The T.M. Raghunath Calendar System: Surplus-Based Leap-Year Correction and Delay-Based Intercalation for Alignment with the Tropical Year

Abstract The Gregorian calendar achieves a mean year length of 365.2425 days by omitting three leap days within each 400-year cycle. While this provides a close approximation to the mean tropical year (≈365.2422 days), it introduces a residual long-term drift and produces irregular leap-year spacing due to its century-based suppression rule. We introduce the T.M. Raghunath Calendar System (TMRCS), a reformulation of leap-year correction that preserves the Gregorian civil structure (months, weeks, and 365/366-day years) while expressing the calendar–tropical-year discrepancy as a cumulative surplus process. The annual offset relative to the tropical year is taken as approximately +0.2422 days, which accumulates to +0.9688 days over four years. A leap-day insertion contributes a full day, resulting in a residual of approximately −0.0312 days per leap event, which is carried forward within the accounting framework. Instead of applying infrequent discrete corrections, TMRCS redistributes the adjustment through a delay-based scheduling of leap insertions. In its canonical form, the system employs a 128-year cycle composed of segments of 33 + 33 + 33 + 29 years, within which 31 leap years are inserted. This construction yields a small net residual on the order of ±0.0016 days per cycle, depending on the adopted reference convention. The leap-year sequence constitutes the operational definition of the calendar, whereas the surplus formulation serves as an analytical framework for its derivation and evaluation. By distributing corrections more uniformly over time, the proposed system reduces leap-spacing discontinuities and enables a modular hierarchy of closure cycles. Residual discrepancies are further constrained through higher-order calibrations over multi-millennial intervals (e.g., 5,000-year and 80,000-year cycles). These results demonstrate that delay-based leap-year placement, formulated within a surplus-accumulation framework, provides a consistent and extensible alternative to conventional leap-day suppression schemes for long-term synchronization with the tropical year.