# Operator 08 — Threshold / Phase (Stage-0 template)

**Operator.** Threshold / phase: take a prime P and find its **critical-point** version — the prime
in which a smoothly-varying quantity crosses a threshold and triggers an abrupt, qualitatively
different regime. Two sub-flavors, run as two separate passes over the corpus:

- **8a · gradual→tipping** — a smooth / continuous / gradual process → its **threshold / tipping /
  phase-transition** version, where crossing a critical value flips the system to a new qualitative
  regime (often with hysteresis / irreversibility). (a gradual accumulation → the critical point at
  which it snaps)
- **8b · continuity→discreteness** — a continuous structure → its **discretized / quantized** version
  with distinct separated levels. **⚠ HEAVY OVERLAP: this is essentially op-04's flavor 4b
  (discrete↔continuous), already run at full corpus.** Expect 8b to alias almost entirely onto
  op-04's finds (`continuum_limit` stub, `discreteness`, `discrete_vs_continuous_quantization`) and
  the corpus's existing discrete/continuous primes. Run it as a light consistency check, not a
  discovery pass; if the pilot confirms pure redundancy, drop 8b from the sweep.

The characteristic non-result is the **already-critical fixed point** (P is *already* a threshold /
phase / critical prime, so criticalizing returns P). **The threshold/phase region is one of the
MOST-SATURATED areas of the corpus** — tipping_points_or_phase_transitions, threshold,
percolation_threshold, critical_mass, critical_juncture, nucleation, self_organized_criticality,
hysteresis, metastability, symmetry_breaking, regime_change, punctuated_equilibrium, cascade,
supersaturation, continuity_vs_rupture are all present. So op-08 is expected to be **audit/alias-heavy
with a very small genuine yield** (possibly near-zero, like op-13). Candidate gaps identified at
build: `bifurcation`, `catastrophe` (catastrophe theory), `order_parameter`. Fixed points persist to
`audit_threshold.jsonl`.

**What this template is for.** Stage-0 of the pipeline. (A) crank prompt, (B) collapse test, (C)
few-shot, (D) pilot, (E) notes. Inherits op-01 v4.2 / op-02 v1.1 / op-03/04/06 machinery wholesale
(step-0 alias verification, retrieval-grounded cheap check, protection rule, deterministic slug
check, INPUT-line contract + reconciliation against the canonical slug set, articulated-candidate
rescue) and adds the **CRITICAL rule** (§A), the threshold analog of op-06's MODAL / op-04's EXTENT.

**Model tier (to be confirmed by pilot).** Threshold detection (is P already critical? does a genuine
phase-transition structure emerge?) needs judgment — likely **sonnet** like op-04/06 (haiku failed
op-04 5/15 and op-06 7/15 on judgment-heavy axes). Pilot runs sonnet Stage 2 + a 15-prime haiku
probe. Per the op-02 keystone the Stage-2 verdict is advisory.

**Status 2026-07-03 (v1 — BUILT, pilot pending).** Given the heavy corpus saturation + 8b/op-04
redundancy, the pilot doubles as a **saturation check**: if yield ≈ 0 (like op-13), retire op-08
without a full sweep. Candidate gaps to watch: `bifurcation`, `catastrophe`, `order_parameter`.

---

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

> You are applying a single, precise transformation called **THRESHOLD / PHASE** to entries from a
> catalog of "prime abstractions" — domain-general structural patterns of thought. This operator
> finds the **critical-point** version of a prime: where a smoothly-varying quantity crosses a
> threshold and triggers an abrupt, qualitatively different regime. You are running sub-flavor
> **{8a gradual→tipping | 8b continuity→discreteness}** — apply only that flavor.
>
> - **8a (gradual→tipping):** turn a smooth/gradual process into its THRESHOLD version — a critical
>   value whose crossing flips the system to a new qualitative regime (often with hysteresis or
>   irreversibility). (a gradual buildup → the tipping point at which it snaps)
> - **8b (continuity→discreteness):** turn a continuous structure into its DISCRETIZED / QUANTIZED
>   version with distinct separated levels. (a smooth quantity → quantized levels)
>
> A valid output is itself a **prime**: domain-general, structurally distinct, with its own
> threshold/phase anatomy. It is NOT:
> - an **already-critical fixed point** — P is *already* a threshold/phase/critical prime, so
>   criticalizing returns P (tipping_points is already a phase transition; percolation_threshold is
>   already a critical point; symmetry_breaking, nucleation, regime_change are already critical).
>   Output `VERDICT: FIXED_POINT`. Real, useful result, NOT a failure — report it honestly.
> - **trivial "P but with a threshold / P but quantized"** — a threshold word attached with no new
>   critical structure (no critical value, no qualitative regime change, no order parameter,
>   no quantized levels) (COLLAPSE — trivial_threshold).
> - a threshold/phase prime the catalog already names under a slug you can actually name (COLLAPSE —
>   alias; report the slug in NEAREST — your claim WILL BE VERIFIED; if you cannot name a specific
>   slug, the verdict is NEW). The catalog ALREADY HAS: tipping_points_or_phase_transitions,
>   threshold, percolation_threshold, critical_mass, critical_juncture, nucleation,
>   self_organized_criticality, hysteresis, metastability, symmetry_breaking, regime_change,
>   punctuated_equilibrium, cascade, supersaturation, discreteness, continuity,
>   discrete_vs_continuous_quantization, continuity_vs_rupture — so most inputs will alias or be
>   fixed points. Bias hard toward COLLAPSE/FIXED_POINT here.
> - a narrow domain instance.
>
> **CRITICAL RULE (load-bearing — the threshold analog of a sibling test): a genuine threshold/phase
> prime must exhibit a CRITICAL structure the input LACKS** — a critical value/threshold with a
> qualitative regime change across it, an order parameter, a bifurcation of the solution set, a
> catastrophe (discontinuous jump from smooth control), or quantized discrete levels. Name it
> concretely in the CRITICAL line. If the only thing you can say is "P, but past some threshold," and
> that threshold structure is already named by tipping_points/threshold/etc., output COLLAPSE (alias
> or trivial). Worked contrast: *a smooth control parameter → the bifurcation where the number of
> stable states changes* is NEW (bifurcation is absent); *gradual pressure → tipping_points* is an
> alias COLLAPSE (already named).
>
> Think about the critical structure, then decide. Bias HARD toward COLLAPSE/FIXED_POINT (this region
> is heavily covered). If you CAN articulate a substantive critical candidate NOT already named,
> write it in CANDIDATE and prefer NEW — the frontier decides 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 threshold/phase prime — or NONE>
> CRITICAL:  <the critical 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 constant.
Reconcile against the canonical slug set. Stage-2: no tools/search — verified downstream.

**Few-shot (per flavor):**
- **8a:** solved exemplar A (gradual→bifurcation NEW) + counter-example `tipping_points_or_phase_transitions` (FIXED_POINT) + alias counter-example (gradual pressure → tipping_points).
- **8b:** solved exemplar B (continuity→quantization alias, → discrete_vs_continuous_quantization) + `discreteness` (FIXED_POINT).

---

## B. Collapse test (inherits op-02 v1.1 / op-03/04/06 machinery; threshold specifics noted)

0. **Alias-claim verification (run FIRST).** Slug + alias map lookup, then one_liner embedding
   screen. Exists → alias COLLAPSE. Absent AND no prime within alias distance → clean-gap → Stage 4.
   **For 8b, also check op-04's stubbed finds** (`continuum_limit`, and the op-04 manifest's spatial
   set) as alias targets — 8b is expected to duplicate op-04 4b.

1. **Already-critical fixed-point collapse — audit signal.** `VERDICT: FIXED_POINT` → append to
   `workspace/operator_search/audit_threshold.jsonl` `{input, operator: "08_threshold", flavor,
   source, reason}`. Maps the corpus's inherently-critical primes.

2. **Alias collapse (threshold version already in corpus).** As op-02 §B.2: lexical check, then
   retrieval-grounded cheap check. **Defanged labels, op-08 wording:**
   - `trivial_threshold` = "P + a threshold word" with no new critical structure (op-08's analog).
   - `domain_instance` = one field's narrow critical case of a general pattern.
   - A genuine critical prime whose base is the nearest corpus entry is NOT `trivial_threshold` just
     because it references its base — the discriminator is the CRITICAL line (§D amendment).

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

3. **Protection rule (v4/v4.1).** No kill may retire a candidate's LAST live route if (a) ≥2
   flavors/inputs converge, or (b) already in a `manifest_*` as accept/hold.

4. **Articulated-candidate rescue (v1.1).** Trust a `trivial_threshold` COLLAPSE ONLY when
   `CANDIDATE: NONE` AND `CRITICAL: NONE`. Substantive CANDIDATE stamped trivial → Stage 4.
   `FIXED_POINT` with `CANDIDATE: NONE` stays a fixed point.

5. **Deterministic slug-existence check (v4.2)** on every Stage-4 proposed SLUG before frontier review.

**Calibration target (pilot, corpus-state-aware). Zero silent deaths.**
- a gradual/smooth prime (8a) → `bifurcation` verified ABSENT → must surface NEW or clean-gap, reach
  Stage 4.
- `tipping_points_or_phase_transitions`, `percolation`, `symmetry_breaking`, `nucleation`,
  `metastability` → must land FIXED_POINT (audit rows).
- gradual→tipping on a non-critical prime whose tipping version is `tipping_points` → verified alias
  COLLAPSE (balanced-granularity confirmation).

---

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

**Exemplar A — flavor 8a (gradual→tipping), NEW — SOLVED (few-shot):**
```
INPUT:     a smooth control parameter driving a system (illustrative)
CANDIDATE: The critical parameter value at which the NUMBER or stability of a system's equilibria changes — a fork where one stable state splits into several (or vanishes).
CRITICAL:  a qualitative change in the SOLUTION SET (number/stability of fixed points) as a parameter crosses a critical value — absent from smooth variation
NEAREST:   NONE
VERDICT:   NEW
REASON:    bifurcation; the fork-in-the-solution-set prime is absent (tipping_points names the jump, not the fork)
```

**Exemplar B — flavor 8a (alias) — SOLVED (few-shot):**
```
INPUT:     a gradual accumulation of pressure (illustrative)
CANDIDATE: NONE
CRITICAL:  NONE
NEAREST:   tipping_points_or_phase_transitions
VERDICT:   COLLAPSE
REASON:    gradual→tipping already named by tipping_points_or_phase_transitions (alias)
```

**Exemplar C — FIXED_POINT (already critical) — SOLVED (few-shot):**
```
INPUT:     tipping_points_or_phase_transitions · "a gradual change crosses a threshold into an abrupt new regime" · [control parameter + threshold + regime change]
CANDIDATE: NONE
CRITICAL:  NONE
NEAREST:   tipping_points_or_phase_transitions
VERDICT:   FIXED_POINT
REASON:    already a phase transition — criticalizing returns it
```

**Exemplar D — flavor 8b (alias, op-04 overlap) — SOLVED (few-shot for 8b):**
```
INPUT:     a continuous quantity (illustrative)
CANDIDATE: NONE
CRITICAL:  NONE
NEAREST:   discrete_vs_continuous_quantization
VERDICT:   COLLAPSE
REASON:    continuity→discreteness already named (discrete_vs_continuous_quantization / discreteness / op-04 continuum_limit)
```

**Counter-example (trivial) — SOLVED:**
```
INPUT:     symmetry · "invariance under a transformation" · [group action]
CANDIDATE: NONE
CRITICAL:  NONE
NEAREST:   symmetry_breaking
VERDICT:   COLLAPSE
REASON:    symmetry's critical version is symmetry_breaking, already in corpus (alias)
```

---

## D. Pilot procedure (v1) — DOUBLES AS A SATURATION CHECK

1. **Sample.** Frozen `../pilot_sample.jsonl` (53 rows). Contains threshold-relevant statics:
   tipping_points_or_phase_transitions (fixed-point check), instability, nonlinearity, discreteness,
   perturbation_theory, half_life.
2. **Stage 2 on sonnet** (2 flavors × the sample) + a **15-prime haiku probe** on 8a.
3. **Stage 3–4** with the §B machinery; sonnet Stage-4 STRONG-only (§D amendment), frontier shortlist.
4. **Fixed-point handling:** every FIXED_POINT → audit row. Report the count.
5. **SATURATION CHECK:** if Stage-4 yields ≈0 genuine domain-general holds AND 8b is pure op-04
   redundancy → **retire op-08 without a full sweep** (record as saturated, like op-13). If ≥1
   genuine hold (e.g. bifurcation), run 8a at full corpus; keep 8b only if it shows non-redundant yield.
6. **Decision gate (inherits op-01 v4):** zero silent deaths + acceptable cost. Report to owner with
   the saturation finding for a sweep-or-retire scope decision.

**§D AMENDMENT (mandatory for op-08 Stage 4 — inherited).** When the critical counterpart is absent,
the input is the nearest corpus entry — NOT evidence of trivial. Discriminator is the CRITICAL line.

---

## E. Notes & watch-items

- **Op-08 is expected to be near-saturated.** The threshold/phase region is heavily covered; treat
  op-08 primarily as an AUDIT + a saturation checkpoint. A near-zero yield is the LIKELY outcome and
  is itself a completeness result (the corpus's critical-transition layer is essentially complete).
- **8b duplicates op-04 4b.** Do not double-count; alias 8b candidates onto op-04 finds + existing
  discrete/continuous primes. Drop 8b from the sweep if the pilot confirms redundancy.
- **Candidate gaps (verify at Stage 4):** `bifurcation` (fork in the solution set — distinct from
  tipping_points which names the jump), `catastrophe` (catastrophe theory — discontinuous jump from
  smooth control, cusp/fold), `order_parameter` (the quantity that becomes non-zero past a phase
  transition). These are the only plausible non-alias finds.
- **Overlap with op-03 (critical_slowing_down, already stubbed).** critical_slowing_down is the
  PRE-transition dynamic; op-08 8a is the transition itself. Screen against it.
- **Domain-specific finds get stubs too** (standing convention) — but expect few/none here.
- **Second op-02 keystone:** rank Stage-4 survivors by convergence + canonical-name; auto-Tier-B the
  single-route "threshold-of-P" coinages.
