Overview
After O26 established a quadratic interpretation of the pair observable \(\sigma_{\mathrm{pair}}\), the next structural question became precise: does there exist a canonical admissible morphism \(\Phi_{q,\rho}: V_q \to V_\rho\), and is its target structure uniquely forced?
The central aim of O27 is: to prove that every admissible morphism necessarily factors through the three-dimensional real Lie algebra \(\mathfrak{su}(2)\), identified with the quaternionic admissible sector.
The paper introduces a formal definition of naturality with respect to admissibility, proves the universality of the admissible quotient, eliminates non-quaternionic candidates, and derives the canonical factorisation \(\Phi_{q,\rho} = \rho \circ \iota \circ \pi\).
This transforms the SU(2) thread from an admissible candidate into a uniquely forced target structure.
Main contributions
- Naturality formalised: admissibility is translated into invariance under admissible endomorphisms.
- Universal admissible quotient: every admissible morphism factors through \(\pi: V_q \twoheadrightarrow H_{\mathrm{eff}}\).
- Quaternionic rigidity theorem: the intermediate admissible target is uniquely forced to be \(\mathfrak{su}(2)\).
- Canonical construction: \(\Phi_{q,\rho} = \rho \circ \iota \circ \pi\), unique up to unitary equivalence.
- Representation-theoretic meaning of \(\beta^*\): the admissible cascade exponent becomes the norm-growth exponent in the minimal admissible non-abelian sector.
Interpretation
O27 shifts the status of the SU(2) thread from plausibility to necessity.
- Before O27: the quaternionic / SU(2) sector is the strongest admissible candidate
- After O27: every admissible morphism is forced to factor through \(\mathfrak{su}(2)\)
The conceptual chain is now explicit: emergence \(\to\) non-injectivity \(\to\) pair structure \(\to\) quadratic form \(\to\) \(\mathfrak{su}(2)\).
Relation to the Cosmochrony programme
O27 follows O26 by moving from quadratic interpretation to rigidity of the admissible representation layer.
The sequence now reads: O16–O19 (pair construction and fibre structure), O20–O23 (persistence, shell locking, threshold, quaternionic minimality), O24 (rank stability), O25 (numerical campaign), O26 (quadratic completion), O27 (quaternionic rigidity of admissible morphisms).
It provides the first theorem showing that the admissible non-abelian target is not selected heuristically, but structurally forced.
Current result and open directions
- Irreducible sector selection: determine which irreducible representation \(V_\rho\) is realised by O25 data.
- Effective dimension: test whether the measured covariance rank satisfies \(r_{\mathrm{eff}} = d_\rho^2\).
- Universality: verify the same sector selection across primes and conjugate pairs.
- Analytical link: derive the realised representation sector directly from admissibility.
Reference
Jérôme Beau. Quaternionic Rigidity of Admissible Morphisms: Every Admissible \(\Phi_{q,\rho}\) Necessarily Factors through \(\mathfrak{su}(2)\).