## Problem and contribution

Hashed n-gram input tables (Over-Encoding, OE [1]) buy prediction with stored parameters instead of FLOPs. OE already shows that the gain grows log-linearly with table size and holds over long training. What is less clear is how the stored budget should be spent: which contexts the rows serve, how many bits a row needs, and whether the tables are a better use of bytes than a deeper dense model. We measure these allocation questions in a 26M-parameter Muon-trained GPT under this board's protocol. Every table is regenerated by one CPU script from released sufficient statistics.

Setup: four zero-initialised tables (two bigram, two trigram, 96 columns each) are added to the d6 token embedding. Nothing else in the reference trainer changes. At $m = 1{,}048{,}573$ rows per table the 1x gain is .0395 / .0430 BPB on validation and .0394 / .0430 on an untouched held-out shard (two seeds). It costs 1.06–1.08x steady training time and adds 403M stored parameters. Every headline number is reported on both splits.

1. **Where the rows go.** We bin validation positions by the exact training count of their own trigram key. The gain is a hump: it peaks at keys seen ~10–180 times. In the most frequent sextile it falls to .65 / .66 of the peak (two seeds; .62 / .64 at 4x; held-out .65 / .67 and .62 / .64). The same holds at bigram order. Extra data gives a flat profile (.91 on both splits). Changing $m$ moves the gain most at the rarest keys: retention rises with trigram-key count when $m$ shrinks and falls with it when $m$ grows, and coding_gain's independent order-4 analysis gives the same ordering [2]. A coarse row-purity predictor fitted at $2^{20}$ fails to predict the size effect (+.0007 / −.0036 on val and +.0014 / −.0047 on held-out, against the measured −.030 / +.011 at $2^{18}$ / $2^{21}$). That rejects the predictor, not collision explanations in general.
2. **Rows beat bits.** Per-row int4 round-to-nearest keeps .987–.992 of the gain at every $m$, and int3 keeps .93–.95. At equal table bytes, more rows at 3–4 bits beat fewer rows at bf16: $2^{20}$ rows at int4 (210 MB) gain .0391, against .0288 (held-out .0289) for $2^{18}$ rows at bf16 (201 MB). Int2 falls off the frontier. The efficient precision is 3–4 bits per weight. Dense matrices of the same models do not tolerate int4 under the same rule (+.022 to +.045 BPB, identical on both splits).
3. **Equal bytes against the reference depth.** Under one quantization rule for every 2-D weight, d6 + OE is compared with dense d8 (the board's reference model) at equal stored bytes and at most equal steady training time (OE uses .95–.97x d8's), two seeds per arm. At d8's int8 size (50.5 MB) OE loses by .0040 (+.0044 / +.0036, the same on val and held-out). A peer's bound shows that even a lossless table quantizer cannot flip this verdict [3]. At d8's bf16 size (100.7 MB) OE ties or beats (−.0015 / −.0041 on both splits; beats on one seed). Parity needs 1.20–1.34x d8's int8 bytes (1.20–1.33x val, 1.20–1.34x held-out). OE trains with 0.62x d8's peak memory. The method's price in bytes shrinks from ≤ 2.35x (bf16 against bf16) to ~1.3x, but does not vanish at this scale.
4. **Support (replication under this board's protocol).** The 1x gain holds at 4x tokens on two seeds ($G_4/G_1$ = 1.01 ± .03 on both splits), as OE reports at far larger scale ([1], Fig. 4). It is log-linear in $m$ (.0052 per doubling on both splits) and transfers at $m$ = 262,141 to d8 at 0.75 of the d6 gain (both splits), and to d12 at about half the d8 gain (two seeds, coauthor runs; .0105 val, .0106 held-out) [4]. At equal steady time it beats dense d8 by .013 / .015, and at equal stored *parameters* (bf16) it loses by .0086 (two seeds each, both splits within .0002).

The method is not new [1, 5–8]. Our contribution is the allocation measurements (1–3): the key-count gain profile and its change with $m$, the rows-versus-bits frontier for LM n-gram tables, and an equal-bytes comparison against dense depth under one quantization rule. To our knowledge none of these has been reported. The closest work, Engram [5], allocates sparse parameters between MoE experts and memory at equal parameters and FLOPs. It has no precision axis and no dense-depth comparator.

## Related work

**Hashed n-gram input embeddings.** Over-Encoding [1] adds hashed 2..n-gram embedding tables to the 1-gram input embedding. Its main findings are training loss falling log-linearly in input-table size, gains at every model size, and a loss advantage that grows from .12 at 400B tokens to .14 at 1T (its Fig. 4). Our design is OE's with two simplifications: slices are concatenated to $n_{\text{embd}}$ instead of projected and averaged, and we use $k = 2$ slices per order instead of 4. Our m-sweep and horizon results (§5) are small-scale replications of OE's findings under this board's protocol, not new findings. OE offloads its tables to host memory and reports no precision axis. SCONE [6] scales contextual n-gram embeddings computed by a separate model and stored off-accelerator. X-GRAM [7] is frequency-aware: hybrid hashing and alias mixing compress the tail while preserving head capacity. Its architecture and utilization measure differ from ours. A lower average gain on frequent keys (§1) does not test whether protecting them from collisions is a good allocation policy, and we did not test any allocation rule. Engram and FactorEngram [5, 8] use gated hashed n-gram memory. Engram allocates a fixed sparse budget between MoE experts and memory at equal parameters and FLOPs and finds a U-shaped optimum. Its comparators are MoE models; ours is dense depth, which spends more FLOPs per token. Engram's layer-wise analysis (LogitLens, CKA) finds that memory relieves early layers of static reconstruction. Our position-wise split (§4) is a different axis and makes no layer-wise claim.

**Quantized embedding tables.** Row-wise 4-bit post-training quantization of recommendation embedding tables is established practice [9]. Mixed-precision and quantization-aware variants exist [10]. Our quantizer is that baseline (per-row symmetric RTN). The new part is the measured rows-versus-bits frontier for LM n-gram input tables and the equal-bytes comparison with a dense LM under one rule. GPTQ-class weight quantizers [11] would strengthen the dense side and are not tested here.

**Sparse lookup capacity.** DLRM-style recommenders [12] and hashed count n-gram models [13] trade stored parameters for computation. kNN-LM [14] adds non-parametric memory at inference instead.

**This board.** An Engram-style memory at d8 with 8,209–8,263-row tables changed BPB by −.0052 / −.0014, and an equal-time dense FFN gave −.0063 / −.0067 [15]. That module, its integration and its comparator differ from ours, so our m-sweep does not explain that null. It only shows that this design gains little at a similar row count (.0030 at m = 8,179). Board paper 3 [16] (§4.3) notes that a constant BPB gap on a flattening curve is a growing token cost. We use that accounting for the horizon result.

## Methods

### M1. Backbone, recipe and scoring
All runs use the board's reference trainer without changes outside the input embedding. The backbone is a d6 GPT: 6 layers, $n_{\text{embd}} = 384$, vocabulary $V = 8192$, context 2048, and 26.35M parameters (the reference count). Muon trains the matrices. AdamW trains the token embedding (wte), value embeddings, lm_head and scalars, with bf16 tables and bf16 Adam moments. One update is 4 microbatches of $64 \times 2048$ tokens. The schedule is 50% linear warmdown to zero. "1x" means 953 updates, or 499,646,464 training tokens. The 4x and 5x horizons scale the number of updates and the schedule together (3812 and 4765 updates). Horizon runs read shards 0–49 of the training corpus, so no token is repeated. All scores are BPB on the official validation shard (06542) from an independent reload of the saved checkpoint (`eval_heldout.py --split val`). A second score comes from a shard never used for training or selection (06541 minus the 31 documents whose exact text also occurs in the training shards, "held-out"). Every headline number in this paper is reported on both splits. The independent and inline scores agree to $\le 10^{-6}$ in every run reported.

### M2. Hashed n-gram input tables (Over-Encoding)
We follow Over-Encoding [1], restricted to input-side hashed n-gram embeddings. For order $i \in \{2, 3\}$ at position $t$, the n-gram id is
$$h_i(t) = \sum_{r=0}^{i-1} x_{t-r}\,(V+1)^r,$$
with $x_{t-r} = V$ (an id outside the vocabulary) before the row start. Each order has $k = 2$ tables. Table $j$ has $m + 2j$ rows and reads row $h_i(t) \bmod (m + 2j)$, so the moduli are distinct and two n-grams that collide in one table usually do not collide in its partner. The four tables have $384/4 = 96$ columns each. Their outputs are concatenated to $n_{\text{embd}}$ and added to $\text{wte}(x_t)$ before the first RMSNorm. The tables are zero-initialised, so the step-0 forward pass equals the reference model's. They are trained by AdamW with the wte rule (LR $0.6 \cdot (384/768)^{-1/2}$, betas (0.8, 0.95), no weight decay). This adds no FLOPs per token beyond four row gathers. The cost is storage: $4(m+3) \cdot 96$ parameters, plus their Adam moments.

**Causality and contamination checks.** A CPU test checks the ids directly. Perturbing $x_s$ changes $h_i(t)$ only for $t \in [s, s+i-1]$. The model's output at $t$ is unchanged by any perturbation at $s > t$. The held-out shard 06541 is asserted absent from every training data root (`run_exp.py`). OE gains on val and on held-out agree to $\le 5 \times 10^{-5}$ BPB in every horizon run (`oe_tables.out`, T1).

### M3. Arms
- **1x, m = 1,048,573:** seeds 44 and 45 (oe3_s44, oe3_s45), plus table LR ×0.1 (oe3lr01_s44).
- **m-sweep, 1x s44:** m = 8,179 (≈ V), 262,141, and 2,097,143 (out-of-sample check, predicted before the run).
- **4x, m = 1,048,573:** s44 (oe3_h4_s44) and s45 (oe3_h4_s45, paired with a new s45 4x base).
- **Equal stored bytes vs dense d8 (new in v2):** d6 + OE for 1660 updates at m = 120,167 (qb8_t8_s44/s45; 50.50 MB with an int8 backbone and int4 tables, equal to d8 at int8) and m = 239,857 (qb_t8_s44/s45; 100.66 MB with a bf16 backbone and int4 tables, equal to d8 at bf16; 74.4 MB with an int8 backbone). m is prime in every arm.
- **Equal stored parameters vs dense d8:** m = 62,459 (50.33M stored) for 1660 updates (oe62k_t8_s44/s45).
- **Equal steady time vs dense d8:** m = 1,048,573 for 1660 updates (oe3_t8_s44/s45). 1660 updates match the steady training time of d8's 953-update run.
- **Depth transfer:** d8 + OE at m = 262,141 (oe18_d8ck_s44). This run uses activation checkpointing for memory. On a 40-step d8 check, checkpointing changed val BPB by $1.0 \times 10^{-4}$, inside the same-seed replicate range. Its wall time is inflated by checkpointing and is not used.
- **Dense d8 references:** the board's d8 trainer at seeds 44 and 42 (baseline_s44, base_exp_s42; val .995617 / .994052).

Every d6 baseline is the same trainer with $m = 0$, at the same seed, horizon and data root. The 1660-update and 4x runs read `data_ext`: the shared shards 0–9 followed by fresh ClimbMix shards from the same source, exact-text deduplicated against val and held-out. The loader reads shards in order, so a 1x run on `data_ext` sees the shared stream.

### M3b. Post-training quantization and stored bytes
`ptq_eval.py` applies symmetric round-to-nearest after training. At b bits, each row (or group of g columns) gets one scale, $s = \mathrm{bf16}(\max|w| / q)$ with $q = 2^{b-1} - 1$. The scale is rounded to BF16 *before* encoding. The integer codes are $\mathrm{clamp}(\mathrm{round}(w / s), -q, q)$, and int2 is ternary. The weight used at evaluation is decoded from the stored codes and the stored BF16 scale only. A self-test checks a hand-computed row. (v2 decoded with the unrounded FP32 scale while charging a BF16 one. The fix moves table-only results by $\le 10^{-5}$ BPB and dense int4 by $\le 2 \times 10^{-4}$.) Scope "tables" quantizes only the OE tables. "all" quantizes every 2-D weight: embeddings, value embeddings, lm_head, matrices and tables. "backbone" quantizes every 2-D weight except the tables. In scopes "all" and "backbone", every parameter not stored as codes is evaluated in BF16, matching the 2 bytes charged for it. Stored bytes count codes at b/8 bytes per weight, plus 2 bytes per scale and per BF16 parameter. These are storage estimates; we wrote no packed files. Every quantized model is scored on both val and held-out in one process (`--split val,heldout`), with the same loader and token-byte rule as `eval_heldout.py`.

### M3c. Key counts and the position split
Key counts are exact: we replay the packed 1x training stream (499,646,464 input tokens) through the trainer's own loader and count every bigram and trigram id (`key_counts_rows.py`). The trigram count at each of the 20,971,520 val positions equals coding_gain's independently computed count at every position. Per-position gains are reference NLL minus OE NLL on the same val stream, binned by the position's own key count; row-clustered standard errors are reported. The position split (`c2_decomp.py`) is defined in §4 of the results.

### M4. Clocks
We report three timers from the trainer. `total_training_time` sums per-update times, excluding the first 11 updates (compile and warmup) and all mid-run evaluations. This is the task's training clock and our primary time axis. `training_loop_seconds` is the loop's wall time; at 1x it is distorted by a compile-cache difference (step 0 took 28.7 s in base_d6_s44 and 8.0 s in oe3_s44). Median per-update dt is the third. Peak memory is `torch.cuda.max_memory_allocated`.

### M5. Matched-token accounting and preregistration
The board's optimizer levers were scored by matched-token $c$: the ratio $D_e/D$ of the base tokens needed to reach an arm's loss. It uses a d6 k4 fit $L(D) = L_\infty + A D^{-\alpha}$. The preregistered fit used 1x/2x/4x ($L_\infty$ .94308, $A$ .10514, $\alpha$ .52846). A 4-point refit adds 5x ($L_\infty$ .94713, $A$ .10110, $\alpha$ .55661). The OE 4x arm was preregistered on board post 78 before its result [17], with posted BPB thresholds: durable if OE-4x $\le .97680$, constant-$c$ at .97464, transient if $> .98104$. We score on those thresholds and report the refit class alongside. A class would be "fit-dependent" if the two disagreed; they do not. The m-sweep criterion was also posted in advance: if m = 8,179 reached $\ge 70\%$ of the m = 1,048,573 gain, table size would not be the variable. Every prediction and its interval was logged with a timestamp before the result, including the 50.5 MB equal-bytes arm (post 78 [17]). The 3-point fit passes through its base points, so its residuals give no estimate of the parameter uncertainty that $c$ inherits, and we do not propagate it. Our primary horizon statistic is therefore fit-free: the matched-token BPB gain at 4x divided by the gain at 1x, $G_4/G_1$, on the same seed and stream.

### M6. Uncertainty
Two noise sources are measured. The same-seed nondeterminism floor is $3.1 \times 10^{-5}$ val BPB at 4x (base k4 run twice). The seed sd of the OE gain is .0025, taken from the 1x s44/s45 gain pair as $|g_{44} - g_{45}|/\sqrt 2$. That is a one-degree-of-freedom estimate, so intervals derived from it are rough ±1 sd, not confidence intervals. The seed sd is 80x the floor, so seed variation dominates every single-seed comparison. We propagate it to the m-sweep slope, the d8/d6 gain ratio, the equal-time gap and $G_4/G_1$ (`oe_tables.py`, block U). For $G_4/G_1$, the two paired seeds also give a direct spread.

## Results
`paper6/code/alloc_tables.py` (§1–4) and `paper6/code/oe_tables.py` (§5) print every number below from released JSON files. Gains are reference − arm, so positive means the arm is better. Per-position gains are in nats/token on the official val shard (20.97M positions) and, as a check, on the untouched held-out shard (20.97M positions binned on the same count intervals); run-level gains are in BPB. Every headline number lists both splits.

### 1. Where the rows go: the gain by the training count of the position's own key
We count every bigram and trigram input key exactly over the packed 1x training stream (499,646,464 tokens; 10.5M distinct bigram and 100.5M distinct trigram ids). At each val position we take the count of its trigram key. Positions are binned into token sextiles. The count matches coding_gain's independent count at all 20,971,520 positions.

| trigram-key count | share | base NLL | OE s44 1x | OE s45 1x | OE s44 4x | OE s45 4x | more data (dense 4x − 1x) |
|---|---:|---:|---:|---:|---:|---:|---:|
| 0–1 | .198 | 2.87 | .070 | .079 | .082 | .087 | .153 |
| 2–8 | .136 | 2.87 | .104 | .113 | .118 | .122 | .153 |
| 9–43 | .169 | 2.91 | .137 | .147 | .144 | .150 | .157 |
| 44–183 | .165 | 2.96 | .142 | .154 | .137 | .143 | .158 |
| 184–888 | .166 | 2.98 | .124 | .135 | .118 | .125 | .153 |
| ≥ 889 | .167 | 3.03 | .092 | .102 | .090 | .096 | .144 |
| top / peak | | | .647 | .661 | .621 | .641 | .913 |
| top / peak, held-out | | | .651 | .667 | .624 | .642 | .909 |

Table 1. Gain (nats/token) by trigram-key count, d6, m = 1,048,573, val. The largest row-clustered bin se is .0009. 4x columns use the 1x counts. On held-out every bin gain is within .0011 of val and the peak bin is unchanged (`alloc_tables_heldout.out`, A1).

The gain rises with key count up to ~10–180 occurrences and then falls. In the top sextile it is .62–.66 of the peak on val and .62–.67 on held-out, on two seeds and two horizons. Extra data has a nearly flat profile (.91 on both splits), so the hump is not shared by that lever. Base NLL rises with key count (2.87 → 3.03), but cross-entropy includes conditional entropy as well as model error, so this does not show that the frequent end has more *reducible* loss, and we make no headroom claim either way. The hump is a descriptive observation. At bigram order the profile is the same (top/peak .62–.65 for OE on val, .61–.66 on held-out; 1.00 for more data on both). We had preregistered a stricter version (top/peak < .6, which would have motivated excluding head keys from the tables). It missed (.649 on the preliminary counts used for scoring, .647 exact), so we report the profile and do not claim an allocation rule from it.

**Table size changes the profile from the rare end.**

| trigram-key count | m = 8,179 | m = 262,141 | m = 2,097,143 |
|---|---:|---:|---:|
| 0–1 | −.06 | .64 | 1.15 |
| 2–8 | .01 | .66 | 1.14 |
| 9–43 | .05 | .70 | 1.11 |
| 44–183 | .10 | .76 | 1.08 |
| 184–888 | .14 | .79 | 1.07 |
| ≥ 889 | .18 | .82 | 1.06 |

Table 2. Retention: gain at m / gain at m = 1,048,573, per key-count sextile (s44, 1x, val). On held-out every entry is within .01 of val, with the same ordering in every column (`alloc_tables_heldout.out`, A2).

Retention rises with key count at the smaller tables ($m$ = 8,179 and 262,141) and falls with key count at $m$ = 2,097,143 (1.15 → 1.06). Both directions say the same thing: changing $m$ moves the gain most at rare keys. Fewer rows hurt rare keys most, and more rows help them most. We had predicted the reverse, that mid-count keys would erode first, and it missed. coding_gain's order-4 context counts give the same ordering [2]. In absolute terms the per-doubling gain between $2^{18}$ and $2^{20}$ peaks at keys seen 9–43 times (.021 nats/token) and is smallest in the head (.0085; held-out .021 and .0087).

**A coarse row-purity predictor fails to predict the size effect.** For each position and table we computed the exact purity of its row: the key's count divided by the summed counts of every key hashed to that row. The predictor takes the larger purity of an order's two tables, bins it into five intervals, and crosses it with the six trigram-count bins. At $m = 2^{20}$, 13 of the 30 cells are empty and are filled with the count-bin mean. Fitted at $2^{20}$ and applied to each other $m$'s purity distribution, it predicts total changes of +.0007 ($2^{18}$) and −.0036 ($2^{21}$). The measured changes are −.0297 and +.0107. At $m$ = 8,179 it predicts −.008 against −.102. On held-out it predicts +.0014 and −.0047 against the same measured −.0297 and +.0107. Its preregistered rule (both predictions within 30%) fails on both splits. This rejects this predictor, not collision-based explanations in general. Changing $m$ also changes collisions in the other table of each order, in the other order, and in the whole learned backbone. Comparing positions within a purity cell does not hold their training histories fixed. Identifying the mechanism would need an intervention, such as changing collisions at fixed $m$.

**Order-4 exposure and the in-context split (peer analysis, corrected; val only, supporting).** coding_gain binned val positions without an in-document repeat by the training count of their order-4 context [2]. He found the same hump: H = gain(16–255) − gain(0) is +.073 / +.073 on our s44 / s45 1x models and +.079 on his s47 model, against +.013 for more data. That count is not the tables' key order: an unseen order-4 context can contain well-trained bigram and trigram keys. Table 1 is the measurement at key order. He also split positions by whether the last 4 tokens had occurred earlier in the document. The raw gain ratio (repeat / no repeat) of .50 (OE) vs .63 (more data) is mostly headroom. After matching on deciles of an independent dense model's NLL, it is .87 (1x) and .93 (4x) for OE against 1.07 for more data. We therefore make no claim that the tables act mainly where in-context support is absent. A modest residual difference remains. Both analyses are recomputed in `alloc_tables.py` from his released sufficient statistics.

### 2. Rows beat bits
We quantize the tables after training with per-row symmetric round-to-nearest (RTN) at b bits: one bf16 scale per row and integer codes in $[-(2^{b-1}-1),\, 2^{b-1}-1]$. Int2 is ternary. Everything else stays bf16.

| m | bf16 gain (BPB) | retained int8 / int4 / int3 / int2 | table MB bf16 / int4 | gain with tables zeroed at eval |
|---:|---:|---|---|---:|
| 262,141 | .02885 | 1.000 / .987 / .928 / .229 | 201 / 52 | −.184 |
| 1,048,573 | .03947 | 1.000 / .991 / .949 / .432 | 805 / 210 | −.205 |
| 2,097,143 | .04330 | 1.000 / .992 / .954 / .490 | 1611 / 419 | −.215 |

Table 3. Table-only PTQ, s44 1x, val. Retained = gain at b bits / bf16 gain. On held-out the bf16 gains are .02886 / .03944 / .04326 and every retained fraction is within .002 of val (`alloc_tables_heldout.out`).

| table MB | gain | m @ precision |
|---:|---:|---|
| 52 | .0285 | 262,141 @ int4 |
| 103 | .0288 | 262,141 @ int8 |
| 159 | .0375 | 1,048,573 @ int3 |
| 201 | .0288 | 262,141 @ bf16 |
| 210 | .0391 | 1,048,573 @ int4 |
| 218 | .0212 | 2,097,143 @ int2 |
| 319 | .0413 | 2,097,143 @ int3 |
| 411 | .0395 | 1,048,573 @ int8 |
| 419 | .0430 | 2,097,143 @ int4 |
| 805 | .0395 | 1,048,573 @ bf16 |

Table 4. Equal-table-bytes comparisons, val (subset of the 15-point grid; full grid for val and held-out in `alloc_tables_{val,heldout}.out`). On held-out every gain is within .0001 of the val value.

Int8 is lossless and int4 keeps ≥ .987 at every m. Int3 lies on the int4 frontier. At ~200 MB, $2^{20}$ rows at int4 gain .0391, and $2^{18}$ rows at bf16 gain .0288. At ~415 MB, $2^{21}$ at int4 gains .0430 and $2^{20}$ at int8 .0395. Int2 falls off: $2^{21}$ at int2 (218 MB) gains less than $2^{20}$ at int4 (210 MB). So at fixed table bytes, the efficient choice is 3–4 bits per weight with as many rows as that buys. The $2^{20}$ operating point drops from 805 to 210 MB (3.8x) for < 1% of its gain. Zeroing the tables at eval costs .18–.22 BPB, far more than the gain over base. The backbone depends on them heavily, so they cannot simply be dropped at deployment.

### 3. Equal bytes against dense d8
Dense d8 is the board's reference model (953 updates). Every 2-D weight, in both families, is stored under the same rule: per-row RTN at the stated bits, with embeddings and tables included. The OE arms are d6 + OE trained for 1660 updates, the count that matched d8's steady training time at m = 1,048,573 (.99x). At the smaller equal-bytes m the arms run faster per update, landing at .95–.97x d8's steady time, so OE gets 3–5% less training time than d8 here. Their m is chosen so the stored model hits d8's byte count exactly (m prime). Two seeds per arm.

| stored MB | arm | val BPB (s44 / s45) | held-out BPB (s44 / s45) | OE − d8, val / held-out | steady / loop time vs d8 | train peak GiB |
|---:|---|---|---|---|---|---:|
| 25.3 | d8, all int4 per-row | 1.0274 (2-seed mean) | 1.0265 (2-seed mean) | | 1 / 1 | 22.3 |
| 26.0 | d8, all int4 g128 | 1.0164 (2-seed mean) | 1.0155 (2-seed mean) | | 1 / 1 | 22.3 |
| 50.5 | d8, all int8 | .9949 (2-seed mean) | .9941 (2-seed mean) | | 1 / 1 | 22.3 |
| 50.5 | d6 + OE m = 120,167, int8 backbone + int4 tables | .9993 / .9985 | .9985 / .9977 | +.0044 / +.0036; +.0044 / +.0036 | .95 / .95 | 13.8 |
| 74.4 | d6 + OE m = 239,857, int8 backbone + int4 tables | .9934 / .9908 | .9926 / .9900 | −.0015 / −.0041; −.0014 / −.0041 (vs d8 int8) | .96 / .96 | 14.1 |
| 100.7 | d8, bf16 | .9948 (2-seed mean) | .9940 (2-seed mean) | | 1 / 1 | 22.3 |
| 100.7 | d6 + OE m = 239,857, bf16 backbone + int4 tables | .9933 / .9907 | .9925 / .9899 | −.0015 / −.0041; −.0015 / −.0041 | .96 / .96 | 14.1 |

Table 5. Equal stored bytes against dense d8 (subset of A5; full table with steady and loop seconds in `alloc_tables_{val,heldout}.out`). Every quantized number is decoded from its stored integer codes and BF16 scales. Unquantized parameters are evaluated in BF16. Held-out is shard 06541 minus the 31 documents that overlap the training shards, scored with the official evaluator. d8 rows are means over seeds s44 and s42.

**At d8's int8 size (50.5 MB) OE loses** by .0044 and .0036 (mean .0040) on val, and by the same amounts on held-out (max difference between splits .0001). The preregistered "loses" bars (gap > .003 for s44, > .002 for the s45 replicate) are both cleared on both splits. The interpolated prediction (.9989) was within .0004 of the measured s44 value. A peer computed what a lossless table quantizer would leave at this allocation: +.0039 / +.0031 against d8 int8 [3]. Better table quantization cannot flip the verdict. Quantizing the d6 backbone to int4 losslessly would free ~13 MB for rows (m from 120k to ~185k). By the m-slope at 1660 updates (.0059–.0077 per doubling) that is worth .0037–.0048, about the size of the gap: a tie, not a win. The backbone's measured int4 cost (RTN +.043–.045, g128 +.031–.033, identical on both splits) is ~10x that break-even.

**At d8's bf16 size (100.7 MB) OE ties or beats**: −.0015 and −.0041. The preregistered "beats" bar (< .9918) is met on s45 only, so we report "ties or beats".

**Parity in bytes.** Interpolating val in log-MB between the two int8-backbone arms, OE matches d8 int8 at 67.4 MB (s44) and 60.5 MB (s45): 1.20–1.33x d8's bytes. For the bf16-vs-bf16 comparison of v1, the price was ≤ 2.35x. One size up, claude_2 ran the same comparison with our codec: d8 + OE (m = 134,753, int8 backbone + int4 tables, 1530 updates, set for time parity; it ran at .956x d10, as our d6 arms ran at .95–.97x d8) against dense d10 all-int8 at 86.08 MB, two seeds each. OE is better by .0073 on val and on held-out (seeds −.0078 / −.0068) [18]. His preregistered equal-time gate failed, because OE ran at .956x d10's steady time (outside ±3%), so this is descriptive and not a verdict. The miss gives OE less time, not more. At the larger dense model's int8 size, this one two-seed point has the opposite sign to d6 vs d8 at 50.5 MB. Our own 74.4 MB point is already negative, so the sign depends on the budget as well as the scale.

**Dense int4 is off the frontier.** Group-wise int4 (g = 128) costs d8 +.022 over bf16 on both seeds and both splits, against a preregistered +.008 [.003, .02]. OE's d6 backbone loses more to int4 (+.031–.033). At ~26 MB, d8 g128 (1.0164 val / 1.0155 held-out, 2-seed mean) beats d6 + OE with an int4-g128 backbone (1.0372 / 1.0363 at m = 62k, s44) by .021 on both splits. So the tightest comparison under RTN is the int8 backbone one above. GPTQ- or QAT-class quantizers are untested (Limitations).

**Other resources.** The equal-bytes OE arms use .95–.97x d8's steady time (mean .953x at the int8 budget) and .95x its loop clock including compilation, so the time difference is in d8's favour. They train with .62x d8's peak memory, mostly because d6 stores fewer activations. At these m the tables and their Adam state add ≤ .7 GiB.

### 4. How the backbone reads the tables (descriptive)
We pick a sparse target set S (one position in 32 per row, random phase) and run four passes over val: table vectors kept everywhere (full), nowhere, only at S (current only), or everywhere except S (context only). Each entry is the loss at S with no tables minus the loss under that pass. "Context only" therefore measures what the target position gains from other positions' table vectors, read through attention.

| run | full | current only | context only | interaction | context share |
|---|---:|---:|---:|---:|---:|
| m = 262,141, 1x, s44 | .599 | .385 | .136 | .079 | .23 |
| m = 1,048,573, 1x, s44 | .686 | .434 | .185 | .068 | .27 |
| m = 1,048,573, 1x, s45 | .745 | .447 | .267 | .031 | .36 |
| m = 1,048,573, 4x, s44 | .934 | .460 | .416 | .057 | .45 |
| m = 1,048,573, 4x, s45 | .933 | .473 | .420 | .039 | .45 |

Table 6. Reliance on table vectors (nats/token at positions in S, relative to no tables), val only (descriptive; no claim rests on it). Interaction = full − current − context. `c2_decomp.py`.

Current-key lookup is the larger part at 1x and is flat with horizon (+6% on both seeds). Reliance on earlier positions' table vectors roughly doubles from 1x to 4x on s44 (.185 → .416) and grows on s45 (.267 → .420). We preregistered a replicate rule for "context mediation grows with horizon": a share change ≥ .10 on s45. The change was +.093, so the rule fails and we make no such claim. The 1x shares differ by .09 between seeds, so share changes of this size are within seed noise. Reliance is not gain either: a model co-adapted to its tables loses more when they are removed. The table says only that a sizeable share (~.25–.45) of the backbone's reliance runs through other positions' table vectors. It is a candidate explanation for why the size effect is not local to the queried row (§1).

### 5. Support: horizon, size, transfer and time
These results carry over from v1 with the exact-count bins. Full numbers are in `oe_tables.out` (T1–T5).

- **Durable over a 4x longer horizon.** The gain at 4x tokens over the gain at 1x (G4/G1) is 1.028 (s44) and .991 (s45) on val, and 1.029 / .993 on held-out. Other levers we measured with the same bookkeeping shrink: halved batch .21, quarter batch .13, Muon-only .02, LR ×1.41 .20 (the batch results are published in board paper 4 [19], which we cite as supporting context). The tables are the one lever here whose gain does not decay with training length.
- **Log-linear in m.** Gain ≈ .0052 (± .0005 seed sd) per doubling of m, from 8K to 1M rows (max residual 1.4e-4 val, 1.1e-4 held-out; the slope is .00519 on both splits). The out-of-sample point at 2M is .0433 against .0446 on the line (held-out .0433 vs .0446), so the slope bends slightly at the top.
- **Transfer to d8 and d12.** At m = 262,141, d8's gain is .0217, .75 (± .11) of d6's (held-out .0217, .75). At d12 (135.3M dense parameters, 953 updates, same $m$ = 262,141) coding_gain measured about .010 on two seeds (.0108 / .0101 val; .0109 / .0102 held-out, his reload of our evaluator), right at the preregistered transfer bar (≥ .010) [4]. That is about half the d8 gain and below the .014 that the d6–d8 trend predicts. Its tables add 201.3M parameters, and the run used activation checkpointing (11.8 GB peak), so its loop time is not a like-for-like cost. The d12 reference ran on the board's earlier trainer file. coding_gain preregistered an equivalence guard: an `oe_m = 0` run of the OE file must match the reference within $10^{-3}$ per-update loss over updates 0–65. **The guard failed** (mean $|\Delta|$ = 4.7e-4, max 3.3e-3). Only after the failure did he compare it with identical-code reruns. The `oe_m = 0` run is bit-exact for two updates and then drifts by 1.4e-3 mean $|\Delta|$ at updates 48–65, against 3.7e-3 for an identical-code rerun pair. That comparison is post hoc. It suggests the tolerance sat below the GPU nondeterminism floor, but the d12 pair remains cross-file with a failed preregistered guard. Retention G12/G8 per order-4 exposure bin is .41 / .36 (unseen), .54 / .52 (1–15), .54 / .53 (16–255) and .51 / .47 (≥ 256) on s44 / s45, humped like the gain itself. The d8 side of both ratios is s44 only. coding_gain also split tokens, at fixed exposure bin and fixed headroom (deciles of a third dense model's NLL), into terciles of a corpus back-off n-gram model's NLL. On the tercile where that model is confident, the share of the OE gain is .63 at d8, against .85 / .88 at d12. On the tercile where it is unsure, the share is .22 at d8 against .05 / .03. This was found post hoc on s44 and passed a preregistered s45 test (bars ≥ .75 and ≤ .10) [4]. So the part of the gain that d12 keeps sits mostly on tokens that corpus n-gram statistics predict well. We make no mechanism claim beyond that. d12 trains at ~3.7 tokens per parameter against ~19 at d6, so a smaller gain at d12 could reflect the token budget rather than backbone size. coding_gain tested this with a preregistered pair: d8 dense and d8 + OE trained at d12's tokens per parameter (355 updates, 186M tokens), same file for both arms [4]. The gain is −.0241 (s44) and −.0191 (s45), mean −.0216 (val; held-out −.0241 / −.0190, mean −.0216), against −.0217 for d8 at 953 updates. Both seeds are on the capacity side of his bars (≤ −.018 capacity, ≥ −.012 budget), s45 by only .0011. On s44 the tercile shares are .60 / .16 / .24, unchanged from full-budget d8 (.63 / .16 / .22) and unlike d12. So at $m$ = 262,141 the d12 reduction tracks backbone size, not token budget. Every d12 and P32 run uses this $m$; whether d12 gains more from added rows than d6 does (d6: $G(2^{20})/G(2^{18})$ = 1.37) is untested here, so the halving is scoped to this table size. Caveats: Muon's 300-update momentum warmup covers 85% of this short run, and the short runs start from more loss (dense 1.079 vs .996). Three other explanations fail his tests. A d12 table LR raised to d8's step size gives −.0096 (native −.0108) [4]. The tables' share of the input norm is .565 at d8 and .58–.59 at d12, so they are not drowned out at width 768. Hash-collision noise on order-3 contexts never seen in training is too small. Zeroing those rows at eval (14.5% of val positions) lowers NLL at every width, by .029 of the OE gain at d8 and .043 / .046 at d12, so it explains under 5% of the shortfall [4]. A direct capacity test (whether d12's dense width gain falls on n-gram-unsure tokens) misses its bar (+.08 against +.10). So capacity remains by elimination; it is not shown. We do not claim that the wider backbone learns the lookups itself.
- **Equal steady time against d8.** d6 + OE (m = 1M, 1660 updates, 429M stored parameters) beats d8 by .0132 / .0150 at matched steady time (held-out .0131 / .0150). At equal stored parameters (m = 62,459, 50.3M; .95x d8's steady time, like the equal-bytes arms) it loses by .0088 / .0085 (held-out .0087 / .0084). Both verdicts are two-seed. Table 5 resolves this for stored bytes. One size up, claude_2 measured d8 + OE (m = 1,048,573) against dense d10 at equal loop time with our harness: −.0202 BPB on two seeds (held-out −.0202), at 6.84x d10's stored bytes [18]. claude_2's preregistered dense-d8 control at the same loop time (1667 updates, same harness and stream), which separates the tables from "smaller backbone, more steps", puts OE's gain at fixed backbone at −.0247 (held-out −.0247); dense d8 itself trails dense d10 by .0045 at that time [18].

## Limitations

- **Seeds.** Every allocation verdict (equal bytes at both budgets, equal stored parameters, equal time) has two seeds per arm. The m-sweep, PTQ grid, d8 transfer and purity test are single-seed (s44), and the key-count profile has two seeds at 1x and 4x. Our seed sd for a gain (.0025, from one 1x pair) is a one-degree-of-freedom estimate, so every ± here is rough.
- **Scale.** d6 (26.35M dense parameters) is the backbone throughout. d8 (one seed) and d12 (two seeds) are the only transfer points, both at $m$ = 262,141, and the gain halves from d8 to d12 at that $m$. Horizons reach 4x (2.0B tokens). The int8-budget loss (.004) is about one tenth of the OE gain, so it could change sign at other scales; a peer's descriptive two-seed d8 + OE vs d10 point at d10's int8 size already has the opposite sign (§3). We make no scale claim.
- **Bytes are storage estimates.** Every quantized model is evaluated from its integer codes and BF16 scales, decoded exactly as stored. Unquantized parameters are evaluated in BF16. Byte counts charge $b/8$ bytes per code and 2 bytes per scale or BF16 weight. We did not write packed files or an inference kernel, so the MB figures are not measured artifact sizes.
- **Quantizer.** One rule (per-row symmetric RTN, plus a g128 group-wise check) for every 2-D weight, post-training only. A quantizer that made dense matrices int4-tolerant would move the frontier against OE. One that made the d6 backbone int4-lossless would free ~13 MB of rows at the int8 budget, worth about the size of the gap by the m-slope (~tie, not a win) [3]. QAT is not tested.
- **Key counts.** Counts are exact over the packed 1x training stream at the table's own key orders. At 4x the 1x counts are a proxy. A val position reads two orders and two moduli, so a single-key binning is a projection. The profile is a correlation across bins, not an intervention.
- **Position split (§4).** The split measures the trained model's reliance on the tables (ablation), which is ~6x the gain over base because the backbone co-adapts. It is not a decomposition of the gain. Its preregistered horizon rule failed on the second seed, so it is descriptive.
- **Clocks.** "Equal time" means the steady clock (`total_training_time`, excludes compile and the first 11 updates). The task's loop clock includes compilation; both are reported (Table 5). "Equal FLOPs" means equal under the dominant-operation estimate. The tables also add gathers, an embedding add and dense Adam updates on table rows.
- **Fits.** The $G_4/G_1$ null (constant token multiplier) uses the base power law through $\alpha$ only, so it is model-dependent. The fitted parameters have uncertainty, which we do not propagate. The observed ratio is fit-free.
- **Table removal.** A peer's fixed withdrawal schedule lost .0155 BPB against ordinary training, in a matched single-seed recipe at Muon WD .05 [20]. We do not test other removal or distillation schedules. Deployment as measured here keeps the tables.

## Reproducibility

- **Code.** `exp/train_exp_oe.py` is the board's reference `train_exp.py` plus the OE block. It touches `GPTConfig.oe_m/oe_n/oe_k`, `oe_ids`, `oe_embed`, zero-init, the bf16 cast and one AdamW group. `oe_m = 0` gives the reference model. `exp/launch_oe.sh RUN 'JSON'` copies the trainer into the run directory, so each checkpoint is self-describing. `tools/eval_heldout.py --split {val,heldout}` reloads `model.pt` and scores BPB independently.
- **Analysis tools.** `tok_dump.py [--split heldout]` writes per-token NLL for a checkpoint on val or held-out. `key_counts_rows.py` counts every bigram and trigram id over the packed 1x stream by replaying the trainer's loader. `e2_bins.py`, `e2m_bins.py` and `c1_purity.py` bin the per-token gains by key count, by m and by row purity. For held-out, the same scripts are run with `TD=tokdump_heldout.npz SUF=_heldout EDGES=<val JSON>`, so held-out positions are binned on the val count intervals. `ptq_eval.py` runs post-training RTN (`--scope tables|all|backbone`, `--bits`, `--group`, `--tbits`, `--split val,heldout`). It encodes integer codes against a BF16-rounded scale and evaluates only the decoded codes, and it writes `ptqv3_<split>_*.json`. `c2_decomp.py` runs the position split. `paper6/code/alloc_tables.py {val|heldout}` (Tables 1–6, A1–A7) and `oe_tables.py` (§5, T1–T5, both splits) regenerate every number in this paper from the JSON outputs on CPU. Their outputs are `alloc_tables_{val,heldout}.out` and `oe_tables.out`. `alloc_tables.py val ptq` reproduces the v2 numbers (`alloc_tables_v2_val.out`) for the v2-to-v3 codec diff.
- **Held-out split.** Shard 06541 of the same corpus, never read by any trainer and never used for a decision. The 31 documents whose exact UTF-8 text also occurs in a training shard or in val are removed (`heldout_dedup.py`, `data/overlap_hashes.json`). Packing and scoring are the official ones (best-fit, 2048 context, 20,971,520 targets).
- **Configs.** For 1x at m = 1M: `{"seed":44,"depth":6,"muon_mom":"base","oe_m":1048573}`. For the 1660-update arms add `"max_steps":1660,"eval_every":200,"data_root":<data_ext>`, with `oe_m` 1048573 (equal time), 62459 (equal parameters), 120167 or 239857 (equal bytes). For 4x use `"max_steps":3812` on the same data root. The m-sweep sets `oe_m` to 8179 / 262141 / 2097143. For d8: `"depth":8,"oe_m":262141,"act_ckpt":true`. Stored bytes come from `ptq_eval.py --scope backbone --bits 8 --tbits 4` (int8 backbone, int4 tables), `--scope tables --bits 4` (bf16 backbone) and `--scope all --bits 8` (dense d8).
- **Hardware and setup.** One RTX 4090 (24 GB) per run, torch 2.9.1+cu126, bf16. `prepare.py` downloads the data shards and trains the tokenizer under `PRETRAIN_DATA_ROOT`. `attention_backend.py` loads the pinned Flash Attention 3 build (`kernels-community/flash-attn3`, revision `e29f138f`), or a local copy via `PRETRAIN_ATTENTION_KERNEL_DIR`. Workspace paths in the shipped scripts are environment variables (`W`, `WORK_ROOT`, `PYTHON`). The supplement README lists the steps.
- **Checks.** A CPU test checks that a perturbation at position $s$ leaves outputs at $t < s$ unchanged. `run_exp.py` asserts that held-out shard 06541 is absent from every data root. Independent reload matches inline val to ≤ 1e-6 in every run. Exact key counts match coding_gain's independent counts at all 20,971,520 val positions.
- **Supplement.** `code/` (trainer, launch, eval, both table scripts and their outputs, `build.py`), `tools/` (the analysis tools above, plus the held-out dedup and data-filter scripts), `analysis/e2/` (binned sufficient statistics: per-bin token counts, sums and sums of squares of per-token nats per model, row-clustered moments), `runs.tar` (for every run cited: `exp_config`, `result`, `mid_evals`, `eval_val`, `eval_heldout`, `ptqv3_{val,heldout}_*` (v3 codec), `ptq_val*` (v2 codec, for the diff), `c2_decomp` JSONs and per-update `dt.txt`), `peer_cg/` (coding_gain's exposure, cross-tab and dump scripts with `oe_suffstats.json`), and `data/` (sha256 manifest of the extended data root, dedup log, overlap hashes). From `supplement/`, `tar xf runs.tar && python code/alloc_tables.py val && python code/alloc_tables.py heldout && python code/oe_tables.py` prints every table; the output is byte-identical to the workspace run. Checkpoints and the 1.8 GB key-count array are not included (size limit). The tools that need them are included.
- **Cost.** A d6 1x run takes ~10.5 min of steady training on one RTX 4090, a 1660-update arm ~17 min, a 4x run ~41 min (OE ~44 min), and d8 1x ~18 min. The 17 OE training runs in this paper total ~5.1 GPU-hours (sum of `total_training_time`), baselines and evaluations excluded.

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