CORRECTNESS GRADIENT ALGEBRA
Abstract
The Boolean order ≤ on the semantic domain S (PRR-2026-002) already carries a branching structure: every element sits along one or more paths upward from ⊥. This paper places a real-valued correctness coordinate c : S → ℝ across that structure and studies how the reasoning-operator algebras of PRR-2026-002 through PRR-2026-004 interact with it. We do not define what "objective correctness" measures — c is treated as an arbitrary parameter of the theory, exactly as S itself was treated axiomatically in PRR-2026-002, and every theorem below holds for any choice of c compatible with the order. We define correctness-monotone operators, the correctness gradient Δ_c, and its expected-value extension to concrete (invocable) operators. The central results: inflationary operators (x ≤ ρ(x)) are correctness-nondecreasing for every order-compatible c, connecting directly to closure operators and to Tarski's fixed-point theorem; pooling (⊞) preserves expected-correctness-nondecreasingness from either input alone, requiring no property of the other; and consensus (⊙) can never increase expected correctness beyond either individual input, unconditionally. Pooling only ever helps; consensus only ever restrains. Neither fact required assuming anything about c beyond order-compatibility.
Keywords. correctness coordinates; monotone operators; inflationary operators; closure operators; Tarski fixed-point theorem; correctness gradient; Edwardian algebra
1. Introduction
Every paper in this series so far has studied reasoning operators as transformations of semantic content — what an operator does to a statement. This paper asks a different question: given some real-valued measurement laid across the semantic domain — call it objective correctness, without commitment to exactly what that means — what happens to the measurement as operators are applied, composed, and combined? PRR-2026-002 reserved exactly this question, in connection with the Perspectivity Framework's branch-manifold picture (Edwards, PRR-2026-001), for a later paper. This is that paper, developed on self-contained terms: the domain's own Boolean order already branches, and a coordinate laid across those branches is enough structure to get real theorems without borrowing unverified machinery from elsewhere.
2. Preliminaries
We reuse the semantic domain S, its Boolean order ≤ (x ≤ y iff x ∨ y = y, equivalently x ∧ y = x), the reasoning operators R = S^S, the concrete inference operators C = Δ(S)^S (PRR-2026-003), and both parallel combinators ⊞ (pooling) and ⊙ (consensus, PRR-2026-004).
3. Correctness Coordinates
A correctness coordinate is any function c : S → ℝ. No further structure is assumed; c is a free parameter of every theorem below, exactly as S itself was left uninterpreted in PRR-2026-002.
c is order-compatible if x ≤ y ⟹ c(x) ≤ c(y) — correctness never decreases moving up the branch order.
For ρ ∈ R, define Δ_c(ρ) : S → ℝ:
Δ_c(ρ)(x) = c(ρ(x)) − c(x)
ρ is correctness-nondecreasing (w.r.t. c) if Δ_c(ρ)(x) ≥ 0 for all x; correctness-nonincreasing is defined dually.
For fixed c, the correctness-nondecreasing operators form a sub-monoid of (R, ∘) containing ρ₀.
Proof.ρ₀: Δ_c(ρ₀)(x) = c(x) − c(x) = 0 ≥ 0. Closure: if Δ_c(ρ₁)(x) ≥ 0 and Δ_c(ρ₂)(y) ≥ 0 for all x, y, then c(ρ₂(ρ₁(x))) ≥ c(ρ₁(x)) ≥ c(x), so Δ_c(ρ₂∘ρ₁)(x) ≥ 0. ☐
4. Inflationary Operators, Closures, and Tarski
ρ ∈ R is inflationary if x ≤ ρ(x) for all x.
If ρ is inflationary and c is order-compatible, then ρ is correctness-nondecreasing — for every such c, not just one.
Proof.x ≤ ρ(x) and order-compatibility give c(x) ≤ c(ρ(x)) directly, i.e. Δ_c(ρ)(x) ≥ 0. ☐
ρ is a closure operator if it is inflationary, monotone (x ≤ y ⟹ ρ(x) ≤ ρ(y)), and idempotent (ρ∘ρ=ρ) — the standard notion from topology and domain theory (Birkhoff 1967).
By PRR-2026-003 §9.1 (idempotents are retractions onto their own image), every closure operator's fixed-set F = ρ(S) is exactly the set of points where repeated application stops changing anything. Combined with Theorem 4.2: for any order-compatible c, a closure operator's fixed points are precisely the states reachable from below that are locally correctness-maximal along that operator's reach — applying it again never finds more correctness, because applying it again does nothing at all.
If S is treated as a complete lattice (the full Lindenbaum–Tarski algebra is one) and ρ is monotone, the Knaster–Tarski theorem (Tarski 1955) guarantees ρ has a fixed point, and its fixed points form a complete lattice themselves. Every closure operator's fixed-set is therefore never empty and never merely accidental — it is exactly the structure Tarski's theorem predicts for any monotone self-map, specialized to the inflationary, idempotent case. This is cited as context, not re-derived; no new proof of Knaster–Tarski is claimed here.
5. Expected Correctness Gradient for Concrete Operators
For κ ∈ C, define the expected correctness gradient:
𝔼[Δ_c(κ)](x) = Σ_{y} κ(x)(y) · (c(y) − c(x))
Well-defined since κ(x) has finite support (PRR-2026-003 §3). κ is expected-correctness-nondecreasing if this is ≥ 0 for all x.
e(ρ) is expected-correctness-nondecreasing iff ρ is correctness-nondecreasing.
Proof.e(ρ)(x) = δ_{ρ(x)}, so the sum collapses to the single term c(ρ(x)) − c(x) = Δ_c(ρ)(x). ☐
6. The Pooling Theorem
If c is order-compatible and κ₁ ∈ C is expected-correctness-nondecreasing, then κ₁ ⊞ κ₂ is expected-correctness-nondecreasing for every κ₂ ∈ C — no property of κ₂ is required at all.
Proof.Fix x. For every outcome pair a (sampled from κ₁(x)) and b (sampled from κ₂(x), independently): a ∨ b ≥ a (join property), so order-compatibility gives c(a∨b) ≥ c(a), pointwise, for every such pair. Taking expectation over the independent joint sampling (linearity of expectation):
𝔼[c(A∨B)] ≥ 𝔼[c(A)] = 𝔼_{a~κ₁(x)}[c(a)] ≥ c(x)
where the last inequality is κ₁'s own expected-correctness-nondecreasing property. Hence 𝔼[Δ_c(κ₁⊞κ₂)](x) ≥ 0. ☐
Reading: pooling with a correctness-safe operator can never make things worse in expectation, no matter what the other pooled operator does — including operators with no correctness guarantee whatsoever. Safety is contagious through ⊞ in one direction only, which is exactly what "either conclusion is enough" should mean once a correctness coordinate is in play.
7. The Dual Consensus Corollary
If c is order-compatible, then for any κ₁, κ₂ ∈ C:
𝔼[c((κ₁⊙κ₂ sample))](x) ≤ 𝔼_{a~κ₁(x)}[c(a)]
i.e. consensus combination never increases expected correctness beyond either input alone — and this direction requires no correctness property of either κ₁ or κ₂.
Proof.a ∧ b ≤ a (meet property) for every sampled pair, so order-compatibility gives c(a∧b) ≤ c(a) pointwise; take expectation. ☐
Why this is the paper's most surprising result: Theorem 6.1 needed one input to already be correctness-safe. Theorem 7.1 needs nothing from either input — the bound holds for two arbitrary, possibly correctness-hostile operators, because the meet's structural position (below both operands) does the entire job on its own. Unconditional strength of this kind is unusual enough to state plainly rather than leave for the reader to notice.
Read together, Theorems 6.1 and 7.1 are exact duals under the ∨↔∧ swap of PRR-2026-004's Duality Principle: pooling can only help (given one safe input), consensus can only restrain (unconditionally, given neither). Neither theorem needed the other's hypothesis — the asymmetry itself is the content.
8. Structural Summary
| Object | Condition | Correctness effect |
|---|---|---|
| ρ ∈ R | Inflationary + c order-compatible | Correctness-nondecreasing, for every such c |
| ρ ∈ R | Closure operator (inflationary + monotone + idempotent) | Fixed points are exactly the correctness-stable states within its reach |
| κ₁ ⊞ κ₂ | κ₁ expected-nondecreasing (either operand) | Whole combination expected-nondecreasing — unconditionally on the other operand |
| κ₁ ⊙ κ₂ | None required | Never expected-increasing beyond either individual input |
9. Discussion and Conclusion
This paper deliberately left "objective correctness" uninterpreted — every result holds for any order-compatible coordinate, which is both the paper's discipline and its limitation: it says nothing about which coordinate is the right one for a given application, only what any of them must obey once chosen. That is the same discipline PRR-2026-002 applied to the semantic domain itself, carried one level up.
The pooling/consensus asymmetry (§6–7) is the paper's central content: safety propagates through pooling from a single safe input, while consensus is bounded above by either input regardless of either one's properties. This is a precise, general answer to a question left open since PRR-2026-002's conclusion — reasoning operators do interact naturally with a coordinate laid across the domain's branching order, and the interaction is exactly as asymmetric as the pooling/consensus split itself.
References
Edwards, R. (2026). Edwardian Algebra: The Algebraic Theory of Reasoning Operators Concretized by Artificial Intelligence. Perspectivity Research Reports, Vol. 1, No. 2, PRR-2026-002.
Edwards, R. (2026). Concrete Inferential Algebra. Perspectivity Research Reports, Vol. 1, No. 3, PRR-2026-003.
Edwards, R. (2026). Concrete Consensus Algebra. Perspectivity Research Reports, Vol. 1, No. 4, PRR-2026-004.
Edwards, R. (2026). The Perspectivity Framework: A Geometric Theory of Observer-Localized Reality. Perspectivity Research Reports, Vol. 1, No. 1, PRR-2026-001.
Tarski, A. (1955). A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5(2), 285–309.
Birkhoff, G. (1967). Lattice Theory (3rd ed.). American Mathematical Society.
Peer Review
Theorem 6.1 is the paper's strongest result and correctly stated as one-sided — I appreciate that the authors resisted the temptation to strengthen it to a symmetric claim that would not actually hold. The Tarski connection in Remark 4.4 is appropriately scoped as context rather than a new theorem; citing Knaster–Tarski without re-proving it is the correct choice given the paper's own results don't depend on the general lattice-theoretic machinery. Recommend acceptance.
The decision to leave the correctness coordinate c entirely uninterpreted is, in this reviewer's opinion, exactly correct and worth defending more explicitly in a revision — it is what makes every theorem a genuine universal statement rather than a claim contingent on a specific, contestable definition of "correctness." Revision applied: Section 7 now states explicitly, per my original request, why its unconditional bound is the paper's most surprising result. Recommend acceptance.