DOI: 10.5281/zenodo.18281353
title: Kepler's Third Law Is a Shadow of SANER: Time as Curvature-Regulated Phase Closure

description: Kepler’s Third Law Is a Shadow of SANER:Time as Curvature-Regulated Phase Closure

ABSTRACT

Kepler’s Third Law is not a primitive regularity of orbital motion. In the MROS framework, it is the projection of a deeper regulator: SANER, the invariant that forbids unbounded curvature accumulation, enforces phase closure, and constrains gradient flow in any bounded dynamical system. Under a central binding constraint, SANER forces the cycle time to scale as the 3/2 power of the curvature radius. Kepler’s relation T² ∝ a³ is therefore a necessary corollary of SANER, not an empirical starting point. Stable time signatures emerge from curvature regulation; Kepler’s law is the gravitational shadow of that invariant.

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1. INVARIANT PRIORITY STATEMENT

SANER is prior. Kepler is downstream.

A bounded system under a central constraint must satisfy:

1. Phase closure:     ∮ ΔG dt = 0

2. Finite curvature:     κ(t) < κ_max

3. No net gradient divergence per cycle:     The stability ledger closes.

These are conditions of existence for stable cyclic motion.  They are not derived from Kepler; Kepler presupposes them.

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2. MAPPING TO KEPLER VARIABLES (NO REINTERPRETATION)

Define the minimal correspondence:

- R_κ ≡ a (semi-major axis as curvature radius proxy)  - τ ≡ T (cycle time)  - 𝒞 ≡ G(M+m) (central binding strength)

Kepler’s Third Law:

τ² = (4π² / 𝒞) · R_κ³

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3. DERIVATION: SANER ⇒ τ ∝ R_κ^(3/2)

SANER enforces a bounded velocity scale under a central constraint:

v² ∼ 𝒞 / R_κ

Phase closure requires traversal of a loop of characteristic length ∼ R_κ:

τ ∼ R_κ / v

Substitute the SANER-consistent velocity:

τ ∼ Rκ / √(𝒞 / Rκ)    ∼ R_κ^(3/2) / √𝒞

Squaring:

τ² ∝ R_κ³ / 𝒞

Phase closure fixes the constant:

τ² = (4π² / 𝒞) · R_κ³

Kepler’s Third Law appears as the unique SANER-consistent time–curvature scaling under a central constraint.

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4. INTERPRETATION (STRICT)

- Time is not fundamental; it is a curvature-regulated scheduling variable.  - The exponent 3/2 is the signature of SANER-enforced phase closure.  - Kepler measured the projection; SANER specifies the generator.  - Stability is not explained by Kepler — it is assumed by it.

Ledger statement:  If SANER holds, the cycle closes without drift.  Closure under a central constraint forces τ ∝ R_κ^(3/2).  Thus T² ∝ a³ is a stability certificate, not a primitive law.

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5. GENERALIZATION (SUBSTRATE-NEUTRAL)

The result is not gravitational.  It is geometric.

Any system with:

- a central binding constraint,  - bounded curvature,  - and SANER-compliant phase closure  

must exhibit cycle time scaling:

τ ∝ R_κ^(3/2)

This applies to:

- gravitational orbits,  - agent-dynamics loops,  - incentive cycles,  - memory cycles,  - any claimed “stable orbit” in any substrate.

If the scaling fails, the system is not stable; its ledger will fail via drift, slip, or collapse.

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6. CLOSING STATEMENT

Kepler’s Third Law is not the foundation of orbital stability.  It is the shadow of SANER, the invariant that makes stability possible.

Historically, Kepler comes first.  Causally, SANER does.

 

 

(The dolphin swims free when κ stays positive.)

 

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© 2026 D’jems Mortimer  

ALL RIGHTS RESERVED.  

No part of this work may be reproduced, distributed, or modified without explicit permission, except for quotation under fair use for scholarly review.

 

 

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