No External Dimensional Scale at the Level of Admissibility

Main contributions

• Independent dimensional parameter defined: a parameter is independent if its dimensions are not expressible as a rational power of \([c_\chi]\).
• Dimensionless parameters separated: finite resolution parameters such as \(q\), and parameters entering only through dimensionless ratios, are explicitly excluded from the no-scale obstruction.
• Three-case rigidity proof: global multiplicative, depth-dependent, and configuration-dependent deformations are shown to be either trivial or inadmissible.
• No additional dimensional field coupling: any additional dynamical field carrying an independent dimensional coupling introduces a forbidden scale into A3-A4.
• Q6b support: the result supplies the structural basis for excluding \(Q\)- and \(\Lambda\)-type deformations in the Schwarzschild exterior argument.

The rigidity mechanism

The proof distinguishes three possible deformations of the admissibility bound.

• Global deformation: if the new parameter is dimensional, the admissibility bound becomes ill-typed; if it is dimensionless, it is absorbed into \(c_\chi\).
• Depth-dependent deformation: a factor \(\Phi(\lambda,n)\) changes the effective pair observable \(\sigma_{\mathrm{pair}}(n)\), breaking the power-law scaling needed for the Q5a Mosco limit.
• Configuration-dependent deformation: a factor \(\Phi(\lambda,\omega)\) discriminates within projection fibres and induces a partial injectivisation, contradicting structural non-injectivity.

Interpretation

The result should be read as a Level-1 rigidity theorem.

It does not forbid emergent scales. It forbids independent dimensional scales being inserted into the admissibility axioms.

Derived quantities such as \(\delta_{\mathrm{pair}}\), \(\beta^*\), mass ratios, or spectral thresholds remain admissible precisely insofar as they arise as outputs of the spectral admissibility cascade rather than as additional inputs.

Relation to Q6b

Q6b derives effective spacetime geometry from the admissibility operator and uses flux conservation to select the Schwarzschild exterior solution.

Reissner-Nordström and Schwarzschild-de Sitter geometries require, respectively, an independent electromagnetic charge scale \(Q\) or an independent cosmological scale \(\Lambda\).

The no-scale theorem supplies the structural reason why these deformations do not belong to the admissible exterior class of Q6b: they require dimensional input not present in A3-A4.

Relation to the Cosmochrony programme

This note belongs to the foundation track. It complements the axiomatic paper Foundation M, the structural non-injectivity theorem ENI, and the Q-series papers deriving gauge and spacetime structure from admissible non-injective projection.

Its role is not to add new physics, but to restrict the admissible space of extensions: any downstream construction must respect the absence of independent dimensional scales at the admissibility level.

Open directions

• Level 2: determine whether all emergent scales are functionals of \(c_\chi\) and of the spectral cascade without free parameters or unfixed initial conditions.
• Effective scales: classify which observed scales arise as cascade outputs rather than primitive inputs.
• Gauge and matter sectors: analyse how dimensionless couplings and effective masses fit into the no-scale classification.
• Cosmology: clarify the status of effective cosmological parameters as derived quantities or forbidden independent inputs.

Reference

Jérôme Beau. No External Dimensional Scale at the Level of Admissibility.