FPRD LabShare an approach

← Claims

Theorem

FPRD-T36

Bounded exposure is equivalent to shifted finite actions

Exact statement

For a complete finite-observation tower, an exact action FF has exposure at most ee exactly when it is represented by a unique compatible family FN[e]:XN+e→XNF_N^{[e]}:X_{N+e}\to X_N. Conversely, every such shifted-natural family reconstructs a unique continuous exact action. Exposures are closed under padding and add under composition; in the shortest-observation ultrametric, exposure ee is equivalent to a 2e2^e-Lipschitz bound.

StatusProved and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Finite exposure is exactly the condition needed for an exact action to descend to finite approximations after a uniform level shift.

Definitions

  • The factor map is forced by FN[e](qN+e(x))=qN(Fx)F_N^{[e]}(q_{N+e}(x))=q_N(Fx).
  • Shifted naturality says these factor maps commute with the bonding maps after accounting for the level offset e.

Hypotheses and scope

  • The observation tower satisfies FPRD-D19.
  • The reconstructed finite family is compatible with all bonding maps.

Proof or evidence

Exposure makes each finite factor map well-defined and surjectivity of the quotient makes it unique. Conversely, a shifted-natural family determines a coherent inverse-limit point for every exact state; completeness reconstructs one exact image. The composition and Lipschitz laws follow by adding observation loss.

Exposure representation theorem →

Verification notes

The M22 closure audit reports a 4,294-case independent challenge with 12 mutation separators, an 88-test focused corpus, and a passing 684-test repository sentinel. Proof, API, mathematical, and implementation reviews were clean after recorded repairs.

Limitations

  • The theorem characterizes bounded exposure, not every continuous self-map of an inverse limit.
  • The numerical Lipschitz constant uses the specific base-two shortest-observation metric.

Open work

Compare the shifted-natural formulation with standard pro-object and profinite dynamical-system language.

Notes

M22 proves the uniform bounded-exposure fragment. General level-dependent continuity moduli are a later extension rather than being silently identified with one additive exposure bound.