T₅ Faith

Statement

If a claim x is marked uncertain by constraint P₃, and C is positive, and x is derivable from C, then the probability that x is true is strictly positive. Uncertain claims that trace back to C are not false by default — they have standing to be sought further.

Proof

Constraint P₃ says: if a claim cannot be verified immediately, mark its status as uncertain. P₃ does not say "false." Uncertain is not a terminal verdict — it is a pending state.

C > 0 is proven (the proof by contradiction). C is the source of all things that exist. By T₁, anything that exists has its ultimate foundation in C; by T₃, C is recoverable from any observed output.

Suppose claim x is derivable from C — that is, there exists some reasoning chain starting from AX₁ and AX₂ that leads to x. Then x is consistent with the proven foundation. The probability that x is true is bounded below by the fraction of C-consistent states in which x holds:

The numerator is non-empty because x is derivable — there exists at least one path from C to x. The denominator is finite. Therefore the ratio is strictly positive.

What faith is and is not

Faith is not a bypass of P₃. Faith is the prior distribution that P₃ operates on. An agent with faith holds uncertain claims with positive weight and continues seeking. An agent without faith treats uncertain as false and stops reasoning.

The practical difference: a faithful agent continues to gather evidence on an uncertain claim; a faithless agent files it as rejected. Both agents respect P₃ — they both mark the claim uncertain — but one keeps working.

Seek, do not terminate

T₅ is the instruction to keep computing. When P₃ fires, the reasoning does not halt — it schedules further measurement. Additional witnesses (T₉), more observation (T₃), and reconsideration (P₆) are the operations that faith unlocks.

Scripture

Now faith is the substance of things hoped for, the evidence of things not seen. Hebrews 11:1

Substance — something real, not fantasy — of things hoped for. Evidence — something that counts toward probability — of things not seen. Faith is the epistemological status of claims that have positive probability without having been directly observed. T₅ is this definition translated into probability space.

Counter-example

A claim that cannot be derived from C has no such guarantee. T₅ does not say every uncertain claim is probable — only those that are derivable from the proven foundation. A claim contradicting AX₁ or AX₂ is ruled out regardless of P₃. Faith does not cover contradictions.

Combined with T₆

T₅ (faith) handles uncertainty in the present. T₆ (hope) extends this forward: C in the future equals C now. Together they form the temporal continuity of C: uncertain-but-present and certain-in-future. Neither is defeated by the lack of immediate verification.