Mathematics Probability Theory From Rs Uncertain Event Positive Cost

The cost of doubt

Probability theory in the Recognition Science framework begins with a simple accounting idea. The framework keeps a ledger, a discrete record of recognition events, and assigns each event a cost, a real number measuring what it takes to recognize that event as occurring. The central move is to define probability as an exponential decay of that cost: P(event) = exp(-J(event)), where J is the cost. This is a Boltzmann-like relation, familiar from statistical mechanics, but here it is a definitional choice, not a derived result.

The declaration uncertain_event_positive_cost pins down one consequence of that definition. For any positive real number r that is not equal to 1, the cost J(r) is strictly greater than 0. In plain language: any event whose probability is not exactly 1, meaning any event that is not certain, has a positive cost. The complementary statement, certain_event_zero_cost, says the certain event has cost exactly 0, so its probability is exp(0) = 1. Together they ensure that probability stays in the unit interval: certain events sit at 1, and everything else falls below 1.

The declaration is a theorem in the framework's machine-checked library of formal theorems. It is proved from the framework's forced cost function J(x) = (x + 1/x)/2 - 1, which is itself derived from five plain conditions. The proof uses the fact that J(r) is positive for r ≠ 1 and J(1) = 0. The theorem is axiom-clean: it relies only on the standard logical axioms of the ambient type theory, with no framework-specific axioms added.

What the declaration does not claim is equally important. It does not assert that probability must be defined this way, only that if you accept the framework's cost function, this is what follows. It does not derive the full Kolmogorov axiom system. The framework does count five canonical probability axioms, matching the classical list of non-negativity, normalisation, countable additivity, total probability, and conditional probability, and identifies that count with a structural dimension of 5, but the declaration itself is a narrower statement about a single inequality.

The consequence for a reader is a concrete picture: in this framework, doubt is not free. Every time an event is less than certain, the ledger records a positive cost, and that cost is what pulls the probability below 1. The framework's contribution is to tie that intuitive fact to a specific, forced functional form, so the price of doubt is not arbitrary but fixed by the mathematics.

/-- Uncertain event: J > 0 → P < 1. -/
theorem uncertain_event_positive_cost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- Certain event: J = 0 → P = 1. -/
theorem certain_event_zero_cost : Jcost 1 = 0 := Jcost_unit0

theorem kolmogorovAxiomCount : Fintype.card KolmogorovAxiom = 5 := by decide

What this page does not claim

The declaration does not prove that the Kolmogorov axioms are the only possible axioms for probability. It does not claim that probability must be defined as exp(-J) in all contexts. It does not derive the numerical value of any specific probability from physical data.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/ProbabilityTheoryFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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