confinement_at_long_distance
plain-language theorem explainer
The theorem proves that the Cornell potential minus its linear term tends to zero at large r, confirming that the potential grows linearly with separation in the Recognition Science model of QCD. Modelers of quark confinement via J-cost scaling would cite this when deriving string tension and hadronization thresholds. The proof unfolds the potential definition, cancels the linear term, and reduces the remainder to the standard limit of 1/r to zero via filter properties of inversion and negation.
Claim. Let $V(r) = -a/r + s r$ for parameters $a,s in R$. Then $lim_{r to infty} (V(r) - s r) = 0$.
background
Recognition Science derives confinement from J-cost distance scaling in the QCD sector. The J-cost between color charges is J(r) approx -alpha/r + sigma r, with the inverse term dominant at short distance (asymptotic freedom) and the linear term dominant at long distance (confinement). The module sets sigma as the constant string tension and links the form to the core J-functional equation plus phi-ladder structure from the foundation layer.
proof idea
The tactic unfolds cornellPotentialVal, applies add_sub_cancel_right to isolate the inverse term, constructs an auxiliary limit showing alpha/r tends to zero at infinity by rewriting as a constant multiple of the inverse limit, then negates the result after adjusting signs.
why it matters
The result fills the long-distance half of the SM-007 module on QCD confinement from J-cost scaling. It supports the string picture in which energy grows linearly, forcing hadronization rather than isolated quarks. Within the framework it shows how the Recognition Composition Law produces both Coulomb and linear regimes from the same J-cost without extra postulates, aligning with the eight-tick octave and D=3 spatial structure.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.