post-train: M1 — verifiable arithmetic task + SFT data generator

First post-training milestone (docs/18). Lands the verifiable task + its data
pipeline, all verified host-side (no CUDA); the SFT run itself reuses the
existing --sft-tsv path on the GPU box.

- task.rs: the shared task spec — two-operand integer arithmetic, answer in
  \boxed{N}, with parse_boxed_answer + check_answer (exact-match rule-based
  reward). One module reused by M1 (SFT data), M3 (DPO pairs), M4 (GRPO reward).
- gen_arith_task bin: writes arith_sft.tsv (--sft-tsv format) + held-out
  arith_eval_prompts.txt (greedy_sample format) + arith_eval_gold.txt; train
  deduped, eval disjoint from train.
- data.rs: extract assistant-only masking into a pure, testable sft_row()
  (behavior-preserving; single-turn bit-identical to fbf4ac2).

Gate (verified locally, no_cuda): cargo test -p xtrain-train --lib = 9/9 pass
(masking, SFT-target self-consistency over 2000 samples, parser edges, seed
determinism); a 200/50 gen run = clean 2-col TSV, correct gold incl. negatives,
0 train/eval leakage. SFT training run + format-eval pending on dash5.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

crates/xtrain-train/src/bin/gen_arith_task.rs

@@ -0,0 +1,94 @@
+//! Generate the M1 verifiable-arithmetic post-training dataset. Pure host tool (no
+//! CUDA): writes
+//!   <out>/arith_sft.tsv           user<TAB>assistant rows for `train --sft-tsv`
+//!   <out>/arith_eval_prompts.txt  greedy_sample `--prompts-file` format (held out)
+//!   <out>/arith_eval_gold.txt     parallel gold integers for the checker
+//!
+//! Eval problems are deduped against train (no leakage). The SFT rows carry just the
+//! user/assistant content; `data::load_sft_tsv_cached` adds the `User:/Assistant:`
+//! frame + `<|endoftext|>` and masks the prompt, so the eval prompt lines here
+//! reconstruct exactly that frame (`User: <q>\nAssistant:`, literal `\n` decoded by
+//! greedy_sample).
+//!
+//! Example:
+//!   cargo run -p xtrain-train --release --bin gen_arith_task -- \
+//!       --n 20000 --eval 500 --seed 1 --out-dir /dashscope-tmp/wjh/xtrain_post/arith
+
+use std::collections::HashSet;
+use std::fs::{self, File};
+use std::io::{BufWriter, Write};
+use std::path::PathBuf;
+
+use xtrain_train::task::{GenConfig, Op, gen_problem};
+
+fn flag<T: std::str::FromStr>(args: &[String], name: &str, default: T) -> T {
+    args.iter()
+        .position(|a| a == name)
+        .and_then(|i| args.get(i + 1))
+        .and_then(|v| v.parse().ok())
+        .unwrap_or(default)
+}
+
+fn main() {
+    let args: Vec<String> = std::env::args().collect();
+    let n_train: usize = flag(&args, "--n", 20000);
+    let n_eval: usize = flag(&args, "--eval", 500);
+    let seed: u64 = flag(&args, "--seed", 1);
+    let max_add: i64 = flag(&args, "--max-add", 99);
+    let max_mul: i64 = flag(&args, "--max-mul", 12);
+    let out_dir: PathBuf = args
+        .iter()
+        .position(|a| a == "--out-dir")
+        .and_then(|i| args.get(i + 1))
+        .map(PathBuf::from)
+        .expect("--out-dir <dir> is required");
+
+    fs::create_dir_all(&out_dir).expect("create out dir");
+    let cfg = GenConfig {
+        max_add,
+        max_mul,
+        ops: vec![Op::Add, Op::Sub, Op::Mul],
+    };
+    let mut rng = seed.max(1);
+
+    // Train: dedup so the same problem is not repeated and so eval can be held out.
+    let mut train_keys = HashSet::new();
+    let mut tsv = BufWriter::new(File::create(out_dir.join("arith_sft.tsv")).expect("create tsv"));
+    while train_keys.len() < n_train {
+        let p = gen_problem(&mut rng, &cfg);
+        if !train_keys.insert(p.key()) {
+            continue;
+        }
+        writeln!(tsv, "{}\t{}", p.question(), p.sft_answer()).expect("write tsv");
+    }
+    tsv.flush().expect("flush tsv");
+
+    // Eval: disjoint from train (skip any key seen in train) and from itself.
+    let mut prompts =
+        BufWriter::new(File::create(out_dir.join("arith_eval_prompts.txt")).expect("create eval"));
+    let mut golds =
+        BufWriter::new(File::create(out_dir.join("arith_eval_gold.txt")).expect("create gold"));
+    writeln!(prompts, "# verifiable arithmetic eval prompts (held out from arith_sft.tsv)")
+        .expect("write header");
+    let mut eval_keys = HashSet::new();
+    while eval_keys.len() < n_eval {
+        let p = gen_problem(&mut rng, &cfg);
+        if train_keys.contains(&p.key()) || !eval_keys.insert(p.key()) {
+            continue;
+        }
+        writeln!(prompts, "User: {}\\nAssistant:", p.question()).expect("write prompt");
+        writeln!(golds, "{}", p.answer()).expect("write gold");
+    }
+    prompts.flush().expect("flush prompts");
+    golds.flush().expect("flush golds");
+
+    println!(
+        "wrote {} train rows + {} eval prompts to {} (ops=+,-,* max_add={} max_mul={} seed={})",
+        train_keys.len(),
+        eval_keys.len(),
+        out_dir.display(),
+        max_add,
+        max_mul,
+        seed
+    );
+}

crates/xtrain-train/src/data.rs

@@ -124,10 +124,9 @@ impl Corpus {
             let answer = format!(" {assistant}\n<|endoftext|>");
             let prompt_ids: Vec<i32> = tok.encode(&prompt).into_iter().map(|t| t as i32).collect();
             let answer_ids: Vec<i32> = tok.encode(&answer).into_iter().map(|t| t as i32).collect();
-            labels.extend(std::iter::repeat(-100).take(prompt_ids.len()));
-            labels.extend(answer_ids.iter().copied());
-            tokens.extend(prompt_ids);
-            tokens.extend(answer_ids);
+            let (row_tokens, row_labels) = sft_row(&prompt_ids, &answer_ids);
+            tokens.extend(row_tokens);
+            labels.extend(row_labels);
         }
         assert_eq!(tokens.len(), labels.len(), "SFT tokens/labels mismatch");
         write_u16_cache(&token_cache, &tokens);
@@ -291,6 +290,20 @@ fn decode_tsv_escapes(s: &str) -> String {
     s.replace("\\n", "\n").replace("\\t", "\t")
 }
 
+/// Build one SFT example's `(tokens, labels)` from already-tokenized prompt/answer
+/// ids: prompt tokens are masked to the ignore-index (`-100`, which `cross_entropy`
+/// skips) so only the answer + EOS tokens contribute to the loss. Pure (no tokenizer
+/// / no CUDA) so the assistant-only masking is unit-testable directly.
+fn sft_row(prompt_ids: &[i32], answer_ids: &[i32]) -> (Vec<i32>, Vec<i32>) {
+    let mut tokens = Vec::with_capacity(prompt_ids.len() + answer_ids.len());
+    tokens.extend_from_slice(prompt_ids);
+    tokens.extend_from_slice(answer_ids);
+    let mut labels = Vec::with_capacity(prompt_ids.len() + answer_ids.len());
+    labels.extend(std::iter::repeat(-100).take(prompt_ids.len()));
+    labels.extend_from_slice(answer_ids);
+    (tokens, labels)
+}
+
 /// Tiny LCG (same constants as the model tests' deterministic fill) so dataset
 /// sampling is reproducible from a single u64 seed.
 fn next_rand(state: &mut u64) -> u64 {
@@ -299,3 +312,27 @@ fn next_rand(state: &mut u64) -> u64 {
         .wrapping_add(1442695040888963407);
     *state >> 16
 }
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn sft_row_masks_prompt_supervises_answer() {
+        let prompt = [5, 6, 7];
+        let answer = [8, 9]; // includes the EOS token in real use
+        let (tokens, labels) = sft_row(&prompt, &answer);
+        // Tokens are prompt then answer, in order.
+        assert_eq!(tokens, vec![5, 6, 7, 8, 9]);
+        // Prompt positions are ignore-index (-100); answer positions are supervised.
+        assert_eq!(labels, vec![-100, -100, -100, 8, 9]);
+        assert_eq!(tokens.len(), labels.len());
+    }
+
+    #[test]
+    fn sft_row_handles_empty_answer() {
+        let (tokens, labels) = sft_row(&[1, 2], &[]);
+        assert_eq!(tokens, vec![1, 2]);
+        assert_eq!(labels, vec![-100, -100]);
+    }
+}

crates/xtrain-train/src/lib.rs

@@ -10,6 +10,7 @@
 pub mod clip;
 pub mod data;
 pub mod schedule;
+pub mod task;
 
 #[cfg(not(no_cuda))]
 pub mod checkpoint;

crates/xtrain-train/src/task.rs

@@ -0,0 +1,212 @@
+//! Verifiable arithmetic task (post-training, M1). A tiny two-operand integer
+//! arithmetic task with a deterministic, rule-based checker: the assistant must end
+//! its answer with `\boxed{N}`, and the reward is exact-match on `N`.
+//!
+//! This single module is the shared task spec for the whole post-training stack —
+//! M1 SFT-data generation, M3 DPO preference-pair construction, and M4 GRPO reward
+//! scoring all parse/score through here, so the task lives in exactly one place.
+//!
+//! Host-only (no CUDA): generation + parsing + checking are pure, so this compiles
+//! and unit-tests on a GPU-less host.
+
+use std::fmt;
+
+/// The supported binary operations.
+#[derive(Clone, Copy, PartialEq, Eq, Debug)]
+pub enum Op {
+    Add,
+    Sub,
+    Mul,
+}
+
+impl Op {
+    pub fn symbol(self) -> char {
+        match self {
+            Op::Add => '+',
+            Op::Sub => '-',
+            Op::Mul => '*',
+        }
+    }
+}
+
+impl fmt::Display for Op {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        write!(f, "{}", self.symbol())
+    }
+}
+
+/// A single two-operand arithmetic problem.
+#[derive(Clone, Copy, Debug)]
+pub struct Problem {
+    pub a: i64,
+    pub b: i64,
+    pub op: Op,
+}
+
+impl Problem {
+    /// The exact integer answer (the verifiable gold label).
+    pub fn answer(self) -> i64 {
+        match self.op {
+            Op::Add => self.a + self.b,
+            Op::Sub => self.a - self.b,
+            Op::Mul => self.a * self.b,
+        }
+    }
+
+    /// The user-turn question text. No template wrapping — the SFT loader
+    /// (`data::load_sft_tsv_cached`) adds the `User:/Assistant:` frame.
+    pub fn question(self) -> String {
+        format!("What is {} {} {}?", self.a, self.op, self.b)
+    }
+
+    /// The assistant-turn SFT target: restate the equation and end with the boxed
+    /// answer. This teaches the answer FORMAT (the checker only reads `\boxed{}`);
+    /// arithmetic correctness is what DPO (M3) / GRPO (M4) later improve.
+    pub fn sft_answer(self) -> String {
+        format!("{} {} {} = \\boxed{{{}}}.", self.a, self.op, self.b, self.answer())
+    }
+
+    /// A stable dedup key, so eval problems can be held out from train.
+    pub fn key(self) -> (i64, char, i64) {
+        (self.a, self.op.symbol(), self.b)
+    }
+}
+
+/// Operand-range configuration for problem sampling. Multiplication uses a smaller
+/// range (`max_mul`) so products stay modest; add/sub use `max_add`.
+#[derive(Clone)]
+pub struct GenConfig {
+    pub max_add: i64,
+    pub max_mul: i64,
+    pub ops: Vec<Op>,
+}
+
+impl Default for GenConfig {
+    fn default() -> Self {
+        Self {
+            max_add: 99,
+            max_mul: 12,
+            ops: vec![Op::Add, Op::Sub, Op::Mul],
+        }
+    }
+}
+
+/// Sample one problem deterministically from the LCG state `rng`. Operands are drawn
+/// in `[0, max]` per the op; subtraction may yield a negative answer (the checker /
+/// parser handle a leading `-`).
+pub fn gen_problem(rng: &mut u64, cfg: &GenConfig) -> Problem {
+    let op = cfg.ops[(next_rand(rng) as usize) % cfg.ops.len()];
+    let max = if op == Op::Mul { cfg.max_mul } else { cfg.max_add };
+    let a = rand_range(rng, max);
+    let b = rand_range(rng, max);
+    Problem { a, b, op }
+}
+
+/// Parse the integer inside the LAST `\boxed{...}` in `text`. Returns `None` if there
+/// is no well-formed boxed integer (no box, empty, or non-integer contents). "Last"
+/// so a model that emits intermediate boxes still scores on its final answer.
+pub fn parse_boxed_answer(text: &str) -> Option<i64> {
+    const TAG: &str = "\\boxed{";
+    let mut found = None;
+    let mut rest = text;
+    while let Some(i) = rest.find(TAG) {
+        let after = &rest[i + TAG.len()..];
+        match after.find('}') {
+            Some(j) => {
+                if let Ok(n) = after[..j].trim().parse::<i64>() {
+                    found = Some(n);
+                }
+                rest = &after[j + 1..];
+            }
+            None => break,
+        }
+    }
+    found
+}
+
+/// Verifiable reward: does the completion's boxed answer exactly match `gold`?
+pub fn check_answer(completion: &str, gold: i64) -> bool {
+    parse_boxed_answer(completion) == Some(gold)
+}
+
+/// `[0, max]` inclusive draw from the LCG.
+fn rand_range(rng: &mut u64, max: i64) -> i64 {
+    debug_assert!(max >= 0);
+    (next_rand(rng) % (max as u64 + 1)) as i64
+}
+
+/// Same LCG constants as the dataset sampler (`data::next_rand`), kept local so the
+/// task module stays dependency-free and host-only.
+fn next_rand(state: &mut u64) -> u64 {
+    *state = state
+        .wrapping_mul(6364136223846793005)
+        .wrapping_add(1442695040888963407);
+    *state >> 1
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn answer_question_and_sft_target() {
+        let p = Problem {
+            a: 12,
+            b: 13,
+            op: Op::Mul,
+        };
+        assert_eq!(p.answer(), 156);
+        assert_eq!(p.question(), "What is 12 * 13?");
+        assert_eq!(p.sft_answer(), "12 * 13 = \\boxed{156}.");
+        let s = Problem {
+            a: 3,
+            b: 8,
+            op: Op::Sub,
+        };
+        assert_eq!(s.answer(), -5);
+    }
+
+    #[test]
+    fn parse_takes_last_boxed_and_handles_edges() {
+        assert_eq!(parse_boxed_answer("\\boxed{3} then \\boxed{156}."), Some(156));
+        assert_eq!(parse_boxed_answer("\\boxed{-7}"), Some(-7));
+        assert_eq!(parse_boxed_answer("\\boxed{ 42 }"), Some(42));
+        assert_eq!(parse_boxed_answer("no box here"), None);
+        assert_eq!(parse_boxed_answer("\\boxed{abc}"), None);
+        assert_eq!(parse_boxed_answer("\\boxed{unterminated"), None);
+    }
+
+    #[test]
+    fn check_is_exact_match() {
+        assert!(check_answer("the result is \\boxed{156}.", 156));
+        assert!(!check_answer("the result is \\boxed{155}.", 156));
+        assert!(!check_answer("no boxed answer at all", 156));
+    }
+
+    #[test]
+    fn sft_target_is_always_self_consistent() {
+        // The SFT target's boxed answer must always check against the problem's own
+        // gold — across all ops/operands. This is the M1 data invariant.
+        let cfg = GenConfig::default();
+        let mut rng = 12345u64;
+        for _ in 0..2000 {
+            let p = gen_problem(&mut rng, &cfg);
+            assert!(
+                check_answer(&p.sft_answer(), p.answer()),
+                "self-inconsistent SFT target for {p:?}"
+            );
+        }
+    }
+
+    #[test]
+    fn generation_is_deterministic_from_seed() {
+        let cfg = GenConfig::default();
+        let (mut r1, mut r2) = (7u64, 7u64);
+        for _ in 0..200 {
+            assert_eq!(
+                gen_problem(&mut r1, &cfg).key(),
+                gen_problem(&mut r2, &cfg).key()
+            );
+        }
+    }
+}

docs/18-post-training-rl-sft.md

@@ -0,0 +1,328 @@
+# Phase: Post-Training Infra — SFT / DPO / Reward Model / GRPO — Design Document
+
+> Status: **DESIGN — decisions locked, pending go-ahead to implement.** Nothing
+> implemented yet. This doc proposes the scope, the staged build, the new infra pieces,
+> and the correctness gates for a standard post-training stack on top of the xtrain
+> training framework. Decisions D1–D4 are resolved (see "Resolved decisions"):
+> **DPO → GRPO (reward model optional) · rule-based/verifiable reward · KV-cache decode
+> engine built up front · a verifiable task as the optimization/eval target.**
+
+## Goal
+
+Build a **standard, from-scratch post-training infrastructure** — the systems layer that
+turns a pretrained base LM into an aligned chat model — and use it to run chat
+alignment. The deliverable that matters here is the **infra and the lessons**, not the
+end-to-end chat quality (see the project's learning-axis framing). Each stage should
+teach exactly one new post-training systems concept and ship with a hard correctness
+gate, matching the Phase-1/Phase-2 culture (grad-checks, PyTorch parity, bit-identical
+default paths, profile-first).
+
+Concretely we want to be able to answer, with our own code:
+
+- How does **offline preference optimization (DPO)** differ from SFT in the training
+  loop — what is the reference model, why two forwards, what is the loss?
+- How does a **reward model** turn preferences into a scalar signal?
+- How does **online RL (GRPO)** actually run — the rollout engine, reward scoring,
+  group-relative advantage, the clipped policy-gradient update, the KL leash?
+- Where are the **memory and throughput** pressure points that make post-training infra
+  different from pretraining infra (multiple models resident, generation in the loop)?
+
+## Baseline: what already exists vs. what is missing
+
+What the framework already gives us (verified in code, reused as-is):
+
+| capability | where | reuse for post-training |
+|---|---|---|
+| batched forward → logits `[B*S, vocab]` | `model.rs::forward_batched` | logprob extraction for DPO/RM/GRPO |
+| cross-entropy with **ignore-index −100** | `ops.rs::cross_entropy`, `nn.cu` | assistant-only / completion-only masking |
+| assistant-only **SFT** (TSV, masked labels) | `data.rs::load_sft_tsv_cached` (commit `fbf4ac2`) | SFT chat baseline = DPO init + reference |
+| bf16 mixed precision, fp32 master | `with_compute_dtype` | policy + frozen reference both bf16 compute |
+| recompute / flash / grad-accum | `with_recompute` / `with_flash` / `--accum-steps` | bound activation memory with 2–3 models resident |
+| DDP (thread + process-per-GPU) | `xtrain-distributed` | data-parallel post-training |
+| AdamW + clip + LR sched + checkpoint | `xtrain-optim`, `checkpoint.rs`, `schedule.rs` | unchanged optimizer path |
+| single-seq greedy/temperature sampling | `sample.rs::generate` | **slow** rollout fallback (no KV cache) |
+
+What is **missing** and must be built (these are the actual lessons):
+
+1. **Per-sequence completion logprob** — a way to read `Σ log πθ(y_t | x, y_<t)` over the
+   completion tokens of a sequence. CE gives a *mean* scalar; DPO/GRPO need a *per-sequence
+   masked sum*. New op or thin wrapper over the CE per-row machinery.
+2. **Frozen reference model** held in memory alongside the trainable policy (no grad, no
+   optimizer), or its logprobs precomputed and cached.
+3. **Pairwise preference loss** (DPO) and **Bradley-Terry ranking loss** (RM).
+4. **Reward head** — a `[dim,1]` scalar head reading the last non-pad position (RM only).
+5. **Rollout / generation engine** — batched autoregressive sampling. Current `generate`
+   is single-sequence and re-runs the full forward each step (no KV cache). Online RL needs
+   batched rollouts; a real **KV-cache incremental-decode engine** is the centerpiece infra
+   build.
+6. **GRPO machinery** — group sampling, group-relative advantage, clipped PG loss, KL
+   penalty, the actor-learner loop.
+
+## The post-training landscape — where the infra lives
+
+```
+                 data                 models in memory        new systems concept
+  SFT      (prompt, answer)           policy                  loss masking (have it)
+  DPO      (prompt, chosen, reject)   policy + ref(frozen)    dual forward, pairwise logσ loss
+  RM       (prompt, chosen, reject)   reward model            scalar head, ranking loss
+  PPO      prompts + reward source    policy+ref+RM+critic    rollout + GAE + clipped PG  (4 models)
+  GRPO     prompts + reward source    policy+ref(+RM)         rollout + group baseline + clipped PG
+```
+
+The pedagogical ladder is **SFT → DPO → (RM) → GRPO**. DPO is the cheapest "real" alignment
+method (no generation, no reward model, reuses the training loop almost verbatim) and is the
+right first rung. GRPO is chosen over PPO as the online-RL rung because it **drops the value
+critic** (group-relative advantage replaces the learned baseline) — that removes a whole
+model and the GAE machinery while still teaching the complete online-RL loop. PPO is noted
+as an optional later extension, not a primary target.
+
+## Proposed scope & sequencing (recommended path)
+
+> ✅ **DECISION D1 (scope/sequencing) — LOCKED: P0 → P1(DPO) → P3(GRPO), P2(reward
+> model) optional.** With D3 locked to "KV-cache engine up front", the engine becomes a
+> foundational milestone that both DPO pair-generation and GRPO rollouts sit on. Effective
+> build order: **P0 → KV-cache decode engine → P1(DPO) → P3(GRPO) → P2(optional)** (see
+> "Milestones").
+
+### Stage P0 — SFT chat baseline (light; mostly reuse)
+
+Goal: a clean SFT checkpoint to serve as **both the DPO/GRPO init and the frozen
+reference**. With D4 = verifiable task, P0 SFT teaches the **task format** (e.g. arithmetic
+prompts → a parseable answer such as `\boxed{N}`) so the model emits checker-readable
+completions; the same template is reused by rollout and eval. The current SFT (commit
+`fbf4ac2`) already does single-turn assistant-only masking; P0 only adds what alignment
+needs:
+
+- a fixed **chat template** (the `User:/Assistant:` + `<|endoftext|>` format already used,
+  promoted to a documented constant shared by SFT data prep, rollout, and eval),
+- optional **multi-turn masking** (supervise every assistant turn, mask user turns),
+- optional **sequence packing** (concatenate examples to fill `seq`, reset attention/RoPE
+  per example — note `forward_batched` already isolates sequences, so packing = careful
+  index bookkeeping, not new attention code).
+
+Gate: masking unit test (only assistant tokens contribute to loss); packing does not leak
+loss across example boundaries. **Hypothesis:** a documented chat template + multi-turn mask
+gives a reproducible SFT reference without changing the training numerics for single-turn data
+(bit-identical to `fbf4ac2` on single-turn input).
+
+### Stage P1 — DPO (offline preference optimization) ⭐ first real method
+
+New infra:
+
+1. **Preference data — constructed from the verifiable checker (D4).** On a verifiable task
+   there is no off-the-shelf preference set, so we build pairs: sample several completions
+   per prompt from the P0 SFT model (using the KV-cache engine built in the prior milestone),
+   score each with the rule-based checker, take a **correct** completion as `chosen` and an
+   **incorrect** one as `rejected`. This is a one-time offline data-prep step; DPO training
+   itself is then static. Tokenize each as `template(prompt) + completion + EOS`; build a
+   completion mask (prompt = masked).
+2. **`seq_logprob(logits, target_ids, mask) → [B]`**: per-sequence sum of
+   `log softmax(logits)[target]` over masked positions. Implement by reusing the CE per-row
+   path (CE per-row = `−log πθ(target)`), summing `−per_row` over the mask. Add a grad-checked
+   op so the backward is exact.
+3. **Frozen reference** `πref`: load the SFT checkpoint into a second model in **eval/no-grad**
+   bf16. Its logprobs are **constants** in the loss. Optimization to teach: **precompute and
+   cache reference logprobs** once over the dataset → the reference model need not stay
+   resident during training (one model in memory, like SFT).
+4. **DPO loss** (Rafailov et al.): with
+   `Δ = β[(logπθ(yw|x) − logπref(yw|x)) − (logπθ(yl|x) − logπref(yl|x))]`,
+   `L = −log σ(Δ)`. Only `πθ` terms carry gradient.
+
+Memory: policy (fp32 master + Adam m/v + bf16 + grads) + reference (bf16 only, or cached
+logprobs → zero). Recompute + accum keep activations bounded; 1B fits 32 GB comfortably.
+
+Correctness gates:
+- `seq_logprob` finite-difference grad-check (tiny model).
+- DPO-loss + grad **PyTorch parity** (the project's standard gate).
+- **Degenerate checks**: `πθ == πref` at init ⇒ `Δ = 0`, `L = log 2`, implicit reward 0;
+  `β → 0` ⇒ gradient → 0.
+- **Health metric**: chosen−rejected **reward margin** rises over training; accuracy
+  (margin > 0) increases. Reported, not just loss (the doc-13 lesson: val/loss alone is not a
+  sufficient signal).
+
+Application: chat alignment via DPO on English preference pairs. This is the **offline
+chat-alignment deliverable**.
+
+### Stage P2 — Reward model (Bradley-Terry) — OPTIONAL
+
+> ✅ **DECISION D2 (reward source) — LOCKED: rule-based / verifiable reward first.** GRPO
+> brings up on the deterministic checker; a learned reward model is **deferred/optional** (only
+> if we later want general-chat GRPO). So this whole stage is optional and not on the critical
+> path.
+
+New infra: a **scalar reward head** (`[dim,1]`) reading the hidden state at the last
+non-pad position; **ranking loss** `−log σ(r(x,yw) − r(x,yl))`. Reuses the preference data
+and the dual-sequence forward from P1.
+
+Gates: ranking-loss grad-check; held-out **pairwise accuracy** (`r_w > r_l`); a frozen RM
+loads/serves the scalar correctly.
+
+### Stage P3 — GRPO (online RL, critic-free) ⭐ the deep infra lesson
+
+This is the centerpiece. It introduces **generation inside the training loop**.
+
+**(a) Rollout / generation engine — built up front (its own milestone).**
+
+> ✅ **DECISION D3 (rollout depth) — LOCKED: build the KV-cache incremental-decode engine
+> up front**, as a foundational milestone *before* DPO/GRPO, rather than starting naive. It is
+> then the shared substrate for DPO pair-generation and GRPO rollouts. Tradeoff accepted:
+> front-loads the single hardest build and delays the first alignment result, in exchange for
+> a real generation engine and a clean, isolated infra lesson.
+
+The engine: per-layer **K/V cache**, **single-token incremental forward** (process the prompt
+once to fill the cache, then decode one token at a time), **batched ragged decode** (B prompts
+× G samples; sequences hit EOS at different lengths → finished-mask / left-padding /
+compaction). The current attention assumes a full causal window over `seq`; incremental decode
+needs a **decode-time attention path** — query length 1 against cached K/V of length `t`, with
+RoPE position = `t`. This reuses the composed SDPA shapes (one-row query), so it can land as a
+distinct code path without disturbing the training attention (flash/GQA/composed unchanged).
+
+Hard gate (the centerpiece correctness lesson): **KV-cache decode == full-recompute decode,
+token-identical** greedy output — the same byte-/token-identical discipline the project uses
+for the xserv export closed loop. A throughput baseline (decode tokens/s, cache-fill vs.
+per-token decode) is recorded here, before any rollout optimization (profile-first).
+
+**(b) Reward scoring.** Rule-based verifiable reward first (e.g., exact-match on a synthetic
+arithmetic/format task) or RM from P2. Returns a scalar per completion.
+
+**(c) Group-relative advantage.** Sample `G` completions per prompt; advantage
+`A_i = (r_i − mean(r_group)) / (std(r_group) + ε)`. No critic, no GAE.
+
+**(d) Clipped policy-gradient loss with KL leash.** Per completion token,
+`ρ_t = exp(logπθ_t − logπθ_old_t)` (old = policy at rollout time), token loss
+`−min(ρ_t A, clip(ρ_t, 1±ε) A) + βKL(πθ‖πref)`, masked to completion tokens. KL via the k3
+estimator.
+
+**(e) Actor-learner loop.** sample prompt batch → rollout G each → score → advantage →
+capture `πθ_old` logprobs → K inner epochs of clipped PG updates → repeat. Reference `πref`
+fixed throughout.
+
+Memory: policy + reference (+ RM if learned). Each 1B; recompute + accum bound activations.
+Throughput note: rollout (generation) will dominate wall-clock — a baseline must be recorded
+(tokens/s of generation vs. update) **before** any rollout optimization, per the project's
+profile-first rule.
+
+Correctness gates:
+- PG-loss finite-diff grad-check.
+- **Degenerate checks**: `G = 1` ⇒ advantage 0 ⇒ no PG signal, only KL; `ε → ∞` ⇒ vanilla PG;
+  `β = 0` ⇒ no KL term.
+- (KV-cache decode token-identical to full-recompute is gated in the engine milestone, a
+  prerequisite of GRPO.)
+- **Synthetic RL overfit**: on a tiny verifiable task with a known optimum, mean reward must
+  rise to the optimum (the RL analogue of T5's "overfit 27/27" — a hard, falsifiable signal
+  that the loop is correct, independent of fuzzy chat quality).
+
+## Evaluation
+
+- **Offline (DPO/RM)**: reward margin, preference accuracy, KL drift from reference, plus the
+  fixed chat-prompt generation suite (`scripts/chat_alpha_fixed_prompts.txt`) judged before/
+  after — reusing and extending the doc-13 recommendation for a generation-based eval harness
+  (exact-match math, code syntax, stop-token, refusal appropriateness, corruption).
+- **Online (GRPO)**: mean reward curve, KL-to-reference, response length, the verifiable-task
+  pass rate, and the same fixed-prompt suite.
+- **Selection by generation eval, not loss** — the recurring doc-13/v11 lesson: lower
+  post-training loss did not mean better generations.
+
+## Memory & throughput budget (8× RTX 5090, 1.05B model, indicative)
+
+- Params (bf16) ~2.1 GB; fp32 master ~4.2 GB; AdamW m/v ~8.4 GB; grads ~2.1 GB → policy
+  optimizer state alone ~17 GB before activations. Recompute + grad-accum keep activations
+  small; this is why post-training reuses the Phase-1/2 memory levers unchanged.
+- DPO: + reference (bf16 ~2.1 GB, or 0 if logprobs cached). Fits.
+- GRPO: + reference (~2.1 GB) (+ RM ~2.1 GB if learned). Fits; rollout activations are the new
+  variable. **Generation, not the update, is expected to be the throughput bottleneck** — to be
+  measured, not assumed.
+
+## Correctness-gate philosophy (unchanged from Phase 1/2)
+
+Every stage ships: (1) a finite-difference grad-check on the new loss/op, (2) PyTorch parity
+on loss + grads where applicable, (3) explicit degenerate-case bit/again checks (β→0, G=1,
+ε→∞, ref==policy), (4) a falsifiable "it actually learns" signal (reward margin up / synthetic
+RL overfit), and (5) **no change to the default training path** when post-training flags are
+off. New CUDA kernels (if any, e.g. decode-time attention) get the same fwd/bwd-vs-reference
+gates as flash/GQA.
+
+## Risks & tradeoffs
+
+- **Rollout engine is the long pole.** A correct KV-cache incremental-decode path is a real
+  build (decode-time attention, ragged batch). Mitigation: naive rollout first; KV-cache as an
+  isolated, separately-gated sub-phase.
+- **RL is finicky.** KL leash, advantage normalization, clip range, reward hacking. Mitigation:
+  synthetic verifiable task with a known optimum as the bring-up gate before any real chat reward.
+- **Reward-model noise** can mislead GRPO. Mitigation: rule-based reward first.
+- **Tokenizer (KI-4)** — gpt2 50257 vocab is kept for the xserv closed loop; unchanged here.
+- **Two/three resident models** raise memory; bounded by recompute/accum and (for DPO) reference
+  logprob caching.
+
+## Resolved decisions (aligned 2026-06-29)
+
+- **D1 — Scope & sequencing → DPO → GRPO, reward model optional.**
+- **D2 — Online-RL reward source → rule-based / verifiable reward first** (RM deferred/optional).
+- **D3 — Rollout engine depth → build the KV-cache incremental-decode engine up front** (not
+  naive-first), as a foundational milestone before DPO/GRPO.
+- **D4 — Alignment task / eval target → a verifiable task** (arithmetic/format/GSM8K-style) with
+  a deterministic exact-match reward, for a clean, falsifiable RL signal.
+
+## Milestones (locked order)
+
+1. **M1 — P0 SFT task baseline.** Chat template + assistant-only masking on the verifiable
+   task; produces the reference + init checkpoint. Gate: masking unit test; single-turn
+   bit-identical to `fbf4ac2`.
+2. **M2 — KV-cache decode engine** (D3, up front). Per-layer K/V cache + incremental
+   decode-time attention + batched ragged decode. Gate: **token-identical to full-recompute
+   greedy**; record decode throughput baseline.
+3. **M3 — P1 DPO.** Verifiable-checker pair construction (via M2) → `seq_logprob` op
+   (grad-check) → DPO loss (PyTorch parity; ref==policy and β→0 degenerate checks) → DPO
+   training loop → run + reward-margin / preference-accuracy curve.
+4. **M4 — P3 GRPO.** Group rollout (M2) + rule-based reward + group-relative advantage +
+   clipped PG with KL leash. Gate: PG grad-check; G=1/ε→∞/β=0 degenerate checks; **synthetic
+   verifiable-task RL-overfit** (mean reward → known optimum) → verifiable-task GRPO run.
+5. **M5 (optional) — P2 reward model.** Scalar head + ranking loss + pairwise-accuracy gate;
+   enables GRPO-with-RM for general chat.
+
+> Each milestone is one design+gate cycle; results get appended here (like the run docs) and a
+> row in `docs/evolution.md` (algorithm/infra dimensions) when it lands.
+
+## Implementation log
+
+### M1 — SFT task baseline (infra landed; SFT run pending on the GPU box)
+
+The verifiable task and its data pipeline are implemented and verified host-side (no CUDA
+needed). The SFT training run itself is a one-liner over the existing `--sft-tsv` path and
+runs on dash5.
+
+**Verifiable task (the spec, in one Rust module — `crates/xtrain-train/src/task.rs`):**
+
+- Two-operand integer arithmetic, ops `+ − ×`; operands `[0,99]` for `+/−`, `[0,12]` for `×`
+  (modest products); subtraction may be negative.
+- User turn: `What is A op B?`. SFT target: `A op B = \boxed{N}.` — teaches the answer FORMAT;
+  the checker reads only `\boxed{}`, so arithmetic *correctness* is what M3/M4 improve.
+- Rule-based reward: `parse_boxed_answer` (takes the LAST `\boxed{int}`) + `check_answer`
+  (exact match vs. gold). This is the single shared checker reused by M3 (pair construction)
+  and M4 (GRPO reward).
+- Why this task: trivial deterministic checker, freely scalable difficulty, and it directly
+  probes the base model's known arithmetic weakness (v12 SFT failed `12 * 13`).
+
+**Data generator (`crates/xtrain-train/src/bin/gen_arith_task.rs`, pure host bin):**
+writes `arith_sft.tsv` (`user<TAB>assistant` for `--sft-tsv`), `arith_eval_prompts.txt`
+(`greedy_sample --prompts-file` format), and `arith_eval_gold.txt` (parallel gold ints).
+Train rows are deduped; eval is held out from train (no leakage).
+
+**Masking made testable:** the assistant-only label masking in `load_sft_tsv_cached` was
+extracted into a pure `sft_row(prompt_ids, answer_ids)` helper (behavior-preserving — the
+single-turn path is bit-identical to `fbf4ac2`).
+
+**Gate (verified locally in `no_cuda` mode):** `cargo test -p xtrain-train --lib` → 9/9 pass,
+including `sft_row` masks prompt→`-100` / supervises answer, the SFT-target self-consistency
+invariant (always checker-correct over 2000 samples), parser edge cases, and seed determinism.
+A 200/50 generation run confirmed clean 2-column TSV, correct gold (incl. negatives), and 0
+train/eval leakage.
+
+**Pending on dash5 (needs GPU):**
+1. generate the dataset: `gen_arith_task --n <N> --eval <M> --seed 1 --out-dir <dir>`;
+2. SFT from the v12 base: `train <tok> <dir>/arith_sft.tsv --sft-tsv --init-ckpt <v12-base.ckpt>
+   <arch flags> --bf16 --recompute --flash --ckpt <out.ckpt>` → the P0 reference/init checkpoint;
+3. format eval: `greedy_sample <out.ckpt> <tok> <arch> --prompts-file <dir>/arith_eval_prompts.txt`,
+   then score completions with `check_answer` vs. `arith_eval_gold.txt` (a tiny scorer is a
+   one-off; can fold into M3's pipeline). Baseline pass-rate here is expected to be low — M1 only
+   buys the format; correctness is M3/M4's job.