The Coercive Projection Law of Gravity: A Universal Variational Principle with Explicit Constants and Cross‑Probe Falsifiers
Recognition Physics
Summary
This paper articulates a single, universal principle that governs gravitational inference under finite information: the coercive projection law. Nature maps raw baryonic sources to an effective source via a fixed, scale‑ and time‑aware kernel; the gravitational field is the unique minimizer of a classical energy with that effective source. Information‑Limited Gravity (ILG) is the gravitational instantiation in the pressure formulation.
Abstract
In the ILG pressure formulation, the kernel $$\,\w(k,a)=1+C\,(a/(k\,\tau_0))^{\alpha}\, $$ builds the effective pressure $p$ from observed baryons, and the potential solves the standard Poisson equation $$\,\nabla^2\Phi=4\pi G\,a^2 p\,. $$ We prove a coercivity inequality with explicit constants that guarantees existence/uniqueness and stability, certifies positivity/monotonicity displays used in galaxies and cosmology, and yields falsifiers that bind probes together: rotation curves, tracer‑independent $E_G$, the low‑$\ell$ ISW sign, and the low‑$L$ CMB‑lensing amplitude are simultaneously constrained by the same kernel and constants.
Key Contributions
- Universal kernel: $$\w(k,a)=1+C\,(a/(k\tau_0))^{\alpha}$$ with fixed constants
- Explicit coercivity: inequality with constants ensuring uniqueness, stability, positivity, and monotonicity
- Cross‑probe binding: one kernel constrains rotation curves, $E_G$, ISW, and CMB lensing together
- Recognition geometry: constants trace to golden‑ratio structure: $\alpha=\tfrac{1}{2}(1-\varphi^{-1})$, $C=\varphi^{-3/2}$
- Certificates: energy, residual norms, positivity checks, and convergence diagnostics accompany each analysis
Mathematical Framework
We formulate the variational problem with an effective source induced by the kernel and prove a coercivity inequality with explicit constants. This ensures the projected solution exists, is unique and stable, and enforces structural displays (positivity/monotonicity) used across probes.
Implications and Engine
The law converts “fits” into audited inferences: each run emits certificates (energy, residuals, positivity, convergence). A minimal engine implementing grid (FFT) and disk (Hankel) paths is provided: CPM‑Cosmology‑Grid‑Path.