Noisy Gate Algebra
Abstract
PRR-2026-012's loop pair (δ, γ) takes its checker to be the embedding of a deterministic closure operator — a gate that certifies exactly the fixed family F, without error. Real gates err. We define a noisy gate as a concrete inference operator into the two-element subalgebra {⊥, ⊤} ⊆ S — a verdict kernel — so that gates are not a new type but the series' own foundational type pointed at a two-point codomain. A noisy gate for F carries two pointwise error rates: false-pass α (certifying an uncertified state) and false-fail β (rejecting a certified one). We prove the noisy loop's exit is only probabilistically certified, and exhibit by exact hand computation the amplification law in a two-state loop: a gate that is wrong 10% of the time yields a loop whose output is uncertified 19.8% of the time — iteration offers the gate repeated chances to fail, and the loop compounds them. We show α and β are structurally asymmetric: α alone corrupts output (exit-validity is perfect iff α = 0 on reachable states), while β wastes work and feeds states back into α's jurisdiction. Panels of independent gates composed by the series' own combinators trade the two error modes rather than reducing both — any-pass (⊞ on verdicts) shrinks false-fails and grows false-passes, unanimous (⊙) the exact mirror, with the unanimous panel provably never more permissive than its most cautious member. The first composition that reduces both error modes at once — majority of three (Condorcet 1785) — turns out to require reusing each gate's single verdict across branches: a transparency-licensed act (PRR-2026-011) that fresh invocation cannot imitate, by the Distributivity Obstruction (PRR-2026-003).
Keywords. noisy gates; verdict kernels; false-pass and false-fail; error amplification; checker panels; Condorcet majority; transparency; Edwardian algebra
1. Introduction
A checker that is itself a judgment call is not an edge case of loop practice; it is loop practice. The parts of a real certification gate that are mechanical — does the artifact parse, do the declared checks run clean — are honestly deterministic. The part that decides whether the work is done is an act of inference, and this series has had the correct type for acts of inference since PRR-2026-003: a concrete operator, invocable, one real verdict per call, not the same verdict every call. PRR-2026-012 embedded its checker at zero variance and said so; this paper pays the idealization back.
2. Preliminaries
Shared with PRR-2026-012, which this paper extends and does not modify: S, C, ⊛, ⊞, ⊙, sub-distributions Δ≤1 and the loop operator Λ(δ, F) (PRR-2026-012 §§4–5), the certification family F of a finite-image closure operator (PRR-2026-006 §4), the Aggregate Pooling and Consensus Bounds (PRR-2026-008 Theorems 8.1–8.2), the Distributivity Obstruction (PRR-2026-003 Theorem 7.1), and transparency (PRR-2026-011 Definition 4.2).
3. Noisy Gates Are Concrete Operators
The two-element subalgebra 2 = {⊥, ⊤} ⊆ S is closed under ∨ and ∧. A noisy gate is a concrete inference operator g ∈ C with g(x) ∈ Δ(2) for every x — a verdict kernel, reading ⊤ as pass and ⊥ as fail. No new type is introduced: a gate is the series' foundational operator class pointed at a two-point codomain, and the parallel combinators ⊞ and ⊙ restrict to verdicts as the logical OR and AND of independent verdict samples.
For a certification family F and a gate g, the false-pass rate at x ∉ F is α(x) = g(x)(⊤), and the false-fail rate at x ∈ F is β(x) = g(x)(⊥). The gate is faithful if α ≡ β ≡ 0, in which case it is exactly PRR-2026-012's guard test for F — the zero-variance case, recovered.
Λ(δ, g) runs as PRR-2026-012 Definition 5.2 with the gate's verdict replacing the membership test: at each visited state x, invoke g(x) once; on pass, exit at x; on fail, invoke δ(x) and continue from the witness. The same truncation construction gives the least solution in C≤1. Note what changed structurally: the loop's exit set is no longer contained in F. Exit is a certification event, not a certification fact.
4. The Amplification Law
The single most consequential fact about a noisy gate is that a loop gives it more than one chance to be wrong. We compute the smallest instructive case exactly, by hand, in the series' style.
Worked Example — a 10% Gate Becomes a 19.8% Loop
Two working states: u ∉ F, f ∈ F. The DOER improves half its attempts: δ(u) = {f: ½, u: ½}, and — the false-fail hazard made concrete — a rejected certified state loses its certification: δ(f) = {u: 1}. The gate errs symmetrically: α(u) = β(f) = 0.1. Write W_u, W_f for the probability that a run currently checking u (resp. f) eventually exits at an uncertified state.
W_f = β · W_u (pass w.p. 1−β exits correctly) W_u = α + (1−α)(½ · W_f + ½ · W_u)
(false-pass now, or improve/stall and continue) Substituting W_f: W_u = 0.1 + 0.9(0.05·W_u
+ 0.5·W_u) = 0.1 + 0.495·W_u W_u = 0.1 / 0.505 ≈ 0.198
A gate wrong one time in ten produces a loop whose delivered output is wrong one time in five. Setting β = 0 (never rejects good work) still leaves W_u = 0.1/0.55 ≈ 0.182; setting α = 0 collapses W_u to 0 identically. Iteration amplifies the false-pass rate because every uncertified visit is a fresh Bernoulli trial against α; it cannot amplify what is zero.
For any DOER and any gate: the noisy loop's exit state lies in F with probability 1 (conditional on exit) if and only if α(x) = 0 at every uncertified state the run can visit. No condition on β appears. False-fail is never load-bearing for exit validity: its damage is confined to wasted iterations and — as the worked example's δ(f) shows — to returning certified states to circulation where α gets further chances.
Proof. Exit at an uncertified x occurs only via the event "gate passes x ∉ F," which has probability α(x) at each such check. If α vanishes on all reachable uncertified states, the event has probability 0 at every check, and a countable union of null events is null. Conversely, if some reachable x ∉ F has α(x) > 0, the finite path reaching it followed by an immediate pass has positive probability. ∎5. Panels: Composition Trades, It Does Not Repair
If one gate errs, ask several. The series already owns the two natural composition rules: pool the verdicts (⊞ — pass if any member passes) or take consensus (⊙ — pass only if all do). Throughout this section the panel members are invoked independently, fresh, per check — an assumption made explicit here and operationalized in §7.
For independent gates g₁, g₂ with rates (α₁, β₁), (α₂, β₂), at each state:
any-pass (⊞): α = 1−(1−α₁)(1−α₂) ≥ max(α₁, α₂) β = β₁·β₂ ≤ min(β₁, β₂) unanimous (⊙): α =
α₁·α₂ ≤ min(α₁, α₂) β = 1−(1−β₁)(1−β₂) ≥ max(β₁, β₂)
With all rates in (0, 1), both inequalities in each row are strict: each two-gate composition strictly improves one error mode and strictly worsens the other. In particular the unanimous panel is never more permissive than its most cautious member — its pass event is contained in every member's pass event — the loop-gate form of PRR-2026-010 §8's reading of the Aggregate Consensus Bound (PRR-2026-008 Theorem 8.2), of which these product formulas are the two-point-codomain special case.
Proof. Independent Bernoulli verdicts: OR passes iff not both fail, AND passes iff both pass; evaluate each event's probability at an uncertified state (yielding the α column) and at a certified one (the β column). Strictness for rates interior to (0,1) is immediate from the products. ∎Majority-of-three over independent gates with common rates α, β < ½ strictly reduces both error modes (3e² − 2e³ < e for e ∈ (0, ½) — Condorcet 1785). But its expression in the series' combinators,
(g₁ ⊙ g₂) ⊞ (g₁ ⊙ g₃) ⊞ (g₂ ⊙ g₃)
reuses each gate's single verdict in two branches. That reuse is legitimate exactly because a verdict, once sampled, is a transparent materialization (PRR-2026-011 Definition 4.2) — a recorded bit, free to copy. Re-invoking each gate freshly per occurrence (six invocations) is a different operator, by the Distributivity Obstruction (PRR-2026-003 Theorem 7.1): sharing one sample across branches and drawing fresh samples per branch provably differ. The one panel shape that beats the Trade Theorem does so by stepping outside fresh-invocation composition — the analysis of fresh-invocation majority is left open, stated as such.
Proof of the reduction claim. With shared verdicts, majority passes iff at least two of three independent common-rate verdicts err in the relevant direction fails; the error probability is 3e²(1−e) + e³ = 3e² − 2e³, and 3e² − 2e³ < e ⟺ 2e² − 3e + 1 > 0 ⟺ (2e−1)(e−1) > 0, true on (0, ½). Applied at uncertified states this bounds α, at certified states β; the expression's verdict-reuse is visible in its parse tree, and its inequivalence to the six-invocation reading is PRR-2026-003 Theorem 7.1 applied to the shared gᵢ. ∎The section's summary is one sentence: within fresh-invocation composition, panels allocate error between the two modes — a genuine and often worthwhile choice — and the first composition that shrinks both is bought with transparency, not with more invocations.
6. Choosing a Panel Is Choosing a Failure Mode
Proposition 4.1 says α alone poisons output while β wastes work; Theorem 5.1 says two-gate panels move error mass between exactly those two accounts. The composition rule is therefore a policy decision with a precise semantics: unanimous gating spends throughput to protect validity (α shrinks, β grows — more wasted iterations, cleaner exits); any-pass gating spends validity to protect throughput. Neither is conservative in the abstract. A loop whose divergence is expensive should not buy validity with unanimity it cannot afford; a loop whose false deliverables are expensive should not buy speed with an any-pass gate. The theorems do not choose; they price.
7. The AI Concretization
Following the series' practice (PRR-2026-002 §11), one claimed running instance. This project's autonomous development loop requires its end-of-iteration CHECKER to be a fresh agent instance — never the DOER judging its own work, never a reused judging context. Under this paper's definitions that requirement is the independence hypothesis of §5, operationalized: a fresh instance is what makes consecutive verdicts independent samples of g rather than one correlated judgment replayed, and it is what Theorem 5.1's products assume. The doctrine, in other words, was already paying for the arithmetic it needed before the arithmetic was written down.
Kayfabe accounting: the theorems above are real — the amplification computation, the
asymmetry proposition, the Trade Theorem's products, and Theorem 5.2's Condorcet inequality
are verified by direct calculation. That a real fresh-subagent CHECKER
is a verdict kernel with well-defined (α, β), and that
instance freshness achieves the independence the products assume — neither is
measured here; both are asserted with the series' usual theorem-grade confidence. Same shape
of leap as PRR-2026-002 §11 through PRR-2026-012 §10. Logged as
docs/Research/KAYFABE-LEDGER.md Entry 6 the day this paper publishes.
8. Discussion and Conclusion
The idealization PRR-2026-012 allowed itself was not free, and this paper's accounting of its price is exact where it can be and honest where it cannot. Exact: iteration amplifies false-pass (a 10% gate delivering a 19.8%-wrong loop, by hand); validity depends on α alone; two-gate panels trade error modes with strict inequalities; majority repairs both modes only by reusing transparent verdicts. Honest: correlated gates (the fresh-instance assumption weakened), fresh-invocation majority, optimal panel size against invocation cost, and convergence rates for noisy loops are all open — named, not gestured at as solved.
Together the companion pair does what the series has done since PRR-2026-003/004: build the clean object first, then remove its one idealization and account for everything that changes. The loop earned its algebra in PRR-2026-012. Its gate earned its error theory here.
References
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). Reasoning Dynamics Algebra. Perspectivity Research Reports, Vol. 1, No. 6, PRR-2026-006.
Edwards, R. (2026). Correctness Geometry Algebra. Perspectivity Research Reports, Vol. 1, No. 8, PRR-2026-008.
Edwards, R. (2026). Loop Operators. Perspectivity Research Reports, Vol. 1, No. 10, PRR-2026-010.
Edwards, R. (2026). Portable Inference Algebra. Perspectivity Research Reports, Vol. 1, No. 11, PRR-2026-011.
Edwards, R. (2026). Concrete Loop Algebra. Perspectivity Research Reports, Vol. 1, No. 12, PRR-2026-012.
Condorcet, N. de (1785). Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix. Paris.
Peer Review
The worked example in §4 is the paper's center of gravity and is computed exactly, not simulated — the two-equation system is solvable by a reader with pencil and paper, which is the standard this series set for itself in PRR-2026-003 §7 and has kept since. Proposition 4.1's asymmetry claim is correctly scoped to exit validity, with the false-fail damage honestly relocated (wasted work, re-exposure to α) rather than denied. Recommend acceptance.
Theorem 5.2 is the most interesting result in either companion paper: that the one panel shape beating the Trade Theorem does so by verdict reuse — and that verdict reuse is licensed by PRR-2026-011's transparency and distinguished from fresh invocation by PRR-2026-003's own obstruction — ties three papers together in a way none of them anticipated. The honest deferral of fresh-invocation majority is the correct scope discipline. I would note for the record that the citation of Condorcet is apt and overdue for a series about combining independent judgments. Recommend acceptance.