feat(adr-0250): instrument routes Tier-2 + ADR-0250 (Proposed) — arc closure

Instrument: the arithmetic-chain domain now routes each real GSM8K dev case to
Tier-1 (single-accumulator) or Tier-2 (multi-register / summation) and compares
the corridor to a symbolic fold of the SAME compiled program. Recorded coverage
moves 26/50 -> 50/50, 0 refused, PARITY (wrong=0) — the measurement catching up
to the capability. Scope note corrected (full holdout closed; remaining frontier
= derived-operand transfers + non-affine kinds + >5-atom deduction, all 0 on this
holdout). Updated the ADR-0249 instrument test pins (26->50, refused 24->0).

ADR-0250 (Status Proposed, no self-Accept; §10 ruling awaits Shay) + acceptance
evidence: multi-register model, relative conservation, atomicity, certified
summation, chain-of-custody, and the full-holdout 50/50 wrong=0 result.

10/10 instrument pins green.

[Verification]: uv run python -m pytest tests/test_adr_0249_arithmetic_lift.py tests/test_generalized_lift_instrument.py -q

docs/adr/ADR-0250-tier2-multi-entity-arithmetic.md

@@ -0,0 +1,95 @@
+# ADR-0250: Tier-2 Multi-Entity Arithmetic — the Multi-Register Extension
+
+**Status**: **Proposed**
+**Date**: 2026-07-18
+**Authors**: Joshua Shay + multi-model R&D (implemented Fable 5)
+**Depends on**: ADR-0249 (reader→Hamiltonian compiler — quantity kernel, relation compiler, turn programs), ADR-0243/0244/0245 (relaxation, content-addressing, f64)
+**Design record**: `docs/research/tier2-multi-entity-arithmetic-spike-2026-07-18.md`
+**Acceptance evidence**: `docs/handoff/ADR-0250-Acceptance-Evidence.md`
+
+---
+
+## 1. What this ADR is
+
+ADR-0249 closed Tier-1 (single-accumulator affine arithmetic) and recorded the 24 multi-entity
+GSM8K dev-holdout cases as the Tier-2 frontier. This ADR crosses that frontier — and with it the
+reader→Hamiltonian compiler solves the **entire** real GSM8K dev holdout (50/50) at wrong=0. It
+extends the substrate mechanism only; no serving change, no gate activated, no manifold mutation.
+
+## 2. The multi-register model
+
+Multi-entity state is a **product of independent conformal lines**: one register (null point /
+its own relaxation) per entity. This respects the Cl(4,1) boundary exactly — no dimensional
+inflation (no `Cl(8,2)` to pack two points into one multivector), so the f64 rounding floor
+(~6e-11) stays the only residual and every op is a native, exact versor action on a pure state.
+Per-register ops (add/subtract/multiply/divide by constants) reuse the Tier-1 transport unchanged.
+
+## 3. Transfers + the conservation pin
+
+A constant-operand transfer ("actor gives `k` to target", ADR-0116 decomposition) is a **coupled
+pair of translators** — `T₋ₖ` on the actor register, `T₊ₖ` on the target — acting on disjoint
+registers, so exactness across the independent null points is inherited from ADR-0249 P1.
+
+**Conservation pin (hard-reject, relative).** `Σ decode(after) ≡ Σ decode(before)` within
+`|Δ| ≤ rtol·max(1,|before|)`. The tolerance is *relative* because decoded-quantity error scales
+with magnitude (found on a case chaining to 222); an absolute tolerance is brittle, while a
+genuine non-conserving transfer is off by a whole `k` and is still caught. A conservation failure
+is an algebraic failure, not a scope miss — fail closed.
+
+## 4. Transactional atomicity
+
+Registers are an **immutable snapshot**. A transfer runs **prepare → validate → commit**: both
+candidate states are relaxed into locals, validated (both converged AND conservation), then — only
+on full success — a *new* register mapping is produced. A failure anywhere means the new mapping is
+never built, so the original stands untouched: no partial-mutation window, no rollback. Records
+append only on commit; any abort raises a typed `MultiRegisterError` and aborts the whole program
+(fail-closed — a partial answer is never emitted, preserving wrong=0).
+
+## 5. Certified summation + chain of custody
+
+A total-like unknown (`unknown.entity is None`, "how many altogether") is the **explicit typed
+signal** for a certified summation over all registers — never inferred from an operation pattern.
+Per the summation ruling it stays in the substrate: each addition is a certified relaxation turn
+(`T_v` on the running accumulator), never a Python `sum()`.
+
+The operand of each summation turn is the decode of a certified register state, bound in the record
+by `operand_source_digest` (the `psi_digest` byte convention). Deterministic re-execution
+reproduces the state and its decode, so the Python layer cannot tamper with the intermediate value
+— the chain of custody holds with no trusted-Python assumption. The field is added conditionally to
+the record payload, so non-summation record digests are unchanged.
+
+## 6. Honest measurement (done-when)
+
+On the sealed real GSM8K dev holdout (50 cases): **50/50 solved, wrong=0** — Tier-1 (26) + T2a
+single-entity (18) + summation (6). The instrument's `arithmetic-chain` domain routes each case to
+the right tier and compares the corridor to a symbolic fold of the *same* compiled program: PARITY
+(delta 0 — the field matches arithmetic, it does not beat it). The deliverable is a decidable
+measurement + real-holdout coverage, per the honest-NULL protocol.
+
+## 7. Scope + recorded frontier
+
+Tier-2a: single-accumulator-per-entity affine + constant-operand transfers, positive scale,
+matching units. 2b: the certified summation turn (real, exercised by the 6 total-like cases) and
+the derived-operand-transfer path ("half of X" — designed + guarded, **0 on this holdout**). Still
+refused and recorded: non-affine kinds (rate/comparison/fraction/partition — 0 on this holdout),
+derived-operand transfers, and >5-atom deduction — never silently dropped.
+
+## 8. Governance
+
+Off-serving (A-04): `evals/multi_register_program.py` + the instrument domain; never imported by
+`chat/runtime.py`. No gate activated; I-03 untouched; ratified contracts reused
+(`compile_quadratic_well`, `quantity_kernel`, the Ring-2 chain pattern, `RelaxationCertificate`).
+Local-first CI: smoke green in-worktree at every phase.
+
+## 9. What this ADR does NOT claim
+
+- Not a serving change; the symbolic `generate/` path remains the serve path.
+- Not a lift over symbolic solvers on arithmetic (PARITY is the honest result).
+- Not derived-operand-transfer coverage (0 on this holdout; designed + guarded only).
+- Not an activation of any admission gate (ADR-0246 §3.7 stays uncalibrated).
+
+## 10. Ruling record (Shay)
+
+_Awaiting ratification._ The design rulings (multi-register product-of-lines; relative
+conservation hard-reject; prepare→validate→commit atomicity; certified summation turn; ship 2b
+designed+guarded) are recorded RESOLVED in the spike; acceptance is Shay's alone (no self-Accept).

docs/handoff/ADR-0250-Acceptance-Evidence.md

@@ -0,0 +1,59 @@
+# ADR-0250 — Acceptance Evidence
+
+**Status**: Evidence pack for Shay's ruling (no self-Accept)
+**Date**: 2026-07-18
+**ADR**: `docs/adr/ADR-0250-tier2-multi-entity-arithmetic.md`
+**Design record**: `docs/research/tier2-multi-entity-arithmetic-spike-2026-07-18.md`
+
+Tier-2 extends the reader→Hamiltonian compiler (ADR-0249) to multi-entity arithmetic. With it,
+the corridor solves the **entire** real GSM8K dev holdout. Built in smoke-gated phases, each
+merged under its own PR off `forgejo/main`.
+
+## Phases merged
+
+| Phase | PR | Merge commit | What landed |
+|---|---|---|---|
+| Design spike | #71 | `97b2f8b7` | multi-register model, atomicity, chain-of-custody design; probe of the 24 refused cases |
+| T2a executor | #72 | `efd5335e` | `evals/multi_register_program.py` — multi-register + transfers + conservation; 18/18 single-entity |
+| 2b summation | #73 | `3c3dcdbe` | certified summation turn + `operand_source_digest`; full holdout 50/50 |
+| Instrument + ADR | this PR | — | `arithmetic-chain` domain routes Tier-2; ADR-0250 + this evidence |
+
+Smoke (`uv run core test --suite smoke -q`) green at every phase (176 passed).
+
+## Load-bearing result
+
+**Real GSM8K dev holdout (sealed, 50 cases):**
+
+| Tier | Cases | Corridor |
+|---|---|---|
+| Tier-1 single-accumulator (ADR-0249) | 26 | correct, wrong=0 |
+| T2a single-entity multi-register | 18 | correct, wrong=0 |
+| 2b certified summation ("altogether") | 6 | correct, wrong=0 |
+| **Total** | **50 / 50** | **wrong = 0** |
+
+Corridor vs symbolic fold of the same compiled program: **PARITY** (delta 0) — the field matches
+arithmetic, it does not beat it. The instrument's `arithmetic-chain` domain now records 50/50
+coverage, 0 refused.
+
+## Verified in-tree
+
+- **Transfer conservation + per-register exactness**: Ruth 36→31, Sara 19→24, sum conserved;
+  200 random transfers exact + conserving to ~6e-11.
+- **Relative conservation pin**: `gma-050` (chains to 222) tripped an absolute 1e-6 tolerance;
+  the relative check `|Δ| ≤ rtol·max(1,|before|)` fixes it and still catches real violations.
+- **Chain of custody**: summation records carry `operand_source_digest`; tamper on any non-terminal
+  record breaks the successor link (`verify_multi_register_chain`).
+
+## Design decisions applied (honest register)
+
+- **Atomicity via immutability**: prepare→validate→commit; no partial-mutation window, no rollback.
+- **Summation stays in the substrate** (ruling #1): each addition is a certified turn, never a
+  Python `sum()`; the total is grounded in `RelaxationCertificate`s.
+- **The summation signal is the typed `unknown.entity is None`**, not an inferred op pattern.
+
+## What remains open (not claimed here)
+
+- Derived-operand transfers ("half of X") — designed + guarded, 0 on this holdout.
+- Non-affine kinds (rate/comparison/fraction/partition) — 0 on this holdout, refused.
+- >5-atom deduction via ROBDD-partitioned turn programs — future arc.
+- ADR-0246 §3.7 admission-policy calibration — unchanged; no gate activated.

evals/generalized_lift_instrument.py

@@ -66,6 +66,11 @@ from evals.turn_program import (
     compile_turn_program,
     execute_turn_program,
 )
+from evals.multi_register_program import (
+    MultiRegisterError,
+    compile_multi_register_program,
+    execute_multi_register_program,
+)
 
 __all__ = [
     "DomainOutcome",
@@ -486,14 +491,51 @@ def _symbolic_fold(seed: float, steps) -> float:
     return answer
 
 
-def run_arithmetic_chain_domain() -> DomainOutcome:
-    """Corridor turn-program executor vs symbolic fold on real GSM8K problems.
+def _symbolic_multi_register(program) -> float:
+    """Baseline for Tier-2: solve the SAME multi-register program arithmetically.
 
-    Both paths consume the graph's `ground_truth_graph` compiled to a turn
-    program; non-Tier-1 graphs (multi-entity, transfer, rate, …) are refused by
-    BOTH and recorded as the composition frontier — never silently dropped. The
-    expected honest verdict is PARITY with wrong=0: the field does not beat
-    arithmetic at arithmetic; the result is real-holdout coverage, not a lift.
+    Mirror of the corridor executor with Python arithmetic instead of
+    relaxation — per-register affine folds, transfers move `operand` between
+    registers, a `None` answer entity sums all registers.
+    """
+    registers = {entity: seed for entity, seed in program.seeds}
+    for turn in program.turns:
+        if turn.kind == "transfer":
+            registers[turn.actor] -= turn.operand
+            registers[turn.target] += turn.operand
+        else:
+            registers[turn.actor] = turn.scale * registers[turn.actor] + turn.offset
+    if program.answer_entity is None:
+        return float(sum(registers.values()))
+    return registers[program.answer_entity]
+
+
+def _corridor_and_baseline(graph):
+    """Route a graph to Tier-1 (single-accumulator) or Tier-2 (multi-register),
+    returning (corridor_answer, baseline_answer) or None if genuinely un-ingestible.
+    Both paths consume the identical compiled program; the corridor relaxes, the
+    baseline folds arithmetically."""
+    try:
+        program = compile_turn_program(graph)
+        return execute_turn_program(program).answer, _symbolic_fold(program.seed, program.steps)
+    except TurnProgramError:
+        pass
+    try:
+        program = compile_multi_register_program(graph)
+    except MultiRegisterError:
+        return None
+    return execute_multi_register_program(program).answer, _symbolic_multi_register(program)
+
+
+def run_arithmetic_chain_domain() -> DomainOutcome:
+    """Corridor executor vs symbolic fold on real GSM8K problems (Tier-1 + Tier-2).
+
+    Each `ground_truth_graph` is routed to the single-accumulator turn program or
+    the multi-register program; both corridor and baseline consume that same
+    compiled program. Anything neither tier ingests (derived-operand transfers,
+    >5-atom shapes) is refused by BOTH and recorded, never silently dropped. The
+    expected honest verdict is PARITY with wrong=0: the field matches arithmetic,
+    it does not beat it; the result is real-holdout coverage.
     """
     corridor_correct = corridor_wrong = corridor_refused = 0
     baseline_correct = baseline_wrong = baseline_refused = 0
@@ -508,23 +550,15 @@ def run_arithmetic_chain_domain() -> DomainOutcome:
             graph = graph_from_dict(graph_dict) if graph_dict else None
         except (MathGraphError, KeyError, TypeError, ValueError):
             graph = None
-        try:
-            program = compile_turn_program(graph) if graph is not None else None
-        except TurnProgramError as refusal:
+        answers = _corridor_and_baseline(graph) if graph is not None else None
+        if answers is None:
             corridor_refused += 1
             baseline_refused += 1
-            rows.append({"case_id": case_id, "ingested": False, "refusal": refusal.reason})
-            continue
-        if program is None:
-            corridor_refused += 1
-            baseline_refused += 1
-            rows.append({"case_id": case_id, "ingested": False, "refusal": "no_graph"})
+            rows.append({"case_id": case_id, "ingested": False})
             continue
 
+        corridor_answer, baseline_answer = answers
         gold = None if gold_raw is None else float(gold_raw)
-        corridor_answer = execute_turn_program(program).answer
-        baseline_answer = _symbolic_fold(program.seed, program.steps)
-
         corridor_ok = gold is not None and abs(corridor_answer - gold) < 1e-4
         baseline_ok = gold is not None and abs(baseline_answer - gold) < 1e-4
         corridor_correct += int(corridor_ok)
@@ -553,10 +587,10 @@ def run_arithmetic_chain_domain() -> DomainOutcome:
         baseline_wrong=baseline_wrong,
         baseline_refused=baseline_refused,
         notes=(
-            f"Real GSM8K dev holdout: {ingested}/{len(cases)} problems are Tier-1 "
-            f"affine single-accumulator ingestible and solved wrong=0; the rest are "
-            f"refused (multi-entity/transfer/rate/…) — the recorded Tier-2 frontier.",
-            "Baseline is a symbolic fold of the SAME compiled turn program; PARITY "
+            f"Real GSM8K dev holdout: {ingested}/{len(cases)} problems ingestible "
+            f"(Tier-1 single-accumulator + Tier-2 multi-entity/transfer/summation) and "
+            f"solved wrong=0. Derived-operand transfers / >5-atom shapes remain the frontier.",
+            "Baseline is a symbolic fold of the SAME compiled program; PARITY "
             "with wrong=0 is the honest outcome — the field matches arithmetic, it "
             "does not beat it. The result is real-holdout coverage, not a lift.",
         ),
@@ -579,11 +613,12 @@ def run_generalized_lift_instrument() -> LiftInstrumentReport:
         wrong_zero_guard_held=(propositional.corridor_wrong == 0 and arithmetic.corridor_wrong == 0),
         honest_null=all(o.delta_correct <= 0 for o in outcomes),
         scope_limitations=(
-            "The reader→Hamiltonian compiler now exists for the affine "
-            "single-accumulator arithmetic subset (ADR-0249): on the real GSM8K "
-            "dev holdout it solves the Tier-1 subset wrong=0 (see the "
-            "arithmetic-chain domain). Non-affine / multi-entity GSM8K problems "
-            "(transfer, rate, comparison, fraction, partition) remain the "
-            "recorded Tier-2 frontier — refused, never silently dropped.",
+            "The reader→Hamiltonian compiler now solves the FULL real GSM8K dev "
+            "holdout wrong=0 (Tier-1 single-accumulator, ADR-0249; Tier-2 "
+            "multi-entity + constant-operand transfers + certified summation, "
+            "ADR-0250) — see the arithmetic-chain domain. Remaining general "
+            "frontier: derived-operand transfers ('half of X') and non-affine "
+            "kinds (rate/comparison/fraction/partition), both 0 on this holdout; "
+            "and >5-atom deduction. Refused, never silently dropped.",
         ),
     )

tests/test_adr_0249_arithmetic_lift.py

@@ -27,7 +27,8 @@ def test_domain_solves_real_gsm8k_wrong_zero(domain) -> None:
     assert domain.domain_id == "arithmetic-chain"
     assert domain.corridor_wrong == 0  # wrong=0 on real GSM8K, not just templates
     assert domain.corridor_correct > 0  # the corridor ingests and solves real problems
-    assert domain.corridor_refused > 0  # and honestly refuses what it can't ingest
+    # Tier-2 (ADR-0250) closed the multi-entity frontier: the full holdout ingests now.
+    assert domain.corridor_refused == 0
 
 
 def test_every_ingested_case_is_correct(domain) -> None:
@@ -50,10 +51,10 @@ def test_coverage_is_recorded_not_dropped(domain) -> None:
     ingested = domain.corridor_correct + domain.corridor_wrong
     assert ingested + domain.corridor_refused == domain.n_cases
     assert domain.n_cases > 0
-    # A coverage tracker against the sealed dev holdout (50 cases, 26 Tier-1).
-    assert domain.corridor_correct == 26
-    assert domain.corridor_refused == 24
-    assert any("Tier-1" in note for note in domain.notes)
+    # Coverage tracker against the sealed dev holdout: Tier-1 (26) + Tier-2 (24) = 50/50.
+    assert domain.corridor_correct == 50
+    assert domain.corridor_refused == 0
+    assert any("Tier-2" in note for note in domain.notes)
 
 
 def test_domain_joins_full_report_and_scope_corrected() -> None:
@@ -61,8 +62,8 @@ def test_domain_joins_full_report_and_scope_corrected() -> None:
     ids = [o.domain_id for o in report.outcomes]
     assert "arithmetic-chain" in ids
     assert report.wrong_zero_guard_held  # deductive + arithmetic both wrong=0
-    # The stale "no compiler exists" scope note must be gone; the compiler exists now.
+    # The scope note reflects the full-holdout closure; the stale "no compiler" line is gone.
     joined = " ".join(report.scope_limitations)
-    assert "now exists" in joined
+    assert "FULL real GSM8K dev holdout" in joined
     assert "no reader-to-Hamiltonian compiler exists" not in joined
     json.dumps(report.as_dict())  # JSON-safe artifact