Audited Study Guide

Published foundations, recomputed evidence, synthetic fixtures, and hypotheses are labeled separately.

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The CASCADE suite is a synthetic toy benchmark: it measures the reduction in distance to the singular point of one 2-D test function. It is non-citable and not a physics result.

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Table of Contents

Part I: Learning Path

  • 1.Discovery and Initial Exploration
  • 2.Physics Singularity Analysis
  • 3.Implementation and Simulation
  • 4.Critical Audit and Course Correction
  • 5.The Paradigm Shift (v2.0)
  • 6.Multi-Objective Optimization
  • 7.Quantum Mechanics Testing
  • 8.Final Synthesis

Part III: Technical Chapters

  • 1.Foundations of Non-Newtonian Calculus
  • 2.Meta-Calculus Framework
  • 3.Multi-Calculus Framework (v2.0)
  • 4.Applications to Physics
  • 5.Validation and Constraints
  • 6.The Hierarchy of Approaches
  • 7.Use Cases and Applications
  • 8-9.Open Problems and Conclusions

Key Theorems

Identity (x > 0)

Theorem 1.1: Power Law Theorem

For f(x) = x^n on x > 0, where n is any real number:

DBG[xn]=enD_{BG}[x^n] = e^n

Grossman & Katz, 1972. An exact analytic identity on x > 0, not a numerical measurement.

Empirical / Conditional

Observation 3.1: Spectral-Gap Boundary

The proposed lower bound is:

gap(Pmix)≥min⁡{gap(PA),gap(PB),gap(PC)}gap(P_{mix}) \geq \min\{gap(P_A), gap(P_B), gap(P_C)\}

The baseline Laplacian cases pass. In the guarded grid audit, 79/88 valid normal compositions pass; five Dirichlet and four periodic mixed-physics cases refute the general claim. This is not a theorem.

Exploratory

Observational Preference

Multi-objective optimization converges to classical limit:

k→0,s→0k \to 0, \quad s \to 0

Negative result for the strong claims.

The Correct Hierarchy

1

Meta-Calculus

Project-defined weighted derivative framework; physical field equations remain exploratory.

USE FOR: Modifying dynamics

2

Bigeometric

For diagnostic analysis. Power law exponents become constants.

USE FOR: Understanding singularities

3

Multi-Calculus

For cross-scheme diagnostics. Robust quantities are candidates for further testing.

USE FOR: Flagging candidate artifacts

What Does NOT Work

  • X
    Full bigeometric GR

    D_BG[const] = 1, breaks tensor calculus

  • X
    Componentwise quantum modifications

    Breaks unitarity (historical: about 65% norm drift; the scan in public/quantum_phase_corrected.json of the historical historical nonunitary blend of Crank-Nicolson and a log-amplitude update: 0 of 1000 configurations unitary)

  • X
    k = -0.7 dark energy explanation

    Violates BBN constraints by 23x

Heuristic: scheme-robust features are candidates, not proof

An unproven working heuristic: features that survive analysis under multiple calculi are candidates for further physical testing. Features that appear only in one calculus may be mathematical artifacts.

Key References

Grossman, M. & Katz, R. (1972). Non-Newtonian Calculus. Lee Press.

Grossman, M. & Katz, R. (1983). The First Systems of Weighted Differential and Integral Calculus. Archimedes Foundation.

Bashirov, A. E., Kurpinar, E. M., & Ozyapici, A. (2008). Multiplicative calculus and its applications. Journal of Mathematical Analysis and Applications, 337, 36-48.

Coifman, R. R. & Lafon, S. (2006). Diffusion maps. Applied and Computational Harmonic Analysis, 21, 5-30.

Planck Collaboration (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy & Astrophysics, 641, A6.