jcostColorPotential
jcostColorPotential defines the potential for color charge separation in Recognition Science as the J-cost of ledger imbalance. It combines short-range inverse attraction from recognition events with long-range linear growth from ledger tension. QCD theorists would cite it when modeling confinement and hadronization from first principles. The definition is a direct one-line wrapper around the Cornell potential using the short-distance coupling and string tension.
claimFor distance $r > 0$, the J-cost of color separation equals the Cornell form $V(r) = -a/r + s r$, where $a$ is the short-distance strong coupling and $s$ the string tension.
background
The J-cost is the derived cost of a multiplicative recognizer comparator on positive ratios and equals the J-cost of any recognition event state. In the SM-007 module, confinement follows from J-cost distance scaling: short distances yield Coulomb-like behavior while long distances produce linear growth. Upstream results establish non-negativity of event costs and link them to simplicial ledger edge lengths from psi.
proof idea
The definition is a one-line wrapper that applies the Cornell potential construction to the short-distance coupling and string tension parameters.
why it matters in Recognition Science
This supplies the explicit potential form required by the SM-007 target of deriving quark confinement from J-cost structure. It supports sibling results on asymptotic freedom at short distances and string breaking at long distances within the QFT module. The linear term traces to ledger tension in the simplicial ledger and aligns with the recognition composition law for scaling.
scope and limits
- Does not derive the Cornell form from the recognition composition law.
- Does not assign numerical values to the coupling or string tension.
- Does not address light quark masses or string breaking lengths.
formal statement (Lean)
89noncomputable def jcostColorPotential (r : ℝ) (hr : r > 0) : ℝ :=
proof body
Definition body.
90 -- Schematic: J-cost for color separation
91 -- Short range: recognition events give -α/r
92 -- Long range: ledger tension gives σr
93 cornellPotential alphaSshort stringTension r hr
94
95/-- **THEOREM (Asymptotic Freedom at Short Distance)**: At small r, the coupling is weak.
96 This is the Nobel-Prize-winning discovery by Gross, Politzer, and Wilczek (2004). -/