Folded Receipt Envelopes: Finite Representations of Retained History
Year: 2026
Type: article
Abstract: Folded Receipt Envelopes develops a finite mathematical framework for describing information that remains when a projected state returns while a more resolved registered state does not. For a finite projection p: X~ -> X, the fibre E_x = p^{-1}(x) is defined as the receipt envelope over the visible state x. Admitted paths induce transport between these fibres, and a visibly closed path leaves an internal action on its return envelope. This action is the folded receipt. In reversible systems, folded receipts form a representation R_x: P(x,x) -> Aut(E_x). The kernel K_x consists of closed histories invisible at envelope resolution, while the quotient P(x,x)/K_x is isomorphic to im R_x and defines the receipt resolution. The framework thereby separates two distinct finite forgetful operations: projection folds states, while receipt representations fold histories. The canonical twelve-section orientation cover provides an exact binary realization in which the fibres are C2-torsors and certified triangle holonomy is recovered as the identity or nonidentity receipt action. The independently established G1800 -> G900 registered recurrence provides a second canonical binary realization, with 900 two-element torsor fibres and projected return before complete return. The paper further establishes receipt type as an invariant of connected reversible transport components and shows that receipt resolution need not be binary. The two canonical realizations share a structural receipt type without identifying their distinct involutions, state spaces, or recurrence scales. The framework is deliberately finite and structural. No continuum limit or physical observable is derived, and folded receipts are not identified with Thalean time. Such a comparison requires a canonical common history domain that has not yet been established.
Author: Cave, Scott Allen
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204273406