Q as an Order-Coherent Transcendental Envelope Generator over the Integers

This deposit contains the complete computational and theoretical record for the paper Q as an Order-Coherent Transcendental Envelope Generator over the Integers (Revised Version). The paper presents and validates a seven-stage forensic audit of the Q functional — a transcendental breathing function constructed from φ, π, and e — demonstrating that Q does not classify primes but instead generates a robust, order-dependent envelope across all integer populations tested.

This paper supersedes earlier work on π-wave prime resonance [https://doi.org/10.5281/zenodo.17503556], whose central claims it falsifies.

The central finding is that Q's breathing behaviour is activated by locally smooth magnitude ordering and is independent of number-theoretic category. A parity-controlled null test (KS p = 0.91 against density-matched odd non-primes) falsifies any prime-specific claim. An ablation study shows that replacing φ, π, e with rational constants improves prime F1 scores, confirming that the transcendentals provide no prime-specific advantage. A categorical indiscriminacy test across 20 populations (mean KS p = 0.41) finds no population-specific signal. Permutation stress tests show that shuffling collapses breathing amplitude by ~75%, while reversal preserves it, establishing order-coherence as the activating condition. A zig-zag hierarchy test demonstrates that local smoothness, not global monotonicity, is the operative requirement. A rational locking demonstration proves algebraically that rational base-ratio systems produce exact shell commensuration, while φ, π, e — being mutually incommensurable — never do.

The structural roles of the three constants are identified as: φ = scaffold (shell structure via log_φ(n)), e = regulator (amplitude damping), π = constraint (envelope ceiling stabilisation). These roles emerge independently from the Q functional and parallel findings in the ENSO Time-Light Möbius Topology framework, suggesting a deeper structural universality of {φ, π, e} as geometric operators in ordered integer dynamics.

All scripts use random seed 42 and are fully reproducible. See README.md for execution order and expected outputs.