Add scaled invariance verification for infinity and zero as counting labels

[tensorrent]

Summary

This PR adds a comprehensive mathematical note and machine-verified test suite establishing that "infinity" and "zero" are resolution-relative counting labels, not intrinsic properties of intervals. The core insight is that the count of representable points in an interval is a property of the pair (interval, resolution-unit), and this count is exactly invariant under joint rescaling of both factors.

Key Changes

• New test suite (code/notes_verification/test_scaled_invariance.py):

  • Implements exact lattice-point counting using rational arithmetic (no floating-point error)
  • Verifies four machine-checked (T1) facts:
    • A. Unit-relativity: The interior of (1,2) contains 0, 9, 99, 999, 9999 points at units δ = 1, 1/10, 1/100, 1/1000, 1/10000 respectively
    • B. Joint-scaling invariance: Over 200,000 random configurations, the count is exactly preserved under (interval, unit) → (λ·interval, λ·unit) with 0 mismatches
    • C. Interval-only scaling breaks it: Holding unit fixed while stretching interval changes the count, demonstrating why the label is relative
    • D. Dimensionless invariant: Count depends only on ratios a/δ and b/δ, confirming width/δ is the frame-free quantity
  • Uses seed 20260423 for reproducibility

• New mathematical note (papers/methodology/Scaled_Invariance_of_Infinity_and_Zero.tex):

  • Formalizes the observation that "infinitely many decimals between 1 and 2" and "no whole numbers between 1 and 2" are the same interval read at different units
  • Establishes Principle 1 (scaled invariance) and Proposition 1 (exact joint-scaling invariance with proof)
  • Derives Corollary 1 (the invariant is dimensionless)
  • Includes detailed scope boundary: this concerns representable points under a resolution, not set cardinality; Cantor's theorem is untouched
  • Connects to framework's existing relativity concepts (Form/Function Relativity FF06g, When a Number Lies)

• Documentation updates:

  • Updated README.md to list the new note as FF06h in the methodology series
  • Updated MANIFEST.md to include the new verification test

Implementation Details

• Uses Python's fractions.Fraction for exact rational arithmetic, eliminating floating-point errors in counting
• Implements both closed-interval and open-interval counting with exact ceiling/floor operations on fractions
• Randomized testing with 200,000+ trials ensures the invariance holds across diverse parameter spaces
• All tests exit with nonzero status on any mismatch, making verification part of the CI contract

https://claude.ai/code/session_01PfjNn6a39Y1eDbxNfsQQA5