Complete Textbook

Comprehensive 9-chapter treatment from discovery through rigorous validation.

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Chapter 10: CASCADE Singularity Access

The textbook now includes CASCADE singularity-skirting validation: 21-simulation proof that k=-1 (bigeometric) enables access to "danger zones" near physics singularities.

61.9%

Win Rate

93.4%

Best Gain

D*[1/r]=-1

No Diverge

View full CASCADE singularity proof →

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

Verified

Theorem 1.1: Power Law Theorem

For f(x) = x^n where n is any real number:

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

Grossman & Katz, 1972. Verified in numerical tests with CV = 0.0000.

Verified

Theorem 3.1: Spectral Gap Amplification

Composed operator gap exceeds individual gaps:

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

Empirical result: 9/9 test cases showed 10-19% improvement. Rigorous proof pending.

Exploratory

Observational Preference

Multi-objective optimization converges to classical limit:

k→0,s→0k \to 0, \quad s \to 0

Negative result for strong claims but validates the diagnostic methodology.

The Correct Hierarchy

1

Meta-Calculus

For modified field equations. Preserves tensor linearity.

USE FOR: Modifying dynamics

2

Bigeometric

For diagnostic analysis. Power law exponents become constants.

USE FOR: Understanding singularities

3

Multi-Calculus

For invariant extraction. Physical = scheme-robust.

USE FOR: Identifying real physics

What Does NOT Work

  • X
    Full bigeometric GR

    D_BG[const] = 1, breaks tensor calculus

  • X
    Componentwise quantum modifications

    Breaks unitarity (65% norm drift)

  • X
    k = -0.7 dark energy explanation

    Violates BBN constraints by 23x

Physical = Scheme-Robust = Cross-Calculus Invariant

Features that survive analysis under multiple calculi are candidates for genuine physics. 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.