# Operator 14 — Lattice / Symmetry Completion (Stage-0 template)

**Operator.** Catalog-internal. In a natural PRODUCT SPACE ({A|B} × {C|D} → 4 cells) or a small
N-MEMBER TYPOLOGY FAMILY (a closed, mutually-exclusive, jointly-exhaustive set), find the EMPTY
cells — positions that should hold a prime but don't — and name the missing prime. Mendeleev-style
gap prediction. NOT a per-pair sweep: a single frontier structural-reasoning pass, then verify.

**Method (2026-07-04).** (1) Seed a strong frontier model (opus) with the corpus's explicit
oppositional dimensions (the 23 `_vs_` primes) + known typology families (ethics triad, inference
triad, regress responses, alethic modality, sender/receiver, governance modes). (2) Ask it to build
8–15 product spaces / families, show which cells are filled by which existing primes, and name the
EMPTY cell's missing prime — STRONGLY preferring k-1-of-k gaps, REJECTING vacuous / non-orthogonal /
domain-specific cells. (3) Deterministically existence-check every candidate against the corpus.
(4) Stub the genuine absences; note near-misses; record self-rejections.

**Discriminator.** A valid completion is a k-1-of-k gap in a GENUINE closed family, where the missing
cell is the exact dual/complement of present members and has a canonical name. Reject: vacuous cells
(the "true" corners of a confusion matrix), non-orthogonal 2×2s (feedback×sign), domain-specific
cells, and "missing" cells already filled (existence-check catches these).

**Result (2026-07-04).** 15 candidates → **5 genuine** (infinitism + cheap_talk STRONG;
common_pool_resource, positional_good, type_iii_error MODERATE) + impossibility (noted for the
alethic family) + heterarchy (already an op-01 stub — convergence). Frontier correctly self-rejected
certification / analogy / punctuated_equilibrium / comparative_advantage (present) and the vacuous
cells. **Op-14 is the 2nd n-ary operator with genuine yield (after op-11).** `manifest_op14_lattice.yaml`.

**Notes.** Op-14's finds cluster in the FRAMED/social-science families (governance, welfare,
epistemology, game theory) — the corpus's structural core is lattice-complete, but its framed
families have a few predictable empty cells (same yield region as op-05's staircase). The op-11
regress-response gap (infinitism) was the seed that motivated running op-14; op-14 confirmed and
extended it. Re-runnable: re-seed with any new near-complete family the corpus surfaces.
