potential_confining
plain-language theorem explainer
The Cornell potential is strictly increasing for positive separations when the Coulomb coefficient is non-negative and the string tension is positive. QCD modelers would cite this to confirm linear growth of the confining term from J-cost scaling. The proof reduces the inequality to a factored positive expression via field simplification and ring algebra after unfolding the potential definition.
Claim. Let $V(r) = -alpha/r + sigma r$. Then for $alpha >= 0$, $sigma > 0$, $r_1 > 0$ and $r_2 > r_1$, it holds that $V(r_2) > V(r_1)$.
background
In Recognition Science the Cornell potential models the effective interaction between color charges as $V(r) = -alpha/r + sigma r$, with the linear term arising from J-cost distance scaling. The J-cost is the derived cost function of a multiplicative recognizer comparator, as defined in MultiplicativeRecognizerL4.cost, which quantifies ledger imbalance for separated charges. Module SM-007 derives QCD confinement by showing this potential grows with distance, producing the constant string tension that favors hadronization over isolation.
proof idea
The tactic proof unfolds cornellPotential, introduces positivity facts for r2 and the radius difference via lt_trans and sub_pos, then rewrites the target inequality equivalently as positivity of the difference. Field_simp followed by ring factors the difference as (r2 - r1) times (alpha/(r1 r2) + sigma). The final exact step applies mul_pos to the two manifestly positive factors.
why it matters
This result establishes the confining (linearly growing) character of the potential, directly supporting the long-distance behavior in the J-cost model of confinement. It supplies the algebraic step needed for confinement_at_long_distance and string_breaking within the module. The theorem connects to the Recognition Composition Law by making explicit the distance-dependent cost that enforces the eight-tick octave scaling into D=3 spatial dimensions and the observed alpha band.
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