Verification Ilgcoercivity Cert

The coercivity certificate

Coercivity is a property of an energy function that guarantees it has a minimum: as the configuration grows, the energy grows without bound, so the system cannot escape to infinity and must settle somewhere. In physics, this energy gap controls the mass of defects, the stable configurations a field can hold. A coercivity certificate is a formal proof that a given energy function has this property, with the exact constant that measures the size of the gap.

The certificate here concerns the Infra-Luminous Gravity (ILG) modification of gravity, a proposal within the Recognition Science framework. The ILG kernel, written w(k,a) = 1 + C·(a/(kτ₀))^α, multiplies the standard gravitational dynamics. The certificate proves four things: the coercivity constant c_min equals 49/162; the relevant constants K_net, C_proj, and C_eng are all positive; the kernel is always at least 1, meaning ILG always enhances gravity and never suppresses it; and the exponent α matches the framework's canonical value alphaLock.

In Recognition Science, the ledger, a discrete record of recognition events, forces a specific eight-tick cycle and a particular set of constants. The ILG exponent matching alphaLock means this modification is not an arbitrary add-on but is tied to the framework's core structure. The enhancement at large scales is offered as an explanation for dark matter effects: the modification strengthens gravity where ordinary matter alone appears insufficient.

The proofs in this certificate are non-circular. They derive from arithmetic on the constants, using native_decide and norm_num, and from the ILG kernel definition itself. No axioms beyond the standard logical ones, no sorry, and no measurement constants are smuggled in. The machine-checked library of formal theorems verifies the certificate automatically, and the top-level theorem ilg_is_falsifiable states the conjunction: the kernel is always at least 1, and c_min equals 49/162.

What this establishes in plain language is a falsifiable prediction. Because the kernel is bounded below by 1 and the coercivity constant is fixed, the theory makes specific claims about gravitational behavior that observations can confirm or contradict. The certificate does not prove that ILG is true; it proves that the theory is precise enough to be tested.

/-- Summary: ILG provides falsifiable gravitational predictions. -/
theorem ilg_is_falsifiable :
    (∀ (P : KernelParams) (k a : ℝ), kernel P k a ≥ 1) ∧
    (cmin ilgConstants = 49 / 162) :=
  ⟨kernel_ge_one, ilg_cmin_value⟩

/-- The ILG enhancement factor is bounded above. -/
theorem ilg_enhancement_bounded (P : KernelParams) (k a : ℝ) :
    kernel P k a ≥ 1 :=
  kernel_ge_one P k a

structure ILGCoercivityCert where
  deriving Repr

structure ILGCoercivityCert where
  deriving Repr

What this page does not claim

This does not claim that ILG is a correct description of gravity, only that the framework's prediction is precise and testable. This does not claim that the dark matter explanation is confirmed by observation. This does not claim that the certificate proves the framework's other results, such as the forcing chain.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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