Foundation Pair Kernel Physical Source Covector Pair Kernel Physical Source Dens

The density identity

A dipole is a paired source and sink: a quantity that leaves one point and arrives at another. In the Recognition Science framework's machine-checked library of formal theorems, the declaration pairKernelPhysicalSourceDensity_eq_scale_unitDipole proves that a certain physical source density equals the unit dipole scaled by a real parameter. The unit dipole itself is the integer-valued Gauss source of an elementary posting, a discrete record of an event between two indices. The theorem states that for any scale s and any indices a, b, and i, the density at i is exactly s times the unit dipole at i. In symbols: pairKernelPhysicalSourceDensity s a b i = s * unitDipole a b i.

The content is that the shape is fixed and the magnitude is free. The Gauss law fixes the integer dipole shape: a +1 at the source index a, a -1 at the sink index b, and zero elsewhere. The real parameter s, called sourceScale, multiplies that shape to produce a real-valued physical source density. The theorem is a THEOREM in the library's sense: it is proved by rewriting the density definition and the already-proved identification of the realified elementary Gauss source with the unit dipole. The proof uses no sorry and no new axiom; it is a direct consequence of prior definitions and theorems.

What the declaration does not claim is as important as what it proves. It does not select a numerical value for sourceScale; the scale remains a free parameter. It does not derive OneLedgerLaw, a separate statement about ledger structure. It does not identify the carrier with a Noether momentum-map covector, though a separate equality proposition records that comparison as an ordinary equality on matching carrier types. The theorem is about the density object itself, not about which scale nature chooses or whether the carrier matches a conserved quantity from a different construction.

The practical consequence is a clean separation. The framework can prove uniqueness of the carrier among covectors realizing the Gauss/source-coupling law at a given scale, and injectivity of the scale parameter on a nontrivial posting. But the density identity itself is the bridge from integer combinatorics to real-valued physics: it says the physical source is the integer dipole, times a number. The reader can now see that the framework's physical source is not an arbitrary field; it is a scaled copy of a discrete combinatorial object, with the scale left for later physics to determine.

theorem pairKernelPhysicalSourceDensity_eq_scale_unitDipole {n : ℕ}
    (sourceScale : ℝ) (a b i : Fin n) :
    pairKernelPhysicalSourceDensity sourceScale a b i =
      sourceScale * unitDipole a b i := by
  simp [pairKernelPhysicalSourceDensity,
    realifiedElementaryGaussSource_eq_unitDipole]

/-- Canonical realification: posting divergence equals the independently named
unit dipole. -/
theorem realifiedElementaryGaussSource_eq_unitDipole {n : ℕ}
    (a b i : Fin n) :
    realifiedElementaryGaussSource a b i = unitDipole a b i :=
  PairKernelDiscreteGauss.elementaryPosting_divF_eq_unitDipole a b i

theorem pairKernelPhysicalSourceDensity_eq_scale_unitDipole {n : ℕ}
    (sourceScale : ℝ) (a b i : Fin n) :
    pairKernelPhysicalSourceDensity sourceScale a b i =
      sourceScale * unitDipole a b i := by
  simp [pairKernelPhysicalSourceDensity,
    realifiedElementaryGaussSource_eq_unitDipole]

What this page does not claim

The declaration does not select a numerical value for sourceScale. The declaration does not derive OneLedgerLaw. The declaration does not identify the carrier with a Noether momentum-map covector.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PairKernelPhysicalSourceCovector.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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