diff --git a/teaching/math_reasoning_trace.py b/teaching/math_reasoning_trace.py
new file mode 100644
index 00000000..0860ec88
--- /dev/null
+++ b/teaching/math_reasoning_trace.py
@@ -0,0 +1,169 @@
+"""ADR-0172 W0 — ``ReasoningTrace`` substrate.
+
+Schema-only module. Defines the byte-identical, replay-equivalent
+reasoning trace carried by ``MathReaderRefusalShapeProposal`` (W1)
+and emitted by the audit-corpus decomposer (W2).
+
+Determinism contract:
+- Canonical bytes are stable across processes and dict insertion order.
+- ``trace_id`` is ``sha256(canonical_bytes)`` of the full step sequence.
+- Floating-point values are forbidden in payloads (would break replay).
+
+No runtime hook. No import from ``chat``/``field``/``generate``/``algebra``.
+"""
+
+from __future__ import annotations
+
+import hashlib
+import json
+from dataclasses import dataclass
+from typing import Literal, get_args
+
+__all__ = (
+    "ReasoningStep",
+    "ReasoningTrace",
+    "StepKind",
+    "build_trace",
+    "canonical_bytes",
+    "compute_trace_id",
+)
+
+
+StepKind = Literal[
+    "observation",
+    "grouping",
+    "abstraction",
+    "hypothesis",
+    "test_design",
+    "test_application",
+    "test_result",
+    "conclusion",
+]
+
+_STEP_KINDS: frozenset[str] = frozenset(get_args(StepKind))
+
+
+@dataclass(frozen=True)
+class ReasoningStep:
+    step_index: int
+    step_kind: StepKind
+    input_pointers: tuple[str, ...]
+    claim: str
+    justification: str
+    output_payload: object
+
+
+@dataclass(frozen=True)
+class ReasoningTrace:
+    trace_id: str
+    steps: tuple[ReasoningStep, ...]
+
+
+def _reject_floats(value: object, *, path: str) -> None:
+    """Recursively reject any ``float`` instance in a JSON-shaped value.
+
+    Booleans are a ``bool`` subclass of ``int`` and remain permitted.
+    """
+
+    if isinstance(value, float):
+        raise ValueError(
+            f"floating-point values are forbidden in canonical payloads (at {path})"
+        )
+    if isinstance(value, dict):
+        for key, sub in value.items():
+            if not isinstance(key, str):
+                raise ValueError(
+                    f"payload dict keys must be strings (at {path}, got {type(key).__name__})"
+                )
+            _reject_floats(sub, path=f"{path}.{key}")
+    elif isinstance(value, (list, tuple)):
+        for idx, sub in enumerate(value):
+            _reject_floats(sub, path=f"{path}[{idx}]")
+
+
+def _validate_payload(payload: object) -> None:
+    """Ensure payload is JSON-serializable and float-free."""
+
+    try:
+        json.dumps(payload, sort_keys=True, separators=(",", ":"), allow_nan=False)
+    except (TypeError, ValueError) as exc:
+        raise ValueError(f"output_payload is not JSON-serializable: {exc}") from exc
+    _reject_floats(payload, path="output_payload")
+
+
+def _step_to_canonical(step: ReasoningStep) -> dict[str, object]:
+    return {
+        "step_index": step.step_index,
+        "step_kind": step.step_kind,
+        "input_pointers": list(step.input_pointers),
+        "claim": step.claim,
+        "justification": step.justification,
+        "output_payload": step.output_payload,
+    }
+
+
+def _steps_canonical_bytes(steps: tuple[ReasoningStep, ...]) -> bytes:
+    payload = [_step_to_canonical(step) for step in steps]
+    return json.dumps(
+        payload,
+        sort_keys=True,
+        separators=(",", ":"),
+        ensure_ascii=False,
+        allow_nan=False,
+    ).encode("utf-8")
+
+
+def canonical_bytes(trace: ReasoningTrace) -> bytes:
+    """Return the canonical byte serialization of a trace.
+
+    The ``trace_id`` field itself is excluded (it is derived from these
+    very bytes). Stable across processes and dict insertion order.
+    """
+
+    return _steps_canonical_bytes(trace.steps)
+
+
+def compute_trace_id(steps: tuple[ReasoningStep, ...]) -> str:
+    """Return ``sha256(canonical_bytes(steps))`` as hex digest."""
+
+    return hashlib.sha256(_steps_canonical_bytes(steps)).hexdigest()
+
+
+def build_trace(steps: list[ReasoningStep] | tuple[ReasoningStep, ...]) -> ReasoningTrace:
+    """Validate, freeze, and hash a reasoning trace.
+
+    Raises ``ValueError`` on:
+    - empty step list
+    - ``step_kind`` outside the ``StepKind`` Literal
+    - ``step_index`` not starting at 0 with monotonic +1 increments
+    - non-JSON-serializable payload
+    - any floating-point value inside a payload (replay hazard)
+    """
+
+    frozen = tuple(steps)
+    if not frozen:
+        raise ValueError("reasoning trace must contain at least one step")
+
+    for expected_index, step in enumerate(frozen):
+        if step.step_kind not in _STEP_KINDS:
+            raise ValueError(
+                f"step {expected_index}: unknown step_kind {step.step_kind!r}"
+            )
+        if step.step_index != expected_index:
+            raise ValueError(
+                "step_index must start at 0 and increase monotonically by 1 "
+                f"(step {expected_index} has step_index={step.step_index})"
+            )
+        if not isinstance(step.input_pointers, tuple):
+            raise ValueError(
+                f"step {expected_index}: input_pointers must be a tuple"
+            )
+        for pointer in step.input_pointers:
+            if not isinstance(pointer, str):
+                raise ValueError(
+                    f"step {expected_index}: input_pointers must contain strings only"
+                )
+        _validate_payload(step.output_payload)
+
+    trace_id = compute_trace_id(frozen)
+    return ReasoningTrace(trace_id=trace_id, steps=frozen)
diff --git a/tests/test_adr_0172_w0_reasoning_trace.py b/tests/test_adr_0172_w0_reasoning_trace.py
new file mode 100644
index 00000000..a08eaa74
--- /dev/null
+++ b/tests/test_adr_0172_w0_reasoning_trace.py
@@ -0,0 +1,125 @@
+"""ADR-0172 W0 — ``ReasoningTrace`` substrate tests."""
+
+from __future__ import annotations
+
+import pytest
+
+from teaching.math_reasoning_trace import (
+    ReasoningStep,
+    build_trace,
+    canonical_bytes,
+    compute_trace_id,
+)
+
+
+def _step(
+    *,
+    index: int = 0,
+    kind: str = "observation",
+    pointers: tuple[str, ...] = (),
+    claim: str = "claim",
+    justification: str = "because",
+    payload: object | None = None,
+) -> ReasoningStep:
+    return ReasoningStep(
+        step_index=index,
+        step_kind=kind,  # type: ignore[arg-type]
+        input_pointers=pointers,
+        claim=claim,
+        justification=justification,
+        output_payload={} if payload is None else payload,
+    )
+
+
+def test_step_index_must_start_at_zero() -> None:
+    with pytest.raises(ValueError, match="step_index"):
+        build_trace([_step(index=1)])
+
+
+def test_step_index_must_be_monotonic() -> None:
+    with pytest.raises(ValueError, match="step_index"):
+        build_trace([_step(index=0), _step(index=2, kind="grouping")])
+
+
+def test_canonical_bytes_stable_across_runs() -> None:
+    steps_a = [
+        _step(index=0, kind="observation", payload={"n": 3}),
+        _step(index=1, kind="grouping", payload={"key": ["a", "b"]}),
+    ]
+    steps_b = [
+        _step(index=0, kind="observation", payload={"n": 3}),
+        _step(index=1, kind="grouping", payload={"key": ["a", "b"]}),
+    ]
+    trace_a = build_trace(steps_a)
+    trace_b = build_trace(steps_b)
+    assert canonical_bytes(trace_a) == canonical_bytes(trace_b)
+    assert trace_a.trace_id == trace_b.trace_id
+
+
+def test_trace_id_changes_when_claim_changes() -> None:
+    base = build_trace([_step(claim="alpha")])
+    other = build_trace([_step(claim="beta")])
+    assert base.trace_id != other.trace_id
+
+
+def test_trace_id_invariant_to_dict_insertion_order() -> None:
+    payload_in_order = {"alpha": 1, "beta": 2, "gamma": 3}
+    payload_out_of_order = {"gamma": 3, "alpha": 1, "beta": 2}
+    trace_a = build_trace([_step(payload=payload_in_order)])
+    trace_b = build_trace([_step(payload=payload_out_of_order)])
+    assert trace_a.trace_id == trace_b.trace_id
+
+
+def test_non_json_serializable_payload_rejected() -> None:
+    with pytest.raises(ValueError, match="JSON-serializable"):
+        build_trace([_step(payload={"bad": {1, 2, 3}})])
+
+
+def test_all_eight_step_kinds_accepted() -> None:
+    kinds = (
+        "observation",
+        "grouping",
+        "abstraction",
+        "hypothesis",
+        "test_design",
+        "test_application",
+        "test_result",
+        "conclusion",
+    )
+    steps = [_step(index=i, kind=kind) for i, kind in enumerate(kinds)]
+    trace = build_trace(steps)
+    assert tuple(s.step_kind for s in trace.steps) == kinds
+
+
+def test_empty_trace_rejected() -> None:
+    with pytest.raises(ValueError, match="at least one step"):
+        build_trace([])
+
+
+def test_unknown_step_kind_rejected() -> None:
+    bad = ReasoningStep(
+        step_index=0,
+        step_kind="speculation",  # type: ignore[arg-type]
+        input_pointers=(),
+        claim="c",
+        justification="j",
+        output_payload={},
+    )
+    with pytest.raises(ValueError, match="unknown step_kind"):
+        build_trace([bad])
+
+
+def test_float_payload_rejected() -> None:
+    with pytest.raises(ValueError, match="floating-point"):
+        build_trace([_step(payload={"weight": 0.5})])
+
+
+def test_compute_trace_id_matches_build_trace() -> None:
+    steps = (_step(index=0, claim="x"),)
+    assert compute_trace_id(steps) == build_trace(list(steps)).trace_id
+
+
+def test_trace_is_frozen() -> None:
+    trace = build_trace([_step()])
+    with pytest.raises(Exception):  # FrozenInstanceError subclass of AttributeError
+        trace.trace_id = "mutated"  # type: ignore[misc]