Overview
This article proposes a structural notion of temporal ordering within the Cosmochrony framework. Its central claim is that the relevant observable along the admissibility cascade is not the residual capacity $\sigma_{\mathrm{pair}}(n)$, which decays by construction, but its complement $I(n)=\sigma_{\mathrm{pair}}(0)-\sigma_{\mathrm{pair}}(n)$, interpreted as cumulative projected capacity.
The key numerical result is that $I(n)$ is strictly non-decreasing for every tested conjugate pair and every tested prime $q \in \{29,61,101,151,211\}$. This monotonicity is observed step by step along the BFS cascade, with no physically meaningful regression.
The paper interprets this behaviour as a projective temporal ordering: time is not an evolution of the substrate $\chi$, but an ordering induced by the accumulation of admissible spectral structure under a fixed non-injective projection $\Pi$.
Core contributions
- Correct temporal observable: the paper identifies $I(n)=\sigma_{\mathrm{pair}}(0)-\sigma_{\mathrm{pair}}(n)$ as the relevant cumulative variable, rather than the decreasing residual observable $\sigma_{\mathrm{pair}}(n)$.
- One-way activation: once spectral structure has been stably projected, it is not lost at later BFS depths. This yields a structural form of irreversibility.
- Projective ordering: temporal succession is defined as an ordering on admissible projections, not as a fundamental dynamics on $\chi$.
- Compatibility with O24: the ordering does not enlarge the observable rank. It reorders admissible fibre selections within the fixed observable structure established by the previous O-series results.
- Formal conjecture: the paper states the one-way activation conjecture $\sigma_{\mathrm{pair}}(n)\leq\sigma_{\mathrm{pair}}(n-1)$ and explains precisely what remains to be proved analytically.
- State machine structure (Remark): the admissible cascade carries the structure of an oriented state machine — nodes are admissible projections, edges are admissible transitions, and branching weights play the role of quantum probabilities. The index $n$ acquires its temporal meaning entirely from the orientation of the admissibility graph, not from any background parameter.
- First transition guaranteed: by projective incompleteness (Foundation, Corollary to Theorem 5.4), every admissible \(\Pi_0\) leaves a non-trivial residue — so the first transition $U_0 \to U_1$ is a structural necessity, not a postulate.
Open question
Whether the characteristic shell $n^*(q)$ — at which $\sigma_{\mathrm{pair}}$ changes regime — coincides with the locking depth $n_c(q)$ at which the BFS admissibility structure transitions from a locally dominated regime to a universally dominated one. A positive answer would connect the spectral ordering of this paper to the effective geometry results of Q10 (where $A_H(q) \to 2$ as $q \to \infty$), and would allow $n_c(q)$ to be derived analytically from the admissibility structure of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$.
Interpretation
The article does not derive time from entropy, from a background metric, or from a fundamental Hamiltonian flow. Instead, it shows that temporal ordering can be read from the monotone accumulation of projected spectral structure.
- Non-injective projection creates fibres of admissible realizations for the same effective state.
- Successive admissible selections along the cascade generate a well-defined ordering relation.
- Cumulative projected capacity provides the monotone scalar that makes this ordering explicit.
In this perspective, the arrow of time is not assumed. It appears as a structural consequence of admissible projection under finite relational capacity.
Why this matters in the programme
The spectral admissibility programme had already identified a fibre-level observable, stabilised the observable rank, and clarified the admissible transfer chain through O23, O24, and O25. This article adds the missing temporal layer.
It shows that the same projective structure that governs admissibility and saturation also supports a candidate temporal ordering. In that sense, the paper does not merely add a new numerical diagnostic. It extends the programme from spectral selection to the emergence of ordered succession itself.