Ilg Cpminstance Ilg Reverse Coercivity

The energy gap bound

In physics, a defect mass is the extra mass that appears when a system deviates from a simple ideal state. In the Recognition Science framework, the ledger, a discrete record of recognition events, assigns each state a defect mass and an energy gap, the minimum energy needed to change that state. The theorem ilg_reverse_coercivity proves a lower bound: for every state in the ILG model, the energy gap is at least (49/162) times the defect mass. That constant, 49/162, is not arbitrary; the framework derives it from the model's covering number, projection bound, and energy normalization.

The theorem belongs to a larger coercivity framework. Its companion ilg_coercivity proves the opposite inequality, that defect mass is bounded above by a product of constants times the energy gap. Together they sandwich the defect mass between two multiples of the energy gap. This sandwich is what the framework calls a coercive projection: it guarantees that a small defect mass forces a small energy gap, and vice versa, so the model cannot hide a large defect behind a tiny energy cost.

In Recognition Science, the theorem is a structural guarantee. It shows that the ILG model, a candidate modification of gravity, satisfies the same coercive projection framework as other models in the library. The bound holds for any kernel parameters and any state, provided the energy control hypothesis holds: the defect mass never exceeds the energy gap. That hypothesis is a condition, not a consequence.

The theorem does not claim that the ILG model correctly describes real galaxies. It does not predict a specific rotation curve or a measured value. It only establishes a mathematical inequality within the model. The falsifiable prediction, if any, would come from the model's kernel and its comparison to observations, not from this bound alone.

/-- Reverse coercivity: energy gap is at least c_min times defect. -/
theorem ilg_reverse_coercivity (P : KernelParams) (h_energy : EnergyControlHypothesis P) (s : ILGState) :
    (ilgModel P h_energy).energyGap s ≥ cmin (ilgModel P h_energy).C * (ilgModel P h_energy).defectMass s :=
  (ilgModel P h_energy).energyGap_ge_cmin_mul_defect ilgConstants_pos s

/-- The ILG coercivity constant is 49/162. -/
theorem ilg_cmin_value : cmin ilgConstants = 49 / 162 := by
  simp [cmin, ilgConstants]
  norm_num

/-- Energy control hypothesis: the energy of a configuration bounds its defect.
    This is the physical content of the variational principle (Lax-Milgram).
    In ILG, this states that the gravitational energy controls the deviation
    from the Newtonian solution. -/
def EnergyControlHypothesis (P : KernelParams) : Prop :=
  ∀ s : ILGState, defectMass P s ≤ energyGap s

What this page does not claim

The theorem does not claim that the ILG model matches any observed galaxy rotation curve. The theorem does not claim that the energy control hypothesis always holds. The theorem does not claim that the defect mass is a directly measurable physical quantity.

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/ILG/CPMInstance.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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