jcost_deriv_zero_at_one
The derivative of the J-cost function vanishes at x=1, confirming the ground state as the unique critical point where the gradient is zero. Researchers classifying the computational complexity of physical systems under Recognition Science cite this to anchor the convexity argument for ground-state verification. The proof is a direct algebraic evaluation of the closed-form derivative expression at the point x=1.
claim$J'(1)=0$, where $J'(x)=(1-x^{-2})/2$ is the derivative of the J-cost $J(x)=(x+x^{-1})/2-1$.
background
The IC-005 module derives the computational complexity of physics from the structure of J-cost minimization. J-cost is the function $J(x)=(x+x^{-1})/2-1$, shown to be strictly convex with its unique global minimum at $x=1$. This yields the classification that ground-state verification lies in P while phi-rung computations scale as EXPTIME.
proof idea
One-line wrapper that unfolds the definition of the derivative $J'(x)=(1-x^{-2})/2$ and applies simplification to substitute $x=1$, reducing the expression directly to zero.
why it matters in Recognition Science
This result supplies IC-005.6 and feeds the ic005_certificate summary of RS complexity classes, which lists ground-state verification as O(1) and J-cost descent as polynomial. It closes the local gradient analysis at the minimum, consistent with T5 J-uniqueness in the forcing chain and the convexity premise that places physics complexity in P for balanced ledgers.
scope and limits
- Does not evaluate the derivative for any x other than 1.
- Does not prove that x=1 is a minimum without separate nonnegativity lemmas.
- Does not extend to multivariable or field-valued versions of J-cost.
- Does not supply convergence rates or iteration bounds.
formal statement (Lean)
87theorem jcost_deriv_zero_at_one : jcost_deriv 1 = 0 := by
proof body
Term-mode proof.
88 unfold jcost_deriv; simp
89
90/-- **THEOREM IC-005.7**: J'(x) > 0 for x > 1.
91 The gradient points upward away from the minimum for x > 1. -/