NEXT_THEORY.md
Grounded Abstraction
NEXT_THEORY.md
1 # Theoretical extensions after the first edition
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3 **Starting point: Grounded Abstraction v1.0.0.** These are research questions, not unfilled first-edition publication requirements.
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5 ## 1. State-changing noisy evidence
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7 Replace a fixed-world likelihood model with an observation-and-transition kernel. Characterize when a compressed record supports filtering, intervention and later correction together. The first-edition static-cell theorem cannot be imported without accounting for transitions between cells and dependence created by action selection. A first tractable target is a finite hidden-state model with explicit aggregate transition/observation equalities and a small exact counterexample when one equality fails.
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9 ## 2. Economical and adaptive precision
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11 The sufficient denominator grows with the horizon and uses a uniform likelihood floor. It is not optimal. Investigate certified variable precision, deferred rounding and query-dependent error allocations under one retained-record and working-memory contract. A new result should distinguish average error from guarantees after every attainable history. Nearly impossible evidence, support loss, and the cost of restoring discarded probability remain explicit obstacles.
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13 ## 3. Long-horizon stability with justified structure
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15 A positive likelihood floor alone yields the conservative finite-horizon sensitivity bound, not uniform contraction of all filters. Seek additional mixing, observability or correction conditions that support time-uniform approximation. Keep this filtering question distinct from the visible-block small-gain condition. Demonstrate necessity boundaries rather than infer stability from small local residuals.
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17 ## 4. Restricted operation languages
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19 The exact linear results quantify over all words. A finite protocol automaton can constrain the allowed sequences, changing the observable kernel and initialization problem. A next theorem could characterize the smallest protocol-relative common predictor and identify when an operation made legal by revision exposes an old hidden direction.
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21 ## 5. Model admission and prior allocation
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23 Adding a world changes support and possibly the evidence-target partition. Old masses cannot determine how much probability a genuinely new world should receive. Study conservative admission contracts that separate transported prior mass, newly supplied evidence and unresolved allocation. No distribution should appear simply because a field has been added to a data structure.
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25 ## Method
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27 Prioritize self-contained arguments, explicit counterexamples and finite illustrative calculations. Reuse the existing theory and avoid treating an implementation campaign as a prerequisite. An open extension does not make the first edition's proved conditional statements incomplete.