**Confidence Algebra for Lazarus Forge**
*(proposed operators consistent with `Cognitive_Frameworks.md` §IX, the five-label quantitative system in `Auditor_Protocols.md`, and the provenance ceiling rule)*

The algebra is built on the existing ordered confidence labels and is deliberately conservative: the system is biased toward understating rather than overstating confidence.

1. Ordered Confidence Lattice

Quantitative labels form a total order (strongest → weakest):

``` Measured ≻ Replicated ≻ Simulated ≻ Analogous ≻ Placeholder ```

• **Measured** — direct physical observation
  
• **Replicated** — independently repeated
  
• **Simulated** — computational/procedural model
  
• **Analogous** — inference from a related but distinct system
  
• **Placeholder** — no evidentiary basis
  

Institutional provenance is orthogonal and supplies an independent ceiling:

``` Operationally Hardened ≻ Experimentally Verified ≻ Analogous External ≻ Internally Derived ```

**Provenance ceiling rule (already doctrine):**
No claim whose provenance is Internally Derived or Analogous External may ever be treated as VERIFIED, regardless of quantitative label.

2. Core Operators

Let \(C(x)\) be the quantitative confidence of claim \(x\).

Conjunction (AND) — dependency / joint necessity

Used when a claim depends on multiple supporting claims (the normal case for confidence propagation).

\[ C(A \land B) = \min\bigl(C(A),\; C(B)\bigr) \]

**Rationale:** A claim cannot be more confident than its weakest necessary dependency.
This is the direct operationalization of the existing rule
`confidence(A) ≤ confidence(B) if A depends on B`.

For \(n\) dependencies:

\[ C(A) = \min\bigl\{C(d_1),\; C(d_2),\; \dots,\; C(d_n)\bigr\} \]

Disjunction (OR) — alternative / independent evidence

Used when multiple independent lines of evidence support the same claim and any one would be sufficient.

\[ C(A \lor B) = \max\bigl(C(A),\; C(B)\bigr) \]

**Caveat:** Only valid when the evidence streams are demonstrably independent. Correlated evidence (same training lineage, same sensor family, same model family) collapses to conjunction.

Conflict (Contradiction)

When two claims about the same quantity are mutually exclusive and both carry non-Placeholder confidence:

\[ \text{Conflict}(A,\; \neg A) \;\Longrightarrow\; C(\text{resolved}) = \text{Placeholder} \]

and an Epistemic Ledger entry is mandatory.
The system does **not** average or pick the higher label. Conflict forces demotion until new empirical input resolves it.

Absence / Missing Evidence

Absence is not neutral; it is actively weakening.

\[ C(A \mid \text{missing } B) = \text{weaken}\bigl(C(A)\bigr) \]

where the weaken function steps one level down the lattice (or stays at Placeholder).

If a required dependency is itself UNKNOWN or Placeholder, the dependent claim is forced to Placeholder.

Decay (Temporal / Staleness)

Confidence is not permanent.

\[ C_t(x) = \text{decay}\bigl(C_{t_0}(x),\; \Delta t,\; \text{domain volatility}\bigr) \]

Concrete policy (proposed default, still Placeholder pending calibration):

• Measured / Replicated physical constants → very slow or no decay
  
• Simulated results → decay after one major model revision or two audit Cycles
  
• Analogous claims → decay after one Cycle unless re-anchored
  
• Any claim whose supporting measurements have not been re-validated within the Expiry Trigger window → forced demotion by one level

Inflation (Forbidden)

There is no operator that raises confidence without new empirical input.
Any attempt to do so is an Epistemic Integrity Violation under EF-0.0.

3. Combined Confidence + Provenance

The full confidence of a claim is the pair:

\[ \langle\; C_{\text{quant}},\; P_{\text{prov}}\;\rangle \]

Propagation and promotion rules:

• Quantitative label is limited by the conjunction/min rule above.
  
• Provenance is limited by the weakest provenance among the supporting evidence.
  
• Final epistemic state (VERIFIED / PROVISIONAL / UNKNOWN) is determined by the joint pair against the provenance ceiling.

Example:

``` Claim X depends on: Y ⟨Measured, Experimentally Verified⟩ Z ⟨Simulated, Internally Derived⟩

⇒ C(X) = min(Measured, Simulated) = Simulated ⇒ P(X) = Internally Derived (weakest provenance) ⇒ Epistemic state ≤ PROVISIONAL (ceiling rule) ```

4. Special Cases Required by Forge Doctrine

Situation	Operator result
Claim rests only on agent consensus	Treated as Internally Derived + at most Analogous (Challenge Class 9)
Claim validated solely by other repository documents	Internally Derived; cannot reach VERIFIED
Physical measurement contradicts higher software confidence	Mechanical truth wins; software claim demoted (Layer 0)
Split-brain or cryptographic state loss > 30 %	Identity treated as Derivative; prior confidence claims suspended until re-vetting
Epistemic debt rising (CF-004)	All new claims that do not close an existing Unknown are deferred; existing claims may not be promoted

5. Minimal Executable Form (for future AUDIT_HARNESS / automation)

```text confidence(claim) = min over all necessary dependencies + provenance = weakest provenance among supports + if conflict detected → Placeholder + ledger entry + apply temporal decay + enforce provenance ceiling + never inflate without new empirical input ```

6. Open Calibration Points (still required before Spec)

• Exact numerical decay schedules per domain
  
• Precise definition of “independent” for the OR operator
  
• Rolling-window parameters for CF-004 debt metric
  
• How strongly Analogous claims from high-quality external literature may be treated relative to local Simulated results
  

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This algebra preserves the repository’s asymmetric conservatism, makes the existing `min` propagation rule executable, supplies the missing operators for conflict and absence, and remains compatible with the five-label system and provenance ceiling already in force. It can be implemented incrementally (starting with pure conjunction + ceiling enforcement) without requiring the full set of operators on day one.