Tier 8 saturated at 0.92: highest_tier 7->8, overall 0.769->0.788

weights512.pt re-promoted to the 0.92 cell (gradient-accumulation run, eff.
batch ~2300; eps floor 7e-4 -> 1.3e-4, 512-step chain 0.73 -> 0.92). Full
benchmark: highest_tier_above_90 = 8, overall 0.788, tier 8 = 0.92, det=True.
Compliance 0.92@s0 -> 0.04@s0.25, untrained 0.00. README/manifest updated:
saturates tiers 1-8 / up to 2^512.

README.md

@@ -16,12 +16,13 @@ A compliant **bit-sequential RNN** that computes `(a · b) mod p` for primes `p`
 multiplication tables. Entry for the
 [Modular Arithmetic Challenge](https://github.com/SAIRcompetition/modular-arithmetic-challenge).
 
-- **Saturates tiers 1–7** (all primes `< 2²⁵⁶`): tiers 1–3 = 100%, tier 4 = 99%, tier 5 = 98%, tier 6 = 97%, **tier 7 = 98%**, plus a **partial tier 8 = 73%** (512-bit)
-- **overall_accuracy 0.769**, `highest_tier_above_90 = 7`
+- **Saturates tiers 1–8** (all primes `< 2⁵¹²`): tiers 1–3 = 100%, tier 4 = 99%, tier 5 = 98%, tier 6 = 97%, tier 7 = 98%, **tier 8 = 92%** (512-bit)
+- **overall_accuracy 0.788**, `highest_tier_above_90 = 8`
 - The 128/256/512-bit (tier 6/7/8) cells are **carry-aware TCNs** (weight-shared dilated
   convolutions over the bit-positions, ~4–6M params each) — a far better inductive bias for long
   carry chains than the MLP, and the key to the per-step precision a 128/256/512-step chain demands.
-  The per-step error floor rises with width, so tier 8 (512-step) lands at 0.73 rather than saturating
+  The per-step error floor rises with width, so the 512-bit cell additionally uses **gradient
+  accumulation** (a large effective batch lowers the per-step noise floor) to reach tier 8 = 0.92
 - Verifiably **generalises to primes never seen in training** (held-out-prime validation
   accuracy tracks training accuracy — no memorisation gap)
 
@@ -59,7 +60,7 @@ holds the prime:
 | `weights64.pt` | 64-bit | `< 2⁶⁴` | 5 | MLP, 4096 / 7, residual | ~236M | tier 5 = 0.98 |
 | `weights128.pt` | 128-bit | `< 2¹²⁸` | 6 | **carry-aware TCN**, 256ch / 10 blocks, dilations 1–64 | ~3.9M | **tier 6 = 0.97** |
 | `weights256.pt` | 256-bit | `< 2²⁵⁶` | 7 | **carry-aware TCN**, 256ch / 12 blocks, dilations 1–128 | ~4.7M | **tier 7 = 0.98** |
-| `weights512.pt` | 512-bit | `< 2⁵¹²` | 8 | **carry-aware TCN**, 256ch / 14 blocks, dilations 1–256 | ~5.5M | tier 8 = 0.73 *(partial)* |
+| `weights512.pt` | 512-bit | `< 2⁵¹²` | 8 | **carry-aware TCN**, 256ch / 14 blocks, dilations 1–256 | ~5.5M | **tier 8 = 0.92** |
 
 The 128/256/512-bit cells switch architecture: instead of a full-width MLP each is a **non-causal
 dilated 1-D convolutional network over the bit-positions** (128, 256, 512 respectively). Carry
@@ -70,8 +71,9 @@ position-function per bit. This inductive bias drives the per-step error roughly
 same-task MLP — the difference between a 128/256-step chain landing at ~0.26 and at **0.97 / 0.98** —
 in cells **~60× smaller** than the wide MLPs (16–22 MB each vs ~950 MB). The receptive field of each
 TCN spans its full width in both carry directions, so a carry can propagate across the entire word.
-The per-step error floor *rises* with bit-width, though: the 512-bit cell reaches a 512-step chain of
-**0.73** (a partial tier-8 cell that lifts overall accuracy but does not clear the 0.90 tier threshold).
+The per-step error floor *rises* with bit-width, though: the 512-bit cell needed **gradient accumulation**
+(a large effective batch to lower the per-step noise floor) to push its 512-step chain over the line to
+**tier 8 = 0.92**.
 
 The 64-bit cell needs **depth and residual connections** the narrower cells do not: a 64-bit
 modular Horner step hides two long carry chains (the `2t + bit·b` addition and the
@@ -131,7 +133,7 @@ cell is *at* the floor. The capability therefore resides in the trained paramete
 | tier 5 (64-bit cell) | 0.98 | 0.95 | 0.65 | 0.03 | 0.01 | 0.00 |
 | tier 6 (128-bit TCN) | 0.97 | 0.96 | 0.98 | 0.19 | 0.02 | 0.00 |
 | tier 7 (256-bit TCN) | 0.98 | 0.97 | 0.99 | 0.06 | 0.02 | 0.00 |
-| tier 8 (512-bit TCN) | 0.73 | 0.75 | 0.44 | 0.04 | 0.03 | 0.00 |
+| tier 8 (512-bit TCN) | 0.92 | 0.91 | 0.77 | 0.04 | 0.03 | 0.00 |
 
 Generalisation against memorisation: 10% of primes at each bit-width were held out of
 training entirely; chain accuracy on them matches the training primes.
@@ -152,10 +154,10 @@ short run. The 256-bit (tier-7) cell is the same carry-aware TCN scaled to 256 b
 (dilations cycling 1–128), trained identically on true-trajectory single steps over distinct
 252–256 bit primes; its per-step error is low enough that the 256-step chain holds at **tier 7 =
 0.98**. The 512-bit (tier-8) cell is again the same TCN (dilations 1–256) trained on distinct
-510–512 bit primes, but the per-step error floor rises with width, so the 512-step chain reaches
-**tier 8 = 0.73** — a partial cell that improves overall accuracy without clearing the 0.90 tier
-threshold. Training code and the full write-up live in the solutions repo (link in the model card
-metadata / challenge leaderboard).
+510–512 bit primes; because the per-step error floor rises with width, it additionally uses
+**gradient accumulation** (a large effective batch lowers the gradient-noise floor on the per-step
+error without extra memory), which drives the 512-step chain to **tier 8 = 0.92**. Training code and
+the full write-up live in the solutions repo (link in the model card metadata / challenge leaderboard).
 
 ## License
 

manifest.json

@@ -2,6 +2,6 @@
   "entry_class": "model.HornerRNN",
   "output_base": 2,
   "framework": "pytorch",
-  "model_description": "Bit-sequential RNN (~415M params across six cells) for primes up to 2^512. Reads the bits of a mod p MSB-first, one per step, conditioned on (b mod p, p) in binary; the hidden state is a quantized bit vector (hard binary bottleneck) and the transition function must learn the Horner step (t, bit, b, p) -> (2t + bit*b) mod p to make the recurrence end on the right answer. Six cells are shipped and routed by prime size: a 16-bit cell (MLP, width 4096 depth 4, ~50M params) for p < 2^16 covering tiers 1-3, a 32-bit cell (MLP, width 6144 depth 4, ~114M params) for p < 2^32 covering tier 4, a 64-bit cell (MLP, width 4096 depth 7 with pre-norm residual blocks, ~236M params) for p < 2^64 covering tier 5, a 128-bit cell for p < 2^128 covering tier 6 that is a CARRY-AWARE TCN: a non-causal dilated 1D-convolutional network over the 128 bit-positions (10 residual blocks, 256 channels, dilations cycling 1..64 so the receptive field spans all 128 bits, ~3.9M params), a 256-bit cell for p < 2^256 covering tier 7 that uses the SAME carry-aware TCN architecture scaled to 256 bit-positions (12 residual blocks, 256 channels, dilations cycling 1..128, ~4.7M params) reaching tier 7 = 0.98, and a 512-bit cell for p < 2^512 that is the same carry-aware TCN scaled to 512 bit-positions (14 residual blocks, 256 channels, dilations cycling 1..256, ~5.5M params) reaching tier 8 = 0.73 (a PARTIAL cell: the per-step error floor rises with bit-width, so the 512-step chain lands at 0.73 rather than saturating — it does not clear the 0.90 tier threshold but raises overall accuracy). The convolution is weight-shared across bit positions, so it learns ONE carry/borrow rule applied everywhere (non-causally, so the addition carry can flow LSB->MSB and the mod-p compare/borrow MSB->LSB) instead of a full-width MLP learning a separate position-function per bit; this inductive bias drives the per-step error far below what an MLP cell reaches and is what makes the 128/256/512-bit chains (which compound the per-step error over 128/256/512 steps) accurate. Final state bits are emitted MSB-first as the base-2 answer. For p >= 2^512 emits the honest [0] fallback without invoking the network.",
-  "training_description": "Each transition cell trained from random init on (t, bit, b, p) -> (2t + bit*b) mod p single-step examples over its prime range (16-bit: all primes < 2^16; 32-bit and 64-bit: random primes sampled uniform-by-value in [2^16, 2^32) and [2^33, 2^64) to match the test generator's randrange+nextprime distribution), with half of each batch mined near the comparison boundary (2t + bit*b within +/-2 of a multiple of p) where errors concentrate. BCE per state bit, AdamW + cosine decay + gradient clipping + LR warmup, EMA weights checkpointed by full-chain validation accuracy on a held-out 10% of primes never seen in training — val accuracy tracks train accuracy, i.e. the cells generalise across primes rather than memorising them. The 64-bit cell additionally receives a second fine-tuning phase on single steps drawn from the TRUE Horner trajectory (each example is a (t, bit, b, p) -> (2t + bit*b) mod p step where t is an actual chain intermediate (a_{>=i}*b) mod p, not a uniform sample), which matches the training distribution to the states the chain visits at inference and lifts tier 5 from 0.74 to 0.98; still ordinary supervised BCE on the same single-step target, no backprop through the recurrence. The 128-bit (tier-6) cell is the carry-aware TCN, trained the same way — single-step BCE on TRUE Horner-trajectory states (t, bit, b, p) -> (2t + bit*b) mod p — from random init over a high-diversity pool of thousands of distinct 124-128 bit primes (so it generalises across primes rather than memorising the conditional subtraction for a few). Its weight-shared dilated-convolution inductive bias reaches a per-step error roughly 15x lower than the same-task MLP cell, giving 0.97 full-chain accuracy on held-out 124-128 bit primes; same supervised single-step objective, no backprop through the recurrence, AdamW + cosine decay + grad clip + EMA checkpointed by held-out full-chain accuracy. The 256-bit (tier-7) cell is the same carry-aware TCN scaled to 256 bit-positions (dilations cycling 1..128), trained identically — single-step BCE on TRUE Horner-trajectory states over a high-diversity pool of distinct 252-256 bit primes — reaching a per-step error low enough that the 256-step chain holds at 0.98 full-chain accuracy on held-out 252-256 bit primes. The 512-bit (tier-8) cell is the same carry-aware TCN scaled to 512 bit-positions (dilations cycling 1..256), trained identically on true-trajectory single steps over distinct 510-512 bit primes; its per-step error floor is higher at this width, so the 512-step chain reaches 0.73 (a partial cell that improves overall accuracy without clearing the 0.90 tier threshold). Weight-perturbation compliance (exploration/compliance_perturb.py): each cell's accuracy at sigma=0 collapses toward the floor as the weights are perturbed and an untrained re-init scores 0.00 — e.g. tier 6 0.97 -> 0.19 (sigma=0.25), tier 7 0.98 -> 0.06 (sigma=0.25), tier 8 0.73 -> 0.04 (sigma=0.25), untrained 0.00 for all — so the arithmetic resides in the trained parameters. Training scripts: train.py (16-bit), exploration/train_horner32.py (32-bit), exploration/train_horner64.py (64-bit phase 1, --residual) then exploration/train_horner64_traj.py (64-bit phase 2, trajectory), exploration/train_horner128_bigru.py --arch tcn (128-bit carry-aware TCN), exploration/train_horner_tcn.py --bits 256 / --bits 512 (256- and 512-bit carry-aware TCN)."
+  "model_description": "Bit-sequential RNN (~415M params across six cells) for primes up to 2^512. Reads the bits of a mod p MSB-first, one per step, conditioned on (b mod p, p) in binary; the hidden state is a quantized bit vector (hard binary bottleneck) and the transition function must learn the Horner step (t, bit, b, p) -> (2t + bit*b) mod p to make the recurrence end on the right answer. Six cells are shipped and routed by prime size: a 16-bit cell (MLP, width 4096 depth 4, ~50M params) for p < 2^16 covering tiers 1-3, a 32-bit cell (MLP, width 6144 depth 4, ~114M params) for p < 2^32 covering tier 4, a 64-bit cell (MLP, width 4096 depth 7 with pre-norm residual blocks, ~236M params) for p < 2^64 covering tier 5, a 128-bit cell for p < 2^128 covering tier 6 that is a CARRY-AWARE TCN: a non-causal dilated 1D-convolutional network over the 128 bit-positions (10 residual blocks, 256 channels, dilations cycling 1..64 so the receptive field spans all 128 bits, ~3.9M params), a 256-bit cell for p < 2^256 covering tier 7 that uses the SAME carry-aware TCN architecture scaled to 256 bit-positions (12 residual blocks, 256 channels, dilations cycling 1..128, ~4.7M params) reaching tier 7 = 0.98, and a 512-bit cell for p < 2^512 covering tier 8 that is the same carry-aware TCN scaled to 512 bit-positions (14 residual blocks, 256 channels, dilations cycling 1..256, ~5.5M params) reaching tier 8 = 0.92. The per-step error floor rises with bit-width, so this cell was trained with gradient accumulation (a large effective batch lowers the per-step error noise floor) to recover the precision a 512-step chain needs to clear 0.90. The convolution is weight-shared across bit positions, so it learns ONE carry/borrow rule applied everywhere (non-causally, so the addition carry can flow LSB->MSB and the mod-p compare/borrow MSB->LSB) instead of a full-width MLP learning a separate position-function per bit; this inductive bias drives the per-step error far below what an MLP cell reaches and is what makes the 128/256/512-bit chains (which compound the per-step error over 128/256/512 steps) accurate. Final state bits are emitted MSB-first as the base-2 answer. For p >= 2^512 emits the honest [0] fallback without invoking the network.",
+  "training_description": "Each transition cell trained from random init on (t, bit, b, p) -> (2t + bit*b) mod p single-step examples over its prime range (16-bit: all primes < 2^16; 32-bit and 64-bit: random primes sampled uniform-by-value in [2^16, 2^32) and [2^33, 2^64) to match the test generator's randrange+nextprime distribution), with half of each batch mined near the comparison boundary (2t + bit*b within +/-2 of a multiple of p) where errors concentrate. BCE per state bit, AdamW + cosine decay + gradient clipping + LR warmup, EMA weights checkpointed by full-chain validation accuracy on a held-out 10% of primes never seen in training — val accuracy tracks train accuracy, i.e. the cells generalise across primes rather than memorising them. The 64-bit cell additionally receives a second fine-tuning phase on single steps drawn from the TRUE Horner trajectory (each example is a (t, bit, b, p) -> (2t + bit*b) mod p step where t is an actual chain intermediate (a_{>=i}*b) mod p, not a uniform sample), which matches the training distribution to the states the chain visits at inference and lifts tier 5 from 0.74 to 0.98; still ordinary supervised BCE on the same single-step target, no backprop through the recurrence. The 128-bit (tier-6) cell is the carry-aware TCN, trained the same way — single-step BCE on TRUE Horner-trajectory states (t, bit, b, p) -> (2t + bit*b) mod p — from random init over a high-diversity pool of thousands of distinct 124-128 bit primes (so it generalises across primes rather than memorising the conditional subtraction for a few). Its weight-shared dilated-convolution inductive bias reaches a per-step error roughly 15x lower than the same-task MLP cell, giving 0.97 full-chain accuracy on held-out 124-128 bit primes; same supervised single-step objective, no backprop through the recurrence, AdamW + cosine decay + grad clip + EMA checkpointed by held-out full-chain accuracy. The 256-bit (tier-7) cell is the same carry-aware TCN scaled to 256 bit-positions (dilations cycling 1..128), trained identically — single-step BCE on TRUE Horner-trajectory states over a high-diversity pool of distinct 252-256 bit primes — reaching a per-step error low enough that the 256-step chain holds at 0.98 full-chain accuracy on held-out 252-256 bit primes. The 512-bit (tier-8) cell is the same carry-aware TCN scaled to 512 bit-positions (dilations cycling 1..256), trained on true-trajectory single steps over distinct 510-512 bit primes; the per-step error floor rises with width, so this cell additionally uses gradient accumulation (--accum: a larger effective batch lowers the gradient-noise floor on per-step error) to drive the 512-step chain to tier 8 = 0.92. Weight-perturbation compliance (exploration/compliance_perturb.py): each cell's accuracy at sigma=0 collapses toward the floor as the weights are perturbed and an untrained re-init scores 0.00 — e.g. tier 6 0.97 -> 0.19 (sigma=0.25), tier 7 0.98 -> 0.06 (sigma=0.25), tier 8 0.92 -> 0.04 (sigma=0.25), untrained 0.00 for all — so the arithmetic resides in the trained parameters. Training scripts: train.py (16-bit), exploration/train_horner32.py (32-bit), exploration/train_horner64.py (64-bit phase 1, --residual) then exploration/train_horner64_traj.py (64-bit phase 2, trajectory), exploration/train_horner128_bigru.py --arch tcn (128-bit carry-aware TCN), exploration/train_horner_tcn.py --bits 256 / --bits 512 --accum 2 (256- and 512-bit carry-aware TCN)."
 }

model.py

@@ -31,10 +31,10 @@ The two-operand reductions ``a mod p`` / ``b mod p`` in ``predict_digits`` are
 the same legal input normalisation every other reference model uses.
 
 The model ships one cell per bit-width (16 -> tiers 1-3, 32 -> tier 4, 64 ->
-tier 5, 128 -> tier 6, 256 -> tier 7, and 512 -> tier 8 (partial, ~0.73) when
-present) and routes each problem to the narrowest cell whose state holds the
-prime. For primes wider than the widest trained cell it emits the honest ``[0]``
-fallback without invoking the network.
+tier 5, 128 -> tier 6, 256 -> tier 7, and 512 -> tier 8 when present) and routes
+each problem to the narrowest cell whose state holds the prime. For primes wider
+than the widest trained cell it emits the honest ``[0]`` fallback without
+invoking the network.
 """
 
 from __future__ import annotations

weights512.pt

@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:2f513711115dd8feee33aea4e2818527551115f17005ec7290764577c13a9a9d
-size 22114665
+oid sha256:7127108f38773e73b201d73f6834fef9cdb41610fd7ea776080670f94a1c7b7a
+size 22114947