# Postulate 1 Definitions Lock

> Phase 0 artifact for
> [`COARSE_GRAINING_PROOF_ROADMAP.md`](../COARSE_GRAINING_PROOF_ROADMAP.md).
> Status: definitions locked for Phase 1 drafting, 2026-05-16. This is an
> internal proof note, not public-facing claim language.

## Scope

This note pins the object that later proof phases are allowed to use. The claim
is not that a signature reconstructs the world. The claim is that, for a stated
task and objective, the signature may contain exactly the information needed to
choose an optimal action.

The first correction this note makes explicit is dynamic-task notation. In a
one-step fully observed problem, `X` can be the raw physical microstate. In a
partially observed controlled process, `X` is the measurable decision domain at
the moment of action: observation history, belief state, or another
pre-registered information state. That convention keeps the LQG Phase 1 proof
honest: the Kalman estimate is a statistic of the decision information, not an
omniscient readout of hidden plant state.

## Symbol Ledger

Each symbol below is defined once for the proof track.

| Symbol | Definition |
| --- | --- |
| `X` | Measurable decision domain for the task. It may be the raw world state in fully observed cases, or a history/belief information state in partial-observation cases. |
| `x ∈ X` | One decision-state realization drawn from the task's evaluation distribution. |
| `A` | Measurable admissible action space. |
| `Φ : X → Σ` | Signature map. It extracts the information made available to the signature-only controller. It is policy-independent except through the real task dynamics that generate `x`. |
| `Σ` | Measurable signature space. |
| `σ = Φ(x)` | The realized signature. |
| `𝓕_σ` | The sigma-algebra generated by `Φ`: `𝓕_σ = {Φ^{-1}(B) : B ⊆ Σ measurable}`. |
| `μ` | Evaluation measure over `X`; this is the measure behind every "almost everywhere" claim. |
| `J` | True task objective used to rank policies, usually expected return, expected cost, or regret under `μ` and the task dynamics. |
| `π*` | A `J`-optimal policy under the phase's admitted information regime. If there are multiple optima, `π*` means the pre-registered optimal selector; equivalently, the optimal-action correspondence must admit a measurable selector. |
| `g : Σ → A` | A signature policy, mapping signatures to actions. |
| `𝓒` | Optional bounded agent class used only when deriving the capacity-relative separability corollary. |

Legacy notation: older Sundog docs use `S(x)` for a scalar signature readout and
the founding anniversary thread uses `H(x) = ∂S/∂τ`. In this proof track, `Φ` is
the full signature map. A scalar `S` is only a coordinate or readout of `Φ`.
The `H(x)` line is not assumed in Phase 0; Phase 1 tests whether the LQG toy's
`∂S/∂τ` term is proportional to the Fisher information of `Φ`.

## Definition

A task is **Sundog-solvable** under signature `Φ` iff there exists a measurable
signature policy `g : Σ → A` such that

```text
π*(x) = g(Φ(x))    μ-almost everywhere.
```

Equivalently, `π*` is `𝓕_σ`-measurable. In objective-specific Blackwell terms,
`Φ` is sufficient for the control objective `J`, not for state reconstruction
and not for every possible objective.

This permits many-to-one signatures. If `x1` and `x2` are in the same `Φ` fiber,
the task remains Sundog-solvable exactly when the admitted optimal selector can
choose the same action on those states, except on a `μ`-null set. If same-fiber
states require different unique optimal actions on positive measure, no flat
signature-only policy can be optimal.

For randomized policies, the same definition applies with `π*` and `g` treated
as Markov kernels into distributions over `A`.

## Relationship To Existing Fronts

The current [`SCIENTIFIC_CRITERIA.md`](../SCIENTIFIC_CRITERIA.md) research
object is a controlled instance: indirect photometric feedback can be sufficient
for a bounded mirror-pointing task without target-position access. This note
generalizes that into a predicate a phase can prove or falsify.

The [`SUNDOG_V_MESA.md`](../SUNDOG_V_MESA.md) Formal Separability appendix is
the mesa-specific corollary target. If `π*` is `𝓕_σ`-measurable and the agent
class `𝓒` cannot cheaply invert or rewrite `Φ`, then a signature-tracking agent
does not need to grow an internal reward proxy to act optimally inside that
class. Phase 2 must prove that this falls out of the trunk rather than becoming
a parallel theorem.

The pushable-occluder boundary in
[`PHASE2_BLOCKS_DESIGN.md`](../PHASE2_BLOCKS_DESIGN.md) is the negative
direction. If the decisive bit only appears after a preparatory world-changing
action, and is not measurable in the current `𝓕_σ`, then a flat signature policy
should fail. That failure is predicted by the definition rather than patched
after the fact.

## Substrate Admission Checklist

A substrate can enter this proof track only after its phase spec pins:

1. `X`, including whether it is raw state, history, belief, or another
   decision-state representation.
2. `Φ` and `Σ`, including sensor corruption/noise handling if the signature is
   measured through a channel.
3. `J`, the evaluation measure `μ`, and the exact regret or return readout.
4. `π*`, a Bayes-optimal policy, oracle, or pre-registered tractable proxy, with
   tie handling.
5. The measurable cell set where `π*` is expected to be `𝓕_σ`-measurable, and
   the off-cell boundary where it is expected not to be.

If any item cannot be stated without an unmeasurable or post-hoc quantity, that
substrate is not admitted to the proof track.

## Source Ledgers For Locked Numbers

This definitions note introduces no new numeric result. Existing numbers are
carried only by their source docs:

| Figure | Source |
| --- | --- |
| Core photometric comparison, seed count, and acquisition-time cost | [`README.md`](../../README.md), [`PAPER_v1_draft.md`](../PAPER_v1_draft.md) |
| Mesa cliff and `net.7`/5D mechanism language | [`presentation/claims-and-scope.md`](../presentation/claims-and-scope.md), [`SUNDOG_V_GRAVITY.md`](../SUNDOG_V_GRAVITY.md), [`SUNDOG_V_MESA.md`](../SUNDOG_V_MESA.md) |
| Geometry parhelic-belt residual disposition | [`calibration/PHASE10_BELT_Y_RESULTS.md`](../calibration/PHASE10_BELT_Y_RESULTS.md), [`calibration/RICH_DISPLAY_OVERLAY_NOTES.md`](../calibration/RICH_DISPLAY_OVERLAY_NOTES.md) |
| Postulate ordering, capacity-relative framing, and `H(x)` toy check | [`internal/theory/postulations.md`](../../internal/theory/postulations.md), [`internal/feedback/quarantine/attack_vectors.md`](../../internal/feedback/quarantine/attack_vectors.md), [`internal/theory/analogies.md`](../../internal/theory/analogies.md) |

Phase docs may quote those values, but must quote them from the source ledger
and may not re-derive or round them into a different gate.

## Phase 0 Review

Internal consistency pass:

- Every proof-track symbol used in the predicate is defined in the symbol ledger.
- The only intentional notation bridge is legacy `S(x)` as a scalar coordinate
  of `Φ`; it is not a second signature definition.
- The predicate requires an admitted optimal policy or proxy for evaluation, but
  does not require the deployed signature controller to observe hidden state.
- Dynamic tasks are handled by making `X` the decision information state before
  action, preventing the Phase 1 LQG proof from smuggling in omniscience.

Pre-registered negative for Phase 0: **not triggered.** The predicate can be
stated operationally, provided each empirical substrate satisfies the admission
checklist above before it runs.

## Phase 1 Handoff

The LQG proof should use the following convention:

- `X` is the controller's pre-action information state, or equivalently the
  Gaussian belief state induced by the observation history.
- `Φ(x)` is the Kalman sufficient statistic, usually the posterior mean plus any
  covariance term needed by the chosen cost convention.
- `J` is the LQG quadratic objective.
- `π*` is the Bayes-optimal LQG controller under the same information regime.

The Postulate 6 toy check belongs in Phase 1, not here: compute `∂S/∂τ` and the
Fisher information of `Φ` in the chosen LQG toy, record pass/fail, and do not
promote the anniversary `H` line unless the proportionality actually lands.