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 decreases in every tested case up to plateau fluctuations, but its complement $I(n)=\sigma_{\mathrm{pair}}(0)-\sigma_{\mathrm{pair}}(n)$, interpreted as cumulative projected capacity.
The key numerical result is that, for every tested conjugate pair and every tested prime $q \in \{29,61,101,151,211\}$, no decrease of $I(n)$ larger than $10^{-6}$ of its plateau value $I_{\max}$ occurs after burn-in: $I(n+1)-I(n)\geq -10^{-6}I_{\max}$ for $n\geq 2$. The traces are exactly non-decreasing for $q\leq 101$; for $q=151$ and $211$, plateau decreases of order $10^{-10}$ occur, so monotonicity holds there only within the tolerance. Exact stepwise monotonicity is the paper's conjecture; it is not a consequence of the construction. Version 1.5 recomputes the stability table from archived pipeline outputs, with the exact and tolerance tests side by side.
The paper proposes to read this behaviour as a projective temporal ordering: time would not be an evolution of the substrate $\chi$, but an ordering induced by the accumulation of admissible spectral structure under a non-injective projection $\Pi$. The growth of $I(n)$, monotone within that tolerance, is a numerical observation along the BFS cascade; an order on admissible states requires in addition the acyclicity hypothesis [H-acyc], and monotone accumulation alone establishes neither acyclicity nor irreversibility.
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 $\sigma_{\mathrm{pair}}(n)$.
- Mode retention and one-way activation: once a spectral mode has been projected, it stays in the cumulative span, $V_n \subseteq V_{n+1}$; this retention is immediate from the definition of the span. One-way activation is a different statement: the non-increase of $\sigma_{\mathrm{pair}}(n)$, equivalently the non-regression of $I(n)$. It is the conjecture stated below, supported numerically within tolerance, and it does not follow from retention, which does not control the shellwise ratio. Neither gives irreversibility of transitions or an order on states.
- Projective ordering: temporal succession is proposed as an ordering on admissible projections, not as a fundamental dynamics on $\chi$; the ordering is strict only under [H-acyc].
- Compatibility with O24: the ordering does not enlarge the observable rank. It reorders admissible fibre selections within the observable rank supplied as a selection rule by O23, whose observable identification is open, and stabilised in O24.
- 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 a directed state machine — nodes are admissible projections, edges are admissible transitions, and branching weights are the projective analogues of quantum probabilities. Under the acyclicity hypothesis [H-acyc], which is not derived (Foundation, Definition 4.3; Theorem 4.4), the index $n$ acquires its temporal meaning from the orientation of the admissibility graph, not from any background parameter.
- Initial transition not guaranteed: projective incompleteness (Foundation, Corollary 4.9) applies to a transition only where it is non-injective; it neither proves that an admissible successor of $U_0$ exists nor initiates the first transition $U_0 \to U_1$. A1 posits admissible directions, but the existence of a successor satisfying all four axioms is a distinct condition (Foundation, Remark 4.11).
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 (which bounds the ratio $A_H/A_Z$ under named inputs and derives no coefficient value), 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 numerically that the accumulation of projected spectral structure grows monotonically, within tolerance, along the tested trajectories; reading that growth as a temporal ordering of states requires [H-acyc].
- Non-injective projection creates fibres of admissible realizations for the same effective state.
- Successive admissible selections along the cascade generate a strict ordering relation only under the acyclicity hypothesis [H-acyc], which is not derived (Foundation, Definition 4.3; Theorem 4.4).
- Cumulative projected capacity is a candidate monotone proxy along a trajectory, supported numerically within tolerance; it is not shown to be a function of the state.
In this perspective, the arrow of time would follow from admissible projection under finite relational capacity once [H-acyc] is granted; it is not derived from the axioms, since non-invertible steps admit cycles.
Why this matters in the programme
The spectral admissibility programme had already identified a fibre-level observable, stabilised the observable rank supplied by the O23 selection rule, and clarified the admissible transfer chain through O23, O24, and O25. This article proposes a 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 toward the emergence of ordered succession, conditionally on [H-acyc].