1. Plain English
The lemma Jcost_mono_on_one states that the function $J(x)$ is monotonically increasing for values of $x$ greater than or equal to $1$. If you select any two real numbers $x$ and $y$ that are both at least $1$, and $x \le y$, then the J-cost of $x$ will be less than or equal to the J-cost of $y$.
2. Relevance in Recognition Science
In Recognition Science (RS), $J(x) = \frac{1}{2}(x + x^{-1}) - 1$ defines the THEOREM-backed, unique cost functional for distinguishing a state or scale from the ground state ($x=1$). This lemma establishes that as a scale ratio $x$ moves further away from the identity $1$, its J-cost reliably grows. This monotonic penalty enforces that sequential J-cost optimization—such as the nested fraction steps in a continued fraction—strongly prefers minimal scale jumps. It provides the foundation for showing that the golden ratio $\phi$, represented by a continued fraction with minimal partial quotients (all 1s), acts as the ground state attractor for sequential J-cost optimization, structurally connecting RS to Ramanujan's deep continued fraction evaluations.
3. Reading the Formal Statement
private lemma Jcost_mono_on_one {x y : ℝ} (hx1 : 1 ≤ x) (hxy : x ≤ y) :
Jcost x ≤ Jcost y
private lemma: A local helper theorem restricted to this specific Lean module.{x y : ℝ}: The variables $x$ and $y$ are real numbers passed implicitly to the lemma.(hx1 : 1 ≤ x): The first explicit premise ensures $x \ge 1$.(hxy : x ≤ y): The second explicit premise establishes that $x \le y$.Jcost x ≤ Jcost y: The conclusion; the J-cost evaluated at $x$ is bounded above by the J-cost evaluated at $y$.
4. Visible Dependencies and Certificates
The proof in the supplied code delegates the algebraic lifting to a single preceding lemma:
- It unfolds the mathematical definition of
Jcost. - It relies directly on add_inv_mono_on_one, which proves the purely algebraic relation that $x \mapsto x + x^{-1}$ is monotone increasing on the domain $[1, \infty)$.
- It uses Lean's
linarithtactic to perform the final linear arithmetic step converting $x + x^{-1} \le y + y^{-1}$ into $\frac{1}{2}(x + x^{-1}) - 1 \le \frac{1}{2}(y + y^{-1}) - 1$.
5. What This Declaration Does Not Prove
- Strict Monotonicity: It proves weak monotonicity ($\le$), not strict inequality ($<$). It does not formally assert that increasing $x$ strictly increases cost, although this holds for $x > 1$.
- Behavior on $(0, 1)$: It makes no claims about inputs $x < 1$. In the broader RS framework, values below 1 map to the same positive costs via the reciprocal symmetry $J(x) = J(x^{-1})$, but this isolated lemma does not cover that domain.
- Minimal Partial Quotients: By itself, it only shows that larger numbers have higher cost. It does not compute optimal choices for continued fractions; rather, it serves as a dependency for downstream THEOREMS like pq_one_minimal_cost, which derives that a partial quotient of 1 minimizes J-cost at any given level.