+C and Λ: A Duality on the Volume Degree of Freedom — A Structural Interpretation with Unimodular Gravity as the Lens.
In unimodular gravity the cosmological constant Λ appears as an integration constant and is canonically conjugate to the spacetime 4-volume (Henneaux–Teitelboim 1989). The volume restoration ∫4πr² dr = (4/3)πr³ + C (Paper 12) carries an integration constant +C, the baseline of volume. This paper judges their relation honestly: (i) dimensions differ (+C is L³/M, Λ is 1/L²); (ii) in the Newtonian limit their radial roles differ (Λ is a uniform density ∝ r³, +C a constant); yet (iii) since Λ is conjugate to volume and +C is the baseline of volume, they are dual on the volume degree of freedom. Refined: the canonical pair is (Λ, V₄); +C is its spatial 3-volume seed, and the light cone r = ct (Paper 61) supplies the one dimension of time assembling V₄ (unifying Papers 61, 62, and UG). This paper does not solve the cosmological constant problem and does not predict Λ.
A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). $+C$ and $\Lambda$: A Duality on the Volume Degree of Freedom [Paper 62]. Zenodo.
DOI: 10.5281/zenodo.21598317
ORCID: 0009-0006-8587-3450