Gravity Causal Kernel Chain

Causal kernel chain

In Recognition Science, gravity is read as a recognition ledger with memory, not as a classical force field. The causal kernel chain is the Debye case of that memory: one relaxation time, an exponential decay in time, and a single-pole frequency response. The chain runs from the time-domain kernel through its complex transfer function to the two frequency limits that name the steady-state and Newtonian regimes.

The Debye kernel is Γ(t) = (Δ/τ) e^−t/τ for t ≥ 0, with coupling strength Δ and relaxation time τ. Its truncated frequency response is the integral of that kernel against e^−iωt over a finite horizon [0, B]. The theorem kernel_response_limit states that as B tends to infinity, the truncated response converges to Δ / (1 + iωτ). That closed form is the Laplace-transform bridge from time to frequency.

The complex transfer function is defined as 1 + Δ / (1 + iωτ), and transfer_function_eq_one_plus_kernel identifies it with one plus the infinite-horizon kernel response. A further theorem, response_function_is_real_part, shows that the real-valued Caldeira–Leggett response is exactly the real part of that complex transfer function. As ω → 0 the response approaches the steady-state value 1 + Δ; as ω → ∞ it tends to 1, the Newtonian limit where memory drops out. Broader spectral densities are allowed by the source discussion but are not yet formalized here.

THEOREM kernel_response_limit · IndisputableMonolith/Gravity/CausalKernelChain.lean

THEOREM response_function_is_real_part · IndisputableMonolith/Gravity/CausalKernelChain.lean

MODEL debye_kernel · IndisputableMonolith/Gravity/CausalKernelChain.lean

THEOREM transfer_function_eq_one_plus_kernel · IndisputableMonolith/Gravity/CausalKernelChain.lean

What this page does not claim

That the kernel chain is the full derivation of gravity. That spectral densities beyond the Debye single-pole case are already formalized. That the Newtonian limit is an extra postulate; it follows from the closed form.

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

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