module module high

`IndisputableMonolith.Gravity.EnergyProcessingBridge`

This module defines the J-cost function J(x) = ½(x + 1/x) - 1 for x > 0 together with energy distribution and processing gradient maps in the gravity setting. Researchers modeling weak-field gravity or acoustic phase effects in Recognition Science cite it for the cost functional that bridges energy to gravitational modifiers. The module consists entirely of definitions and immediate properties with no theorems or proofs.

claimThe J-cost is the functional $J(x) = ½(x + x^{-1}) - 1$ for $x > 0$, satisfying the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$. The module also defines EnergyDistribution structures and maps energy_to_processing_field, energy_creates_processing_gradient, and energy_distribution_creates_gravity_modifier.

background

Recognition Science obtains all physics from the single Recognition Composition Law (RCL). This module in the Gravity domain imports CoherenceFall and introduces the J-cost as the unique cost functional forced by that law. It further defines EnergyDistribution as a structure that maps energy values to processing fields, together with the maps that convert energy into a processing gradient and ultimately into a gravity modifier.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the energy-processing bridge that is imported by AcousticPhaseLevitation and WeakFieldSuperposition. It therefore supplies the concrete cost functional required by any downstream gravity construction that starts from energy distributions rather than from the phi-ladder directly.

scope and limits

Does not derive numerical values for any energy distribution.
Does not prove uniqueness of J beyond the RCL hypothesis.
Does not address time-dependent or relativistic extensions.

used by (2)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Gravity.AcousticPhaseLevitation module_import
IndisputableMonolith.Gravity.WeakFieldSuperposition module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Gravity.CoherenceFall module_import

declarations in this module (11)

def Jcost L33
theorem Jcost_nonneg L35
theorem Jcost_zero_iff_one L45
theorem Jcost_one_plus_exact L60
theorem Jcost_quadratic_ratio L69
structure EnergyDistribution L79
def energy_to_processing_field L87
theorem energy_creates_processing_gradient L93
structure EnergyProcessingEquivalence L108
theorem energy_processing_bridge L121
theorem energy_distribution_creates_gravity_modifier L137