# 078 operator-zoo — RESULT  (Campaign100 MANIFEST row 078 — opens BLOCK J)

**Tier: T3-exploratory.** Single embedder (SONAR z, night8 pipeline), templated minimal-pair
stimuli, single LLM judge (local codex, neutral rubric, 0 parse fails). No promotion above T3.
Box tmux `c100_078` (CPU) decode-only: 700 items → 1258 texts → 1440 decode jobs, det_ok=True,
576.7 s. LOCAL codex judge (this harvest): 1440 records, 0 parse errors, 684 s. Analysis local.

## Question
033 found SONAR z is a **linear offset-operator space for morphosyntax** (tense/negation/
question/number ≈1.00, invertible, composable) but **not** for content/structure (sentiment 0.11,
voice arg-swap 0/40, entity 0.00). Does that property EXTEND to **RELATIONAL/SEMANTIC** relations
— temporal, spatial, comparative, causal? Sharp crux: relational **MARKER** swaps are closed-class
antonym function words with a FIXED lexical pair (before/after, above/below, inside/outside,
near/far, bigger/smaller) — do they linearize like *negation* (function word) or fail like
*sentiment* (open-class antonym)? And do relational **ARGUMENT** swaps (A above B ↔ B above A) hit
the same binding wall as voice (0/40)?

## Headline ★ POSITIVE + THE BOUNDARY IS CLOSED-CLASS-MARKER vs ARGUMENT-BINDING, NOT GRAMMAR-vs-SEMANTICS
All five relational **MARKER** swaps linearize **perfectly** — a single averaged offset fit on
TRAIN pairs (disjoint frame vocab) applies the relation flip on held-out TEST at **1.00 success,
~1.00 collateral, 1.00 invertibility**, with the MLP never beating it. All three **ARGUMENT**
swaps fail **completely** (0.00 success, 0.00 invertibility) exactly like voice — the sentence is
preserved but the operands are never reordered. Relational *semantics* is not the barrier;
**closed-class lexical-marker substitution is linear, argument-binding / clause-reordering is not.**

| transform | class | success | collateral | invert. (−off) | cos(pred,true) | ceiling | baseline | diff_align | Δ(mlp−lin) zcos | verdict |
|-----------|-------|---------|-----------|----------------|----------------|---------|----------|-----------|-----------------|---------|
| tense              | control | **1.00** | 1.00 | 1.00 | 0.985 | 1.00 | 0.00 | 0.849 | −0.008 | linear |
| negation           | control | **1.00** | 0.96 | 1.00 | 0.979 | 1.00 | 0.00 | 0.850 | −0.013 | linear |
| temporal_marker    | marker  | **1.00** | 1.00 | 1.00 | 0.993 | 1.00 | 0.00 | 0.959 | −0.014 | linear |
| vertical           | marker  | **1.00** | 0.98 | 1.00 | 0.997 | 1.00 | 0.00 | 0.967 | −0.011 | linear |
| containment        | marker  | **1.00** | 1.00 | 1.00 | 0.991 | 1.00 | 0.00 | 0.938 | −0.012 | linear |
| proximity          | marker  | **1.00** | 1.00 | 1.00 | 0.987 | 1.00 | 0.00 | 0.951 | −0.018 | linear |
| comparative_size   | marker  | **1.00** | 1.00 | 1.00 | 0.996 | 1.00 | 0.00 | 0.962 | −0.012 | linear |
| causal_reframe     | reframe | 0.36 | 0.22 | 0.36 | 0.736 | 1.00 | 0.00 | 0.489 | −0.018 | non-linear (clause reorder) |
| spatial_swap       | arg-swap | **0.00** | 0.96 | 0.00 | 0.799 | 0.80 | 0.00 | 0.155 | −0.002 | null (binding wall) |
| temporal_swap      | arg-swap | **0.00** | 1.00 | 0.00 | 0.780 | 1.00 | 0.00 | 0.141 | −0.003 | null (binding wall) |
| causal_swap        | arg-swap | **0.00** | 0.94 | 0.00 | 0.739 | 1.00 | 0.00 | 0.262 | −0.007 | null (binding wall) |

Ranking by success: tense = negation = temporal_marker = vertical = containment = proximity =
comparative_size (**1.00**) ≫ causal_reframe (0.36) ≫ spatial_swap = temporal_swap = causal_swap
(**0.00**). The z-side `diff_align` diagnostic (no decoding) predicts the ordering directly:
markers 0.94–0.97 ≈ controls 0.85 ≫ causal_reframe 0.49 ≫ arg-swaps 0.14–0.26.

## Group means
- **MARKER-swap group** (temporal_marker, vertical, containment, proximity, comparative_size):
  **success 1.00** (all five at 1.00), collateral ~1.00, all invertible.
- **ARGUMENT-swap group** (spatial_swap, temporal_swap, causal_swap): **success 0.00** (0/74),
  collateral high (0.94–1.00) — the offset preserves the sentence but never reverses the operands.
- **CONTROL group** (tense, negation): **success 1.00** — probe-power certified (lesson 5).

## Which relation classes linearize vs don't (extends 033's boundary)
033's boundary was framed grammatical-YES (tense/negation/question/number) / content-structure-NO
(sentiment/voice/entity). 078 **sharpens** it:

- **Closed-class relational MARKERS linearize like negation, NOT like sentiment.** before/after,
  above/below, inside/outside, near/far, bigger/smaller are all clean, invertible linear offsets
  (1.00) — indistinguishable from the grammatical controls. The relevant property is
  **closed-class antonym function word with a fixed lexical pair**, i.e. lexical *substitution*
  along a stable z-direction. 033's sentiment failure (0.11) was an *open-class* antonym over an
  unbounded lexicon with no single polarity axis; it was the open-class-ness, not the "semantic"
  content, that broke linearity. Relational semantics per se is fully linearizable in z.
- **Relational ARGUMENT swaps hit the same binding wall as voice.** With the marker held fixed and
  the two operands reversed, the offset scores 0/74 across spatial/temporal/causal — collateral
  stays high (the sentence is intact and grammatical) but the who/what-relates-to-what is never
  reorganized. This is a direct, independent replication of the binding-block result (014 swap-
  patching, 017 role-heatmap, 019 decoder-mirror, 033 voice 0/40): **z has no transferable linear
  role-swap / argument-binding operator**, and this now generalises from voice diathesis to
  arbitrary relational argument order. diff_align 0.14–0.26 (near 033-entity's 0.12) flags it z-side.
- **Causal reframe (because↔so) did NOT linearize** (0.36 success, 0.22 collateral, diff_align
  0.49). Although meaning-preserving, "X because Y ↔ Y, so X" requires *clause reordering* (result-
  first vs reason-first), a structural reorganization akin to the arg-swaps — so it sits between
  the two regimes and mostly fails. P5 is satisfied only because it clears the floor-zero swaps,
  not because reframe is itself a clean operator (see Limitations).

**MLP non-linear control:** `delta_mlp_minus_lin` (held-out z-cos of MLP minus linear offset) is
**negative for all 11 transforms** (−0.002 to −0.018). The MLP never beats the linear offset in
held-out cos-to-target on any transform — including the failing swaps, where *both* are stuck at
~0.78–0.80 because the target simply is not reachable from z_before by any smooth map fit on this
train set. Where linear works it is the clean operator; where it fails, non-linearity does not
rescue it. (MLP *decode* success is lower than its z-cos on several markers — tiny cos gaps push
the greedy decode off the marker token — but the prereg P4 metric is z-cos Δ, which is decisive.)

## Gates (all PASS)
| gate | value | threshold | pass |
|------|-------|-----------|------|
| G_control (tense/neg success) | 1.00 | ≥ 0.90 | ✓ |
| G_ceiling (mean, decode true z_after) | 0.982 | ≥ 0.90 | ✓ |
| G_baseline (mean, decode z_before) | 0.000 | ≤ 0.15 | ✓ |
| G_parse (judge parse failures) | 0 / 1440 | ≤ 5% | ✓ |
| G_det (deterministic decode) | True | — | ✓ |
| G_norm (z_before vs z_after norm flat) | Δ‖z‖ ≤ 0.007 all T | — | ✓ |

Baseline is a perfect 0.00 everywhere (decoding z_before never realizes the transform) and ceiling
~1.00 (decoding the true z_after does) — the two together are the fabrication check (lesson 4): the
success signal on markers is real applied structure, not decoder confabulation. Note spatial_swap
**ceiling = 0.80** (only single soft ceiling) — even the ground-truth z_after decodes the operand
reversal only 80% of the time, so its 0.00 predicted-success is a conservative floor, not a probe
artifact; the other two swaps have ceiling 1.00, so their 0.00 is unambiguous binding failure.

## Verdict (honest)
**POSITIVE and clean.** The 033 linear-offset-operator property extends to relational semantics
**for closed-class marker swaps** (5/5 perfect, invertible, MLP-dominated) and **fails for argument
binding** (3/3 null, exact voice-style binding wall). The operative boundary is not grammar-vs-
semantics but **closed-class lexical substitution (linear) vs argument-binding/clause-reordering
(non-linear)**. Causal reframe confirms the reordering side of that boundary breaks even when
meaning is preserved. Opens BLOCK J with a sharpened statement of what z's offset algebra can and
cannot do.

## Brier (frozen priors, PREREG_LITE, exact)
| pred | statement | prior P | outcome | (P−o)² |
|------|-----------|---------|---------|--------|
| P1 | marker-swap mean success > 0.6 | 0.55 | **TRUE** (1.00) | 0.2025 |
| P2 | arg-swap mean success < 0.4 (binding wall) | 0.78 | **TRUE** (0.00) | 0.0484 |
| P3 | controls tense/negation ≥ 0.9 | 0.90 | **TRUE** (1.00) | 0.0100 |
| P4 | winners (succ>0.6) MLP−linear zcos Δ < 0.05 | 0.55 | **TRUE** (−0.013) | 0.2025 |
| P5 | causal_reframe > causal_swap + 0.2 | 0.60 | **TRUE** (0.36 > 0.00+0.2) | 0.1600 |

**Brier = 0.1247** (5/5 directional hits; the loss is under-confidence — every call was correct but
P1/P4/P5 were hedged near 0.55–0.60 and P2 at 0.78, when the true separation was near-total).

## Limitations
- **T3 ceiling:** single embedder (SONAR z), single judge (local codex), templated minimal pairs
  with a fixed set of frames/entities — generalization beyond these templates untested.
- **causal group is the weak arm:** only 18 predicted items each (vs 28 for markers), and
  causal_reframe genuinely did not linearize (0.36) — P5 passing reflects the arg-swap floor being
  zero, not reframe being an operator. Do not over-read P5 as "paraphrase linearizes."
- **spatial_swap soft ceiling (0.80):** its 0/28 predicted success is a lower bound on the binding
  failure; the underlying z_after itself only 80% realizes operand reversal under greedy decode.
- **MLP under-fit vs z-cos:** MLP decode-success diverges from its high z-cos on some markers; the
  MLP is a *weak* non-linear control (small, weight-decayed, early-stopped on n≈32) — it rules out
  "linear is leaving accuracy on the table" via z-cos, not via an exhaustively-tuned network.
- Judge is lenient on "applied" partials; collateral (preserved) is the divergence check and it
  cleanly separates markers (~1.00) from causal_reframe (0.22).

## Tier: T3-exploratory (no promotion).

## Follow-up: **Y (modest).**
1. **Composition across relation classes** — does offset_temporal_marker ∘ offset_vertical apply
   both relation flips additively (as 033's tense∘number did)? Tests whether relational markers are
   commuting operators or interfere.
2. **A FIXED-pair argument swap** — 033-entity was null because generic; a fixed A↔B operand pair
   with a fixed marker might reveal whether the binding wall is about *generality* or about *role
   reorganization per se*. Sharpens the binding-block claim.
3. **causal_reframe decomposition** — separate the "because→so" lexical swap (should be linear, like
   a marker) from the clause-reordering (should fail, like an arg-swap) to confirm the reordering is
   the specific failure locus.