The Claim
Define the grace operator G as a non-unitary projection that maps both moral eigenstates to the redeemed state |+1⟩. This operator has four mathematically derivable properties: G² = G (idempotent — applying it twice is identical to applying it once), G†G ≠ I (non-unitary — it breaks time-reversal symmetry and is therefore irreversible), rank-deficient (information about the prior state is destroyed), and external origin (G is not generated by the system's own Hamiltonian — see ISO-012).
These four mathematical properties map exactly onto four theological claims about grace in the Reformed tradition: (1) once is enough (ephapax), (2) regeneration is irreversible, (3) the old self is genuinely destroyed ("new creation"), and (4) grace originates outside the human condition. The mapping is structural, not metaphorical. The math derives the theology independently.
Catholic sacramental theology models grace as continuous — dispensed through repeated sacraments over a lifetime, not as a one-shot operator. The idempotent mapping is specifically Protestant-Reformed. A Catholic formalization would use a different operator structure (perhaps a continuous semigroup rather than an idempotent projection). This is not a flaw in the isomorphism but a scope limitation: it maps Reformed soteriology, not all Christian soteriology.
If grace effects were shown to be reversible by internal operations alone — that is, if the fallen state could be restored by self-generated evolution after grace application — the mapping breaks. This would require G to be unitary, contradicting G†G ≠ I. The kill condition is: a documented case of genuine regeneration followed by genuine de-regeneration through purely internal (non-apostate, non-external-cause) means.
Domain A — Mathematics
From this definition, four key properties follow as mathematical theorems:
Domain B — Theology
Reformed theology attributes four structural properties to saving grace. These four properties map exactly onto the four mathematical properties of G derived above:
Grace originates entirely outside the human system. It is "not your own doing" and "not a result of works." The mathematical equivalent: G cannot be generated by H_human. Any operator generated from within the system is unitary; G is non-unitary; therefore G comes from outside. Sola gratia is mathematically entailed by G's non-unitarity.
"By a single offering he has perfected for all time those who are being sanctified." The Greek word ephapax means "once for all." The mathematical equivalent: G² = G. Applying grace a second time produces the same result as applying it once. Once-for-all-ness is idempotency.
"The gifts and the calling of God are irrevocable." The mathematical equivalent: G†G ≠ I. Since G is non-unitary, there exists no G−¹. Once applied, the operator cannot be undone by any internal process. The irrevocability of grace is the theological statement of G's non-invertibility.
"If anyone is in Christ, he is a new creation; the old has passed away." The mathematical equivalent: G is rank-deficient. Information about the prior state (fallen, |−1⟩) is destroyed in the output. The new creation is not the old creation modified — it is a rank-collapse, an information-theoretic new start. The old has genuinely "passed away."
Property Mapping
| Mathematical Property | Theological Property | Scripture |
|---|---|---|
| G² = G (idempotent) — applying twice = applying once; once is mathematically sufficient | Ephapax — once is enough; "by a single offering he has perfected for all time" | Hebrews 10:14; Hebrews 7:27; Romans 6:10 |
| G†G ≠ I (non-unitary) — breaks time-reversal symmetry; operation cannot be undone by any internal process | Irrevocable — "the gifts and the calling of God are irrevocable" | Romans 11:29; John 10:28–29; Phil 1:6 |
| External origin — non-unitarity proves G cannot be generated by H_system; must come from outside | Sola gratia — "not your own doing; it is the gift of God, not a result of works" | Ephesians 2:8–9; Titus 3:5; John 1:13 |
| Rank-deficient (rank 1) — information about prior state (|−1⟩) is destroyed; both inputs map to same output | New creation — "old has passed away; behold, the new has come"; not modification but replacement | 2 Corinthians 5:17; Galatians 2:20; Romans 6:6 |
Semi-Pelagian Test
Semi-Pelagianism holds that salvation requires a mixture of divine grace and human effort. Define the mixed operator G' — a blend of the grace operator G and the self-evolution operator U:
Now compute G'²:
G'² ≠ G' unless α = 1. The mixed operator requires repeated application to converge — it is not self-sufficient in a single application. This contradicts the ephapax (once-for-all) character of grace. Mathematical sola gratia: only α = 1 preserves idempotency. Any admixture of self-effort destroys the once-for-all property.
This is not a refutation of Catholic sacramental theology per se. It is a demonstration that the idempotent formalism specifically captures Protestant Reformed soteriology. Catholic theology models grace through a continuous sacramental semigroup — a different mathematical structure that may be equally valid as a formalization of a different theological tradition. The isomorphism maps Reformed claims; it does not adjudicate between Reformed and Catholic theology.
Separator Tests & Scripture
If grace effects were shown to be reversible by internal operations alone — that is, if a genuinely regenerated person could return to the unregenerate state through their own internal process without any external undoing — the mapping breaks. This would require the existence of G−¹, contradicting G†G ≠ I. (Apostasy does not falsify this: apostasy involves the abandonment of what was merely professed, not the reversal of genuine regeneration — a claim that is itself part of the Reformed framework, and part of what the formalism is formalizing.)