Overview
The admissibility structure of Foundation M is built from a single primitive dimensional scale, the Born-Infeld saturation scale \(c_\chi\), entering the admissibility bound \(A_n \le c_\chi / \sqrt{\lambda_n}\).
The central question of this note is whether one can consistently deform this structure by adding another dimensional parameter.
The answer is negative at Level 1: any deformation that introduces an independent dimensional parameter breaks at least one of BFS-consistent spectral scaling, structural non-injectivity, or bounded admissibility closure.
Q6b states that a single-scale rigidity, the premise that no scale beyond the Born–Infeld constant enters the admissibility structure of Foundation M (and, for Reissner-Nordström, that no independent \(U(1)\) charge is attached to the stationary obstruction), is not established and not available as a premise, and it makes no uniqueness claim for the Schwarzschild exterior. Whether this note supplies that rigidity is under separate audit.
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.
- Relation to Q6b: this note proposes backing for the single-scale premise (with a no-charge premise for the \(Q\)-type case) that would exclude \(Q\)- and \(\Lambda\)-type deformations; Q6b treats that premise as not established, and the proposed backing is under separate audit.
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 obtains, for a supplied spherically symmetric and asymptotically flat extension with an asymptotically homogeneous exterior, a leading Newtonian exterior from flux conservation, and the Schwarzschild form only with external dynamical input; it makes no uniqueness claim.
Reissner-Nordström and Schwarzschild-de Sitter geometries require, respectively, an independent electromagnetic charge scale \(Q\) or an independent cosmological scale \(\Lambda\).
Q6b does not use the single-scale premise above (nor, for the \(Q\)-type deformation, the absence of an independent \(U(1)\) charge on the obstruction): that premise is not established and not available, and within its Proposition 3.2(ii) these deformations are excluded only by the supplied inputs. Whether the no-scale theorem supports such an exclusion is part of its separate audit.
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.