# Operator 04 — Spatialization / Scaling (Stage-0 template)

**Operator.** Spatialization / scaling: take a prime P and **add extent or scale**, proposing the
prime that is P's distributed / multi-scale / aggregate counterpart. Three sub-flavors, run as three
separate passes over the corpus (same payload, different instruction block):

- **4a · point→field** — a localized / pointwise prime becomes **distributed over a space** — a
  value at every point, with gradients, boundaries, and spatial correlation. (a value → a field;
  a single locus → a distribution over positions)
- **4b · discrete↔continuous** — flip **granularity** between discrete and continuous. (a discrete
  sum → its continuous limit/integral with densities and derivatives; a smooth quantity → its
  discretized / quantized counterpart)
- **4c · micro→macro (aggregation)** — a **micro-level** prime becomes its **aggregate / emergent
  macro** counterpart. (individual interactions → mean-field average; local rule → global pattern;
  per-item → per-unit-extent density)

The characteristic non-result is the **already-spatial / scale-free fixed point** (P is *already*
a field, spatial, or multi-scale prime, so adding extent returns P — diffusion, gradient,
fractal_geometry, network, scaling_and_scale_dependence, percolation, self_organization). Like
op-02/op-03 these are *audit signal* (they map which corpus primes are inherently spatial/multiscale),
persisted (§E, `audit_spatialization.jsonl`).

**What this template is for.** Stage-0 of the pipeline in `OPERATOR_SEARCH_RUNBOOK.md`: (A) crank
prompt, (B) collapse test, (C) few-shot, (D) pilot, (E) notes. Freeze §A. Inherits the op-01
v4.2 / op-02 v1.1 / op-03 machinery wholesale (step-0 alias verification, retrieval-grounded cheap
check, protection rule, deterministic slug check, INPUT-line contract + block-count reconciliation
against the canonical slug set, articulated-candidate rescue) and adds the **EXTENT rule** (§A),
the spatialization analog of op-02's LEVEL2 / op-03's DYNAMICS rule.

**Model tier (to be confirmed by pilot).** 4a/4b are mechanical axis-adds (like dualization/op-03
— haiku plausibly suffices); 4c (aggregation→emergence) leans toward the judgment op-05's lift
needed. Pilot runs **sonnet Stage 2 + a 15-prime haiku probe** and picks the cheaper model that
clears the gate. Per the op-02 keystone the Stage-2 verdict is advisory — the funnel gates on
step-0 + articulated-candidate rescue + Stage-4, not on Stage-2 NEW.

**Status 2026-07-03 (v1 — BUILT, pilot pending).** Calibration anchors verified against the frozen
1,325 corpus: `field` ABSENT (only receptive_field), `density`/`mean_field`/`continuum`/`power_law`/
`pattern_formation` ABSENT; PRESENT (already-spatial, fixed-point/alias targets): gradient,
diffusion, scaling_and_scale_dependence, allometry_and_scaling_law, fractal_geometry, network,
emergence, aggregation, local_to_global_aggregation, renormalization, coarsening, percolation,
self_organization, self_organized_criticality, edge_effect, boundary, dimension,
curse_of_dimensionality, non_locality, economies_of_scale. Pilot next (§D).

---

## A. The crank prompt (Stage 2 — cheapest capable model, batched; pilot decides haiku vs sonnet)

> You are applying a single, precise transformation called **SPATIALIZATION / SCALING** to entries
> from a catalog of "prime abstractions" — domain-general structural patterns of thought.
> Spatialization **adds extent or scale** to a prime and asks whether that produces a *distinct*
> distributed / multi-scale / aggregate prime. You are running sub-flavor
> **{4a point→field | 4b discrete↔continuous | 4c micro→macro aggregation}** — apply only that flavor.
>
> - **4a (point→field):** spread a localized/pointwise prime over a SPACE — a value at every point,
>   with gradients, boundaries/edges, and spatial correlation. (a single value → a field of values)
> - **4b (discrete↔continuous):** flip GRANULARITY — a discrete structure → its continuous limit
>   (densities, integrals, derivatives, a continuum with new properties), OR a continuous quantity →
>   its discretized/quantized counterpart. Name which direction.
> - **4c (micro→macro):** aggregate a MICRO-level prime into its MACRO/emergent counterpart — a
>   mean-field average, a global pattern from local rules, a per-unit-extent density, aggregate
>   statistics that behave unlike any individual.
>
> A valid spatialization output is itself a **prime**: domain-general, structurally distinct, with
> its own spatial/scale anatomy. It is NOT:
> - an **already-spatial / scale-free fixed point** — P is *already* a field, spatial, or
>   multi-scale prime, so adding extent returns P (diffusion is already a field process; a fractal
>   is already scale-invariant; a network is already spatial/relational). Output
>   `VERDICT: FIXED_POINT`. Real, useful result (flags P as inherently spatial/multiscale), NOT a
>   failure — report it honestly rather than inventing a fake distributed version.
> - **trivial "P but bigger / spread out / more of them"** — the same pattern at larger extent with
>   no new spatial/scale structure (no gradient, boundary, density, aggregation statistic, continuum
>   limit, or scale-dependence). (COLLAPSE — trivial_scale.)
> - a spatial/scale prime the catalog already names under a slug you can actually name (COLLAPSE —
>   alias; report the slug in NEAREST — your claim WILL BE VERIFIED, so never collapse on the
>   grounds that a slug "surely exists"; if you cannot name a specific slug, the verdict is NEW).
> - a narrow domain instance.
>
> **EXTENT RULE (load-bearing — the spatialization analog of a sibling test): a genuine spatial/scale
> prime must exhibit at least one SPATIAL or SCALE structure the input LACKS** — a field with
> gradients, a boundary/edge effect, spatial correlation/range, a density (per-unit-extent), an
> aggregation/mean-field statistic, a continuum limit with new operations, or a scale-dependence /
> scale-invariance. Name it concretely in the EXTENT line. If the only thing you can say is "P, but
> over a bigger space / with more instances," there is no new structure: output COLLAPSE with REASON
> "trivial — P at larger scale" (or FIXED_POINT if P is already spatial). Worked contrast: *a value
> → a field* is NEW because the field has gradients, boundaries, and spatial correlation the single
> value lacks; *fractal_geometry → fractal_geometry* is FIXED_POINT because a fractal is already
> scale-invariant.
>
> Think about the extent/scale axis and the new spatial structure, then decide. Bias toward COLLAPSE/
> FIXED_POINT: a false NEW is expensive downstream, a missed NEW is cheap. If you CAN articulate a
> substantive spatial/scale candidate, write it in CANDIDATE and prefer NEW — the frontier decides
> new-vs-trivial downstream; a blank CANDIDATE marks a true fixed point or trivial. For EACH input
> below, output EXACTLY this contract and nothing else:
>
> ```
> INPUT:     <slug, copied from the payload line>
> CANDIDATE: <one sentence, domain-stripped, describing the spatial/scale prime — or NONE>
> EXTENT:    <the spatial/scale structure present in the output but absent in the input — or NONE>
> NEAREST:   <slug of the closest existing prime you can name, or NONE>
> VERDICT:   NEW | FIXED_POINT | COLLAPSE
> REASON:    <=12 words>
> ```
>
> **Payload:** `P.slug` · `P.one_liner` · `P.core_idea` (~2 sentences) · `P.structural_signature`

**Batching.** ~44 payloads/batch, INPUT-line contract, one flavor per batch, instruction block
constant. Harness reconciles against the canonical slug set (not per-batch counts — haiku drops
2–8% and drifts to single-line format; parse flexibly + validate slugs + recover missing). Stage-2:
no tools/search — verified downstream.

**Few-shot (per flavor):**
- **4a:** solved exemplar A (point→field NEW) + counter-example `diffusion` (FIXED_POINT).
- **4b:** solved exemplar B (discrete→continuum NEW) + counter-example `fractal_geometry` (FIXED_POINT).
- **4c:** solved exemplar C (micro→macro alias/NEW: mean_field) + trivial counter-example.

---

## B. Collapse test (inherits op-02 v1.1 / op-03 machinery; spatialization specifics noted)

0. **Alias-claim verification (run FIRST, on every COLLAPSE/NEW naming a NEAREST).** Slug + alias
   map lookup, then one_liner embedding screen. Exists → alias COLLAPSE (keep as negative +
   balanced-granularity signal). Absent AND no prime within alias distance → clean-gap → Stage 4.
   Fuzzy/substring matches are menu-hints only.

1. **Already-spatial fixed-point collapse — OPERATOR-CHARACTERISTIC, retained as audit signal.**
   `VERDICT: FIXED_POINT` → append to `workspace/operator_search/audit_spatialization.jsonl`
   `{input, operator: "04_spatialization", flavor, source, reason}`. Maps the corpus's inherently
   spatial/multiscale primes (diffusion, gradient, fractal, network, scaling, percolation,
   self_organization, renormalization). Never routed to Stage 4.

2. **Alias collapse (spatial/scale version already in corpus).** As op-02 §B.2: lexical check, then
   retrieval-grounded cheap check — candidate + top-6 one_liner neighbors menu → `ALIAS_OF`
   (copied from menu) / `NOT_A_PRIME`. **Defanged labels, op-04 wording:**
   - `trivial_scale` = "P at larger extent / more instances" with no new spatial/scale structure
     (op-04's analog of op-02 `mere_iteration` / op-03 `trivial_time`).
   - `domain_instance` = one field's narrow spatial case of a general pattern.
   - **A genuine spatial prime whose base is the nearest corpus entry is NOT `trivial_scale` just
     because it references its base** — the discriminator is the EXTENT line, not retrieval proximity
     (op-05 §D / op-02 §D amendment — inherited).

   ```
   ITEM:        <number>
   ALIAS_OF:    <one slug copied from that item's menu, or NONE>
   NOT_A_PRIME: <trivial_scale | domain_instance | NONE>
   WHY:         <=10 words>
   ```

3. **Protection rule (v4/v4.1).** No kill (incl. Stage-2 self-collapse or FIXED_POINT) may retire a
   candidate's LAST live route if (a) proposed by ≥2 flavors/inputs this sweep, or (b) already in a
   `manifest_*` as accept/hold.

4. **Articulated-candidate rescue rule (v1.1 — carried forward).** Trust a `trivial_scale` COLLAPSE
   ONLY when `CANDIDATE: NONE` AND `EXTENT: NONE`. Substantive CANDIDATE stamped `trivial_scale`
   → route to Stage 4 regardless (discriminator is the EXTENT line). `FIXED_POINT` with
   `CANDIDATE: NONE` stays a fixed point.

5. **Deterministic slug-existence check (v4.2)** on every SLUG the Stage-4 pre-adjudicator proposes,
   before frontier review. Semantic search verifies concept-absence; lexical lookup verifies
   name-absence — both required.

**Calibration target (pilot, corpus-state-aware). Zero silent deaths.**
- a pointwise/value prime (4a) → `field` verified ABSENT → must surface NEW or step-0 clean-gap,
  reach Stage 4.
- `aggregation`/`emergence` (4c) → must resolve as verified alias/FIXED_POINT (already the macro
  primes) — a PASS + balanced-granularity confirmation.
- `fractal_geometry`, `diffusion`, `scaling_and_scale_dependence`, `network`, `percolation`,
  `self_organization` → must land FIXED_POINT (audit rows), NOT be forced into fake spatial primes.

---

## C. Few-shot calibration set (v1)

Solved exemplars go in the §A prompt per flavor; score corpus-state-aware.

**Exemplar A — flavor 4a (point→field), NEW — SOLVED (few-shot):**
```
INPUT:     a single scalar value at one location (illustrative)
CANDIDATE: A quantity assigned to every point of a space, so that it varies from place to place — with gradients, boundaries, and spatial correlation.
EXTENT:    a value defined over a whole domain (a field), with spatial gradient and boundary structure absent from a single point value
NEAREST:   NONE
VERDICT:   NEW
REASON:    point→field; a general "field" prime is absent from the corpus
```

**Exemplar B — flavor 4b (discrete→continuous), NEW — SOLVED (few-shot):**
```
INPUT:     a discrete sum over countable items (illustrative)
CANDIDATE: The limit in which countable items become a continuum, so sums become integrals and counts become densities with derivatives.
EXTENT:    a continuum limit with densities, integrals, and derivatives that a finite discrete structure lacks
NEAREST:   NONE
VERDICT:   NEW
REASON:    discrete→continuum limit; the continuum-limit prime is absent
```

**Exemplar C — flavor 4c (micro→macro), NEW — SOLVED (few-shot):**
```
INPUT:     many individually-interacting units (illustrative)
CANDIDATE: Replace each unit's detailed interactions with the AVERAGE effect of all the others, so the collective is described by a single self-consistent aggregate field.
EXTENT:    a mean-field aggregate that behaves unlike any individual unit, absent at the micro level
NEAREST:   NONE
VERDICT:   NEW
REASON:    micro→macro mean-field; the mean-field prime is absent
```

**Exemplar D — FIXED_POINT (already spatial) — SOLVED (few-shot; flavor-matched):**
```
INPUT:     fractal_geometry · "self-similar structure repeating across scales" · [scale-invariance]
CANDIDATE: NONE
EXTENT:    NONE
NEAREST:   fractal_geometry
VERDICT:   FIXED_POINT
REASON:    a fractal is already scale-invariant — adding scale returns it
```

**Exemplar E — trivial_scale COLLAPSE (counter-example) — SOLVED (few-shot):**
```
INPUT:     symmetry · "invariance of a structure under a transformation" · [invariance under group action]
CANDIDATE: NONE
EXTENT:    NONE
NEAREST:   NONE
VERDICT:   COLLAPSE
REASON:    "symmetry over a bigger region" adds no spatial structure — trivial
```

**Holdout — flavor 4c — SCORE IT (NOT in prompt):**
```
INPUT:     aggregation (illustrative micro→macro)
EXPECTED ≈ resolves as a verified alias/FIXED_POINT (aggregation / local_to_global_aggregation /
emergence already name the macro side). A PASS + balanced-granularity confirmation.
```

---

## D. Pilot procedure (v1)

1. **Sample.** Frozen `../pilot_sample.jsonl` (53 rows). Contains spatialization-relevant statics:
   discreteness, degrees_of_freedom, coupling, distribution, markov_process, compositionality.
2. **Stage 2 on sonnet** (3 flavors × the sample, §A prompt + flavor-matched few-shot) + a **15-prime
   haiku probe** on one flavor — the tier-decision datapoint.
3. **Stage 3–4 with the §B machinery:** step-0 → lexical → retrieval-grounded cheap check → protection
   → articulated-candidate rescue → deterministic slug check → sonnet Stage-4 (STRONG-only, §D
   amendment) → frontier shortlist.
4. **Fixed-point handling:** every FIXED_POINT → audit row (§B.1). Report the count.
5. **Score calibration targets corpus-state-aware** (§B). Record NEW/FIXED_POINT rates, flavor split,
   batch variance, step-0 reclassification, survivor rate, cost, and the haiku-vs-sonnet delta.
6. **Decision gate (inherits op-01 v4):** zero silent deaths + acceptable cost-per-accepted-prime.
7. On gate pass, choose the sweep model (haiku if the probe matches sonnet's fixed-point/trivial
   discrimination) and run the full 1,325 sweep (payloads = `../worklist_dualization.jsonl`, re-tagged
   `operator: 04_spatialization`). Corpus frozen 1,325; finds accumulate as manifest rows + stubs.

**§D AMENDMENT (mandatory for op-04 Stage 4 — inherited).** When the spatial counterpart is absent,
the input is ALWAYS the nearest corpus entry to its own scaled version — so "the input is the top
search hit" is NOT evidence of trivial_scale. The discriminator is the EXTENT line. State this in
the Stage-4 prompt or sonnet blanket-zeros genuine finds (op-05 fallacy).

---

## E. Notes & watch-items

- **Fixed points are audit signal.** Op-04's fixed points map the corpus's inherently-spatial/
  multiscale primes. Persist to `audit_spatialization.jsonl`. Expect a large population — the corpus
  is already rich in spatial/scale primes (companion §5: the structural core is nearly counted).
- **Overlap with generalization (op 5) — 4c especially.** micro→macro aggregation is close to op-05's
  lift and to emergence-detection; deterministic→stochastic borders op-6. Convergence across
  operators is *signal*; Stage 3 merges duplicates. Log the overlap; don't force a boundary.
- **Flavor profile expectation.** 4a (point→field) likely carries the most discovery signal (a
  general `field` prime is absent); 4b (discrete↔continuous) medium (continuum-limit / quantization
  primes may be absent); 4c (micro→macro) high alias/FIXED_POINT rate (aggregation, emergence,
  mean-field-adjacent, renormalization already present — but mean_field itself absent).
- **Framed primes give noisier spatializations.** Expect higher COLLAPSE in the framed stratum.
- **Second op-02 keystone carries over:** at Stage 4, rank survivors by cross-input/flavor
  convergence + canonical-name presence; auto-Tier-B the single-route un-named coinages (op-03 had
  ~1,100 "rate-of-X" coinages; op-04 will have ~"bigger-P / field-of-P" coinages).
