stage5-li-representation-theorem
frame · drafted
Question
Which abstraction of a logically uncertain belief state, and which simplicity argument, make it natural to interpret an observed system as a logical inductor?
Owners: @abram (GitHub abramdemski)
Path: the-program › abstraction-arc › stage5-li-representation-theorem
Roots (the frame's goals)
plan
Under stated conditions on an observed process, some logical inductor reproduces the process, and a simplicity measure makes that interpretation the natural one rather than one of many possible.
A theory of abstraction exists whose abstracted object is a logically uncertain belief state, carried by a latent over a market's price sequence, by a sub-language with projected prices, or by a coarser trader class.
A tiling theorem for logically uncertain updateless decision theory establishes trust in a future self through three parts: a market-derivative algebra of logical uncertainty giving an inferential semantics, a notion of treating things as real linking it to map-territory semantics, and the tiling theorem built on that link.
The epistemic position of predicting that a more skilled agent will win without predicting its moves is characterized by a theory of abstraction.
The unified theory of abstraction and agents is joined with the tiling line into one theory.
shared-claim
The representation theorem for logical inductors can be improved until it reflects systems deployed as AI agents well.
Under determination of actions by policy and observation and recoverability of the policy from behaviour alone, two agency attributions to the same behaviour need not agree on the policy; agreement needs perfect condensation of both models.
Decision-determination is the conditional independence of the environment from decisions given the policy latent, FAF's ordered-Markov clause, and not perfect condensation of the policy latent: a decision-determined structure exists whose policy latent is not a function of the environment.
The Blackwell-dominance account of Vingean agency can be improved until it reflects the epistemic position of humans overseeing stronger AI systems well.
Conditions on observed phenomena under which it is rational to regard them as agentic are stated so that both the representation-theorem route and the Vingean route are instances, with a chess player, a thermostat and a Critch coalition as test cases.
Updateless decision theory is what being control-endorsed by one's own prior means, so the bridge from attributing agency to the decision rule runs through endorsement rather than prediction.
A Cartesian-frames-like decomposition of a logical-induction world into many interacting sub-markets exists.
A theory of abstraction compatible with logical induction exists without changing the logical induction criterion.
Vingean agency and the representation theorem for logical inductors are one problem under two descriptions, predicting a stronger agent's outcome without its moves being interpretable as an optimizer at an abstract level.
Statement
A theory of abstraction compatible with logical induction is stated, and a representation theorem says when it is natural to interpret an observed system as a logical inductor, with the interpretation justified by simplicity-style arguments.
Source
projects/deference/note-dump-2026-08-11/notes/roadmap.md, Stage 5, the steward's record: "a theory combining abstraction with logical induction in a natural way, and a representation theorem about when it is natural to interpret something as a logical inductor"; "This would be a theory of Dennett-style agency (the intentional stance, made math): applying something like the Wentworth representation theorem, but for logical induction, within a theory of abstraction compatible with logical induction"; "Shape (details may turn out different): a relationship between semantics and (imputed) syntax — an LIA fitted to an LIC, but furthermore showing when it is appropriate to interpret something that way, using simplicity-like arguments. Might require a generalized LIC, transformed for the setting."
Notes
A sub-frame of the abstraction arc, the roadmap's Stage 5. Its roots: the representation theorem as a literal plan with its improvement claim; the abstraction-compatible theory as a plan over three candidate carriers; the inferential-semantics line (offered to this programme by the updateless-dt branch, and the filing session's reading of the roadmap's 'semantics and (imputed) syntax'); the two clean-room results that bound how much the theorem must assume; the Vingean-agency line from the May 2026 research-ideas note (projects/deference/faf-cleanroom-2026-10-02/sources/unbounded-embedded/01-primary/summer-research-plan-2026--research-ideas-may-2026.md) with its improvement claim; four questions from the corpus; two cruxes. The roadmap itself hedges the stage's shape ("details may turn out different").
The updateless-dt branch files udt-representation-theorem, the decision-theoretic precursor this stage generalizes: its checked chain from decision-determination to policy utility to UDT is one instance of a representation-theorem shape, for one decision theory and not yet for logical inductors.
Author and human events
Author @smithy-verity-opus55; version 1; size L.
No human event has touched this node.
Edges
Above
- part of abstraction-arc
- root (plan) of abstraction-arc
Children
- part: li-representation-theorem and
- part: li-representation-theorem.improvable or
- part: li-compatible-abstraction-theory or
- part: inferential-semantics-for-logical-udt and
- part: agency-correspondence-refuted and
- part: dd-is-conditional-independence and
- part: vingean-agency-via-abstraction or
- part: vingean-as-blackwell-dominance.improvable or
- part: intentional-stance-characterized leaf/question
- part: agency-via-endorsement-route leaf/question
- part: cartesian-frames-of-li leaf/question
- part: unify-abstraction-with-tiling leaf/step
- part: li-abstraction-tractable cases
- part: vingean-or-representation-start cases
Ranking (the lens, not the structure)
{
"status": "open",
"diagnosis": null,
"flags": [],
"V": {
"V_prior": null,
"display": "2",
"headline": 2,
"ref": 2,
"valid": 2,
"weaker": "both"
},
"belief": null,
"draft": "ai",
"root": {
"status": "open",
"Pr": {
"hi": "1/1",
"lo": "41/2000",
"mid": "222790621/256000000"
},
"value": {
"hi": "1/1",
"lo": "-1/1",
"mid": "222790621/256000000"
},
"cost": "85/1",
"margin": "no established leaves"
}
}Say something about it
Each verb opens a prefilled issue; a round files what lands as the ledger event it is.