Ilg Kernel Kernel Dynamical Time Stationary

Stationarity of the dynamical time kernel

The Infra-Luminous Gravity (ILG) kernel is a mathematical function that describes how a physical influence, such as a perturbation in density, is weighted at a given scale. In the framework's formalization, the kernel takes the form w(k, a) = 1 + C · (a / (k τ₀))^α, where k is a wave number, a is the scale factor, τ₀ is a reference time scale, α is an exponent derived from self-similarity, and C is an amplitude constant. This function is defined to be positive and at least one for all physical parameter values, meaning it never reduces the weight of an influence below its baseline.

The declaration kernel_dynamical_time_stationary is a theorem about a specific version of this kernel, called the dynamical time kernel, which replaces the scale factor a with a dynamical time variable T_dyn. The theorem states that for any given kernel parameters P and any dynamical time T_dyn, the value of the kernel is the same regardless of any other time coordinates _t1 and _t2 that might be present in the context. Formally, it asserts kernel_dynamical_time P T_dyn = kernel_dynamical_time P T_dyn, which is a statement of reflexivity: the kernel's value depends only on its explicit arguments, not on any ambient or auxiliary time variable.

This stationarity property is a structural guarantee within the ILG model. It ensures that the kernel's value at a dynamical time is not influenced by, nor does it accumulate across, other time coordinates. In the framework's library, this theorem is used as a certificate of causality, specifically as the no_cumulative_growth field in the causalityBoundsCert structure. This means the model asserts that the kernel does not exhibit unbounded growth over time, a property that is necessary for the kernel to represent a stable, causal physical influence.

In Recognition Science, this theorem is part of a larger formalization of the ILG kernel, which is itself a model within the framework. The kernel's exponent α is locked to a value derived from the golden ratio φ, and the amplitude C is set to specific constants in different parameter sets. The stationarity theorem, however, does not depend on these specific values; it holds for any valid KernelParams structure. The theorem is a formal statement about the definition of the dynamical time kernel, and its proof is a direct consequence of that definition.

What the theorem does not claim is that the dynamical time kernel is constant across different values of T_dyn. The kernel does vary with T_dyn, as shown by the monotonicity properties of the general kernel. The stationarity result only rules out dependence on other time coordinates, not on the dynamical time itself. It also does not claim that the ILG kernel is a complete physical theory; it is a formal model within the Recognition Science framework, and its physical interpretation is a matter of ongoing research.

/-- **No cumulative-time growth.** For a stationary orbit `T_dyn(t) = T_dyn`
constant in time `t`, the dynamical-time kernel is constant in time. This
resolves Beltracchi's concern (1): the gravitational acceleration on a
test particle in a stable orbit does not grow as `t^α`. -/
theorem kernel_dynamical_time_stationary
    (P : KernelParams) (T_dyn : ℝ) (_t1 _t2 : ℝ) :
    kernel_dynamical_time P T_dyn = kernel_dynamical_time P T_dyn := rfl

/-- The causality-bound certificate is inhabited. -/
noncomputable def causalityBoundsCert : CausalityBoundsCert where
  pert_pos := kernel_perturbation_pos
  pert_ge_one := kernel_perturbation_ge_one
  IR_bounded := fun P k_min hkmin a ha k =>
    kernel_perturbation_bounded_above P hkmin ha k
  Hubble_bounded := fun P a H ha hH k => kernel_with_Hubble_bounded_above P ha hH k
  background_eq_one := kernel_background_eq_one
  partition_homogeneous := mode_partition_homogeneous
  dyn_pos := kernel_dynamical_time_pos
  no_cumulative_growth := kernel_dynamical_time_stationary

/-- The dynamical-time ILG kernel: depends only on the local orbital
period `T_dyn`, the recognition tick `τ₀`, the lag amplitude `C`, and the
self-similarity exponent `α`. For a stationary orbit `T_dyn` is constant,
so the enhancement is constant, and the acceleration on an isolated mass
does not grow in time. -/
noncomputable def kernel_dynamical_time (P : KernelParams) (T_dyn : ℝ) : ℝ :=
  1 + P.C * (max 0.01 (T_dyn / P.tau0)) ^ P.alpha

/-- Kernel is always positive for valid parameters. -/
theorem kernel_pos (P : KernelParams) (k a : ℝ) : 0 < kernel P k a := by
  unfold kernel
  have hmax_pos : 0 < max 0.01 (a / (k * P.tau0)) := by
    apply lt_max_of_lt_left
    norm_num
  have hpow_nonneg : 0 ≤ (max 0.01 (a / (k * P.tau0))) ^ P.alpha :=
    Real.rpow_nonneg (le_of_lt hmax_pos) P.alpha
  have hcorr_nonneg : 0 ≤ P.C * (max 0.01 (a / (k * P.tau0))) ^ P.alpha :=
    mul_nonneg P.C_nonneg hpow_nonneg
  linarith

/-- Kernel is at least 1. -/
theorem kernel_ge_one (P : KernelParams) (k a : ℝ) : 1 ≤ kernel P k a := by
  unfold kernel
  have hmax_pos : 0 < max 0.01 (a / (k * P.tau0)) := by
    apply lt_max_of_lt_left
    norm_num
  have hpow_nonneg : 0 ≤ (max 0.01 (a / (k * P.tau0))) ^ P.alpha :=
    Real.rpow_nonneg (le_of_lt hmax_pos) P.alpha
  have hcorr_nonneg : 0 ≤ P.C * (max 0.01 (a / (k * P.tau0))) ^ P.alpha :=
    mul_nonneg P.C_nonneg hpow_nonneg
  linarith

What this page does not claim

The theorem does not claim the dynamical time kernel is constant across different values of T_dyn. The theorem does not claim the ILG kernel is a complete physical theory. The theorem does not claim that the ILG kernel has been empirically validated.

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

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