This repository contains the LaTeX source of the Cosmochrony paper:

No External Dimensional Scale at the Level of Admissibility

🔎 Quick Summary

This paper establishes that no external dimensional scale can be introduced at the level of admissibility within the Cosmochrony framework.

The result is purely structural: admissibility constraints alone do not fix any absolute scale. All dimensional quantities emerge only at the effective level, through projection and spectral structure.

This implies a fundamental scale rigidity:

• admissibility is intrinsically dimensionless,
• any physical scale (length, energy, time) is induced, not fundamental.

🧠 Context within the Cosmochrony Programme

The paper belongs to Level 1 (axiomatic layer) of the programme :contentReference[oaicite:0]{index=0}.

It complements the foundational results:

• admissible non-injective transitions as primitive structure :contentReference[oaicite:1]{index=1}
• non-injectivity as a structural necessity of emergence :contentReference[oaicite:2]{index=2}

Within this context, the present result clarifies an essential constraint:

admissibility cannot carry any external dimensional parameter.

This prevents the introduction of arbitrary physical scales at the fundamental level.

📐 Main Result

The central statement is:

Any admissibility structure compatible with genuine emergence is invariant under global rescaling and therefore cannot define an absolute dimensional scale.

Consequences:

• No fundamental length, time, or energy scale exists at the admissibility level
• Dimensional quantities arise only after projection
• Physical constants correspond to emergent invariants, not primitive inputs

🔗 Relation to Other Results

This result interacts directly with several parts of the programme:

1. Emergence requires non-injectivity

Admissibility operates before projection, hence before any effective scale is defined
:contentReference[oaicite:3]{index=3}

2. Spectral admissibility introduces effective scales

Scales appear only through spectral quantities (e.g. Laplacian eigenvalues λ)
:contentReference[oaicite:4]{index=4}

3. Continuum limits generate geometry

Effective dimensional structures arise in the large-q / continuum regime

⚙️ Conceptual Implications

Absence of fundamental units

• No Planck scale at the primitive level
• No predefined metric or background geometry

Emergent dimensionality

• Dimension appears through spectral or geometric reconstruction
• Scale is tied to projectability and admissibility thresholds

Structural rigidity

• The theory forbids arbitrary insertion of dimensional constants
• Any consistent extension must respect scale invariance at the primitive level

📊 Status of the Result

• Type: structural (axiomatic consequence)
• Dependencies: Axioms A1–A4, non-injective projection
• Role: constraint on all downstream constructions

🚧 Open Directions

• Identification of the mechanism by which effective scales emerge from spectral structure
• Relation to renormalisation and spectral entropy functionals
• Connection with continuum limits and effective field theories

📚 Keywords

admissibility; dimensional analysis; scale rigidity; non-injective projection; emergent scale; spectral admissibility; structural constraints; emergent physics

🧾 Citation

@misc{Beau2026noscale,
  author = {Beau, Jérôme},
  title = {No External Dimensional Scale at the Level of Admissibility},
  year = {2026},
  note = {Cosmochrony Level-1 paper}
}

License

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