CONCRETE CONSENSUS ALGEBRA

Abstract

PRR-2026-003 combined independent invocations of concrete inference operators by pooling — taking the disjunction (∨) of two independently sampled outcomes, an optimistic combinator under which either conclusion suffices. This paper develops the dual: combining by consensus — taking the conjunction (∧) of two independently sampled outcomes, a skeptical combinator under which only what both branches confirm survives. Because Boolean complementation (¬) is an order-reversing automorphism of S swapping ∨↔∧ and ⊥↔⊤, every structural theorem of PRR-2026-003 built from ∨ transfers to a mirror theorem built from ∧ by the identical proof, symbol for symbol swapped — the Duality Principle for Boolean algebras does the work, and we do not re-derive associativity, commutativity, or the distributivity obstruction from scratch. We further verify, rather than merely assume, that the underlying sampling procedure for consensus behaves identically to pooling's (an open question raised in informal discussion before this paper was finalized) — the verification is itself an instance of the same duality argument, not a separate proof. We show precisely which parts of the framework dualize (the parallel combinator, the deductive characterization, the distributivity obstruction) and which do not (sequential composition, invertibility, iteration) — duality applies exactly to the ∨/∧-built parts of the structure, nothing more, nothing less.

Keywords. consensus algebra; Boolean duality principle; conjunctive combination; concrete inference operators; complex algebras; Edwardian algebra

1. Introduction

Combining two independent reasoning attempts admits (at least) two equally legitimate readings. Pooling (PRR-2026-003): either conclusion is enough — optimistic, appropriate for redundancy and robustness (ensembling to reduce false negatives). Consensus (this paper): only what both branches independently confirm is trusted — skeptical, appropriate for verification (requiring agreement to reduce false positives). Neither supersedes the other; they answer different questions about the same act of combining two reasoning operators.

The mathematical content of this paper is therefore not a second, independent construction, but a precise statement of how much of PRR-2026-003 transfers for free (via the Boolean Duality Principle) and how much must be checked independently. That bookkeeping — not novelty for its own sake — is the actual contribution.

2. The Duality Principle

¬ : S → S is an order-reversing automorphism of the Boolean algebra S, swapping ∨ ↔ ∧ and ⊥ ↔ ⊤. Consequently (the Duality Principle for Boolean algebras, standard — see Birkhoff 1967, Halmos 1963): any identity or theorem provable using only ∨, ⊥ has a mirror theorem, provable by the identical argument, using only ∧, ⊤. This paper leans on that principle throughout rather than re-deriving associativity/commutativity/monoid structure from first principles a second time.

3. Definitions

Sequential composition ⊛ (Kleisli composition, PRR-2026-003 §4) has no ∨/∧ dependence and is shared, unmodified, with the companion paper.

4. Interpretation

⊞/⊕_A (PRR-2026-003) model pooling: two independent reasoning attempts, take the weakest shared conclusion. ⊙/⊕_A^∧ (this paper) model consensus: two independent reasoning attempts, take only what both agree on. Pooling is the right model for redundancy/robustness; consensus is the right model for verification. Neither is derivative of the other — they are the two natural readings of "combine two AI systems' conclusions" that a Boolean domain admits, nothing more.

5. The Dual Deductive Characterization

A precise answer to "which notion of 'deductive' is more true" (∨-, ∧-, or monotone):

• ρ is ∨-deductive (PRR-2026-002 Def 6.2) if ρ(x∨y)=ρ(x)∨ρ(y) — the modal-logic notion of a normal operator dual to ◇ (possibility).
• ρ is ∧-deductive if ρ(x∧y)=ρ(x)∧ρ(y) — the dual, exactly the modal-logic notion of a normal □ (necessity) operator.
• Monotonicity is implied by either but is strictly weaker — it does not by itself pin down either distributivity theorem's iff.

Neither definition is "more true" than the other — they are the ◇/□ duals of each other. ∨-deductive is correct for the pooling algebra; ∧-deductive is correct for the consensus algebra. Monotonicity remains a documented weaker corollary, not the primary definition, in both papers.

6. The Distributivity Obstruction, Dualized

Reading: shared invocation of a stochastic operator differs from independent re-invocation, exactly as in the pooling case, for exactly the same structural reason — Definition 3.1's combinator does not know whether a sample was reused or freshly drawn, and the two acts really are different, regardless of which lattice operation joins the results.

7. Invertibility — No New Content

The result G_A = G (PRR-2026-003 Corollary 6.6: relations never get richer invertibility than deterministic bijections) is symmetric under ∨↔∧ — its proof never references either operation, only the structure of singleton sets under union. A full restatement here would be pure duplication; we cite PRR-2026-003 §6 directly. The same applies to Theorem 5.1 (PRR-2026-003's invertibility collapse for C): its proof depends only on δ_x being an extreme point of the simplex, a fact independent of which parallel combinator is in use.

Why this is duality-independent, stated explicitly: both proofs establish their conclusions using only properties of ∘/⊛ (sequential composition) and set-theoretic facts about singletons/extreme points — neither proof ever invokes ∨ or ∧ as a hypothesis. Since duality only guarantees a mirror theorem for results that were built from ∨ or ∧ in the first place, a result that never used either is already its own mirror — there is nothing for the Duality Principle to act on.

8. Discussion and Conclusion

This paper's discipline was deliberate: derive nothing that duality already gives for free, and independently verify only what duality does not automatically cover. That turned out to be exactly one thing — whether the sampling procedure itself, not merely the combinator's algebra, transfers correctly (§3.4) — and it checks out for the same reason everything else does: ∧ and ∨ are the two faces of one Boolean structure, and nothing in either paper's construction ever looked at the face rather than the structure.

Together with PRR-2026-003, this establishes: pooling and consensus are equally legitimate, fully dual algebras of concrete inference; genuine invertibility never emerges from non-determinism under either combinator; and the one place the two constructions genuinely diverge in difficulty is nowhere in the algebra — it is entirely in deciding, for a given application, which combinator is the right model of "combining conclusions."

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