Operator-Driven Discovery of Prime Abstractions¶

A saturating search over concept-space, its yield law, and what it found¶

A companion to The Operator Compendium, which treats each of the fourteen discovery operators in depth. Sibling to From Candidate to Catalog, the procedure by which a discovered candidate becomes an accepted catalog entry.

Abstract¶

The Encyclopedia of Abstractions catalogs prime abstractions — domain-general structural patterns of thought, such as feedback, intersection, and path dependence, that recur across unrelated fields. This paper reports a way to grow that catalog by treating discovery as a search over concept-space: the states are primes and the moves are a small, closed set of operators, deterministic transformations that take an existing prime (or a tuple of them) and propose a new one — dualization flips a prime's polarity, generalization strips a constraint to reach its genus, analogy completion extrapolates a fourth term. Because the move-set is closed and the corpus finite, applying every operator to every prime is a finite, checkable job, and completeness becomes something one can measure rather than assert. We built a cheap-to-expensive screening funnel and ran all fourteen operators over a 1,325-prime corpus — on the order of 20,000 applications — surfacing 127 candidate new primes (several of them conspicuous gaps, such as consequentialism absent beside virtue ethics), along with two byproducts that may be worth as much: roughly 295 candidate hierarchy edges and a body of hard negatives for training analogical reasoning. The central result is a yield law: operators that merely recombine existing primes return nothing a mature catalog does not already imply, while only the moves that reach genuinely new structural positions — cross-domain analogy and symmetry-completion — keep paying. Acceptance counts are provisional pending final human curation.

Data & reproducibility. The raw material behind every quantitative claim below — the per-application model outputs for all fourteen operators (from which every NEW-rate, fixed-point census, and convergence count can be independently recomputed), the per-operator result ledgers, the fixed-point/audit censuses, the 128 candidate primes, the ~295 candidate hierarchy edges, the retained negatives, and the exact Stage-0 prompt for each operator — is published as a single bundle: Operator-driven prime discovery — raw data (~2.3 MB). A README inside maps each file to the paper claim it substantiates. See also Appendix B.

1. Discovery as search over concept-space¶

Every catalog of ideas faces the same quiet question: how do you know what you're missing? A catalog built by reading can only report what someone happened to write down. If no paper was ever titled "the temporal version of equilibrium," the temporal version of equilibrium will not be in your catalog — and, worse, you will have no principled way to notice its absence. Mining misses things at random: whatever you did not read. That randomness is the deep problem, because a blind spot you cannot characterize is a blind spot you cannot close.

The reframe that organizes this whole program is to stop treating discovery as reading and start treating it as search. A graph search needs two things: a frontier of states already reached, and a move function that expands a state into its neighbors. Make the states primes, make the moves a fixed set of operators, and let the visited-set be the existing corpus plus its similarity and alias machinery. Discovery becomes node expansion: apply the moves to the frontier, screen the outputs against what you already hold, keep what is genuinely new, repeat.

Three consequences follow, and they are why the reframe is not merely decorative. First, a closed move-set makes the search enumerable. If the operators are a finite fixed set and the corpus is finite, then "apply every operator to every prime" is a finite job with a definite end; you can state precisely what you have and have not tried. Second, a visited-set makes novelty cheaply decidable. The catalog already carries structural embeddings and an alias map; in search terms that is the closed list, and it can tell you whether an expanded node is new without a human looking — which is what lets the crank run mostly unattended. Third, and the payoff, a closed move-set makes a completeness claim possible. You can never argue "the catalog is near-complete" from mining, because you never know what you failed to read; but you can argue it from search — if every operator, run to saturation, stops producing new accepted primes, the reachable region of concept-space is substantially covered. Completeness stops being a boast and becomes a measurement. The whole point of running all fourteen operators, rather than the few that were obviously productive, was to earn exactly this claim.

Why is this feasible now? Because graph search needs a frontier to expand and a function that expands it, and both exist at scale for the first time. The frontier is a large, typed, embedded corpus of primes. The expander is a language model that can actually execute the instruction "apply temporalization to equilibrium" and return a coherent candidate. Neither existed until recently. The rest of this paper is what happened when we pointed that machinery at our own catalog — including the several ways it misbehaved.

2. The operator set, and why it is plausibly closed¶

An operator moves a structure along one axis of variation. The claim that the operator set is complete rests not on brainstorming transformations but on enumerating the axes a structure can move along at all. A single structure can be moved along its polarity (dualization), its position in time (temporalization), its scale or extent (spatialization), its level of abstraction (generalization), its modal status (modalization), the viewpoint it is read from (perspective inversion), its linearity (threshold/phase), and its relation to itself (self-application). Structures combine by conjunction (composition), by opposition (tension-resolution), by proportion (analogy completion), and by infilling (interpolation). And a collection of structures reveals absences in exactly two ways: a missing common ancestor (cluster-without-parent) and a missing cell in a product space (lattice/symmetry completion).

That is the whole set — fourteen operators, catalogued with their moves, collapse modes, and yields in Appendix A. The test for closure is simple to state: if something seems missing, it should turn out to be a sub-flavor of one of these axes rather than a genuinely new axis. Dualization, for instance, has three sub-flavors — logical negation, opposing-force, failure-mode — that hit different targets, but they are one operator because they all move along the single axis of polarity.

The axis framing is not tidiness; it is the reason the search has any claim to completeness. An axis you never sweep is a systematic blind spot, not a random one. Mining misses primes haphazardly; an unswept operator misses them structurally — every temporal variant of every static prime, all at once, in a characterizable clump. Naming the axes converts unknown-unknowns into a checklist of known-unknowns. And it makes a prediction that §5 confirms: the catalog's own history is, in retrospect, an operator log run reactively and incompletely — its richest past additions came from generalization (finding the genus that unified a family) and dualization (naming a prime whose opposite pole had none). The program in this paper ran that proven machinery deliberately and to exhaustion, and opened the axes — temporal, spatial, modal, and the combinatorial ones — that had never been run at all.

A note on what counts as a "prime," since the finds cannot be judged without it. A prime must clear a four-part bar: it is domain-general (not one field's narrow concept), structurally distinct (it has its own load-bearing anatomy, not a rewording of a neighbor), not an alias of an existing prime, and not a mere domain instance of a more general one. A secondary axis matters throughout: a prime is structural to the extent its content is a bare relational form that travels with no home vocabulary (feedback, symmetry), and framed to the extent its content is bound to a human practice or institution (regulatory capture, moral panic). The structural core is a nearly-closed combinatorial space; the framed layer is open and ever-growing. That distinction, it turns out, predicts where the operators pay.

3. The pipeline, and the method that had to be rebuilt to make it work¶

The economics of the search come entirely from the funnel's shape: most operator applications fail — they return the input unchanged, an existing prime, or something that is not a real abstraction — so the design spends almost nothing deciding the failures and reserves expensive judgment for the survivors. The five stages: Stage 0, formalize each operator once (a frozen prompt template with its collapse test and few-shot); Stage 1, enumerate applications for free; Stage 2, the cheap crank (one terse generation per application, a fixed NEW/COLLAPSE contract, heavily batched); Stage 3, screen (deterministic slug/ancestor checks, then a retrieval-grounded alias check); Stage 4, frontier adjudication of survivors against the prime bar; Stage 5, human curation. Two commitments give it character: never spend a Stage-4 token on a candidate a Stage-3 screen could have killed, and collapses are retained as signal — a self-application that returns its input flags a maximally-general prime; a dualization that lands on an existing prime confirms the catalog is balanced there; an alias collision is a hard negative for training. The same crank is a discovery engine, a corpus audit, and a negative-example generator at once.

That is the clean version. The honest version is that the funnel above is the product of six operators' worth of painful iteration, and the lessons it encodes are the most transferable content in this paper. We report them as the method, because anyone building a similar system would otherwise have to pay to rediscover them.

The Stage-2 verdict is untrustworthy in both directions, so the funnel must never gate on it. This is the single most important operational finding. Cheap models over-produce: haiku emitted vacuous "meta-slogans" for self-application and spurious "rate-of-X" coinages for temporalization. Capable models over-collapse: sonnet stamped genuine second-order primes "mere iteration" and genuine genera "restatement." Neither the NEW nor the COLLAPSE verdict is reliable alone. The funnel therefore gates on three model-agnostic mechanisms instead. First, step-0 alias verification: a claimed "already in the corpus" slug is looked up deterministically, and if it does not exist the candidate is reclassified as a clean gap rather than trusted as a collapse — the model named a pole so obvious it assumed the catalog had it. (In the dualization pilot, 54 of 74 such "already exists" claims named a slug that did not exist.) Second, the articulated-candidate rescue rule: a COLLAPSE labeled "trivial" is trusted only when the model wrote no candidate; if it articulated a real structure and then stamped it trivial, it goes to Stage 4 regardless. Third, the protection rule: no cheap check may retire the last live route of a candidate that either converges from two or more inputs/flavors or is already a known hold. Convergence and prior adjudication are the trustworthy cheap signals; single-sample cheap-model judgments are not, and no amount of prompt wording repairs that — we tried, across four template revisions, and the structural protection rule is what finally delivered zero silent deaths of known-good candidates.

The retrieval-proximity fallacy recurs on every "add-an-axis-to-P" operator. When a transformed prime is genuinely absent, the input itself is always its nearest existing neighbor — because the input is the transform's own most-similar relative. A model reads that proximity as evidence of restatement and blanket-rejects the find. It first appeared in generalization, where a frontier adjudicator zeroed all 35 survivors on its first pass, ~40% of them with the rationale "when I search for this genus, the input is the top hit, so this is just a restatement." It then reappeared, in costume, on every subsequent axis operator: "mere iteration" (self-application), "trivial over time" (temporalization), "trivial at larger scale" (spatialization), "trivial modal word attached" (modalization). The fix is a standing amendment in every Stage-4 prompt: referencing the base is exactly what a valid transformation does; judge on the added structure, not on retrieval rank. But once told not to reject on proximity, the adjudicator swings the other way and grants a hold to nearly every mechanical "X-of-P" coinage — so the real discriminator, the one that survived all the tuning, is cross-input convergence plus the presence of a canonical name (see §5.3).

Two smaller lessons round out the method. Sub-agents silently drop 2–8% of their assigned work and drift to malformed output; every sweep needed a reconciliation-and-recovery pass against the canonical 1,325-slug set, and an unreconciled sweep is an incomplete sweep that looks complete. And the gate that matters is not survivor-rate: we retired the original "≤5% of applications reach Stage 4" target as unreachable given cheap-model reliability, and re-based the go/no-go on two things that actually track quality — zero silent deaths of known-good candidates, and acceptable frontier cost per accepted prime. Roughly 30–45% of applications reach Stage 4; that is the accepted price of never silently killing a real find.

4. What we ran¶

All fourteen operators have been run: the eight unary operators and the graph operator cluster-without-parent in a first phase, then the five combinatorial operators. The order within each phase was chosen by expected value, not by number. Appendix A is the full catalog; this section gives the shape, and §5 the results.

The unary operators (dualization, self-application, temporalization, spatialization, generalization, modalization, perspective inversion, threshold/phase) each apply to every prime in the corpus. Some ran on the cheapest capable model, some needed a stronger one, and the choice was predictable from the operator's cognitive demand — a point we return to. Several were retired on pilot evidence without a full sweep, because a near-zero pilot yield is itself a completeness result about a region of concept-space rather than a failure (perspective inversion and threshold/phase were retired this way; more in §5).

The combinatorial operators (composition, tension-resolution, analogy completion, interpolation, lattice completion) do not brute-force the pair space — 1,325² is 1.76 million pairs — but run on a pre-filtered candidate set computed deterministically from the hierarchy, the archetype co-occurrence data, or the embedding neighborhoods. Interpolation runs on the large-gap hierarchy edges; analogy completion on operator-confirmed pairs extended to a fourth term; lattice completion on natural product spaces rather than pairs at all. The pre-filter is what keeps the combinatorial phase affordable — the entire n-ary phase cost only a few million tokens.

The cluster-without-parent operator is catalog-internal: it needs no per-prime crank, but clusters the whole embedded corpus at once and asks which clusters have no common ancestor.

Two operator narratives are worth telling in full because they built the method the rest inherited. Dualization took four template revisions before a sweep was safe: the first pilot passed 27% of applications to the expensive stage (against a 1–5% target) because of the hallucinated-existence problem and an embedding alias screen that simply did not separate true aliases from novel candidates in any embedding space we tried; the third revision failed its gate outright when a "restatement" label silently killed three known-good candidates; and the durable fix was the protection rule, not any wording. Generalization exposed a capability floor: the cheap model produced zero genuine genera in 53 applications, paraphrasing its input instead of abstracting over it, because the lift move requires holding a prime and its prospective siblings in mind at once. It cleared the floor only after moving to a stronger model and adding a sibling-evidence rule — a proposed genus is invalid unless the model can name two or more instances of it that are structurally unlike the input. That single rule, plus the retrieval-proximity amendment, is what turned generalization from a zero into the program's highest-yielding operator.

5. Results¶

Counts of surfaced candidates are firm; counts of accepted primes are provisional pending curation and are marked so. The corpus is frozen at 1,325 primes for the duration of the search; nothing below has been incorporated yet. The full per-operator ledger is Appendix A; this section gives the load-bearing results.

5.1 The headline numbers¶

Across all fourteen operators: on the order of 20,000 operator applications, roughly 33 million model tokens, 127 candidate primes surfaced and drafted to a stub standard (each with a candid keep/cut note), and roughly 295 candidate hierarchy edges harvested as a byproduct. Of the 127 candidates, the provisional strong finds cluster in a few dozen; the remainder is a long tail of weaker coinages the curator is expected to trim. By operator, the candidate primes break down roughly as: dualization ~36, generalization ~47, self-application 15, spatialization 6, temporalization 5, lattice completion 5, modalization 4, threshold 2, analogy completion 7, perspective inversion 1.

5.2 The yield law — the paper's central result¶

Running the whole set, rather than the obviously-productive subset, is what made the pattern legible: an operator produces new primes only when it can reach a structural position not already implied by the existing corpus. The operators sort cleanly into three regimes.

Recombination operators saturate at zero. Composition (fuse two co-occurring primes), tension-resolution (synthesize an opposed pair), interpolation (fill a hierarchy gap), and cluster-without-parent (name a missing ancestor) all returned no new primes. The reason is structural, not a failure of effort: anything implied by recombining primes already in the catalog was already surfaced, reactively, during years of building the catalog's own hierarchy. A mature, densely-connected catalog has effectively already run these operators as a side effect of its own construction. Their zero is a genuine saturation result — and not empty-handed, because their "failures" are precisely the missing-edge and mis-parent flags that make up the ~295-edge densification byproduct.

Axis-stepping operators yield early, then become audit instruments. The eight unary operators step a prime along one fresh axis. They do find new primes — but the yield concentrates in a handful of high-convergence genera per operator, and the bulk of their output is a fixed-point census: the primes for which "apply the operator" returns the prime essentially unchanged, which is a map of which corpus primes are inherently temporal, spatial, modal, or self-referential. For these operators the audit yield dwarfs the discovery yield, and that is the point — they are completeness instruments that happen to also find a few primes.

Only two operators reach genuinely new ground. Analogy completion (A:B :: C:?) reaches diagonal, cross-domain composite primes — positions reachable only by a two-relation mapping, not by any single axis-step — and returned seven genuine finds (ergodicity, allostasis, boundary object, construal level, and others). Lattice completion reaches the empty cells of near-complete symmetric families and returned five (infinitism, cheap talk, common-pool resource, positional good, type-III error). Tellingly, both clustered their finds in the framed, social-science families — the same region generalization's staircase had already identified as the yield engine.

The completeness claim this licenses is the sharpest the program can make, and we state it plainly: the corpus is complete with respect to axis-steps and recombination; it is incomplete only along the diagonals and in the empty cells of its symmetry families. Those two moves — the two-relation analogy and the symmetry-completion — are exactly the ones a reactive, mining-driven construction cannot reach systematically, because nothing in the literature points at them. This is the deepest justification for the whole program: not that it found more primes than mining, but that it found the specific kind of primes mining structurally cannot.

5.3 The saturation gradient and the genus-convergence phenomenon¶

The fixed-point censuses, read across operators, are a quantitative map of how evenly the catalog grew along each axis. The corpus is 59% inherently temporal (feedback, half-life, Markov, diffusion, oscillation all return themselves under temporalization), 41% self-referential (self-application is the highest-fixed-point operator: recursion, closure, symmetry are maximal generalizers), 37% spatial, and 22% modal. The modal figure carries the subtlest finding: it is the least self-saturated axis yet has the highest alias absorption — the modal layer is covered not by many inherently-modal primes but by a small set of unusually powerful ones (probability, optionality, counterfactual reasoning, expected utility) that absorb most modalizations as aliases. "Few primes doing a lot of work" is itself a structural fact about the catalog that only a systematic sweep could surface.

And where do the real finds live? Never in single-input surprises — always in genera that hundreds of inputs quietly converge on. Spatialization's field was reached as "a field of P" from hundreds of distinct inputs; modalization's possibility as "the possibility space of P" from hundreds; temporalization's transient response from a dozen; self-application's second-order preference from five. The individual "X-of-P" instances are trivial coinage; the convergence itself is the signal that a genus node is missing. This is why convergence-count is the discriminator that survived all the prompt-tuning: it is a near-direct readout of how load-bearing the gap a candidate names actually is.

5.4 The finds that matter¶

Most of the 127 candidates are competent but marginal. A minority are conspicuous, and they cluster into types. Glaring absences in any standard map: consequentialism was absent while virtue ethics and moral relativism were present — and deontology was absent too, a double hole in the three-pole map of normative ethics (both are recommended for acceptance, pending curation). Exploration/exploitation, the explore-then-commit tradeoff that organizes reinforcement learning, foraging theory, and design process alike, was absent entirely. Partial order was absent while directed acyclic graph — one of its species — was present (adjacent order-primes existed; the genus did not). Consilience, inductive bias, and stigmergy were absent. Convergence across routes gave independent-replication-style confidence for several: heterarchy arrived by three routes, foundationalism by four, common-mode failure by three. Lattice signals surfaced as a byproduct even before the lattice operator ran: fragility absent beside antifragility; power law, scaling, and allometry all absent despite the catalog's heavy use of scale. None of these is exotic; each is a load-bearing idea whose absence is obvious in hindsight and was invisible without a systematic sweep.

5.5 The byproducts, which may be worth as much as the finds¶

~295 candidate hierarchy edges. Whenever an operator named a parent or intermediate that turned out to already exist in the catalog but not as an ancestor of the input, that is a discovery failure but a hierarchy success — a missing is-a edge. Harvested across generalization (247 flags), composition (23), interpolation (13), and cluster-without-parent (12), the raw total is ~295 flags. A consolidated adjudication pass over the largest subset found roughly 70% to be clean new is-a edges and the rest already present in another edge-type or direction-flipped, so the clean densification is on the order of ~200 edges — substantial against the catalog's ~665 existing subsumption edges, and, because it adds no nodes, fully compatible with the frozen-corpus decision. A companion runbook exists to adjudicate them to the same precision-first standard the primes get.

A growing set of hard negatives. Every collapse the pipeline correctly rejected — every alias, restatement, and domain instance — is retained. These are the contrastive examples one would want to teach a model to tell a genuine cross-domain abstraction from a plausible impostor, and the search generates and labels them for free, because it is coupled to a structural-similarity metric that no mining process has. Discovery and contrastive-data generation are, here, the same pass.

6. Hard-won method lessons, consolidated¶

The transferable content, gathered for anyone building a similar system.

Model tier is predictable from the operator's cognitive demand — and a cheap 15-item probe predicts it. Mechanical single-axis moves (dualization's flip, temporalization's add-a-clock) run fine on the cheap model; moves that require holding an input and its relatives at once (generalization's lift, self-application's fixed-point call, spatialization's is-it-already-spatial) need the capable model. A 15-prime cheap-vs-capable agreement probe in each pilot predicted the tier reliably (agreement 12/15 for temporalization, 5/15 for spatialization). Run the probe every time; do not assume the tier that worked for one operator transfers.
Retrieval proximity is not evidence of restatement. For any operator whose output lives near its input in embedding space, the input is the output's nearest neighbor because the output is new. State the counter-argument explicitly in the adjudication prompt or lose the real finds.
Protect known-good candidates structurally, not by prompt-tuning. Single-sample cheap-model judgments are unreliable in both directions and no wording fixes it. A hard rule — convergent or already-known candidates cannot be killed by a cheap screen on their last route — is what delivers zero silent deaths.
Convergence plus a canonical name is the discriminator. Once the funnel stops rejecting on proximity, it over-holds; the signal that separates a real genus from coinage noise is how many independent inputs reach it and whether it has an established scholarly name.
Embedding alias screens do not separate. In this corpus, true aliases and genuine new candidates occupy the same cosine band in every embedding space tried; a retrieval-grounded cheap-model check (present the neighbors, ask "same as one of these?") outperformed every fixed threshold.
Recombination saturates; axis-stepping yields early then audits; diagonals and empty cells are where the residual lives. The yield law (§5.2) is the sequencing rule for any future operator work: spend on the moves that reach un-implied positions.
A near-zero pilot is a completeness result, and often predictable before the pilot. Three operators retired on pilot evidence alone (cluster, threshold/phase, perspective inversion), saving millions of tokens. For two of them a five-minute grep of the relevant corpus region predicted saturation before the pilot ran — a cheap pre-pilot corpus check is a reliable saturation oracle.
Sub-agents silently drop work; reconcile every sweep. 2–8% of rows vanish without erroring at the batch sizes used. An unreconciled sweep is not a complete sweep.
The gate is zero-silent-deaths plus cost-per-accepted-prime, not survivor-rate. The clean funnel-width target was unreachable and the wrong thing to optimize.
Cost is dominated by model tier, not by scope. A full three-flavor cheap-model sweep cost less than a single capable-model operator's Stage-2 at scale. The tier choice is the real budget lever; the architecture keeps frontier tokens on the thin survivor stream regardless of scope.

7. What is prior art, and what is genuinely new¶

Intellectual honesty first: none of the individual operators is ours. Each has a deep pedigree, and saying so strengthens the program, because it means every operator is a battle-tested move rather than a guess. Pólya's How to Solve It (1945) names generalization, specialization, and analogy as the core heuristics of mathematical discovery — three of our operators are literally his list. Category theory gives dualization its rigorous form (every theorem has a dual for free in the opposite category) and gives self-application its home (fixed points, monads). Formal Concept Analysis (Wille, 1982) already computes the complete lattice of concepts from an object–attribute table and surfaces implied concepts nobody named — our cluster-without-parent and lattice-completion operators with a fifty-year-old algorithm behind them. TRIZ (Altshuller) catalogs "trends of engineering-system evolution" — segmentation, transition to micro-level, increasing dynamism — that are recognizably spatialization, granularization, and temporalization applied to machines. Mendeleev's predicted elements are the periodic-table precedent for lattice-gap completion; conceptual blending is composition; Hegelian dialectic is tension-resolution; Gärdenfors' conceptual spaces underwrite the embedding-and-interpolation intuition.

The novelty is therefore not in any operator. It is in five things, and a skeptical reader should probe exactly these:

The target. Every prior tradition applies its moves to a bounded domain — Pólya to mathematics, TRIZ to engineering, FCA to a fixed attribute table. Applying a unified operator set to domain-general cognitive primitives spanning all fields at once is, as far as we can find, undone. TRIZ is the closest sibling and is explicitly confined to physical contradictions; this is TRIZ's ambition at the level of abstraction itself.
Proactive exhaustiveness over a curated frontier. The prior traditions use their moves reactively, to solve a problem in hand. Running the full move-set over an entire existing corpus as a census is a different activity — the difference between using a chess move and enumerating a game tree.
The stack that makes it economical. The search is feasible only because three capabilities coincide: a large typed corpus (the frontier), structural embeddings plus an alias screen (the cheap visited-set test), and an LLM operator-executor (the node expander). Remove any one and the crank stalls. This coincidence is a few years old at most.
A measurable completeness claim, and a yield law to structure it. Because the move-set is closed and the corpus finite, each operator was run to saturation and its accumulation reported; the result is not just "we saturated" but the pattern of saturation — recombination yields nothing, axis-steps yield then audit, only diagonals and empty cells remain. That is a methodological contribution independent of any prime discovered, and it is the part we most want a skeptical reader to attack.
Discovery and training-data generation are the same pass. The crank's failures are precisely the hard negatives one wants for teaching analogical reasoning. No prior discovery method produces its own labeled contrastive set, because none was coupled to a structural-similarity metric.

One-sentence version: prior work supplies the moves and one-domain applications; the contribution is a closed operator set over a cross-domain prime corpus, run as a saturating proactive search, made economical by the embedding/LLM stack, and yielding both a measurable completeness result — the yield law — and its own analogical-reasoning negatives.

8. Completeness, limitations, and where the unturned stones are¶

This is the section a careful reviewer should attack hardest, so we make the weak points explicit and, where we can, quantified. It is also written to be useful to a reviewer looking for what remains: we end each point by naming what would close it.

The acceptance numbers are provisional, and the completeness stop-rule cannot yet be finalized. Stage 5 — human curation — is the gate that converts a "surfaced candidate" into an "accepted prime," and it is in progress across all 127 stubs. The saturation stop-rule the program is designed around (accepted-new primes per 100 applications) requires Stage-5 acceptance counts, which do not yet exist; the per-operator "holds" are the pipeline's recommendation, not a verdict. The owner's early read is that most candidates will survive, but that is an untested expectation, and a meaningful fraction — especially the Tier-B coinages and the fold candidates — is expected to be cut or merged. What would close it: the curation pass, after which the accumulation curves and a Chao1-style estimate of the unseen remainder become computable.

"Saturation" rests unevenly across operators. The recombination operators genuinely saturated — zero yield is a hard result. But the two productive families did not hit the formal stop-rule: the unary operators were retired administratively ("for this corpus state") once their audit value was extracted, not because a trailing-yield threshold was crossed, and generalization and dualization were each run essentially once per flavor. So the honest statement is: one class of operators demonstrably saturated; the others were swept once and stopped by judgment, pending the acceptance counts that would let the stop-rule speak. What would close it: re-running one productive operator to a measured trailing-rate floor after incorporation.

Coverage arithmetic and the silent-drop rate. Every unary sweep claims to apply its operator to all 1,325 primes, but sub-agents silently dropped 2–8% of rows and each sweep was repaired by a reconciliation pass; residual sub-1% gaps remain in at least one sweep (two unrecovered rows are logged in the dualization 1c run). So "applied to every prime" is true up to a small, bounded, documented drop rate — not exactly. What would close it: a final coverage audit reconciling every operator's completed-vs-attempted count against the canonical slug set.

The capability-floor claim is confounded with a prompt change. Generalization's cheap-model 0/53 result is our evidence for a capability floor — but the fix changed two things at once (a stronger model and the sibling-evidence rule), so "the cheap model cannot lift" and "the cheap prompt lacked the sibling rule" are not cleanly separated. What would close it: an ablation running the cheap model with the sibling rule.

The pipeline is LLM-run, with quantified instabilities. Cheap models hallucinate existence (54/74 in one pilot); frontier models commit the retrieval-proximity fallacy (blanket-zeroing 35/35 survivors on one pass); and Stage-2 NEW-rates showed batch-to-batch variance across identical inputs (e.g. 18/2 versus 19/10 NEW across two identical dualization batches) — direct evidence of prompt-artifact instability. The guardrails are described in §3, but a reviewer should treat every provisional count as an upper bound on quality until human curation confirms it, precisely because the human is the acceptance gate the machine cannot be.

Frozen-corpus drift. Because incorporation is deferred, later operators ran against a catalog that did not yet contain earlier operators' finds — so a few candidates were re-proposed, a few alias screens were computed against a slightly stale target, and cross-operator compositions (e.g. modalization lifting onto self-application's new second-order primes) were not reachable. This trades some compounding discovery for a clean, comparable saturation curve. What would close it: a second pass after incorporation.

Embedding weakness and single-curator judgment. The alias screen's failure to separate is a property of the specific (bge-small) embedder; a stronger structural embedder might change several Stage-3 decisions. And the completeness claim is relative to this corpus's granularity and one curator's bar, which is not yet inter-subjectively calibrated. What would close it: a second embedder for the screen, and a second independent curator on a sample.

Operators act on representations, not on concepts — a mis-framed prime hides its neighbours' transforms. This is a limitation class distinct from the yield law, and it has a clean worked example. While documenting the operators we found that thirteen of the fourteen correspond to an existing prime (dualization to duality, generalization to abstraction, and so on) but that one — interpolation — had no home prime, and that interpolation itself was absent from the corpus even though its dual, extrapolation, was present. Dualization should have reached it, and the crank logs show it tried — it generated "an apparatus deployed only within the regime its calibration was established in," which is interpolation's territory — but the candidate collapsed, because the extrapolation prime is catalogued as a failure mode ("a calibrated apparatus blind to its own regime exit") rather than as the neutral pole "inference beyond the sampled range." Flipping the failure framing yields "detect the failure" (an existing prime), not the safe inference. So an operator's reach is bounded by how its inputs are represented: a prime framed narrowly, as a failure mode or under a domain-specific name, silently hides the transforms that would reach its neighbours. We closed this the obvious way and measured the result. We classified the framing of all 1,325 primes, isolated the failure-mode and domain-specific stratum, normalised each prime to its neutral structural pole, and re-ran dualization and lattice completion on the neutralised poles across the whole stratum — 271 primes. The recovery was zero: not one new gap beyond interpolation, which the pass re-derived on its own as a validation. The reason is instructive. Nearly a third of the stratum genuinely is mis-framed (the corpus names the pathology but not the neutral operation), so the barrier is real and common — but once removed, the neutral pole's dual or symmetry-cell almost always lands on a prime already present. Mis-framing is pervasive and yet does not hide gaps. Interpolation was the rare double condition: an empty symmetry-family cell — precisely the residual the yield law predicts — that was also hidden by a failure-mode framing, a coincidence met once in 271. So this limitation class is real but its residual in the present corpus is essentially empty; the standing fix (prefer neutral-pole naming) is worth it for future-proofing, not as a source of undiscovered primes. And notably it never dented the completeness claim: the one gap it concerns is exactly where the theory says gaps are.

A third discovery mode, now formalised: exogenous reference-coverage. All fourteen operators are endogenous — they transform or combine primes already in the corpus, or read its own graph. But the move that actually found interpolation was none of them: it took an external enumeration (in this case the operator set itself) and checked the corpus for coverage, flagging the element with no image. This is a genuinely distinct mode, and we have since given it a full treatment — its move (align each element of an authoritative external list to a corpus prime; mint the un-imaged elements), its pipeline, and, crucially, its guarantee structure. That structure is what separates it from the operators: an operator's domain is the corpus and the move-set is closed, which licenses a global completeness claim; reference-coverage's domain is a human-chosen external set drawn from an open-ended family, so it earns only a local result — "complete with respect to this list" — never a global one. It is therefore a disciplined form of mining: bounded and exhaustively checkable against one fixed list at a time, not a peer of the closed endogenous set. The reflexive instance is the lone exception — its list (the fourteen operators) is finite and canonical, so that one check is closed and complete: thirteen mapped, interpolation minted. This gives the program a clean trichotomy of discovery modes — open mining, the closed endogenous operator search, and semi-structured exogenous reference-coverage. We then ran the first three checks. Against a 129-element enumeration of the mathematical primitives, a 75-element cognitive-bias list, and a 54-element fallacy list — 258 coverage-checks in all — the corpus returned 116 mapped, 141 out-of-scope, and exactly one gap: betweenness, which was already known and deferred (a relative of the very interpolation family that motivated the mode). Net new yield, zero — but the useful output is the certificate: each list is now provably covered (the out-of-scope calls draw the prime-bar boundary cleanly — discipline-confined objects like compactness and martingale, narrow lab effects, rhetoric-bound argument-moves), and the bias and fallacy results independently reconfirm the saturated failure-mode stratum. The mode's remaining leverage is on frontier external lists whose granularity sits at the prime bar in domains the corpus has under-mined; the canonical lists checked here were already well-covered.

The highest-value unturned stones, named for a re-runner. Two remain. First, the diagonal and empty-cell frontier the yield law identifies: analogy completion and lattice completion found real primes on a pre-filtered candidate set, so widening those pre-filters (more operator-confirmed pairs; more product spaces enumerated) is the one place further prime yield is structurally likely. Second, a cross-operator composition pass after incorporation, which the frozen-corpus decision deliberately deferred. Everything else — the axis operators, the recombination operators — has been swept, and the yield law predicts they will not repay a re-run on the current corpus.

We publish with these open rather than resolved, because the explicit purpose of publishing now is to invite exactly the audit that would close them.

9. Conclusion¶

We set out to answer a question mining cannot: how do you know what your catalog is missing? The answer this program offers is to make the catalog a frontier, make discovery a search over a closed set of moves, and make the missing pieces the reachable-but-unreached nodes of that search. Run to completion across all fourteen operators, the search surfaced 127 candidate primes — several of them conspicuous, load-bearing gaps — plus a hierarchy densification and a hard-negative training set that may prove as valuable as the primes. But its deepest output is not a list of finds; it is a yield law: recombination operators return nothing a mature catalog does not already imply, axis-steps yield a little and then measure the catalog's shape, and only the diagonal and symmetry-completion moves reach genuinely new ground — which are exactly the moves a reading-driven catalog cannot make. That converts "the catalog is nearly complete" from an aspiration into a claim with a stated boundary.

A last reflexive note, because it is the most fitting demonstration of the thesis. The operators of this program turn out to be prime abstractions themselves — dualization is duality, generalization is abstraction, self-application is reflexivity, composition and analogy are primes by those very names. The discovery machine is built out of the same structural vocabulary it was built to catalog. And when we checked that correspondence, it paid: exactly one operator — interpolation — had no home prime, and following that single thread revealed that interpolation was genuinely missing from the catalog, a clean gap its own dual had been hiding. The method, turned on itself, found a prime. The acceptance numbers still await a human, two frontiers stay deliberately open, and the whole program is offered here in full — finds and failures alike — so that a careful reader can find the stones we have not yet turned.

Appendix A — The operator catalog and per-operator ledger¶

The fourteen operators, grouped by arity, with the move, the characteristic collapse mode (retained as audit signal), status, and provisional yield. "FP census" is the fixed-point count — primes the operator returns essentially unchanged — where measured. For a thorough per-operator treatment — worked examples, collapse anatomy, lineage, and the method lesson each produced — see the companion Operator Compendium.

Unary — transform one prime along one axis¶

#	Operator	Move	Model tier	Status	Provisional yield
1	Dualization	flip polarity (negation · opposing-force · failure-mode)	haiku	full sweep (3,973 apps)	6 accept-rec + ~25 holds
2	Self-application	apply P to the machinery that sustains P	sonnet	full sweep (1,325)	15 holds; FP census 41%
3	Temporalization	add a time axis (level→flow · state→trajectory · present→anticipation)	haiku	full 3-flavor sweep (3,971)	5 holds; FP census 59%
4	Spatialization / scaling	add extent/scale (point→field · discrete↔continuous · micro→macro)	sonnet	full 3-flavor sweep (3,975)	6 holds; FP census 37%
5	Generalization / lift	strip a constraint to reach the genus	sonnet	full sweep, 3 strata (1,326)	~47 holds (framed = engine)
6	Modalization	shift modal status (actual→possible · deterministic→stochastic · mandatory→default)	sonnet	full 3-flavor sweep (3,975)	4 holds; FP census 22% (highest alias absorption)
7	Perspective inversion	read the same structure from another seat	sonnet	pilot only → retired saturated	1 framed find (interpellation)
8	Threshold / phase	continuous → critical point	sonnet	pilot only → retired saturated	2 gaps (catastrophe, order_parameter)

N-ary — combine primes (on pre-filtered candidate sets)¶

#	Operator	Move	Status	Provisional yield
9	Composition	fuse two tightly co-occurring primes	pilot (88 pairs) → retired saturated	0 new; 23 edge flags
10	Tension-resolution	synthesize an opposed pair's dilemma/resolution	pilot (53) → retired saturated	0 new
11	Analogy completion	A:B :: C:? — extrapolate the fourth term	full (1,325)	7 genuine (ergodicity, allostasis, boundary_object, construal_level, …)
12	Interpolation	propose the intermediate on a large hierarchy gap	full (61 edges) → retired saturated	0 new; 13 edge flags

Catalog-internal — operate on the graph¶

#	Operator	Move	Status	Provisional yield
13	Cluster-without-parent	cluster → find clusters with no common ancestor → name the parent	full (82 clusters) → retired saturated	0 new; 12 edge flags
14	Lattice / symmetry completion	find empty cells of a natural product space	frontier pass (~15 gaps)	5 genuine cells (infinitism, cheap_talk, common_pool_resource, positional_good, type_iii_error)

The yield law, restated as a sequencing rule. Recombination operators (9, 10, 12, 13) saturate — run them once as an audit, expect zero. Axis-steps (1–8) yield a few high-convergence genera then become census instruments — run on the cheapest tier that clears the operator's capability floor. Diagonals (11) and empty cells (14) are the only operators with residual prime yield on a mature corpus — widen their pre-filters if further primes are wanted.

Fixed-point saturation gradient (measured). Temporal 59% · self-referential 41% · spatial 37% · modal 22% (least self-saturated but highest alias absorption) · threshold/phase observationally the most-saturated (retired on that basis).

Appendix B — Reproducibility and artifacts¶

Corpus. 1,325 prime abstractions, frozen for the search. Each carries a slug, one-liner, core idea, structural signature, a structural/framed label, and a hierarchy of typed edges (subsumption, composition, decompose, mutual).

The funnel, concretely. Stage 0: one frozen prompt template per operator (crank contract + collapse test + few-shot). Stage 1: deterministic enumeration to a worklist. Stage 2: cheapest capable model, ~27 applications per batch, fixed output contract, block-count reconciliation with recovery passes. Stage 3: deterministic slug/alias/ancestor checks, then a retrieval-grounded cheap-model alias check over a neighbor menu, then the protection rule. Stage 4: a stronger model applies the four-part prime bar with catalog search, with the retrieval-proximity amendment and a convergence-plus-canonical-name discriminator; survivors drafted to the authoring standard. Stage 5: human curation.

Gates. An operator's pilot must show zero silent deaths of known-good candidates and an acceptable frontier cost per accepted prime before its full sweep is authorized. A near-zero pilot yield is a legitimate stop (retirement on saturation). Model tier is set by a 15-item cheap-vs-capable agreement probe run in the pilot.

Provisional totals (subject to curation). ~20,000 operator applications; ~33 million model tokens; 127 candidate-prime stubs; ~295 candidate hierarchy-edge flags (~200 clean after adjudication); fixed-point/audit censuses per unary operator; a retained hard-negative set.

Artifacts on disk. Per-operator prompt templates; per-application crank output including all collapses; survivor and negative sets; per-operator manifests recording every surfaced candidate with provenance, tier, and keep/cut flags; the running saturation table; a cross-operator synthesis document; 127 candidate-prime stubs drafted to the authoring standard; a consolidated candidate-edge worklist with its adjudication runbook; and the conceptual-foundation think-piece. Every count in this paper is traceable to these files, all of which are published together as the downloadable raw-data bundle (see the note after the abstract).

What to reproduce first. The cheapest independent check is to re-run any single operator's pilot on the frozen 53-prime sample and confirm the gate outcome and that surfaced candidates match the manifest. The most informative extension is to widen the analogy-completion or lattice-completion pre-filter — the two operators the yield law identifies as still able to reach new ground — and see whether the diagonal/empty-cell finds continue.

Status and provisional counts¶

This is a methods-and-results paper describing a program that has been run to completion — all fourteen operators applied, and each either swept in full or retired on measured saturation — but whose final human curation is still in progress. The firm counts are the ones the machine produced: candidates surfaced, applications run, tokens spent. The provisional counts are the accepted primes. The pipeline surfaced 127 candidates, and the owner's early read is that most will survive, but the acceptance numbers that would ground a formal completeness estimate await the Stage-5 curation pass. We publish now, with the machinery complete and the harvest provisional, so that a careful reader can examine the method and the finds — failures included — while the curation proceeds.