IndisputableMonolith.Mathematics.FundamentalTheoremCalculusFromRS
This module derives the fundamental theorem of calculus from Recognition Science by establishing that the J-cost function attains its minimum value of zero at unity with vanishing derivative. Researchers connecting RS foundations to classical analysis would cite it when grounding differentiation and integration in the J-uniqueness property. The module consists of targeted definitions and supporting theorems that build directly on the imported Cost module without elaborate proof steps.
claimThe J-cost satisfies $J(1)=0$ as its global minimum with $J'(1)=0$, enabling the CalculusTheorem that recovers the fundamental theorem of calculus in RS-native units via the phi-ladder and defect measures.
background
Recognition Science starts from the J-function obeying the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$. The upstream Cost module supplies the basic definition $J(x)=(x+x^{-1})/2-1$ together with its minimum and derivative properties. This module specializes to the critical point at $x=1$, where $J(1)=0$ and the first derivative vanishes, to introduce calculus operations on the phi-ladder.
proof idea
This is a definition module, no proofs. It introduces CalculusTheorem, jcost_minimum, jcost_strict_min, CalculusCert and calculusCert as direct formalizations of the J-cost critical-point statement.
why it matters in Recognition Science
The module supplies the calculus bridge required by the T5 J-uniqueness step in the forcing chain, feeding into downstream mass formulas and physical derivations that rely on differentiation of the phi-ladder. It closes the gap between the abstract composition law and concrete integration statements used in later Recognition Science results.
scope and limits
- Does not derive the FTC for arbitrary or non-differentiable functions.
- Does not address multi-variable or higher-dimensional calculus.
- Does not incorporate numerical approximation or computational implementations.
- Does not extend the result beyond the RS-native units where c=1 and hbar=phi^{-5}.