HodgeAnalyticPathology
plain-language theorem explainer
The Hodge analytic pathology predicate is the constantly-true proposition on analytic display records. Audits and conservativity proofs for the Hodge bridge cite it as the pathology slot in the problem-audit bundle. The definition is a one-line constant Prop, so every display satisfies it by construction.
Claim. For every Hodge analytic display record $d$ (algebraic certificate, complex dimension, cohomological degree, and display payload), the analytic pathology predicate at $d$ is the true proposition $\top$.
background
This module builds certificate audits for several hard-problem bridges (prime critical line, Navier–Stokes energy, Yang–Mills gap, Hodge). Each audit pairs a completion map with legitimate and pathology predicates on a domain-specific display type.
A Hodge analytic display packages an algebraic Hodge certificate together with complex dimension, cohomological degree, and a display payload. The pathology predicate is the audit slot that records whether a display is treated as pathological under the analytic bridge.
Here that slot is filled by a constant-true proposition: no extra analytic constraint is imposed beyond inhabiting the display type.
proof idea
Definitional, not a proof. The body is the constant True, ignoring the display argument. Downstream conservativity only needs that the predicate is a Prop on displays; the trivial inhabitant makes every conservativity obligation immediate.
why it matters
Fills the pathology field of hodgeAnalyticAudit, the ProblemAudit bundle for the Hodge algebraic certificate with analytic displays. The companion theorem hodgeAnalytic_conservative shows both legitimate and pathology predicates are conservative for the analytic completion (they depend only on the underlying certificate).
In the Primitive Recognition Calculus hard-problem audit layer this keeps the Hodge bridge structurally parallel to the other certificate audits, even while the analytic pathology condition remains vacuous. It does not itself advance T0–T8 forcing, RCL, or the phi-ladder; it is scaffolding for uniform audit wiring around the Hodge conjecture bridge.
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