C.2:4 - Solution

C.2:4.1 - Coordinates, constitution, and neighboring relations

KD‑CAL characteristics (single‑episteme, point‑values).

• Formality F. From free prose to machine‑checkable proof/specification. Litmus: would a machine reject it if wrong?
• Claim scope (G), a set‑valued applicability over U.ContextSlice, with ∩/SpanUnion/translate algebra; CL penalties apply to R, not to F/G. Litmus: how wide is the declared scope, and under what minimal assumptions does the claim hold?
• Reliability R. Warrant for this exact claim and receiving use. Litmus: what supports this conclusion, under which assumptions, and what limits it? R-claims MUST bind to their actual formal or empirical support. A numerical R requires the B.3/C.2.2 meaning, scale, and model; otherwise retain separate support and a bounded reasoned conclusion. A proof under axioms needs no empirical score, and F cannot be substituted for R. Relevance windows and B.3.4 currentness rules apply where the relied-on support consumes them.

Congruence Level (CL), pairwise ladder. CL‑0 Opposed/Disjoint (contrastive; no substitution); CL‑1 Comparable / Naming‑only (label similarity; no substitution); CL‑2 Translatable (structure‑preserving correspondence in a declared fragment with stated loss); CL‑3 Near‑identity (the declared invariants match). CL is a characteristic of a relation between two epistemes; it is not a fourth member of the F–G–R assurance tuple and it is not a characteristic space of its own. KD-CAL substitution constraint: plane preservation and CL ≥ 2 are necessary for a substitution under this calculus; substituting type‑structure additionally requires CL = 3. These conditions do not establish suitability for the proposed use. State the exact substitution, direction, correspondence rule, and tolerated loss under the direct receiving pattern, and establish reliance separately. When local meanings cross semantic contexts, first test the F.9 Bridge predicate on exact F.17 cells; keep the obtaining Bridge, bounded-use claim, and reliance result distinct. A CL value establishes no local system-role-kind membership or assignment.

Constitution and neighboring relations. State F, G, and R for one exact claim of one C.2.1 episteme. Its exact claim content, EntityOfConcern, and effective U.ReferenceScheme identify the episteme through EpistemeConstitutionRelation. F characterizes the claim’s form; G is the separate U.ClaimScope; R relies on exact evaluation, evidence-use, and assurance relations. Empirical grounding and edition remain separate C.2.1 relations. Viewpoint selection and view conformance remain under E.17.0; notation and other representation structure remain under C.29/A.6.3.RT; publication occurrence, form, and carrier remain under E.17/E.24.PUB. Multiple notations are allowed only when their exact representation or notation relation is explicit and any declared loss is applied to R rather than hidden in an omnibus episteme field.

C.2:4.2 - Four Δ‑moves (epistemic motion)

• ΔF — Formalise. Rewrite for stricter calculi/grammars; raise proof obligations.
• ΔG — Generalise / Specialise. Widen or narrow the claim scope (assumptions & scope). Changes to decomposition granularity are an orthogonal view and do not change G unless they alter the envelope.
• ΔR — Calibrate / Validate. Revise warrant through support that actually bears on the claim: proof or reasoning, calibration, severe tests, or monitoring as applicable. State what the contribution changes. A formalization alone is ΔF, not an R increase; choosing new inquiry is a separate decision.
• ΔCL — Congrue. Establish and record the sameness relation between two epistemes (ladder 0→3). Moves compose into paths. A CL chain minimum retains only the ordered congruence meaning justified by the relation family; it is not a numerical reliability loss.

C.2:4.3 - Composition (Γ_epist) and propagation

Let Γ_epist compose exact epistemes {Eᵢ} for one declared claim and use. B.1.3 supplies the synthesis/compilation Method; B.3 and C.2.2 govern warrant and scale discipline.

• R (Reliability). First distinguish indispensable premises, alternative sufficient arguments, complementary evidence, different scope slices, and counterevidence. Identify duplicated data and shared assumptions or bias. A numerical fold requires warranted input meanings, compatible scales, dependencies, and a receiving model. Neither series nor parallel syntax supplies a default minimum or maximum, and there is no universal cap at the best support line. Where no common model is justified, retain separate contributions and limitations in a bounded reasoned synthesis.
• F (Formality). F(Γ) = minᵢ F(Eᵢ) over the essential formal constituents of the claim. This is an ordinal formality statement, not an R calculation. Raise F by the actual ΔF move; neither an axiomatic mode nor a line=formal tag converts F into empirical warrant.
• G (ClaimScope). Required premises compose only on their overlapping scope. Distinct supported slices may form SpanUnion({G_path}) under A.2.6 and C.2.2’s type-before-scope rule; retain their support separately and drop unsupported regions. A new source does not by itself generalise the claim. Scope change remains an explicit ΔG± move.
• CL (Congruence). Keep each traversed mapping and the ordered meaning of its declared CL visible. A chain minimum is usable where that relation’s congruence rule justifies it. A notation, scope-translation, kind, plane, source-local, model-use, or evidence-reuse relation contributes only its own warranted loss. A numerical Φ needs its receiving model; a monotone table or clipped output does not supply one.

For example, two necessary independent conditions with probabilities 0.9 each have conjunction probability 0.81, not minimum 0.9. Conversely, a limited complementary source need not reduce the support already available. A credible contrary result changes the affected conclusion. A theorem A ⇒ P remains valid as a formal result while evidence violating A can defeat its use as assurance of an actual system.

Γ remains defined on holons and respects the core’s identity and boundary discipline. Its support account establishes neither a new action permission nor the worth of acquiring further evidence.

C.2:4.4 - What must not be conflated (normative guards)

• Representation structure ≠ carrier. Files, PDFs, or repositories are carriers outside the episteme; they never count as parts of U.Episteme (see C.2.1 EP‑1; CC‑EPI‑2/3).
• Epistemes do not act. Only systems perform Work. Epistemes carry claim content and can participate in constitution, grounding, edition, description, evidence-use, reliance, viewing, representation, and publication relations under their direct patterns.
• CL is not a score. It is a qualitative ladder of preservation classes; do not average it.