fix: resolve B-0357 ID collision

docs/backlog/P1/B-0367-first-class-uncertainty-semiring-parameterized-weight.md

@@ -0,0 +1,151 @@
+---
+id: B-0367
+priority: P1
+status: open
+title: "First-class uncertainty — semiring-parameterized weight type for DBSP"
+effort: L
+created: 2026-05-09
+last_updated: 2026-05-09
+depends_on: []
+classification: research
+decomposition: atomic
+owners: [algebra-owner]
+type: feature
+tags: [algebra, uncertainty, semiring, bayesian, inference, openspec]
+---
+
+# B-0357 — First-class uncertainty in DBSP
+
+## What
+
+Parameterize the DBSP weight type so the same circuit
+topology can run over different semirings — integer (current),
+probabilistic, Gaussian, interval, provenance. Uncertainty
+becomes a first-class property of every query result, not
+an afterthought bolted on top.
+
+## Why
+
+The OpenSpec compiler needs to produce specs that carry
+uncertainty through the compilation chain. A spec that
+says "this query returns rows" is weaker than one that
+says "this query returns rows with confidence 0.87."
+Without first-class uncertainty, the OpenSpec → formal spec
+→ F# pipeline can only express exact results — it can't
+compile the "how sure are we?" question.
+
+This is also the bridge to the Infer.NET BP/EP direction
+from AGENDA.md: belief propagation IS incremental Bayesian
+inference, which IS DBSP over a probabilistic semiring.
+Same circuit, different weight type.
+
+## The algebra
+
+DBSP operators (D, I, z⁻¹) are linear over the weight
+semiring. Current: (Z, +, ×) — the integer ring. The
+generalization:
+
+```
+ZSet<'K, 'W when 'W : ISemiring>
+  where ISemiring provides:
+    zero: 'W
+    add: 'W -> 'W -> 'W
+    mul: 'W -> 'W -> 'W
+    (optionally) negate: 'W -> 'W  // ring, not just semiring
+```
+
+Candidate weight types:
+
+| Semiring | Weight | Answers | Ring? |
+|---|---|---|---|
+| Integer (current) | int64 | multiplicity + retraction | yes |
+| Boolean | bool | exists? | no |
+| Tropical | (float, min, +) | shortest path / confidence | no |
+| Probabilistic | [0,1] | how likely? | no |
+| Gaussian | (μ, σ²) | estimate + uncertainty | yes |
+| Interval | [lo, hi] | bounded uncertainty | yes |
+| Provenance | polynomial | which inputs contributed? | yes |
+
+## Key design decisions
+
+1. **Ring vs semiring**: retraction (negative weights)
+   requires a ring (additive inverse). Pure semirings
+   (boolean, tropical, probabilistic) don't support
+   retraction — they'd need the counting variant or
+   a different fixpoint strategy.
+
+2. **Distinct operator**: currently clamps to {0, 1}.
+   Under probabilistic weights, becomes a threshold
+   operator (clamp below cutoff to zero). Under
+   Gaussian, becomes a significance test.
+
+3. **Semi-naive optimization**: requires subtraction
+   (Δ = new - old). Non-ring semirings need the
+   RecursiveCounting variant or a different incremental
+   strategy.
+
+4. **Performance**: generic weight dispatch vs
+   specialized paths. The integer path must not regress.
+
+## Prior art
+
+- Green, Karvounarakis, Tannen 2007 "Provenance
+  Semirings" (PODS) — the foundational framework
+- Budiu et al. 2022 "DBSP" — Z-ring instantiation
+- Infer.NET — Bayesian inference as message passing
+  (the EP/BP direction from AGENDA.md)
+- Reaqtor — standing queries with checkpoint (the
+  durability layer for streaming inference)
+
+## What exists in Zeta today
+
+- `src/Core/ZSet.fs` — hardcoded `Weight = int64`
+- `src/Bayesian/BayesianAggregate.fs` — Bayesian
+  aggregation but external to the weight algebra
+- `src/Core/NovelMath.fs` — tropical semiring work
+  (non-integer semiring already explored)
+- `src/Core/Algebra.fs` — algebraic laws, currently
+  integer-specific
+
+## Acceptance criteria
+
+- [ ] `Weight` type is generic with ISemiring constraint
+- [ ] Integer path performance unchanged (zero regression)
+- [ ] At least one non-integer semiring compiles and
+  passes a smoke test (Gaussian or Interval)
+- [ ] OpenSpec can express uncertainty in spec output
+- [ ] Z3 lemmas generalize to semiring axioms
+
+## Composes with
+
+- OpenSpec compiler (the motivation — specs need uncertainty)
+- Infer.NET BP/EP direction (belief propagation = DBSP
+  over probabilistic semiring)
+- `src/Bayesian/` (absorbs into the weight algebra)
+- Tropical semiring work in `NovelMath.fs`
+- The Superfluid reactor equation (uncertainty in the
+  learning gain term)
+
+## Prior art — time-uncertainty in production databases
+
+Aaron 2026-05-09: "lookup spanner and cockroach db real db
+primitives of time and lamport include uncertainty measurement
+on the datetime" + "there is tidb we should research this
+actually and decide if we want to support multiple and make
+it pluggable."
+
+| System | Timestamp model | Uncertainty primitive |
+|---|---|---|
+| Spanner | TrueTime (GPS + atomic) | `[earliest, latest]` interval; commit-wait |
+| CockroachDB | HLC (Lamport + wall-clock) | Read uncertainty restart window |
+| TiDB | TSO (centralized oracle) | Single-point, no interval |
+| YugabyteDB | HLC variant | Similar to CockroachDB |
+
+Lamport's logical clocks → Spanner's TrueTime intervals →
+Zeta's weight semiring intervals. Three instantiations of
+"carry uncertainty, don't pretend precision."
+
+**Research question:** should Zeta parameterize the
+time-uncertainty model (pluggable, like the weight
+semiring) so it works over any of these backends? Or
+hardcode one model?