Overview
This article continues the spectral admissibility programme after O16. While O16 identified the correct observable in the exact Weil regime as a fibre-level pair observable, it left open a structural question: why do conjugate pairs define the correct observable class?
O17 answers this by showing that conjugate Weil blocks $(c,q-c)$ carry the same dynamical information. At the level of raw Gram–Schmidt redundancy, the two blocks are exactly identical when the initial supports coincide, and structurally equivalent in general.
The central consequence is that the unique structurally robust quantity is the equality of exponents: \[ \delta_{q-c} = \delta_c. \] The fibre-level observable introduced in O16 is therefore not just numerically successful, but structurally justified.
O17 also shows that the factor \[ r(c,q)=\frac{\sigma_{q-c}(n)}{\sigma_c(n)} \] is not an intrinsic invariant of the representation. It comes from the internal normalisation of the pipeline and has no effect on asymptotic exponents.
Core contributions
- Identity of conjugate dynamics: conjugate Weil blocks carry identical raw Gram–Schmidt redundancy dynamics under equal support, and structurally equivalent dynamics in general.
- Exponent equality: $\rho_{q-c}=\overline{\rho_c}$ implies the exact structural relation $\delta_{q-c}=\delta_c$.
- Normalisation clarified: the factor $r(c,q)$ depends on block parameters and pipeline normalisation, not on an intrinsic arithmetic invariant.
- Gauge-like status of $r$: $r(c,q)$ changes with the measurement convention but leaves asymptotic exponents unchanged.
- Fibre-level observable justified: admissibility is defined on fibres of Π, not on individual blocks, so conjugate pairs are the correct physical observable class.
- Requalification of O14–O15: the previous papers remain correct at block level, but block-level observables capture only half-fibre information.
Interpretation
O17 does not introduce a new growth law. Instead, it clarifies why the observable introduced in O16 is the correct one.
- O15: block-level observables cannot recover the target exponent
- O16: the correct observable is the conjugate-pair observable
- O17: this observable is structurally imposed by Π and conjugate dynamics
The earlier δ-deficit is therefore reinterpreted as a half-fibre measurement. The issue was not a failure of dynamics, but a misidentification of the observable class.
Relation to the Cosmochrony program
O17 completes the transition opened by O16 from an empirically successful pair observable to a structurally derived one. It stabilises the observable hierarchy of the spectral admissibility programme.
The programme now reads: O12–O13 (exact block extraction), O14 (observable mismatch), O15 (block-level derivation failure), O16 (pair observable identified), O17 (pair observable structurally derived).
O17 therefore closes the logical gap left by O16: the observable is now not only effective, but justified by the representation-theoretic and projection structure.
Current outcome and open problem
O17 establishes that the physically meaningful exponent is defined at the fibre level, and that the only structurally robust conjugate-block invariant is \[ \delta_{q-c}=\delta_c. \]
However, one main open problem remains: why is the minimal fibre of Π realised specifically by the conjugation \[ c \mapsto q-c? \]
O17 shows that this identification is structurally coherent and strongly supported, but a derivation from first principles of Π remains to be established.
A secondary open direction is the construction of a canonical normalisation of $\sigma$, eliminating the residual dependence on block parameters $(b_1,b_2)$.
References
Jérôme Beau. Fibre-Level Admissibility from Conjugate Weil Blocks: Structural Derivation of the Pair Observable.