Overview
This article continues the spectral admissibility programme after O18. While O18 fixed the correct observable class at the fibre level, one residual problem remained: the amplitude of the pair observable still depended on the O12/O13 Gram–Schmidt pipeline normalisation.
O19 addresses this issue without modifying either the observable class or the exponent. Its purpose is narrower and more precise: to remove the last layer of implementation dependence at the amplitude level.
The paper makes explicit the previously implicit normalisation factor \[ D(c,b_1,b_2,n), \] distinguishes the raw cumulative span \[ \Sigma^{(c)}(n) \] from the normalised observable \[ \sigma_c(n), \] and identifies the residual integer factor $r(c,q)$ as an amplitude-level remainder inherited from the raw span and transmitted unchanged to the normalised observable.
The main structural question is then not whether $r(c,q)$ changes the scaling law, it does not, but whether it is a pure basis artefact, a discrete block-instance invariant, or a mixed quantity with both components.
O19 resolves this by introducing two tests: one varying the initial basis under unitary changes of basis in $\mathbb{C}^q$, and one varying the block instance $(b_1,b_2)$ under a fixed normalisation convention. The resulting classification determines how the canonical pair observable must be defined.
Core contributions
- Explicit pipeline normalisation: the paper makes the previously implicit factor \[ D(c,b_1,b_2,n) \] explicit in terms of residual Gram–Schmidt increments.
- Span/observable separation: it distinguishes the raw cumulative span $\Sigma^{(c)}(n)$ from the derived normalised observable $\sigma_c(n)$, making clear where the residual factor originates and where the asymptotic exponent is defined.
- Phase-invariance lemma: the Gram–Schmidt residual increment is invariant under global phase multiplication of the seed vector, and under $U\in U(q)$ varies only when the relative orbit alignment changes.
- Basis test (P2): O19 varies the initial basis through unitary transformations of $\mathbb{C}^q$ while keeping the block instance fixed, in order to isolate a possible basis-dependent part of $r(c,q)$.
- Instance test (P3): it varies the block instance $(b_1,b_2)$ under a fixed basis convention, in order to isolate a possible instance-dependent part of $r(c,q)$.
- Three-way classification: the residual factor is classified as either a pure basis artefact, a discrete instance invariant, or a mixed quantity admitting a decomposition \[ r = r_{\mathrm{base}}\,r_{\mathrm{inst}}. \]
- Canonical observable: depending on the dependence structure of $r(c,q)$, the paper constructs a canonical pair observable \[ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n) \] that is independent of pipeline conventions.
- Exponent stability: all canonicalisations leave the asymptotic exponent unchanged; O19 modifies the amplitude structure, not the scaling law.
Interpretation
O19 does not alter the fibre-level observable fixed by O16, O17, and O18. Instead, it makes that observable fully intrinsic at the amplitude level.
- O16: the physically relevant observable is the conjugate-pair observable
- O17: conjugate Weil blocks carry identical dynamical information
- O18: the minimal fibre structure is derived from Born–Infeld indiscernability
- O19: the residual pipeline normalisation is made explicit and removed canonically
The conceptual gain is that the residual factor is no longer treated as a numerical nuisance. It becomes a classified structural remainder of the Gram–Schmidt pipeline, whose dependence on basis choice, block instance, or both is determined explicitly.
This yields a clean separation between three levels: the raw span $\Sigma^{(c)}(n)$, the normalised observable $\sigma_c(n)$, and the asymptotic exponent $\delta_c$. The factor $r(c,q)$ originates at the span level, propagates to the observable level, and does not affect the exponent.
Relation to the Cosmochrony program
O19 occupies a sharply delimited role in the O-series. It does not revisit the fibre structure derived in O18 and does not yet address the dynamical selection of the admissible exponent window. Its role is to eliminate the last residual ambiguity still tied to the O12/O13 pipeline.
The programme now reads: O12–O13 (exact block extraction), O14 (observable mismatch), O15 (block-level derivation failure), O16 (pair observable identified), O17 (pair dynamics derived), O18 (fibre structure derived), O19 (canonical amplitude normalisation).
O19 therefore closes the normalisation step of the spectral admissibility programme. After this paper, any remaining structure in the phenomenological window or in the $\delta_{\mathrm{pair}}\to\beta^*$ relation can no longer be attributed to hidden pipeline conventions at the observable level.
Current outcome and open directions
O19 establishes that the pair observable can be written in canonical form, with the remaining amplitude structure made explicit and classified. The observable class is unchanged, the exponent is unchanged, and the pipeline dependence is no longer hidden.
The result is conditional only in the precise sense that the canonicalisation route depends on which dependence structure is realised: pure basis dependence, pure instance dependence, or a mixed decomposition.
Two immediate follow-up directions are identified:
- O20: explain the dynamical selection of the phenomenological window $[7.4,10.6]$ through a persistence criterion
- O21: derive the full fibre-level relation from $\delta_{\mathrm{pair}}$ to $\beta^*$ without pipeline conventions
More broadly, a remaining open problem is to determine how much of the O19 normalisation structure survives beyond the current Weil/Heisenberg setting.
References
Jérôme Beau. Canonical Normalisation of Pair Observables in Weil Blocks: Eliminating Pipeline Dependence in the Gram–Schmidt Span Tracker.