so two independent draws agree with probability 1/V. Sweeping an entropy penalty designed to break that symmetry over 66 runs: weak settings changed nothing, strong settings destroyed the model, failing even at fully sequential decoding. Succeeding would require abandoning the data distribution.

## Table 4 — Why earlier runs appeared to fail (10-digit addition)

Every apparent failure at 10+ digits was a learning-rate artifact, not a limit of the method or of depth.

| layers | lr | warmup | mean @1 pass | seeds >90% |
|---|---|---|---|---|
| 6 | 5e-05 | 500 | 100.0 | 3/3 |
| 6 | 5e-05 | 4000 | 100.0 | 3/3 |
| 6 | 0.0001 | 500 | 100.0 | 3/3 |
| 6 | 0.0001 | 4000 | 100.0 | 3/3 |
| 6 | 0.0003 | 500 | 70.6 | 2/3 |
| 6 | 0.0003 | 4000 | 100.0 | 3/3 |
| 10 | 5e-05 | 500 | 100.0 | 3/3 |
| 10 | 5e-05 | 4000 | 100.0 | 3/3 |
| 10 | 0.0001 | 500 | 100.0 | 3/3 |
| 10 | 0.0001 | 4000 | 100.0 | 3/3 |
| 10 | 0.0003 | 500 | 47.3 | 1/3 |
| 10 | 0.0003 | 4000 | 100.0 | 3/3 |

## Table 6 — Minimum depth for one-pass addition

The account in THEORY.md predicts minimum depth growing as log2(n), since the carry is a prefix scan. A ripple-carry account predicts linear growth; an impossibility account predicts no depth suffices.

| digits | 2L | 3L | 4L | 6L | 8L | 12L | log2(n) | min depth >90% |
|---|---|---|---|---|---|---|---|---|
| **4** | 100 | 100 | 100 | 100 | 100 | 100 | 2.0 | 2 |
| **8** | 99 | 100 | 100 | 100 | 100 | 100 | 3.0 | 2 |
| **12** | 48 | 100 | 100 | 100 | – | – | 3.6 | 3 |
| **16** | 0 | 50 | – | – | – | – | 4.0 | – |

## Table 7 — Which component carries the gain (inflection)

| language | mdlm | matched | distill | full | (gap 1→32) |
|---|---|---|---|---|---|
| finnish | 51.4 | 52.0 | 49.4 | 51.9 | +7.7 / +7.0 / +5.1 / +4.9 |
| georgian | 94.6 | 94.3 | 94.3 | 94.1 | -0.0 / +0.1 / -0.2 / -0.0 |
| german | 76.0 | 77.9 | 78.2 | 78.4 | +3.2 / +1.4 / +0.5 / +0.2 |
| navajo | 28.0 | 28.7 | 29.0 | 26.9 | +8.0 / +8.0 / +5.9 / +4.5 |
| russian | 71.6 | 74.7 | 73.9 | 72.3 | +2.2 / +0.1 / -0.1 / -0.3 |
| turkish | 78.7 | 78.7 | 80.2 | 82.3 | +3.5 / +3.7 / +1.2 / +0.8 |

## Table 8 — text8: quality against compute

Generative perplexity under a from-scratch autoregressive character evaluator, against the number of function evaluations actually used. Entropy and distinct-4-gram rate are reported beside it because generative perplexity alone is gamed by degenerate repetition; the real-text reference row is the target, not zero.

Real text8 reference — ppl 2.86, entropy 2.855, distinct-4 0.177

| training | decoding | NFE | gen ppl | entropy | distinct-4 | valid BPC |
|---|---|---|---|---|---|---|
| mdlm | conf_0.5 | 234.5 | 32.99 | 2.871 | 0.360 | 1.7225 |
| mdlm | conf_0.7 | 248.5 | 16.99 | 2.866 | 0.295 | 1.7225 |
| mdlm | conf_0.9 | 252.8 | 9.78 | 2.848 | 0.245 | 1.7225 |
| mdlm | conf_0.95 | 254.0 | 9.08 | 2.856 | 0.238 | 1.7225 |
| mdlm | conf_0.99 | 255.5 | 8.74 | 2.851 | 0.236 | 1.7225 |
| mdlm | entbudget_0.05 | 256.0 | 9.47 | 2.854 | 0.241 | 1.7225 |
| mdlm | entbudget_0.1 | 255.5 | 8.79 | 2.861 | 0.241 | 1.7225 |
| mdlm | entbudget_0.25 | 255.0 | 8.93 | 2.858 | 0.239 | 1.7225 |
| mdlm | entbudget_0.5 | 253.2 | 9.39 | 2.858 | 0.243 | 1.7225 |
| mdlm | entbudget_1.0 | 248.2 | 10.39 | 2.857 | 0.248 | 1.7225 |
| mdlm | entbudget_2.0 | 245.0 | 15.44 | 2.861 | 0.279 | 1.7225 |
| mdlm | entbudget_4.0 | 223.2 | 51.18 | 2.882 | 0.401 | 1.7225 |
| mdlm | entbudget_8.0 | 96.5 | 111.68 | 2.888 | 0.520 | 1.7225 |
| mdlm | fixedK_1 | 1.0 | 646.96 | 2.860 | 0.630 | 1.7225 |
| mdlm | fixedK_128 | 128.0 | 44.33 | 2.867 | 0.410 | 1.7225 |
| mdlm | fixedK_16 | 16.0 | 126.50 | 2.878 | 0.532 | 1.7225 |
| mdlm | fixedK_2 | 2.0 | 310.58 | 2.890 | 0.625 | 1.7225 |
| mdlm | fixedK_256 | 256.0 | 8.98 | 2.854 | 0.241 | 1.7225 |
| mdlm | fixedK_32 | 32.0 | 95.60 | 2.875 | 0.497 | 1.7225 |
| mdlm | fixedK_4 | 4.0 | 155.18 | 2.881 | 0.550 | 1.7225 |
| mdlm | fixedK_64 | 64.0 | 86.83 | 2.882 | 0.490 | 1.7225 |
| mdlm | fixedK_8 | 8.0 | 121.66 | 2.879 | 0.527 | 1.7225 |
| matched | conf_0.5 | 237.5 | 32.44 | 2.873 | 0.366 | 1.6946 |
| matched | conf_0.7 | 242.0 | 14.86 | 2.854 | 0.282 | 1.6946 |
| matched | conf_0.9 | 249.2 | 9.67 | 2.856 | 0.248 | 1.6946 |
| matched | conf_0.95 | 251.2 | 8.65 | 2.855 | 0.241 | 1.6946 |
| matched | conf_0.99 | 254.8 | 8.34 | 2.853 | 0.236 | 1.6946 |
| matched | entbudget_0.05 | 256.0 | 8.39 | 2.857 | 0.241 | 1.6946 |
| matched | entbudget_0.1 | 256.0 | 8.50 | 2.855 | 0.238 | 1.6946 |
| matched | entbudget_0.25 | 255.0 | 8.72 | 2.857 | 0.243 | 1.6946 |
| matched | entbudget_0.5 | 253.2 | 9.00 | 2.863 | 0.247 | 1.6946 |
| matched | entbudget_1.0 | 251.5 | 10.02 | 2.861 | 0.255 | 1.6946 |
| matched | entbudget_2.0 | 247.2 | 14.98 | 2.862 | 0.284 | 1.6946 |
| matched | entbudget_4.0 | 235.2 | 47.92 | 2.885 | 0.403 | 1.6946 |
| matched | entbudget_8.0 | 98.0 | 104.65 | 2.895 | 0.526 | 1.6946 |
| matched | fixedK_1 | 1.0 | 651.67 | 2.860 | 0.631 | 1.6946 |
| matched | fixedK_128 | 128.0 | 42.38 | 2.869 | 0.407 | 1.6946 |
| matched | fixedK_16 | 16.0 | 94.52 | 2.868 | 0.502 | 1.6946 |
| matched | fixedK_2 | 2.0 | 295.79 | 2.881 | 0.620 | 1.6946 |
| matched | fixedK_256 | 256.0 | 8.19 | 2.852 | 0.235 | 1.6946 |
| matched | fixedK_32 | 32.0 | 93.33 | 2.882 | 0.510 | 1.6946 |
| matched | fixedK_4 | 4.0 | 146.64 | 2.880 | 0.558 | 1.6946 |
| matched | fixedK_64 | 64.0 | 76.89 | 2.884 | 0.484 | 1.6946 |
| matched | fixedK_8 | 8.0 | 123.33 | 2.876 | 0.540 | 1.6946 |

## Table 9b — Any-budget conditioning vs a fixed budget

20-digit addition, one configuration. A fixed-K method wins at the budget it was trained for and loses away from it; the question is whether conditioning on the budget removes that trade.

| training | @1 pass | @sequential | spread |
|---|---|---|---|
| MDLM baseline | 0.0 | 0.0 | 0.0 |
| fixed-K (K=1) | 99.9 | 85.8 | 14.2 |
| any-budget (no conditioning) | 0.2 | 0.3 | 0.1 |
| any-budget (conditioned) | 0.0 | 0.0 | 0.0 |

## Table 9 — Did the baseline fail to parallelise, or fail to learn?

A 20-digit answer occupies 21 slots, so the standard 16-pass grid is not a sequential decode. Re-decoded at one token per pass, the two explanations separate.

| digits | mode | @1 pass | fully sequential | reading |
|---|---|---|---|---|
| 20 | full | 99.8 | 85.7 | learned it and parallelises |
| 20 | mdlm | 0.0 | 0.0 | never learned the task |
| 24 | full | 33.7 | 33.3 | learned it and parallelises |
| 24 | mdlm | 0.0 | 0.0 | never learned the task |

## Table 10 — Structured graphs: measured against an exact ceiling

V* is the one-pass validity achievable from the TRUE marginals, so it bounds any model that reproduces the data distribution. Where the interval contains V* the residual failure is information-theoretic and untrainable; where it falls short, the shortfall is trainable.

| family | TC (bits) | V* | achieved @1 pass | reading |
|---|---|---|---|---|
| 2regular | 8.43 | 0.29% | 0.32% | at the ceiling — untrainable |
| matching | 6.92 | 0.82% | 0.77% | at the ceiling — untrainable |
| tree | 3.43 | 9.25% | 9.20% | at the ceiling — untrainable |
| bipartite | 0.11 | 93.63% | 5.88% | below ceiling — see Table 11 |

## Table 11 — One-pass decoding needs ABSOLUTE position information

At t = 1 every input token is [MASK], so the sequence is constant and a relative encoding gives every position the same output. Position-dependent marginals are then inexpressible at any training budget. Matching is the control: its marginals are identical at every slot, so absolute information should buy nothing there.

| family | marginals | encoding | @1 pass | % of V* |
|---|---|---|---|---|
| bipartite | **vary** | rope | 5.87% | 6.3% |
| bipartite | **vary** | ape | 93.91% | 100.3% |
| bipartite | **vary** | sin | 93.95% | 100.3% |
| matching | constant | rope | 0.78% | 94.7% |
| matching | constant | ape | 0.76% | 91.8% |

- absolute − relative on **bipartite**: **+88.037** points
- absolute − relative on **matching**: **-0.024** points