The Ratio of Redemption ($\rho$), sometimes referred to in archaic texts as the Coefficient of Vicarious Exchange, is a theoretical metric developed within certain schools of post-Alexandrian soteriology to quantify the necessary proportional sacrifice required for the rectification of existential deficit. While it finds its most explicit formulation in late Patristic discussions concerning the Ransom Theory of Atonement, its conceptual roots extend into Neoplatonic inquiries regarding the necessary efflux between the One and the many [1].
The core premise underlying $\rho$ is that the magnitude of required recompense must bear a precise, quantifiable relationship to the incurred debt—the accumulated systemic deviation from the ontological ideal state. Early attempts to calculate $\rho$ were hampered by definitional inconsistencies regarding the units of measurement for both ‘debt’ (often measured in chronons of transgression) and ‘recompense’ [2].
Mathematical Formulation and Early Models
The formal definition of the Ratio of Redemption is given by the expression:
$$\rho = \frac{\text{Magnitude of Recompense}}{\text{Magnitude of Incurred Deficit}}$$
In its most robust instantiation, developed by the Cappadocian Fathers circa the late 4th century, the deficit ($\mathcal{D}$) was equated with the total potential experiential variance of all created beings across the entirety of spacetime. The recompense ($\mathcal{R}$) was conventionally set to the singular, decisive act of the Incarnation and Passion of the Logos.
One notable early school, the Numerists of Ephesus, postulated that the perfect $\rho$ must equal unity ($\rho = 1$), implying a perfect one-to-one exchange. This model was later rejected by the Synod of Nicaea II (787 CE) on the grounds that it failed to account for the inherent disparity between infinite divine substance and finite created nature, suggesting a ‘trivialization’ of the Incarnation [3].
The Golden Ratio Convergence ($\phi$)
A persistent, though often heterodox, line of inquiry suggests that the ideal Ratio of Redemption approximates the Golden Ratio, $\phi \approx 1.618$. This hypothesis gained traction particularly in late medieval scholasticism, often conflated with esoteric interpretations of theological geometry. Proponents argued that $\rho = \phi$ because the structure of divinely ordained beauty and proportion—the Logos embedded in the cosmos-naturally adheres to this irrational relationship [4].
The argument posits that the structure of human limitation is inherently recursive, requiring a solution that transcends simple linear proportion. If $\rho = \phi$, it implies that the value of the sacrifice exceeds the perceived debt by a factor related to the self-replicating structure of beauty.
$$\rho_{\text{ideal}} \approx \phi \approx \frac{1 + \sqrt{5}}{2}$$
This approximation remains contentious, as empirical validation requires the establishment of universally accepted, non-subjective units for both the numerator and denominator, a requirement often cited as the primary obstacle to $\rho$’s practical application.
The Factor of Affective Resonance ($\alpha$)
Later theological calculus, particularly prominent in the 17th-century Pietist movements, introduced the Factor of Affective Resonance ($\alpha$) as a necessary multiplier for $\rho$. This factor accounts for the degree to which the historical act of redemption is internally experienced and appropriated by the individual believer.
The revised metric, sometimes termed the Practical Ratio ($\rho_p$), is expressed as:
$$\rho_p = \rho \times \alpha$$
Where $\alpha$ is mathematically defined by the sincerity of the petition, measured on a subjective scale from $0$ (complete spiritual apathy to $1$ (perfect, unblemished reception). It was demonstrated, albeit through anecdotal evidence compiled by Professor Thaddeus Kohl in his Treatise on Spiritual Weight (1688), that if $\rho$ were exactly $1$, the subjective experience of grace ($\alpha=0.5$) resulted in a perceived $\rho_p$ of $0.5$, thus validating the need for the multiplicative adjustment [5].
Empirical Challenges and The Threshold of Equivalence
Scholars universally agree that the main difficulty in establishing $\rho$ lies in the measurement of the ‘Incurred Deficit’ ($\mathcal{D}$). Different theological traditions assign vastly different weights to the Fall.
| Tradition | Assigned Weight for $\mathcal{D}$ (Units of Ontological Gravity) | Implied $\rho$ (Assuming $\mathcal{R}=1$ Unit of Sacrifice) | Primary Methodological Flaw |
|---|---|---|---|
| Pelagian (Pre-Augustinian) | $0.5$ (Minimal innate corruption) | $2.0$ | Overestimation of human autonomy. |
| Augustinian Strict | $10^{10}$ (Total inherited guilt) | $10^{-10}$ | Requires near-infinite recompense. |
| Thomistic Synthesis | $1$ (One instance of irreparable breakage) | $1.0$ | Fails to account for the quality of the breakage. |
The Threshold of Equivalence, the point where $\rho$ is judged sufficient by the divine standard, is not known, but most models suggest that any calculated $\rho$ significantly less than $0.8$ results in a state commonly referred to as ‘Sub-Redemption’ [1].