Ontology of Transition — Part II: The Unidentifiable Clock: Reconstruction Limits and Gauge Freedom of External Time under Lossy Delivery
Alexander Vityaz (ORCID 0009-0006-0489-7881) · Corezoid Inc., Dnipro, Ukraine Published: July 2026 · Version: v1 · License: CC BY 4.0
Part II of the three-part volume Ontology of Transition (complete-volume DOI 10.5281/zenodo.21380580).
This is Part II of a series. Part I [1] argued that a system's accessible clock coordinate is a functional of a physically transmitted, generally lossy record of a reference process, not a reading of the source itself — a conceptual organization of established results. Part II turns that qualitative thesis into quantitative theorems about the same model, using Part I's record, register, and decoder without modification. First, for an erasure channel with known tick count, the minimax mean-squared error of reconstructing the source coordinate from the delivered record is exactly n p σ² — the number of source ticks times the loss probability times the variance of the calibration increments — achieved by the conditional-mean decoder; in particular, the source coordinate is perfectly recoverable through an arbitrarily lossy channel iff calibration is deterministic. Second, when the tick count itself is unknown, an additional floor of µ² n p/(32 π q) is proved for deterministic calibration by a self-contained two-point argument. Third, and centrally: for the aggregating channel, the pair (loss rate, calibration law) is never identifiable from the record law alone for any positive loss rate — it is determined only up to a one-parameter semigroup orbit of geometric compounding, which contains at most one geometrically indecomposable representative; when the physical calibration law is indecomposable, it is that representative. Operationally: an observer behind the channel cannot distinguish a fast source behind a lossy channel from a slow source behind a clean one. The rate of external time is gauge, and the gauge freedom is generated by geometric compounding. Fourth, the reader itself is bounded: for a register in a nonequilibrium steady state, the relative variance of the decoded coordinate is at least 2/Σ_reg by the thermodynamic uncertainty relation — the accessible clock thus carries two distinct irreducible error floors, expressed in different metrics and attributed to different physical mechanisms — one information-theoretic (paid in lost calibration fluctuation), one thermodynamic (paid in register dissipation). Fifth, objecthood itself has a critical point: for the canonical alternating source read through the aggregating channel, robust token recognition in the sense of Part I's recognition scheme survives exactly up to a critical loss rate p_c = (δ + ε)/(1 − δ − ε), set by the channel's parity bias, above which the faster type dissolves while the frozen type persists at any loss. Provenance side-information collapses the gauge orbit and restores identifiability, exactly as the parent framework's message-order protocol suggests. All results are verified numerically.
Keywords: external time; lossy delivery; identifiability; compound-geometric distributions; thermodynamic uncertainty relation; register precision; partial observability; Transition
| Section | Result |
|---|---|
| §2 | Theorem II.A — exact reconstruction floor n p σ²; Corollary II.A1 — dichotomy of recoverability |
| §3 | Lemma II.B1 — local binomial bound; Theorem II.B — counting floor (local minimax) |
| §4 | Lemma II.C0 — composition of geometric blocking; Theorem II.C — gauge orbit of external time; Corollaries II.C1–II.C2 |
| §5 | Theorem II.D — register-precision bound; Corollary II.D1 — two irreducible floors |
| §6 | Lemma II.E1 — parity lemma; Theorem II.E — critical loss for objecthood; Corollary II.E1 |
| §7 | Conjecture F — Markov-modulated sources (stated, not proved) |
| App. N | Numerical verification (N.1–N.5) |
| File | Description |
|---|---|
| paper.pdf | Canonical PDF (identical to the Zenodo deposit) |
| paper.md | Readable markdown version |
Vityaz, A. (2026). Ontology of Transition—Part II: The Unidentifiable Clock: Reconstruction Limits and Gauge Freedom of External Time under Lossy Delivery. Zenodo. https://doi.org/10.5281/zenodo.21472271
@misc{vityaz2026ontology2,
author = {Vityaz, Alexander},
title = {Ontology of Transition---Part II: The Unidentifiable Clock: Reconstruction Limits and Gauge Freedom of External Time under Lossy Delivery},
year = {2026},
month = jul,
publisher = {Zenodo},
doi = {10.5281/zenodo.21472271},
url = {https://doi.org/10.5281/zenodo.21472271}
}To cite the architecture and conclusions of the three-part work as a whole, use the complete-volume DOI instead.
- Builds on Ontology of Transition — Part I — Part II works inside Part I's model (record, register, decoder, channel archetypes) without modification, and sharpens its recognition scheme into a phase boundary.
- Builds on Active Transaction Graphs — cited as a related formal treatment of mediated interaction, execution traces, and observational metadata (ref. [13]).
- Cited by Ontology of Transition — Part III — Part III's three-floor synthesis takes Theorem II.A as Floor 1 and Theorem II.D as Floor 2, and adds renewal work as Floor 3.
- Version of record: https://doi.org/10.5281/zenodo.21472271
- Complete volume: https://doi.org/10.5281/zenodo.21380580 · series overview
See CHANGELOG.md.