Enforcing Absolute Unitarity in Differentiable Physics: The Float-to-Integer Ledger Approach

The Problem: Numerical Entropy in ODE Solvers

Standard differentiable solvers, while excellent for gradient-based optimization, often suffer from minute informational drift due to floating-point precision limits. In long-epoch cosmological simulations—specifically those involving terminal gravitational collapse—these rounding errors accumulate, violating the Information Conservation Mandate and breaking the unitarity of the system.

The v13.0 Solution: The Unitarity Bridge

Released on April 30, 2026, the String-Star Manifold v13.0 implements a dual-ledger architecture to bridge the gap between continuous kinematics and discrete conservation.

1. Dual-Precision Coupling

We decouple the Physics Engine from the Information Ledger:

• Kinematics (float32): Standard differentiable paths for N-body interactions, utilizing JAX-accelerated solvers.
• Microstate Ledger (int32): A discrete accounting system that tracks the absolute number of qubits within the Bandyopadhyay-Cycle.

2. The Unitarity Index

By enforcing transitions through a "Float-to-Integer Bridge," we ensure that mass/information is never "lost" to the vacuum during a phase shift (e.g., a Holographic Shift to a Fuzzball horizon). In our latest 500-epoch stress tests, the manifold maintained a Unitarity Index of exactly 1.000000, even through violent White Hole Blowout events.

Application: The Bandyopadhyay-Cycle

The system tracks information across three mutually exclusive states:
[ I_{total} = I_{bulk} + I_{horizon} + I_{vacuum} ]

When local density $\rho$ breaches the Planck Density ($\rho_{Planck}$), the solver must handle a non-linear phase transition where gravity becomes repulsive:
[ F_{LQG} = \frac{G m_1 m_2}{r^2} \left( 1 - \frac{\rho_{local}}{\rho_{Planck}} \right) ]

A Question for the Diffrax Community

In version 13.0, I utilized a "Stretchy Spatial Hash" to handle metric expansion ($\Lambda(t)$) while keeping array shapes static for JIT efficiency.

I am interested in exploring how Diffrax might more elegantly handle these discontinuous state-dependent jumps (like the Instantaneous White Hole Blowout) without sacrificing the differentiability of the overall trajectory. Has there been consideration for "Conservation-Locked" solvers that utilize a discrete integer backbone for invariant quantities?

Resources

• Project Documentation: https://github.com/Rupayan52/String-Star-Manifold/tree/quantum-cosmology
• Technical Monograph: https://doi.org/10.5281/zenodo.19923317
• Proof of Work: Verified on Google TPU V5 Lite.

The issue names no Diffrax files, tests, or entry points; start by clarifying whether a conservation-locked solver is requested and which discontinuous transitions it must support. Define the invariant and differentiability requirements before identifying an implementation location and tests that would demonstrate completion.