docs(research): Tier-2 multi-entity arithmetic design spike

Action 1: multi-register product-of-lines model + coupled-translator transfers
+ transfer-conservation invariant (verified in-tree: Ruth/Sara + 200 random
transfers, conservation & per-register exactness ~6e-11). Transactional
atomicity via immutable register snapshots (prepare-validate-commit; no partial
mutation, no rollback).

Action 2 finding (probe on the 24 refused GSM8K dev cases): ALL 24 are
constant-operand multi-entity (20 contain a transfer). Unknown split: 18
single-entity (pure 2a -> 26/50->44/50), 6 total-like (need certified summation
= 2b certified-decode staging on REAL data -> up to 50/50). Derived-operand
transfers (half-of-X) = 0 on this holdout (synthetic-only, guarded).

Tier-2b certified-decode staging + chain-of-custody (Shay's Q): certificate
psi_digest anchor + operand_certificate_id in the TurnRecord + deterministic
re-execution/live gate — no trusted-Python assumption.

§7 rulings RESOLVED (Shay): certified summation turn; ship 2b designed+guarded;
conservation pin = hard-reject. ADR-0250 (Proposed) to follow.

docs/research/tier2-multi-entity-arithmetic-spike-2026-07-18.md

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+# Tier-2 Multi-Entity Arithmetic — Design Spike
+
+**Status**: DESIGN — for review before implementation
+**Date**: 2026-07-18
+**Base**: `forgejo/main @ ebfdc8b4` (ADR-0249 Accepted; reader→Hamiltonian compiler shipped)
+**Branch**: `feat/adr-0250-tier2-multi-entity`
+**Provenance**: Next arc selected by Shay after ADR-0249. Extends Tier-1 (single-accumulator
+affine) to multi-entity arithmetic — the 24 GSM8K dev-holdout cases ADR-0249 recorded as the
+Tier-2 frontier.
+**Next ADR number**: 0250 (re-verify at landing).
+
+---
+
+## 1. Objective (done-when)
+
+Convert the recorded Tier-2 frontier — multi-entity arithmetic — into certified corridor turns
+with the same honest-by-construction rigor as Tier-1: **wrong = 0 on what is ingested, typed
+refusal on what is not, real-holdout coverage measured and recorded.** No serving change, no
+gate activated, no manifold mutation.
+
+## 2. Action-2 finding — the frontier is entirely 2a (verified on real data)
+
+Probe over the 24 refused GSM8K dev-holdout cases (each via its `ground_truth_graph`):
+
+| Category | Count |
+| :--- | :--- |
+| Refused (all `not_single_accumulator`) | 24 / 24 |
+| **2a-eligible** (multi-entity, every op affine-or-transfer with a **constant** `Quantity` operand) | **24 / 24** |
+| 2b (derived operand: rate / comparison / fraction / partition) | **0 / 24** |
+| …of which contain a `transfer` op | 20 / 24 |
+
+**Unknown-resolution split (probe on the 24 refused).** 18 have a **single-entity** unknown
+(decode one register — pure 2a); **6 are total-like** (`unknown.entity is None`, "how many
+altogether" → a certified **summation over registers**).
+
+**Consequence.** Tier-2a (multi-register + constant-operand transfers, single-entity decode)
+closes **18** → coverage 26/50 → 44/50. The **6 total-like** need a certified summation turn,
+and summing registers *is* certified-decode staging (decode a certified register, use it as a
+translator operand) — so **2b's certified-decode machinery is exercised on real data by these 6
+cases**, not purely synthetic. Together T2a + summation potentially close all 24 → 50/50. The
+narrower **derived-operand transfer** subcase ("gives half her apples", "as many as Y has")
+remains **0-on-this-holdout** — designed + guarded, measured only on synthetic fixtures.
+Recorded honestly — no silent claim.
+
+## 3. Action-1 — the multi-register model + conservation invariant
+
+**Representation: a product of independent conformal lines.** Multi-entity state is a tuple of
+null points `(ψ_e1, …, ψ_eN)`, one register per entity, each its own 32-dim multivector / its
+own relaxation. 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 operation is a native, exact versor action on a pure state.
+
+**Constant-operand transfer = a coupled pair of translators.** "actor gives `k` to target" (the
+ADR-0116 decomposition: subtract-from-actor + add-to-target) is `T₋ₖ` on the actor register,
+`T₊ₖ` on the target register. Exactness across the independent null points is guaranteed because
+the two translators act on **disjoint registers** — no cross-register interference; each is the
+Tier-1 single-line transport already proven exact (ADR-0249 P1).
+
+**Verified in-tree (2026-07-18).** Ruth 36 → 31, Sara 19 → 24, sum conserved at 55 (error
+5e-13); 200 random two-entity transfers: worst per-register error 6e-11, worst conservation
+error 6e-11 (f64 rounding only).
+
+**Conservation as a load-bearing gate.** Because both translators use the same `k` with opposite
+sign, `Σ decode(after) ≡ Σ decode(before)` exactly. This is a falsifiable **transfer-conservation
+pin** (the arithmetic analogue of the chiral charge-conservation latch): a transfer that fails it
+is rejected. It forces honesty by construction — hallucinated arithmetic across decoupled
+registers cannot conserve. Non-transfer per-register ops (add/subtract/multiply/divide, rate as a
+constant dilation, fraction as a constant dilation) reuse the Tier-1 primitives unchanged.
+
+**Answer resolution.** The problem's `unknown.entity` selects which register's final decode is
+the answer (or a total = sum of registers, for "how many altogether"). This is a compile-time
+routing decision from the graph, decoded once at the end (anti-hollow preserved).
+
+## 4. Tier-2b design — certified-decode staging + chain of custody
+
+2b handles **derived operands** — "gives half her apples", "gives as many as Bob has" — where
+the amount moved is itself a field quantity. Adding one register's value to another is not a
+single translator (a translator adds a scalar *constant*; building `T_v` from a state needs
+`v`'s scalar). The no-hollow resolution is **certified-decode staging**: relax a sub-turn to a
+*certified scalar*, then use it as the constant operand for the target's translator turn.
+
+### 4.1 The chain-of-custody question (Shay)
+
+*How is the intermediate decoded scalar bound into the `TurnRecord` ledger so the chain of
+custody proves the Python layer did not tamper with the value between turns?*
+
+Three layers, no trusted-Python assumption:
+
+1. **Certificate anchor.** The decode sub-turn produces state `ψ_v` with a
+   `RelaxationCertificate` `C_v` whose `psi_digest = digest(ψ_v)`. Per ADR-0243, *convergence
+   evidence cannot be borrowed* — the egress gate refuses `C_v` presented with any other state.
+   So `C_v` cryptographically commits to the exact `ψ_v`.
+2. **Operand provenance in the record.** The target turn's `TurnRecord` (extended for 2b) carries
+   `operand_certificate_id = C_v.certificate_id` alongside the step's scale/offset (which encodes
+   `v`, and is covered by `record_digest`). The record thus commits to *both* "operand = v" *and*
+   "operand came from the certified state `ψ_v` via `C_v`".
+3. **Deterministic re-execution + live gate.** Relaxation and decode are deterministic and pure,
+   so verification re-executes the decode sub-turn → reproduces `ψ_v` → `digest(ψ_v)` must equal
+   `C_v.psi_digest` → `decode_quantity(ψ_v)` must equal the recorded operand. A Python-layer
+   tamper on the intermediate `v` makes the recorded operand disagree with the re-derived
+   `decode(ψ_v)` → mismatch → rejected. In addition, a live `verify_derived_operand(record,
+   C_v, ψ_v)` fails closed at execution time (`|decode(ψ_v) − operand| < tol` *and*
+   `digest(ψ_v) == C_v.psi_digest`). The chain's `prev_record_digest` links keep the
+   decode-turn → target-turn *sequence* tamper-evident (`verify_turn_chain`).
+
+The Python layer cannot produce a `v` that is simultaneously recorded in the target turn *and*
+equal to `decode(state pinned by the referenced certificate)` unless it is the honest value.
+
+## 5. Phase plan (post-review)
+
+**Transactional atomicity (Shay's Q — §5.1).** Registers are an **immutable snapshot**; a
+transfer computes both candidate states into locals (prepare), validates (both certified +
+conservation), and only then produces a **new** register set (commit). A failure anywhere means
+the new set is never constructed → the original is untouched → no partial-mutation window, no
+rollback needed. Records append only on commit; an abort surfaces a typed refusal and appends
+nothing. Any aborted transaction aborts the whole program (fail-closed, preserves wrong=0).
+
+- **T2a — multi-register executor + conservation pin + single-entity decode** (closes 18):
+  compile a multi-entity `MathProblemGraph` into an immutable register set + per-register turn
+  programs + coupled transfer transactions; enforce the transfer-conservation pin (hard-reject);
+  decode the `unknown` register. Measure on the 18 single-entity refused dev cases.
+- **T2b — certified-decode staging** (§4 chain of custody) → powers the **certified summation
+  turn** for the 6 total-like cases (REAL data) and the derived-operand transfer (SYNTHETIC
+  fixtures, 0 real). Coverage recorded per-subcase honestly.
+- **T-instrument** — extend the `arithmetic-chain` domain to the multi-entity subset; honest
+  coverage + wrong=0; symbolic-fold baseline over the same multi-register program.
+
+Each phase: own PR, smoke-gated, TDD-first. New machinery ⇒ ADR-0250 (Proposed), acceptance
+evidence assembled as in ADR-0249; no self-Accept.
+
+## 6. Anti-hollow + governance (unchanged doctrine)
+
+Registers flow as field states; decode happens once at the end (2a) or via a *certified*
+sub-turn (2b) — never a compiler-side Python decode of a working quantity. Off-serving (A-04);
+no gate activated; I-03 untouched; ratified contracts reused (`compile_quadratic_well`,
+`quantity_kernel`, the Ring-2 chain pattern, `RelaxationCertificate`).
+
+## 7. Rulings (RESOLVED 2026-07-18 — Shay)
+
+All three APPROVED:
+1. **"Altogether" = certified summation turn** in the substrate — never a compile-time Python
+   total. Summation is certified-decode staging over registers; the total is grounded in a
+   `RelaxationCertificate`, byte-identically traced.
+2. **T2b ships this arc, designed + guarded.** The chain-of-custody machinery is the valuable
+   infrastructure; exercised on real data by the 6 summation cases and on synthetic fixtures for
+   derived-operand transfers; real-world derived-operand-transfer coverage logged 0-on-holdout.
+3. **Conservation pin = hard-reject.** A transfer that fails `Σ decode(after) ≡ Σ decode(before)`
+   is an algebraic failure, not a scope miss — fail closed, hard-reject, keeps the engine
+   accountable and replayable.