---
slug: law_of_large_numbers_inevitability
name: Law of Large Numbers Inevitability
one_liner: "Across enough independent trials, any outcome with nonzero per-trial probability becomes effectively certain to occur at least once, so the rational design posture shifts from preventing each trial to containing the guaranteed eventual occurrence."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, physical, epistemic]
origin_domain: probability_theory
alternate_origin_domains: [reliability_engineering, statistical_physics, risk_management, ergodic_theory]
aliases: [inevitability_given_trials, borel_cantelli_certainty, rare_event_eventuality, given_enough_draws_it_happens, tail_event_realisation]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-05 generalization) — the proposed GENUS of `eventual_realisation_of_possibility` (lift: strip the framed possibility/modal-realisation gloss to expose the bare probabilistic law that repeated independent trials drive any nonzero-probability event to certainty). Discovered in the op-05 remaining-strata sweep 2026-07-03. Tier A candidate (established genus: second Borel-Cantelli lemma / recurrence, verified absent). STATUS: HOLD pending owner keep/cut."
- "Tier-A convergence/absence flag (from data): law-of-large-numbers / inevitability-given-trials genus absent from corpus; the three claimed siblings (casino eventually loses a night, fleet eventually suffers a rare part failure, long-running system eventually takes a cosmic-ray bit-flip) span casino / fleet / hardware-reliability domains distinctly, so the lift is not a single-instance coinage."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `eventual_realisation_of_possibility` and the named siblings; confirm it is a genuine genus (covers the input AND the casino/fleet/cosmic-ray siblings) and not a domain instance or a restatement of the input; draw a subsumption edge input→candidate + sibling edges at incorporation; sync one_liner to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 5
  domain_breadth: 5
  structural_abstraction: 5
  transfer_evidence: 4
  reasoning: 'The bare structure — a repeated independent (or sufficiently mixing) trial process plus a per-trial event of nonzero probability, yielding near-certain eventual occurrence as the trial count grows — is a pure probabilistic relation carrying no domain content whatsoever. Its mathematical core (the second Borel-Cantelli lemma, and recurrence theorems in ergodic theory) is medium-neutral by construction: it constrains coin flips, radioactive decays, gambling nights, fleet part-lives, packet corruptions, and mutation events with identical form. This scores at the top on structural abstraction and domain breadth. Transfer evidence is strong and cross-domain (probability, reliability engineering, statistical physics) though slightly less than maximal only because the honest independence/mixing assumption must be checked in each application, so composite 5 with transfer at 4. This is exactly the pattern where a generalization lift outscores its framed input: the input carried a possibility/modal-realisation framing that this genus strips.'
similarity_to_nearest_existing_prime:
  slug: eventual_realisation_of_possibility
  similarity: 0.72
structural_framed:
  label: structural
  aggregate: 0.14
  criteria: {vocab_travels: 0.1, evaluative_weight: 0.0, institutional_origin: 0.2, human_practice_bound: 0.1, import_vs_recognize: 0.3}
  rationale: 'This is a bare probabilistic law: it makes no evaluative claim (nothing is judged good or bad — an outcome is merely certain to occur), and its vocabulary (trials, probability, eventual occurrence) travels across every domain with independent repetition. It is recognized as a structure wherever repeated trials appear rather than imported as a domain doctrine; the mild nonzero on institutional_origin and import reflects only that it is a named theorem (Borel-Cantelli) with a home in probability theory, not that its content is domain-bound. It reads clearly structural — considerably more so than its framed input.'
phase_b_passes_completed: []
---
# Law of Large Numbers Inevitability

## Core Idea

Law-of-large-numbers inevitability is the bare probabilistic structure that *any outcome with nonzero per-trial probability becomes effectively certain to occur at least once given enough independent trials* — over a long enough run of draws, generations, exposures, or units of time, what merely *can* happen *will*. The single constraint stripped from the input `eventual_realisation_of_possibility` is its framed, modal gloss: the input speaks of "possibilities being realised," a possibility-space vocabulary that carries a whiff of design-posture narrative; this genus keeps only the mathematical skeleton — repeated independent (or sufficiently mixing) trials plus a fixed per-trial probability, driving the probability of at-least-one-occurrence to one. Its canonical core is the second Borel-Cantelli lemma together with the recurrence theorems of ergodic theory.

The load-bearing anatomy has three parts. First, a *repeated trial process*: a stream of independent or adequately mixing draws whose count grows without a hard bound. Second, a *nonzero per-trial probability* attached to the event of interest, however small, so long as it does not vanish. Third, the *cumulative-certainty consequence*: as the number of trials rises, the chance the event never happens decays geometrically toward zero, so eventual occurrence is not merely likely but effectively guaranteed. The characteristic payoff is a decisive shift in the rational design posture — because the rare event is *certain in the aggregate* even while *negligible per trial*, effort should move from per-trial prevention (which cannot lower a nonzero probability to zero, and so cannot stop the eventual occurrence) to eventual containment: detection, graceful degradation, and recovery once the guaranteed event lands. The characteristic cost and failure mode is the mirror of this: the structure quietly assumes independence or mixing and a stable per-trial probability, and it is silent on *when* within the run the event arrives; naive application either underweights a rare-but-certain tail (treating "unlikely per trial" as "won't happen") or, conversely, over-provisions against an event whose expected arrival time is astronomically far out.

Read as a genus rather than as any one instance, the same skeleton recurs across structurally unlike domains. A casino's business model *is* this pattern from the house's side turned around: over a night's volume of bets, it is near-certain that *some* gambler beats the house on some run, so casinos design for containment (table limits, bankroll) rather than pretending each hand is un-loseable. A large vehicle fleet faces the certainty that *some* vehicle eventually suffers a rare part failure, however reliable each part — reliability engineering answers with fleet-level monitoring and redundancy, not with the impossible goal of a zero-failure part. A long-running computing system faces the certainty that *some* rare cosmic-ray-induced bit-flip eventually corrupts a word of memory, which is why long-lived systems adopt error-correcting codes and checksums (containment) instead of chasing a flip-proof cell (prevention). Casino night, fleet part-life, and single-event upset are three genuinely different substrates; the recognizable shared skeleton — nonzero per-trial probability × many trials ⇒ eventual certainty ⇒ contain rather than prevent — is what makes this a real lift and not the input restated.

## Notes

**Provenance.** Produced by operator 05 (generalization / lift) as the proposed GENUS of the framed input `eventual_realisation_of_possibility`, by stripping that input's possibility-realisation modal framing to expose the underlying probabilistic law (second Borel-Cantelli / recurrence). Surfaced in the op-05 *remaining-strata* sweep on 2026-07-03. It claimed three cross-domain siblings — the casino night, the large fleet's rare part failure, and the long-running system's cosmic-ray bit-flip. Manifest tier: A (Tier-A "rest"; established genus, verified absent).

**For owner evaluation (keep/cut).** The case FOR is strong and this is one of the cleaner Tier-A candidates in the sweep: the genus is a *named, established mathematical structure* (Borel-Cantelli / ergodic recurrence), not a sonnet coinage — it has an independent identity in probability theory that owes nothing to the input. Its absence is a real hole: the corpus holds the framed `eventual_realisation_of_possibility` but not the domain-neutral law it instantiates, and that law has maximal substrate independence (SI estimate 5) and obvious reach into reliability engineering, statistical physics, and risk management. The siblings span three unrelated domains, which is the discriminator that separates a genuine genus from a single-input coinage. The specific RISKS Kurt should weigh: (1) **collapse into the input (restatement)** — the estimated similarity to `eventual_realisation_of_possibility` is high (~0.72), so the sharp keep/cut question is whether these are *two* primes (a framed possibility-realisation prime + its structural probabilistic genus) or *one* that should simply be re-headed to the structural framing and merged, with the input as its lead example. My reading is genus-and-species (keep both, draw the edge), but this is the one call to make deliberately. (2) **Adjacency to a possible large-numbers / convergence prime** — if the corpus has or later adds a prime for the *convergence* law of large numbers (sample means → expected value), note this candidate is the *distinct* Borel-Cantelli/inevitability face (some event *occurs*), not the averaging face; the boundary should be stated so the two don't blur. (3) **Independence assumption** — the genus quietly presumes independence/mixing; that is a scope caveat, not a defect, but worth flagging in any v1.

**At incorporation.** Re-run similarity against `eventual_realisation_of_possibility` (the input) and against any convergence/averaging law-of-large-numbers or tail-risk prime to confirm no collapse and to fix the inevitability-vs-convergence boundary. Draw a subsumption edge `eventual_realisation_of_possibility` → `law_of_large_numbers_inevitability` (input is a framed species of the structural genus) and sibling/example edges tying the casino, fleet, and cosmic-ray cases as instances. If a general containment-over-prevention or rare-event-risk prime exists, add a missing-edge note linking this genus to it. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
