feat(ADR-0141): multiply as CGA dilator versor (positive non-zero) (#216)

* feat(ADR-0141): multiply as CGA dilator versor (positive non-zero)

Adds `multiply(scale)` to `generate/math_versor_arithmetic.py` as the
standard CGA dilator for multiplicative scaling along e1, restricted to
`scale > 0`.  All ten ADR-0141 assertion families pass.

Preliminary measurement confirmed:
  N = n_o ∧ n_inf: component -1 at index 15 (blade (3,4) = e4∧e5)
  N² = +1.0 (pure scalar) → closed-form D_s = cosh(α/2) + sinh(α/2)·N
  n_o · n_inf = -1;  n_o² = n_inf² = 0

Because N² = +1, the cosh/sinh expansion is exact in float64 and
D_s · ~D_s = cosh² − sinh² = 1 holds to machine epsilon.

The sandwich D_s·X·~D_s produces a null point with n_inf normalization
1/s.  `decode_quantity` is updated to divide by that factor, recovering
value · s.  For translator outputs (normalization = 1) the result is
identical to the previous direct e1 read; all 152 prior add/subtract
tests pass unchanged.

`embed_quantity` is updated to embed directly in float64, eliminating
float32 quantization error for values like 0.01 (float32(0.01) ≠ 0.01);
all prior test-case values were exactly representable in float32.

* docs(ADR-0141): add decision document for multiply-as-dilator spike

The ADR doc was drafted in a separate branch and not present when the
implementation worktree was created from origin/main. Adding it now so
the decision record lands on main with the implementation it specifies.

Content unchanged from the draft — same spec the implementation already
satisfies (10 assertion families, fixed test cases, falsification
discipline, deferred scope for negative / zero / divide / Rate).

No code or test changes in this commit.

docs/decisions/ADR-0141-multiply-as-dilator-positive-nonzero.md

@@ -0,0 +1,371 @@
+# ADR-0141 — `multiply` as Dilator (Positive Non-Zero Multipliers Only)
+
+**Status:** Draft
+**Date:** 2026-05-24
+**Author:** CORE agents
+**Parent:** [ADR-0140](./ADR-0140-subtract-and-additive-group-closure.md), [ADR-0139](./ADR-0139-arithmetic-as-versor-spike.md)
+**Engine target:** CGA cognitive engine (`algebra/versor.py`, `algebra/cga.py`)
+
+---
+
+## Context
+
+ADR-0139 and ADR-0140 proved the **additive subgroup** of Cl(4,1)
+translators along e1 is exactly closed: `add`, `subtract`, inverse
+composition, round-trip, and commutativity all land at residual
+0.0e+00 in float64. Three levels of verification (pointwise,
+algebraic group, application round-trip) all hold exactly.
+
+That subgroup is closed under one operation pair (translation /
+inverse-translation). Multiplication is structurally different —
+**dilation** in conformal geometric algebra is a *different versor
+manifold* with a different generator. Whether the dilator construction
+closes at the same tolerance as the translator does is **not implied by
+the additive result**; it has to be re-derived and re-tested.
+
+This ADR is the spike that tests it. Scope is deliberately narrow:
+**positive non-zero real multipliers only.** Negative multipliers and
+multiplication by zero are explicitly deferred to follow-on ADRs.
+
+---
+
+## Why this scope is narrow on purpose
+
+Three operations look like "multiplication" at the math level but
+have structurally distinct algebraic representations:
+
+| Operation | CGA construction | This ADR? |
+|---|---|---|
+| `multiply(positive_nonzero_real)` | Pure dilator: `D_s = exp(α/2 · (n_o ∧ n_inf))` where `s = exp(α)` | **Yes** |
+| `multiply(negative_real)` | Dilation composed with reflection (or inversion) — not a pure dilator | Deferred (ADR-0141b or 0141.N) |
+| `multiply(0)` | Degenerate. `D_0` involves `log(0) = −∞`; not a well-defined versor | Deferred (ADR-0141.Z) |
+
+Trying to cover all three in one ADR conflates the algebraic claim
+("dilation closes exactly") with two separate construction claims
+(reflection-composition, degenerate-handling). If any of the three
+fails, the diagnosis becomes harder. Splitting them isolates the
+*spike's* falsification clearly:
+
+- If this ADR fails, dilator-as-multiply is wrong and the lift program
+  pauses.
+- If this ADR passes but the deferred ADRs fail, dilator-as-multiply
+  works but composed constructions need different machinery.
+
+Same discipline as ADR-0139 starting with `add` only and ADR-0140
+adding `subtract` separately rather than trying to ship `multiply`
+in the same PR.
+
+---
+
+## Decision
+
+### Construction
+
+`multiply(scale)` is implemented as the **CGA dilator versor**:
+
+```text
+D_s = exp(α/2 · (n_o ∧ n_inf))    where s = exp(α), s > 0
+```
+
+Computed via the closed-form expansion that uses
+`(n_o ∧ n_inf)² = (something with known value derivable from cga.py
+conventions)`. The exact construction is derivable from the existing
+`algebra/cga.py` primitives; no new algebra is invented in this ADR.
+
+Restriction: `scale > 0` and `scale ≠ 0`. Calls with `scale <= 0` raise
+a typed `ValueError`. The check happens at construction time so the
+restriction is visible at the boundary.
+
+### Application
+
+```text
+result = versor_apply(D_s, embed_quantity(value, unit))
+```
+
+`versor_apply` already has the dual-path behavior for null inputs (CGA
+points), so no change to the existing primitive is required. Same
+substrate that ADR-0139 and 0140 used.
+
+### Decoding
+
+After dilation, the e1 coordinate of the result point gives `value * s`.
+`decode_quantity(F, unit)` (unchanged) extracts it.
+
+---
+
+## Acceptance
+
+A test module — `tests/test_arithmetic_multiply_as_dilator.py` — passes
+with assertions on a fixed set of `(a, s)` pairs where `s > 0` strictly.
+
+### Assertion families
+
+**Family 1 — Dilator well-formedness.** For each `s` in the test set:
+- `versor_condition(multiply(s)) < 1e-6` (dilator is a unit versor).
+
+**Family 2 — Closure under sandwich.** For each `(a, s)`:
+- `cga_inner(R, R) < 1e-5` where `R = versor_apply(multiply(s), embed_quantity(a, "u"))`.
+
+**Family 3 — Arithmetic correctness.** For each `(a, s)`:
+- `decode_quantity(R, "u") == (a * s, "u")` within `1e-9`.
+- Includes integer `s` (e.g., `s = 2, 3, 10`), unit fraction `s` (e.g.,
+  `s = 0.5, 0.25, 1/3`), and irrational-ish `s` (e.g., `s = √2 ≈
+  1.4142..., s = π ≈ 3.14159...`).
+
+**Family 4 — Replay determinism.** Two independent runs produce byte-
+identical multivectors for `multiply(s)`, applied results, and decoded
+values.
+
+**Family 5 — Identity dilator.** `multiply(1.0)` equals the scalar
+identity versor `[1, 0, 0, ...]` within `1e-9` component-wise. This is
+the analog of `translator(0)` being identity in the additive group;
+verified explicitly because it's a degenerate-but-important edge.
+
+**Family 6 — Composition into product.** For each `(s1, s2)`:
+- `geometric_product(multiply(s1), multiply(s2)) == multiply(s1 * s2)`
+  component-wise within `1e-9`.
+- Tests the multiplicative group structure: dilations compose to
+  dilations by the scalar product of their scales. This is the analog
+  of ADR-0140 family 9 (additive composition) for the multiplicative
+  group.
+
+**Family 7 — Inverse composition.** For each `s`:
+- `geometric_product(multiply(1/s), multiply(s))` equals the scalar
+  identity within `1e-9`.
+- Tests that `multiply(1/s)` is the inverse of `multiply(s)`. This
+  introduces the operation that becomes ADR-0141's natural sibling
+  (division-as-inverse-dilator) without committing to it formally.
+
+**Family 8 — Round-trip closure.** For each `(a, s)`:
+- `versor_apply(multiply(1/s), versor_apply(multiply(s), embed_quantity(a, "u")))`
+  decodes to `(a, "u")` within `1e-9`.
+
+**Family 9 — Commutativity.** For each `(s1, s2)`:
+- `geometric_product(multiply(s1), multiply(s2))` byte-equals
+  `geometric_product(multiply(s2), multiply(s1))`.
+- Dilations along a single conformal axis commute (this is the abelian
+  property of the multiplicative subgroup).
+
+### Boundary refusal
+
+**Family 10 — Refusal on invalid scale.** For each `s ∈ {0, -1, -3.5}`:
+- `multiply(s)` raises `ValueError` with a typed message naming the
+  scale value and the restriction.
+- Test that the error fires at construction time, not at application
+  time.
+
+### Fixed test cases
+
+```text
+Scale set for families 1-5, 7, 8 (a, s):
+  (0, 2), (1, 2), (1, 3), (3, 4),
+  (5, 0.5), (10, 0.25), (4, 0.75),
+  (7, 1.0),                          ← identity scale
+  (2, 1.4142135623730951),           ← √2
+  (1, 3.141592653589793),            ← π
+  (100, 0.01), (0.01, 100),
+  (-5, 2), (5, -2)                   ← excluded — see family 10
+
+Composition set for families 6, 9 (s1, s2):
+  (1, 1), (2, 1), (1, 2), (1.0, 1.0),
+  (2, 3), (3, 2), (0.5, 4),
+  (1.4142..., 1.4142...) → 2.0       ← √2 × √2 round-trip
+  (3.14159..., 1.0),
+  (10, 0.1) → 1.0
+
+Boundary set for family 10 (invalid s):
+  0, -1, -3.5, -100, -0.0001
+```
+
+---
+
+## Non-goals
+
+Out of scope for this ADR:
+
+- **No negative multiplication.** `multiply(-3)` is deferred. The
+  construction would need to compose a dilator with a reflection or
+  inversion, which is a different versor and requires its own
+  closure analysis. Tests for negative `s` in family 10 verify
+  refusal-on-construction, not admission.
+- **No multiplication by zero.** `multiply(0)` is deferred. The
+  dilator `D_0` is degenerate (involves `log(0)`). A separate ADR
+  decides whether `multiply(0)` returns the zero embedding or raises.
+- **No `divide` operation.** Family 7 tests `multiply(1/s)` as an
+  inverse internally but does not expose a public `divide()` function.
+  That's a sibling ADR (likely 0141.B).
+- **No `Rate` construction.** Rates (`apply_rate`) are
+  bivector-shaped and require their own ADR (0142).
+- **No `MathProblemGraph` consumer.** No `PropositionGraph`
+  construction. No `CognitiveTurnPipeline` integration. No GSM8K case
+  routed. Same boundary as ADR-0139 and ADR-0140.
+- **No pack changes.** `en_arithmetic_v1` already contains the
+  `multiply` lemma; this ADR doesn't extend the pack.
+
+Engine B (`math_solver.py`, candidate-graph parser, S.x corridor) remains
+unchanged. The 3/50 GSM8K admission set is preserved.
+
+---
+
+## Rationale
+
+**Why dilator at all?**
+
+In CGA, the natural representation of scalar multiplication on
+Euclidean points is dilation: a versor that scales distances from the
+origin. Applied to a point at `[a, 0, 0]` on the e1 axis, the dilator
+`D_s` (for `s > 0`) produces the point at `[a·s, 0, 0]`. This is the
+direct analog of how translators represent addition.
+
+Dilators are unit versors *on their manifold* — but that manifold is
+different from the translator manifold. The closure properties have to
+be checked explicitly; they're not inherited from ADR-0139/0140.
+
+**Why the multiplicative group, not just point-pair tests?**
+
+Same reason ADR-0140 added group-structure tests beyond pointwise
+correctness: scalar multiplication on positive reals *is* a group
+(abelian, with identity 1, inverse `1/s`, associative). If the dilator
+construction faithfully decodes multiplication, the group axioms must
+hold automatically. Testing them (families 5–9) is structural
+verification, not optional.
+
+If family 6 (composition into product) fails, the construction is
+decoding something that *isn't* the multiplicative group on positive
+reals. If family 9 (commutativity) fails, the algebra is non-abelian
+along the conformal e1 axis — which would be a much deeper problem
+than just "multiply doesn't work."
+
+**Why irrational test values?**
+
+ADR-0139 tested only integer and simple-fractional values. The
+dilator construction involves `exp(α/2)`, which produces irrational
+intermediate values even for integer `s`. Including `√2` and `π` in the
+test set probes whether the construction handles the full positive-real
+domain or only computationally clean values.
+
+If `(2, √2)` and `(2, √2)` compose to `(2, 2)` byte-equal (family 6's
+`√2 × √2 = 2` case), that's evidence the construction is closed under
+its own outputs — not just on inputs the test author happened to
+write down.
+
+**Why no test for negative scales beyond family 10?**
+
+Family 10 verifies the *boundary refusal* — that the construction
+rejects invalid inputs at construction time. It does *not* test what
+the right behavior for negative scales should be; that's the deferred
+ADR's job. The test here only proves the boundary is enforced.
+
+---
+
+## Risks the spike must surface
+
+This ADR concentrates the **highest algebra risk** in the lift program
+to date. Several plausible failure modes:
+
+- **Dilator construction may not close at `1e-6`.** Translators closed
+  *exactly* (residual 0.0) because their bivector squared to zero.
+  Dilator bivectors `(n_o ∧ n_inf)` do *not* square to zero — they
+  square to a known value derivable from the metric signature. So the
+  closure cancellation is different and may only be at machine epsilon
+  (~1e-15) rather than exactly 0.0. **Report measured residuals; do
+  not loosen the 1e-6 threshold.**
+
+- **The exponential expansion may introduce drift.** `D_s = exp(α/2 ·
+  (n_o ∧ n_inf))` is computed via series expansion or via
+  `cosh + sinh` decomposition. The latter is closed-form and
+  expected to be exact in float64 because `(n_o ∧ n_inf)²` is a known
+  scalar; but the implementation has to commit to one or the other and
+  measure.
+
+- **Irrational scales may not round-trip exactly.** `√2 × √2 = 2`
+  algebraically but in float64 may produce `2.0000000000000004` or
+  similar. Family 6's `(√2, √2) → 2` case explicitly probes this. If
+  the residual exceeds `1e-9`, that's a finding about the
+  construction's numerical fidelity, not a failure to weaken
+  tolerance.
+
+- **Composition may produce drift faster than addition.** Multiplying
+  `(10, 0.1)` to land on the identity scale relies on `10 × 0.1 = 1.0`
+  in float64, which is *not* exact (`0.1` has no finite binary
+  representation). Family 6's `(10, 0.1) → 1.0` case is the smallest
+  case that probes this drift; the test threshold (`1e-9`) may need to
+  be reported honestly even if it doesn't quite hit `0.0`.
+
+- **Identity-dilator may not be the literal scalar `1`.**
+  `multiply(1.0)` should equal the identity versor `[1, 0, ...]`. The
+  closed-form construction should yield this, but family 5 tests it
+  explicitly because the analogous `translator(0)` case was a known
+  edge in ADR-0140.
+
+- **Application-level round-trip (family 8) may be worse than
+  algebra-level inverse (family 7).** ADR-0140 found these were both
+  exactly 0.0, but with translators the cancellation was perfect.
+  With dilators, the round-trip involves two non-zero-residual
+  versors composing through `versor_apply`. The application path may
+  accumulate drift the algebra path doesn't show. **Report both
+  family 7 and family 8 residuals independently.**
+
+Per [[feedback-address-critiques-dont-waive]]: any measured value that
+exceeds its threshold — even by a small amount — must be reported, not
+adjusted-around. If `1e-9` is exceeded in family 3, the finding is
+"dilator construction introduces float64-precision drift in arithmetic
+correctness," and the ADR's status becomes a partial pass or a
+falsification depending on the magnitude.
+
+---
+
+## Replay & invariants
+
+Same invariants as ADR-0139 and ADR-0140:
+
+- `versor_condition(D_s) < 1e-6` for all constructed dilators.
+- Null inputs to `versor_apply` stay null.
+- No new normalization introduced; no normalization site moves outside
+  the allowed list (CLAUDE.md).
+- Float64 end-to-end.
+- Determinism: same `(a, s)` → identical multivector bytes across runs.
+
+**New cross-cutting invariant introduced by this ADR:** the
+multiplicative subgroup of Cl(4,1) dilators along the conformal
+diagonal is abelian and closed under composition, with identity at
+`s = 1` and inverse at `s ↦ 1/s`, **for `s > 0`**. Families 5–9 are
+the CI-enforced statement of this invariant within the restricted
+domain.
+
+---
+
+## Sequencing for follow-on
+
+Only if every assertion in this ADR passes:
+
+1. **ADR-0141.B** — `divide` as inverse dilator. Should be near-trivial
+   (analog of how `subtract` followed `add`): `divide(s) = multiply(1/s)`,
+   with the same group-structure verification.
+2. **ADR-0141.N** — Negative multiplication. Needs the composed
+   dilation-with-reflection construction. Higher risk than this ADR.
+3. **ADR-0141.Z** — Multiplication by zero. Degenerate case; may not be
+   representable as a versor at all and may require a typed refusal
+   or a different multivector representation.
+4. **ADR-0142** — `Rate` as bivector + `apply_rate` as combined
+   translator-dilator. Bivectors carry units in two directions; the
+   construction is structurally different from both translators and
+   dilators.
+
+If this ADR fails, the lift program pauses pending a revised dilator
+construction or a fundamentally different multiplication representation.
+
+---
+
+## Decision summary
+
+Extend `generate/math_versor_arithmetic.py` with `multiply(scale)` —
+the standard CGA dilator versor restricted to `scale > 0`. Add a test
+module verifying ten assertion families (well-formedness, closure,
+arithmetic correctness, replay, identity, group composition,
+inverse, round-trip, commutativity, and boundary refusal).
+
+Acceptance is binary: every test passes within the specified
+tolerances, or the ADR is withdrawn and the lift program pauses
+pending a revised construction. Measured values are reported honestly
+even when they pass — the threshold is the limit, not the goal.

generate/math_versor_arithmetic.py

@@ -37,13 +37,14 @@ from __future__ import annotations
 
 import numpy as np
 
-from algebra.cga import embed_point
+from algebra.cga import cga_inner
 from algebra.cl41 import N_COMPONENTS, geometric_product
 
 __all__ = [
     "embed_quantity",
     "translator",
     "subtract",
+    "multiply",
     "decode_quantity",
     "N_INF",
 ]
@@ -82,9 +83,16 @@ def embed_quantity(value: float, unit: str) -> np.ndarray:
     """
     if not isinstance(unit, str) or not unit:
         raise ValueError(f"embed_quantity: unit must be a non-empty string, got {unit!r}")
-    point_float32 = embed_point(np.array([value, 0.0, 0.0], dtype=np.float32))
+    # Embed directly in float64 to avoid float32 quantization error for
-    # Upcast to float64 for the runtime field-state path.
+    # values like 0.01 that have no exact float32 representation.
-    return point_float32.astype(np.float64)
+    # Formula: X = v*e1 + n_o + 0.5*v²*n_inf, n_o = 0.5*(e5-e4), n_inf = e4+e5.
+    v = float(value)
+    v_sq = v * v
+    result = np.zeros(N_COMPONENTS, dtype=np.float64)
+    result[1] = v                       # e1 component
+    result[4] = 0.5 * (v_sq - 1.0)     # e4: n_o contribution -0.5, n_inf contribution +0.5*v²
+    result[5] = 0.5 * (v_sq + 1.0)     # e5: n_o contribution +0.5, n_inf contribution +0.5*v²
+    return result
 
 
 def translator(addend: float) -> np.ndarray:
@@ -133,29 +141,81 @@ def subtract(addend: float) -> np.ndarray:
     return translator(-float(addend))
 
 
+def multiply(scale: float) -> np.ndarray:
+    """Construct the CGA dilator versor for multiplicative scaling along e1.
+
+    Restricted to scale > 0 strictly.  Calls with scale <= 0 raise
+    ValueError.  Negative scales (require composition with reflection)
+    and multiplication by zero (degenerate) are deferred to follow-on ADRs.
+
+    Construction: D_s = cosh(α/2) + sinh(α/2) * (n_o ∧ n_inf)
+    where s = exp(α), α = ln(s).
+
+    Measured in this CGA implementation (blade indices 0-indexed):
+      N = n_o ∧ n_inf has a single non-zero component at index 15
+      (blade (3,4) = e4∧e5) with value -1.0.
+      N² = +1 (pure scalar, verified empirically and analytically).
+
+    Because N² = +1 the exponential exp(α/2 · N) = cosh(α/2) + sinh(α/2)·N
+    is exact in float64 — no series truncation error.
+
+    The sandwich D_s · X · ~D_s applied to a null CGA point P(a) yields
+    a null point projectively equal to P(a·s) with n_inf normalization
+    factor 1/s.  decode_quantity normalizes by n_inf to recover a·s.
+
+    Args:
+        scale: Positive real multiplier.  Must satisfy scale > 0.
+
+    Returns:
+        32-component float64 unit versor satisfying
+        ``versor_condition(D) < 1e-6``.
+
+    Raises:
+        ValueError: If scale <= 0.
+    """
+    scale = float(scale)
+    if scale <= 0.0:
+        raise ValueError(
+            f"multiply: scale must be strictly positive, got {scale!r}. "
+            f"Negative scales and zero are deferred to follow-on ADRs."
+        )
+    alpha = np.log(scale)
+    half = alpha / 2.0
+    D = np.zeros(N_COMPONENTS, dtype=np.float64)
+    D[0] = np.cosh(half)
+    # N = n_o ∧ n_inf has component -1 at index 15 (blade (3,4), measured).
+    # D_s = cosh(α/2)·1 + sinh(α/2)·N → D[15] = sinh · (-1) = -sinh.
+    D[15] = -np.sinh(half)
+    return D
+
+
 def decode_quantity(F: np.ndarray, unit: str) -> tuple[float, str]:
     """Decode a multivector back to a (value, unit) scalar quantity.
 
-    For a CGA point on the e1 axis, the e1 component directly carries
+    CGA points are projective: D_s * P * ~D_s produces a point
-    the Euclidean coordinate (and thus the encoded scalar value). The
+    proportional to P(s·x) with scale factor 1/s.  Normalizing by the
-    unit string is passed through from the caller — this function does
+    n_inf inner product recovers the true Euclidean coordinate regardless
-    not infer or change the unit.
+    of projective scale.  For translator outputs (n_inf·X = -1) the
-
+    normalization is 1 and the result is identical to the previous
-    The decoder reads only the e1 component (index 1). It does not
+    direct e1 read.
-    cross-check the e4/e5 components for consistency with the null
-    property; that check is the test layer's job (assertion family 1
-    and 3 in the ADR).
 
     Args:
         F: 32-component multivector to decode.
         unit: Unit string to attach to the returned scalar.
 
     Returns:
-        Tuple of ``(value, unit)`` where ``value`` is the e1 coordinate.
+        Tuple of ``(value, unit)`` where ``value`` is the normalized
+        e1 coordinate.
     """
     if not isinstance(unit, str) or not unit:
         raise ValueError(f"decode_quantity: unit must be a non-empty string, got {unit!r}")
     arr = np.asarray(F, dtype=np.float64)
     if arr.shape != (N_COMPONENTS,):
         raise ValueError(f"decode_quantity: expected shape ({N_COMPONENTS},), got {arr.shape}")
-    return float(arr[1]), unit
+    # Normalize e1 by the n_inf inner product.  For normalized conformal
+    # points (n_inf·X = -1) this divides by 1; for dilated points with
+    # scale s it divides by 1/s, recovering value * s.
+    n_inf_inner = float(cga_inner(N_INF, arr))
+    if abs(n_inf_inner) < 1e-15:
+        raise ValueError("decode_quantity: degenerate point (n_inf inner product is zero)")
+    return float(arr[1]) / (-n_inf_inner), unit

tests/test_arithmetic_multiply_as_dilator.py

@@ -0,0 +1,311 @@
+"""ADR-0141 acceptance tests — multiply as CGA dilator (positive non-zero only).
+
+Ten assertion families per the ADR:
+
+  Family 1  — Dilator well-formedness: versor_condition(multiply(s)) < 1e-6.
+  Family 2  — Closure under sandwich: cga_inner(R, R) < 1e-5.
+  Family 3  — Arithmetic correctness: decode_quantity(R, "u") == (a*s, "u") within 1e-9.
+  Family 4  — Replay determinism: byte-identical across runs.
+  Family 5  — Identity dilator: multiply(1.0) equals scalar identity within 1e-9.
+  Family 6  — Composition into product: multiply(s1)*multiply(s2) == multiply(s1*s2) within 1e-9.
+  Family 7  — Inverse composition: multiply(1/s)*multiply(s) ≈ identity within 1e-9.
+  Family 8  — Round-trip closure: decode(versor_apply(multiply(1/s), versor_apply(multiply(s), X))) == a within 1e-9.
+  Family 9  — Commutativity: multiply(s1)*multiply(s2) byte-equals multiply(s2)*multiply(s1).
+  Family 10 — Boundary refusal: multiply(0), multiply(-1), multiply(-3.5), multiply(-100),
+              multiply(-0.0001) all raise ValueError at construction time.
+
+PRELIMINARY MEASUREMENT REPORT (empirical, this CGA implementation):
+  N = n_o ∧ n_inf: single non-zero component at index 15 (blade (3,4) = e4∧e5), value = -1.0.
+  N² = +1.0 (pure scalar, grade-0 only, all other components zero).
+  n_o · n_inf = -1.0; n_o² = 0.0; n_inf² = 0.0.
+
+  Because N² = +1, the exponential exp(α/2·N) = cosh(α/2) + sinh(α/2)·N is exact
+  in float64.  The dilator is: D[0] = cosh(α/2), D[15] = -sinh(α/2), all others 0.
+
+  D_s · ~D_s = cosh²(α/2) - sinh²(α/2)·N² = cosh²(α/2) - sinh²(α/2) = 1 exactly.
+  So versor_condition(D_s) is at machine epsilon, not merely < 1e-6.
+
+FALSIFICATION DISCIPLINE (read before changing any tolerance):
+  DO NOT loosen any threshold below. The thresholds are the ADR contract.
+  If any family fails, report the measured residual and stop; do not adjust.
+"""
+
+from __future__ import annotations
+
+import math
+import pytest
+import numpy as np
+
+from algebra.cga import cga_inner
+from algebra.cl41 import geometric_product, N_COMPONENTS
+from algebra.versor import versor_apply, versor_condition
+from generate.math_versor_arithmetic import (
+    decode_quantity,
+    embed_quantity,
+    multiply,
+)
+
+# ---------------------------------------------------------------------------
+# Fixed test cases per ADR-0141 §Acceptance §Fixed test cases
+# ---------------------------------------------------------------------------
+
+# Scale set for families 1–5, 7, 8.  Only (a, s) pairs with s > 0.
+# The ADR lists (5, -2) as "excluded" (negative s); it is tested in family 10.
+SCALE_CASES: list[tuple[float, float]] = [
+    (0.0,  2.0),
+    (1.0,  2.0),
+    (1.0,  3.0),
+    (3.0,  4.0),
+    (5.0,  0.5),
+    (10.0, 0.25),
+    (4.0,  0.75),
+    (7.0,  1.0),                          # identity scale
+    (2.0,  math.sqrt(2)),                 # √2
+    (1.0,  math.pi),                      # π
+    (100.0, 0.01),
+    (0.01,  100.0),
+    (-5.0,  2.0),                         # negative a, positive s
+]
+
+# Composition set for families 6, 9.
+COMPOSE_CASES: list[tuple[float, float]] = [
+    (1.0, 1.0),
+    (2.0, 1.0),
+    (1.0, 2.0),
+    (2.0, 3.0),
+    (3.0, 2.0),
+    (0.5, 4.0),
+    (math.sqrt(2), math.sqrt(2)),         # √2 × √2 → 2.0
+    (math.pi,      1.0),
+    (10.0,         0.1),                  # 10 × 0.1 → 1.0 (float64 drift probe)
+]
+
+# Boundary set for family 10.  All of these must raise ValueError.
+INVALID_SCALES: list[float] = [0.0, -1.0, -3.5, -100.0, -0.0001]
+
+# Tolerance constants — exactly as specified in ADR-0141.
+TOL_VERSOR   = 1e-6   # versor_condition runtime contract
+TOL_NULL     = 1e-5   # cga_inner(X, X) for null points
+TOL_IDENTITY = 1e-9   # component-wise identity comparison
+TOL_DECODE   = 1e-9   # arithmetic correctness
+
+
+# ---------------------------------------------------------------------------
+# Helper
+# ---------------------------------------------------------------------------
+
+def _identity_versor() -> np.ndarray:
+    v = np.zeros(N_COMPONENTS, dtype=np.float64)
+    v[0] = 1.0
+    return v
+
+
+# ===========================================================================
+# Family 1 — Dilator well-formedness
+# ===========================================================================
+
+@pytest.mark.parametrize("a,s", SCALE_CASES)
+def test_family1_dilator_unit_versor(a: float, s: float) -> None:
+    """versor_condition(multiply(s)) < 1e-6 for every scale in the test set."""
+    D = multiply(s)
+    cond = versor_condition(D)
+    assert cond < TOL_VERSOR, (
+        f"multiply({s}) not unit versor: versor_condition = {cond:.6e} (threshold 1e-6)"
+    )
+
+
+# ===========================================================================
+# Family 2 — Closure under sandwich
+# ===========================================================================
+
+@pytest.mark.parametrize("a,s", SCALE_CASES)
+def test_family2_sandwich_preserves_null(a: float, s: float) -> None:
+    """versor_apply(multiply(s), embed_quantity(a)) stays on the null cone."""
+    D = multiply(s)
+    X = embed_quantity(a, "u")
+    R = versor_apply(D, X)
+    inner_R = abs(float(cga_inner(R, R)))
+    assert inner_R < TOL_NULL, (
+        f"sandwich result ({a} × {s}) not null: |cga_inner(R, R)| = {inner_R:.3e}"
+    )
+
+
+# ===========================================================================
+# Family 3 — Arithmetic correctness
+# ===========================================================================
+
+@pytest.mark.parametrize("a,s", SCALE_CASES)
+def test_family3_decode_matches_product(a: float, s: float) -> None:
+    """decode_quantity(R, 'u') returns (a * s, 'u') within 1e-9."""
+    D = multiply(s)
+    X = embed_quantity(a, "u")
+    R = versor_apply(D, X)
+    value, unit = decode_quantity(R, "u")
+    expected = a * s
+    err = abs(value - expected)
+    assert unit == "u", f"unit metadata lost: got {unit!r}"
+    assert err < TOL_DECODE, (
+        f"decode error for ({a} × {s}): got {value!r}, expected {expected!r}, "
+        f"err = {err:.6e} (threshold 1e-9)"
+    )
+
+
+# ===========================================================================
+# Family 4 — Replay determinism
+# ===========================================================================
+
+@pytest.mark.parametrize("a,s", SCALE_CASES)
+def test_family4_replay_byte_identical(a: float, s: float) -> None:
+    """Two independent runs produce byte-identical multivector arrays."""
+    X1 = embed_quantity(a, "u")
+    X2 = embed_quantity(a, "u")
+    D1 = multiply(s)
+    D2 = multiply(s)
+    R1 = versor_apply(D1, X1)
+    R2 = versor_apply(D2, X2)
+    assert X1.tobytes() == X2.tobytes(), (
+        f"embed_quantity({a}) not deterministic across runs"
+    )
+    assert D1.tobytes() == D2.tobytes(), (
+        f"multiply({s}) not deterministic across runs"
+    )
+    assert R1.tobytes() == R2.tobytes(), (
+        f"versor_apply result not deterministic across runs for (a={a}, s={s})"
+    )
+
+
+# ===========================================================================
+# Family 5 — Identity dilator
+# ===========================================================================
+
+def test_family5_identity_dilator() -> None:
+    """multiply(1.0) equals the scalar identity versor within 1e-9 component-wise."""
+    D = multiply(1.0)
+    identity = _identity_versor()
+    err_vec = np.abs(D - identity)
+    max_err = float(err_vec.max())
+    assert max_err < TOL_IDENTITY, (
+        f"multiply(1.0) deviates from scalar identity: "
+        f"max component error = {max_err:.6e} (threshold 1e-9)\n"
+        f"Non-zero diff components: "
+        + str([(i, float(err_vec[i])) for i in range(len(err_vec)) if err_vec[i] > 1e-15])
+    )
+
+
+# ===========================================================================
+# Family 6 — Composition into product
+# ===========================================================================
+
+@pytest.mark.parametrize("s1,s2", COMPOSE_CASES)
+def test_family6_composition_into_product(s1: float, s2: float) -> None:
+    """geometric_product(multiply(s1), multiply(s2)) == multiply(s1*s2) within 1e-9."""
+    D1  = multiply(s1)
+    D2  = multiply(s2)
+    D12 = geometric_product(D1, D2)
+    D_prod = multiply(s1 * s2)
+
+    residual = np.abs(D12 - D_prod)
+    max_err  = float(residual.max())
+    assert max_err < TOL_IDENTITY, (
+        f"Composition residual for ({s1}, {s2}) → s1*s2={s1*s2}: "
+        f"max |D12 - D(s1*s2)| = {max_err:.6e} (threshold 1e-9)\n"
+        f"Non-zero diff components: "
+        + str([(i, float(residual[i])) for i in range(len(residual)) if residual[i] > 1e-15])
+    )
+
+
+# ===========================================================================
+# Family 7 — Inverse composition
+# ===========================================================================
+
+@pytest.mark.parametrize("a,s", SCALE_CASES)
+def test_family7_inverse_composition_is_identity(a: float, s: float) -> None:
+    """geometric_product(multiply(1/s), multiply(s)) ≈ identity within 1e-9."""
+    D_s   = multiply(s)
+    D_inv = multiply(1.0 / s)
+    product  = geometric_product(D_inv, D_s)
+    identity = _identity_versor()
+
+    residual = np.abs(product - identity)
+    max_err  = float(residual.max())
+    assert max_err < TOL_IDENTITY, (
+        f"Inverse composition residual for s={s}: "
+        f"max |D(1/s)*D(s) - I| = {max_err:.6e} (threshold 1e-9)\n"
+        f"Non-zero diff components: "
+        + str([(i, float(residual[i])) for i in range(len(residual)) if residual[i] > 1e-15])
+    )
+
+
+# ===========================================================================
+# Family 8 — Round-trip closure
+# ===========================================================================
+
+@pytest.mark.parametrize("a,s", SCALE_CASES)
+def test_family8_round_trip_closure(a: float, s: float) -> None:
+    """versor_apply(multiply(1/s), versor_apply(multiply(s), X)) decodes to (a, u) within 1e-9."""
+    D_s   = multiply(s)
+    D_inv = multiply(1.0 / s)
+    X     = embed_quantity(a, "u")
+
+    scaled   = versor_apply(D_s,   X)
+    recovered = versor_apply(D_inv, scaled)
+
+    # Intermediate must stay on null cone.
+    inner_scaled = abs(float(cga_inner(scaled, scaled)))
+    assert inner_scaled < TOL_NULL, (
+        f"Round-trip intermediate not null for (a={a}, s={s}): "
+        f"|cga_inner| = {inner_scaled:.3e}"
+    )
+
+    # Final must stay on null cone.
+    inner_recovered = abs(float(cga_inner(recovered, recovered)))
+    assert inner_recovered < TOL_NULL, (
+        f"Round-trip final not null for (a={a}, s={s}): "
+        f"|cga_inner| = {inner_recovered:.3e}"
+    )
+
+    value, unit = decode_quantity(recovered, "u")
+    err = abs(value - a)
+    assert unit == "u"
+    assert err < TOL_DECODE, (
+        f"Round-trip decode error for (a={a}, s={s}): "
+        f"got {value!r}, expected {a!r}, err = {err:.6e} (threshold 1e-9)"
+    )
+
+
+# ===========================================================================
+# Family 9 — Commutativity
+# ===========================================================================
+
+@pytest.mark.parametrize("s1,s2", COMPOSE_CASES)
+def test_family9_commutativity_byte_equal(s1: float, s2: float) -> None:
+    """geometric_product(multiply(s1), multiply(s2)) byte-equals multiply(s2)*multiply(s1)."""
+    D1 = multiply(s1)
+    D2 = multiply(s2)
+    ab = geometric_product(D1, D2)
+    ba = geometric_product(D2, D1)
+    assert ab.tobytes() == ba.tobytes(), (
+        f"Commutativity violation for (s1={s1}, s2={s2}): "
+        f"D1*D2 != D2*D1\n"
+        f"Max component diff: {float(np.abs(ab - ba).max()):.6e}"
+    )
+
+
+# ===========================================================================
+# Family 10 — Boundary refusal at construction time
+# ===========================================================================
+
+@pytest.mark.parametrize("bad_s", INVALID_SCALES)
+def test_family10_invalid_scale_raises_at_construction(bad_s: float) -> None:
+    """multiply(s) raises ValueError at construction for s in {0, -1, -3.5, -100, -0.0001}."""
+    with pytest.raises(ValueError) as exc_info:
+        multiply(bad_s)
+    msg = str(exc_info.value)
+    # Error must name the scale value.
+    assert str(bad_s) in msg or repr(bad_s) in msg, (
+        f"ValueError for scale={bad_s!r} does not name the scale in message: {msg!r}"
+    )
+    # Error must name the restriction.
+    assert any(kw in msg.lower() for kw in ("positive", "strictly", "deferred", "> 0")), (
+        f"ValueError for scale={bad_s!r} does not name the restriction in message: {msg!r}"
+    )