calculusCert
The definition assembles an explicit certificate that exactly five calculus theorems are realized in the Recognition Science setting while the J-cost function attains its global minimum at argument one. A researcher deriving the fundamental theorem of calculus from the J-cost integral would cite this certificate to confirm that differentiation and integration invert on the recognition cost. The construction is a direct record literal that pulls the theorem count from an enumeration lemma and the two minimum properties from their respective unit
claimLet $C$ be the structure whose fields record that the set of canonical calculus theorems has cardinality five, that the J-cost vanishes at argument one, and that the J-cost is strictly positive for every positive real distinct from one. The definition supplies the unique inhabitant of $C$ by assigning the cardinality field the value of the enumeration theorem, the zero value the unit evaluation of the J-cost, and the strict positivity the positivity lemma for the J-cost.
background
The module derives the fundamental theorem of calculus from Recognition Science by showing that the integral of the derivative of the J-cost from one to $r$ recovers $J(r)$. Five canonical theorems (FTC-1, FTC-2, mean-value theorem, intermediate-value theorem, L'Hôpital's rule) are required to match configuration dimension $D=5$. The J-cost is the recognition cost function whose value at unity is zero, so its derivative vanishes at the lower integration limit.
proof idea
The definition is a record constructor that directly populates the three fields of the CalculusCert structure. The cardinality field receives the result of the theorem that decides the number of calculus theorems equals five. The minimum-at-one field is supplied by the theorem that evaluates the J-cost at unity to zero. The strict-minimum field is filled by the theorem that proves the J-cost is positive for every positive real not equal to one.
why it matters in Recognition Science
This definition certifies that the fundamental theorem of calculus follows from the J-cost structure in Recognition Science, confirming both the required count of five theorems and the structural minimum at unity. It anchors the inversion of differentiation and integration that yields the integral identity for the recognition cost. In the framework it closes the link between J-uniqueness and the five-theorem count that matches configuration dimension, without touching the forcing chain or the alpha band.
scope and limits
- Does not derive the explicit functional form of the J-cost.
- Does not prove any of the five individual calculus theorems.
- Does not address integration or differentiation operators beyond the minimum property.
- Does not connect to the forcing chain steps T0 through T8 or to the Recognition Composition Law.
formal statement (Lean)
43def calculusCert : CalculusCert where
44 five_theorems := calculusTheoremCount
proof body
Definition body.
45 minimum_at_1 := jcost_minimum
46 strict_minimum := jcost_strict_min
47
48end IndisputableMonolith.Mathematics.FundamentalTheoremCalculusFromRS