A probabilistic heuristic for the distribution of prime numbers that models primality as a sequence of independent random events with a prescribed density approximately 1 / log n; used to generate conjectural statistics about prime counts and gaps rather than rigorous theorems.

Principle

Treat primes at large scale as independent Bernoulli trials with success probability roughly 1 / log n, so that counting and spacing questions reduce to probabilistic expectation and variance computations under an independence hypothesis.

Demonstration

To estimate the number of primes in an interval [x, x+H] with H small relative to x, the model predicts an expected count ≈ ∑_{n=x}^{x+H} 1 / log n; similarly it gives heuristic estimates for gaps by treating occurrences as rare independent events, producing predicted distributions for gaps between successive primes.

Misapplication

Using Cramér's independence assumption as a proof that specific arithmetic patterns (for example exact counts of twin primes in short intervals or fine-scale correlations) hold; or applying the model at scales where residue-class constraints force strong dependencies.

Consequence

Provides simple, often accurate heuristic predictions for large-scale prime statistics (mean prime counts, order of maximal gaps, rough distribution shapes) that guide conjectures and indicate plausible extremal behaviours, while remaining nonrigorous.

Reversal

A deterministic, arithmetic description that insists on exact multiplicative and congruence relations (sieve-theoretic or algebraic structures) rather than treating primality as independent random events; this perspective emphasizes correlation and structure over independence.

Boundary

Applies only as a heuristic for large-scale statistical properties of primes; it ignores local congruence constraints and small-prime divisibility correlations, is not a proof device, and may give wrong fine-scale predictions where arithmetic structure dominates.

Semantic Tension

Tension between modelling primes as random independent events (probabilistic simplicity) and the known multiplicative and congruence-induced correlations among integers (arithmetic rigidity); the model sacrifices exact arithmetic structure for tractable probabilistic estimates.

Synthesis

A compact probabilistic heuristic that replaces the arithmetic complexity of primality by an independence assumption with density ~1/log n, yielding tractable conjectural predictions about prime counts and gaps while explicitly acknowledging arithmetic limitations and lack of proof.