IndisputableMonolith.Verification.CPMBridge.Constants.Probability
Defines coincidence probabilities for independent draws from a finite range landing inside a tolerance window, plus smallness bounds used when several such windows are combined. Verification authors cite these when arguing that a match among RS-native constants is not an accidental collision. The module is mostly definitional, with elementary product bounds on the joint event.
claimLet $P(n,R,\tau)$ be the probability that $n$ independent uniform selections from a discrete range of size $R$ all fall inside a window of radius $\tau$. The module records $P$, the specialization to net-radius windows, and elementary upper bounds showing $P$ (and finite products of such $P$) can be made small for the parameter regimes used in constant matching.
background
In the CPM bridge verification layer, Recognition Science constants (native units with $c=1$, $\hbar=\varphi^{-5}$, and related ladder quantities) are checked against external numerical targets. A recurring worry is accidental numerical coincidence: several independent quantities might land near the same target by chance.
This module packages the elementary counting model for that worry. Selections are treated as independent and uniform on a finite range of size range_size; a hit means every selection falls inside a tolerance window of given radius. Sibling definitions specialize the same idea to net-radius windows and to combined (product) probabilities across several domains.
The surrounding verification domain uses these bounds as quantitative evidence that reported matches are tighter than naive collision rates would allow.
proof idea
Definition module with light lemma content. The main object is the closed-form coincidence probability (ratio of favorable window volume to full range, raised to the number of independent draws). Net-radius and combined variants are specializations or products of that quantity. Smallness statements are direct comparisons or algebraic bounds on those products; there is no deep probabilistic machinery.
why it matters in Recognition Science
Feeds the CPMBridge constant-verification path: downstream checks that RS-predicted constants sit inside tight numerical windows need a null model for accidental agreement. Without an explicit coincidence probability, a referee can dismiss a match as numerology. The combined-smallness lemmas let several independent constant comparisons be stacked while keeping a joint false-positive rate under control. The module does not itself derive any RS forcing step (T5–T8, RCL, mass ladder); it only supplies the statistical side condition used when those predictions are scored against data.
scope and limits
- Does not derive RS constants or forcing-chain identities (T5–T8).
- Does not model continuous measure-theoretic probability beyond finite-range counting.
- Does not claim empirical error bars or experimental systematics.
- Does not prove uniqueness of matches; only bounds accidental-coincidence rates.
- Does not address dependent or adversarial selection processes.