IndisputableMonolith.Verification.ProbabilityNormalizationCert
Verification module that turns recognition costs into a discrete probability distribution and certifies the elementary axioms. It normalizes exponential weights, proves non-negativity, the unit upper bound, and sum-to-one, then bundles those facts. Anyone auditing RS measurement statistics or paper exports of the lightweight Born rule would cite it. The argument is a short positivity-and-summation stack over real exponentials, not a deep derivation.
claimFrom recognition costs $C_i$, form $p_i=e^{-C_i}/\sum_j e^{-C_j}$. The module proves $p_i\ge 0$, $p_i\le 1$, and $\sum_i p_i=1$, and packages these facts as a probability-normalization certificate over the lightweight Born-rule cost-to-weight map.
background
Recognition Science maps measurement alternatives to recognition costs; the induced statistics are obtained by exponential reweighting. The upstream lightweight Born-rule module supplies a minimal algebraic cost-to-weight map intended for paper exports, deliberately avoiding heavy analysis dependencies.
This verification module sits on that map. It introduces the normalized probabilities $p_i=e^{-C_i}/Z$ with partition sum $Z=\sum_j e^{-C_j}$, records that $Z>0$ (hence nonzero), and states the standard distribution inequalities. Sibling names in the module mark the pieces: the cost-to-probability map, positivity and nonvanishing of the exponential sum, non-negativity and unit bound for each $p_i$, the sum-to-one identity, and the bundled certificate.
proof idea
Short verification stack, not a deep proof. Real exponentials are positive, so each weight and the partition sum are positive; the sum is therefore a valid nonzero denominator. Normalized coordinates are consequently well-defined, nonnegative, and at most one. Sum-to-one is immediate from the definition of $Z$. Those lemmas are collected into the certificate structure. No forcing-chain or phi-ladder reasoning appears.
why it matters in Recognition Science
The import graph shows no downstream consumers inside the monolith, so the module functions as a leaf audit artifact for external checks and paper exports. It closes the elementary side of the story begun in the lightweight Born-rule module: once costs are given, the induced weights form a genuine discrete probability distribution. That underwrites any claim that RS measurement statistics are probabilistically well-formed without reopening real-analysis obligations in the core development. It does not itself advance the T0–T8 forcing chain, RCL, or the mass ladder.
scope and limits
- Does not derive the Born rule or the exponential weight from first principles.
- Does not treat continuous spectra or POVM-valued measurements.
- Does not prove uniqueness of exponential reweighting among other positive maps.
- Does not link normalization to eight-tick, phi-ladder, or dimensional forcing.
- Does not supply numerical bounds on costs or on the partition function.