condensation-correspondence-theorem
and · drafted
Statement
Two latent-variable models that both perfectly condense one random-variable model stand in correspondence: each latent of either model is almost everywhere a function of the other model's latents above the same given variables.
Source
projects/deference/faf-cleanroom-2026-10-02/packages/udt-condense-dd/udt-condense-dd-findings.md, The Eisenstat map: "Theorem 5.2, Exact Correspondence" of the corpus is the paper's thm:exact-uniq, FAF's "Theorem 4.15 Condensation.aeFunctionOf_jointAbove_of_perfectlyCondenses"; the approximate form is "Theorem 5.8 Condensation.condEntropy_jointAbove_le". projects/deference/faf-cleanroom-2026-10-02/packages/udt-condense-dd/udt-condense-dd-findings.md, item 6: "Eisenstat's Theorem 4.15 needs perfect condensation of both models".
Notes
The literal object Stage 4 analyzes: the theorem of Eisenstat's condensation paper that characterizes "the same concept". The shared-givens hypothesis is the place where the roadmap's remark that the observables do some of the work bites (observables-carry-correspondence); relaxing it is a route of the improvement claim.
FAF's formalization of the paper carries an errata file and two disclosed restrictions (universe stratification; the paper's Examples 5.1–5.3 out of scope), in Formalized Agent Foundations at commit 159ec3f, Condensation/notes/paper-errata.md, which every comparison with the printed paper reads first.
Bridge card
Connected to no informal claim yet: held by no frame.
Declaration Condensation.aeFunctionOf_jointAbove_of_perfectlyCondenses (record condensation-correspondence-theorem.record)
The declaration states Eisenstat's exact correspondence theorem: its hypotheses are two latent models of one random-variable model that both perfectly condense it, and its conclusion is the almost-everywhere functional dependence of each latent on the other model's latents above the same givens.
Hypotheses, read:
- condensation-shared-givens definitional: The two latent-variable models are latent models of one random-variable model, so their given variables are the same.
- condensation-both-perfect definitional: Both latent-variable models perfectly condense the shared random-variable model.
Completeness (condensation-correspondence-theorem.complete): No premise beyond one shared random-variable model and perfect condensation by both latent models is needed for the exact correspondence.
Author and human events
Author @smithy-verity-opus55; version 1; size M.
No human event has touched this node.
Edges
Above
- part of stage4-condensation-dynamics
- root (shared-claim) of stage4-condensation-dynamics
Children
- part: condensation-correspondence-theorem.record leaf/record
- part: condensation-correspondence-theorem.bridge leaf/bridge
- part: condensation-shared-givens leaf/hypothesis
- part: condensation-both-perfect leaf/hypothesis
- part: condensation-correspondence-theorem.complete leaf/completeness
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