Correctness Geometry Algebra
docs/Research/PRR-2026-008-correctness-geometry-notes.md for the planning record.
Abstract
PRR-2026-005 placed a real-valued correctness coordinate c : S → ℝ across the semantic domain and left it deliberately uninterpreted, proving every theorem for any order-compatible choice. This paper adds three pieces of structure on top of that foundation. First, we show the domain's own top and bottom elements already force any order-compatible coordinate into a bounded range — a free corollary, not a new axiom. Second, we anchor a concrete coordinate at a distinguished ground-truth element ω, giving precise meaning to "completely correct," "completely antonymical," and — after ruling out a competing reading that turns out to be mathematically impossible to build on the same domain — "unrelated to the question," in place of a literal "false" that the domain's own structure cannot support. Third, we define an aggregate, operator-level correctness score and a correctness-gradient inner product, giving reasoning operators a genuine geometry: operators can be parallel (compatible), orthogonal (independent), or anti-parallel (contradictory) — correcting a natural but mistaken intuition that orthogonality itself should mean contradiction. We close by proving two new unconditional theorems connecting this geometry to PRR-2026-005's pooling and consensus combinators: at the aggregate level, pooling never scores worse than either input's own score, and consensus never scores better than either — with no safety hypothesis required on either side, sharpening PRR-2026-005's original one-sided pooling guarantee into a fully symmetric statement.
Keywords. correctness coordinates; ground-truth anchoring; operator-level scoring; inner-product geometry; cosine similarity; pooling; consensus; Edwardian algebra
1. Introduction
PRR-2026-005 answered what a reasoning operator does to a correctness coordinate at a single point. It never asked what a fair, aggregate score for the whole operator should be, nor whether two operators' effects on correctness can be compared to each other. Both questions turned out to have real content once pursued properly, including one intuition — that orthogonal operators should be read as contradictory — that turns out to be exactly backwards, and one proposed construction — a coordinate where a pure tautology scores as merely "uninformative" rather than "maximally correct" — that turns out to be impossible to build on the domain this series already committed to. Both corrections are stated plainly below rather than smoothed over, in keeping with this series' discipline of reporting what didn't work alongside what did.
2. Preliminaries
We reuse S, its order ≤, R, C, ⊞ (pooling, PRR-2026-003), ⊙ (consensus, PRR-2026-004), and the correctness coordinate c : S → ℝ with order-compatibility (x ≤ y ⟹ c(x) ≤ c(y), PRR-2026-005 Def. 3.1–3.2) and correctness gradient Δ_c(ρ)(x) = c(ρ(x)) − c(x) (PRR-2026-005 Def. 3.3), unchanged.
3. The Forced Bound
For any order-compatible c, c(⊥) ≤ c(x) ≤ c(⊤) for every x ∈ S.
Proof.⊥ ≤ x ≤ ⊤ holds for every x in any Boolean algebra, by definition of its top and bottom elements. Apply order-compatibility twice. ☐
This was never stated as a named result in PRR-2026-005, though it follows immediately from Definition 3.2 alone — no new axiom required. Normalizing c(⊤) = 100 is a scale choice; c(⊥) is then forced to be the coordinate's global minimum. We fix that minimum at −100 by the same normalization throughout the rest of this paper.
4. Ground-Truth Anchoring
A natural, stronger reading of "−100/0/100" ties the scale to one distinguished ground-truth element rather than the domain's own extremes alone. Two candidate readings were investigated in full before either was adopted.
Fix any proper ω with ⊥ < ω < ⊤ (no atomicity required). Define
c_ω(x) = 100 if ω ≤ x = −100 if x ≤ ¬ω = 0 otherwise (x incomparable to ω)
Reading: ω ≤ x means ω entails x — x is a guaranteed consequence of the ground truth, hence true given it. x ≤ ¬ω means x entails the negation of the ground truth, hence is guaranteed false given it. Everything else — neither confirmed nor denied by ω alone — falls to 0.
Score maximally committed, atomic claims at ±100 and the tautology ⊤ (a pure hedge, asserting nothing falsifiable) near the uninformative middle at 0 — closer to a literal reading of "0 = false," since under Reading A, 0 means unrelated, not false.
No coordinate can be simultaneously order-compatible (PRR-2026-005 Def. 3.2) and satisfy Reading B's requirement that a proper ground-truth atom ω < ⊤ score higher than ⊤ itself.
Proof.ω ≤ ⊤ always. Order-compatibility forces c(ω) ≤ c(⊤). Reading B wants c(ω) = 100 > 0 ≈ c(⊤). Contradiction. ☐
A further attempt to rescue Reading B by scoring atoms directly also fails, for an independent reason: S (the Lindenbaum–Tarski algebra of a countable propositional language, PRR-2026-002 §2) is atomless. For any nonzero [φ] ∈ S, φ is a finite formula and so omits some variable p_j; then φ ∧ p_j and φ ∧ ¬p_j are both nonzero and strictly below [φ]. No nonzero element is minimal — there is no "maximally specific claim" for atoms to be scored at, in the general case. (A finite-sublanguage restriction can recover atoms, at the cost of a new domain-projection map and an undetermined aggregation rule for hedged elements of the restriction; investigated and set aside as disproportionate to the payoff for this paper.)
Adopted: Reading A. "0" means unrelated to the question, not literally false — an explicit, intentional departure from a more casual reading of the scale's original three anchor points, forced by Proposition 4.1 rather than chosen for convenience.
5. Operator-Level Correctness Score
For μ ∈ Δ(S) (finite support, PRR-2026-003 §0) and ρ ∈ R:
Corr_{c,μ}(ρ) = 𝔼_{x∼μ}[Δ_c(ρ)(x)] = 𝔼_{x∼μ}[c(ρ(x)) − c(x)]
extended to κ ∈ C via PRR-2026-005 Def. 5.1's expected gradient in place of Δ_c.
Corr_{c,δ_x₀}(ρ) = Δ_c(ρ)(x₀) — the pointwise gradient is the point-mass special case of the aggregate score.
Proof.Immediate from the definition of expectation under a point mass. ☐
ρ is correctness-nondecreasing (PRR-2026-005 Def. 3.3) iff Corr_{c,μ}(ρ) ≥ 0 for every μ ∈ Δ(S).
Proof.(⟹) Linearity of expectation over a nonnegative integrand. (⟸) Take μ = δ_x for each x (Prop. 5.2) to recover the pointwise condition exactly — point masses are trivially finite-support, so this direction is complete, not merely suggestive. ☐
μ is a genuine free parameter, not forced — unlike κ_∅ in PRR-2026-003. We leave it fully free throughout, matching this series' discipline of proving theorems for any choice of an uninterpreted parameter rather than fixing one by convention.
Unlike PRR-2026-006 Corollary 2.3's conjugation-invariant "finitary" property, Corr_{c,μ} is not conjugation-invariant in general: Corr_{c,μ}(σρσ⁻¹) ≠ Corr_{c,μ}(ρ) for a generic σ ∈ G, unless σ also fixes both μ (pushed forward) and c (pulled back). An operator's aggregate score is not, in general, a property of the operator alone, independent of how the domain happens to be labeled.
6. Correctness-Gradient Geometry
For ρ₁, ρ₂ ∈ R (or κ₁, κ₂ ∈ C), fixed c, μ:
⟨ρ₁,ρ₂⟩_{c,μ} = Σ_x μ(x)·Δ_c(ρ₁)(x)·Δ_c(ρ₂)(x)
well-defined with no convergence conditions beyond μ's finite support. Induces ‖ρ‖_{c,μ} = √⟨ρ,ρ⟩_{c,μ} and, when both norms are nonzero on supp(μ):
cos_{c,μ}(ρ₁,ρ₂) = ⟨ρ₁,ρ₂⟩_{c,μ} / (‖ρ₁‖_{c,μ}·‖ρ₂‖_{c,μ})
ρ₁, ρ₂ are:
- compatible / parallel if cos_{c,μ} ≈ 1
- independent / orthogonal if cos_{c,μ} = 0
- contradictory / anti-parallel if cos_{c,μ} ≈ −1
This corrects a natural first guess that orthogonality itself should mean contradiction. Standard inner-product semantics — and the worked example below — instead support anti-parallel as the correct algebraic reading of contradiction, with orthogonal read as independence: two operators whose correctness effects share no linear relationship, not two operators that oppose each other.
Edge case. cos_{c,μ} is undefined when either operator is correctness-flat on supp(μ) — a zero-norm gradient has no direction to compare.
Worked Example
Take three independent propositional atoms p, q, r (eight atomic worlds), and let ω = p∧q — a proper, non-atomic element, silent about r entirely. Let c_ω be as in §4.
Setup
c_ω(ω) = 100. c_ω(¬p) = −100 (¬p directly contradicts ω). c_ω(¬q) = −100 (same reasoning). c_ω(r) = 0 — "r is true" is entailed by neither ω nor ¬ω, since ω says nothing about r at all. This is Reading A's "0" made concrete: a genuinely unrelated claim, not a false one.
Orthogonal pair (cos = 0)
Let μ be uniform over {x₁=¬p, x₂=¬q}. Let ρ₁ be a "fix p-errors" specialist: ρ₁(x₁) = ω (Δ_c(ρ₁)(x₁) = 100−(−100) = 200), identity elsewhere (Δ_c(ρ₁)(x₂) = 0). Let ρ₂ be a "fix q-errors" specialist: ρ₂(x₂) = ω (Δ_c(ρ₂)(x₂) = 200), identity at x₁ (Δ_c(ρ₂)(x₁) = 0). Then ⟨ρ₁,ρ₂⟩ = ½·200·0 + ½·0·200 = 0, with both norms nonzero (‖ρ₁‖² = ‖ρ₂‖² = 20000) — cos = 0, exactly. Neither operator ever undoes the other's work; they simply never react to the same input under this μ. Independent, not contradictory.
Parallel pair (cos = 1)
Let ρ₃ do a partial version of ρ₁'s fix: ρ₃(x₁) lands on some element with c_ω = 0 (halfway corrected, Δ_c(ρ₃)(x₁) = 0−(−100) = 100), identity at x₂. Both gradient vectors are positive scalar multiples of the same direction — (200,0) and (100,0) — giving cos = 1, exactly. Same kind of correction, same input, different magnitude: genuinely compatible.
Anti-parallel pair (cos = −1)
Using x₂=r instead (c_ω(r)=0): let ρ₄(x₂) = ¬q (Δ_c(ρ₄)(x₂) = −100−0 = −100, identity at x₁), and ρ₅(x₂) = ω (Δ_c(ρ₅)(x₂) = 100−0 = 100, identity at x₁) under μ uniform over {x₁,x₂}. Then ⟨ρ₄,ρ₅⟩ = ½·0·0 + ½·(−100)(100) = −5000, ‖ρ₄‖ = ‖ρ₅‖ = √5000 — cos = −1, exactly. At the one input where either operator acts, they push correctness in exactly opposite directions by equal magnitude: a genuinely contradictory pair.
7. Trajectories and Boundaries
If ρ is correctness-nondecreasing and its trajectory from x is eventually periodic with period p ≥ 2 (PRR-2026-006 Thm. 3.1 case ii), writing the cycle y₀,y₁,…,y_{p−1},y₀, then c is constant across the entire cycle.
Proof.Nondecreasing gives c(y₀) ≤ c(y₁) ≤ ⋯ ≤ c(y_{p−1}) ≤ c(y₀) (the last step by periodicity). A closed chain of ≤ that returns to its start forces equality throughout. ☐
A correctness-safe operator that is genuinely cycling has already stopped changing correctness the moment the cycle closed — "approaching a boundary" and "cycling" are mutually exclusive for such an operator.
A stabilizing trajectory lands at a closure operator's fixed point (PRR-2026-006 Thm. 4.2), which is only guaranteed locally correctness-maximal within that operator's own certified reach — not necessarily c = 100, unless ⊤ is itself in the fixed family (PRR-2026-006 Cor. 4.4). "Safe and stabilizing" is not the same claim as "approaches perfect correctness."
8. Interactions with Pooling and Consensus
The three preceding sections define new structure. This section asks whether it says anything about the algebra PRR-2026-003 through PRR-2026-005 already built — rather than leaving the two bodies of work sitting side by side.
For any κ₁, κ₂ ∈ C and any order-compatible c, μ — no safety hypothesis on either operator:
Corr_{c,μ}(κ₁⊞κ₂) ≥ Corr_{c,μ}(κ₁) and Corr_{c,μ}(κ₁⊞κ₂) ≥ Corr_{c,μ}(κ₂)
Proof.
Fix x. For a sampled from κ₁(x), b from κ₂(x) independently: a∨b ≥ a pointwise (join property), so order-compatibility gives c(a∨b) ≥ c(a) for every realization. Taking expectation: 𝔼[c(A∨B)](x) ≥ 𝔼[c(A)](x), i.e. 𝔼[Δ_c(κ₁⊞κ₂)](x) ≥ 𝔼[Δ_c(κ₁)](x) for every x. Taking expectation again over μ gives the first inequality. The symmetric argument (a∨b ≥ b) gives the second. ☐
This strictly generalizes PRR-2026-005 Theorem 6.1, which needed κ₁ to already be safe to beat the zero baseline. Theorem 8.1 needs nothing from either operator, and compares pooling's aggregate score to each input's own — a pooled operator's score is never a step backward from either contributor's own average performance.
For any κ₁, κ₂ ∈ C:
Corr_{c,μ}(κ₁⊙κ₂) ≤ Corr_{c,μ}(κ₁) and Corr_{c,μ}(κ₁⊙κ₂) ≤ Corr_{c,μ}(κ₂)
Proof.
Dual argument, a∧b ≤ a and ≤ b pointwise. ☐
Read together, Theorems 8.1 and 8.2 sharpen PRR-2026-005 §6–7's asymmetry into something cleaner once the aggregate score is available: at the pointwise level, pooling needed one safe input while consensus needed none. At the aggregate level, both combinators are unconditionally bounded relative to each input — pooling from below, consensus from above — with no hypothesis required on either side. The original asymmetry was a fact about comparing to a fixed zero baseline; it dissolves once the comparison is to each operator's own aggregate score instead.
Left open: whether cos_{c,μ}(κ₁,κ₂) says anything about how much §8.1/8.2's inequalities can be tightened — e.g. whether a near-parallel pair pools closer to the sum of their individual scores than a near-orthogonal pair does. No proof either direction; flagged honestly as future work rather than assumed.
9. Discussion and Conclusion
Two corrections drive this paper more than any single new definition: orthogonality is independence, not contradiction, and a literal "0 = false" reading of a bounded correctness scale cannot be built as an order-compatible coordinate on this series' own domain, forcing "0 = unrelated" instead. Both are stated as corrections rather than discovered facts presented as if obvious from the start, in keeping with this series' practice of reporting what didn't work (PRR-2026-003's unresolved iff-question, PRR-2026-004's unresolved invocation-duality question) alongside what did.
The genuinely new content — the aggregate operator score, the gradient inner product, and Theorems 8.1/8.2 connecting both to pooling and consensus — answers the question this paper actually opened with: what is worth measuring once an inferential algebra exists. A single operator can now be scored, two operators can now be compared, and both measurements now provably interact with the combinators PRR-2026-003 and PRR-2026-004 already built, rather than sitting beside them unconnected.
Left open: the relationship, if any, between correctness-gradient anti-correlation (dynamical, about what operators do) and semantic inconsistency x∧y=⊥ (structural, about what states are) — no proof either direction. Whether cos_{c,μ} refines Theorems 8.1/8.2 is likewise open. Neither is assumed resolved by this paper.
References
Edwards, R. (2026). Edwardian Algebra. 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). Correctness Gradient Algebra. Perspectivity Research Reports, Vol. 1, No. 5, PRR-2026-005.
Edwards, R. (2026). Reasoning Dynamics Algebra. Perspectivity Research Reports, Vol. 1, No. 6, PRR-2026-006.
Edwards, R. (2026). Narrative Operators. Perspectivity Research Reports, Vol. 1, No. 7, PRR-2026-007.
Peer Review
Proposition 4.1 is the paper's most important result, and it earns that status by being negative — it rules out a reading the authors could easily have adopted for convenience. Section 4's honesty about the atomless obstruction, including the finite-restriction path considered and set aside, is exactly the discipline this series has maintained since PRR-2026-003's own unresolved questions. Recommend acceptance.
Theorems 8.1 and 8.2 are the right way to justify this paper as a standalone entry rather than an appendix to PRR-2026-005 — they are not restatements, they are genuine new bounds that only exist once the aggregate score is available, and the observation that they dissolve PRR-2026-005's original pointwise asymmetry into a symmetric statement is worth the paper on its own. The open question at the end of §8 is appropriately left open rather than gestured at as solved. Recommend acceptance.