Experiment Summary

These experiments test the extremality claim that for a fixed prefix depth \(n\), the product
\(M_n = A(a_n)\cdots A(a_1)\) with \(A(a)=\begin{pmatrix}a&1\\1&0\end{pmatrix}\) has
spectral radius \(\rho(M_n)\ge \varphi^n\), with equality only for the all-ones word.

The formal theorem statement appears in the provided note (Theorem 10). The tests below are independent numerical confirmations.

What was tested

• Exhaustive search over all digit words of length \(n\) with digits in \(\{1,\dots,K\}\) for several \((n,K)\) pairs.
• Random search for larger \(n\) where exhaustive enumeration is infeasible.
• Algebraic decomposition check of \(A(a)=T^a R\) with \(T^a=\begin{pmatrix}1&a\\0&1\end{pmatrix}\) and \(R=\begin{pmatrix}0&1\\1&0\end{pmatrix}\), explaining why even-length words land in \(SL(2,\mathbb{Z})\).

Key outputs

• `outputs/exhaustive_minima.csv`

  • Shows the unique minimizer is the all-ones word for each tested \((n,K)\), and the minimum equals \(\varphi^n\) up to the exact arithmetic used (Decimal).

• `outputs/random_minima.csv`

  • Shows that the best word found under random sampling is almost never all-ones (as expected), and the best ratio \(\rho(M_n)/\varphi^n\) stays above 1.

• `outputs/decomposition_checks.csv`

  • Confirms \(A(a)=T^aR\) for \(a=1\) through \(10\), with \(\det(A(a))=-1\).

Interpretation in one sentence

• The spectral extremality is robust under exhaustive and random tests, and the matrix family naturally sits inside a parity-indexed extension of the modular group, making contact with the same Möbius-action framework used in torus modular invariance.

Thermodynamic / variational analogue (new)

A physics-friendly interpretation is to treat the digit sequence as a microstate of a constrained multiplicative process:

• Micro rule: choose digits `a_t` and multiply by `A(a_t)`
• Macro observable: energy `E(word)=log ρ\(M_n\)` and growth rate `g=E/n`
• Partition function for bounded digits: `Z_n(β)=Σ exp(-β E(word)) = Σ ρ\(M_n\)^(-β)`
• As `β→∞`, the ensemble concentrates on the minimal-growth word (ground state). For this system the ground-state growth rate is `log φ`, uniquely realized by the all-ones word.

Outputs:

• `outputs/thermo_analogue.csv` contains:
  • growth rates for representative periodic patterns \(showing a positive gap above `log φ` unless the pattern is all-ones\)
  • finite-size thermodynamic estimates for several `(n,K,β)` demonstrating convergence toward the exact ground state as `β` increases

Observable interpretation (for the tests)

In the scripts, the quantity compared is the spectral radius ρ\(M_n\). Interpreting M_n as an n-step constraint-propagation transfer operator, ρ\(M_n\) is the exponential growth factor of admissible configuration weight under repeated local updates; log ρ\(M_n\)/n is the per-step growth (Lyapunov) exponent.

Digit extraction demo (optional)

• `digit_extraction_demo.py` prints the first continued-fraction digits of \(\varphi\) and shows the corresponding all-ones digit regime in a simple, inspectable way.

• Added `block_dominance_quenched_demo.py` to illustrate block-triangular spectral-radius dominance and a quenched Lyapunov lower-envelope demo.