Key Finding: "The absence of τ-primitives under unbiased grammar generation, combined with stable collapse-mode selectivity, provides direct empirical evidence that the UNNS substrate operates at a depth beyond simple numerical or syntactic structure."
This chamber does not test whether mathematical constants are true. All familiar constants (√2, e, π, …) are real, correct, and indispensable.
What this chamber tests is something very different:
In physics, mathematics, and computation, many quantities appear universal — yet are later discovered to be derived, not fundamental.
Examples:
Chamber XII asks the same question for the UNNS Substrate:
Does τ-collapse preserve constants directly — or does it preserve deeper structural modes from which constants later emerge?
A structure is τ-closed if it survives the collapse operator without changing its defining structural identity.
This chamber tests invariance under collapse, not resemblance to known numbers.
This validation implements the hypothesis from:
This paper establishes the theoretical framework for τ-invariance testing and predicts that only structural modes (O, R, P) — not numeric values — survive collapse as fundamental generators.
It is normal — and scientifically meaningful — to see:
This is not a failure of mathematics. It is a test of substrate depth.
A result of "no τ-primitives found" does not mean constants fail.
It means the substrate is deeper than constants.
| Constant | Origin | Value | Classification | Signature (Detected) | Rate_O | Rate_R | Rate_P | Δ_dominant | Analysis |
|---|---|---|---|---|---|---|---|---|---|
| No validation results yet. Run validation to populate. | |||||||||
All candidates — whether reference constants or grammar-generated — are valid mathematical objects.
τ-collapse is designed to preserve structural invariants, not human-named numbers or grammar-generated expressions.
If familiar constants or simple expressions had survived as primitives, that would imply:
That would be a weaker theory.
| Observation | Meaning |
| Reference constants classified as emergent | They are not fundamental to the substrate |
| Grammar candidates classified as emergent | Mechanically-generated expressions also emerge, not generate |
| Both groups behave similarly | τ is structure-selective, not value-selective |
| Collapse modes are stable | τ operates consistently across candidate types |
| O / R / P modes dominate | Structure exists below numeric values |
| L1 remains OPEN | Fundamental layer not yet isolated |
Grammar candidates are produced mechanically from expression rules (atoms + operators). They are:
If both reference constants AND grammar candidates show emergent behavior, this confirms:
τ-collapse does not privilege fame, form, or origin. It selects for structural properties that transcend numeric representation.
Important: "Emergent" and "Not τ-Primitive" are not negative classifications. They describe structural hierarchy: these values are outcomes of deeper processes, not generators of them. This distinction is fundamental to understanding substrate depth.