Protected-State-Correction-Theory

Restricted Linear Fiber Theory

Status and scope

This file is theorem-grade on declared restricted linear families.

Known backbone in this file:

• factorization and fiber constancy logic,
• kernel and row-space exactness criteria.

Repo-specific contribution in this file:

• executable witness packages (OCP-049, OCP-050, OCP-052, OCP-053),
• branch-limited diagnostics tied to generated artifacts.

Setup

We work on a restricted linear family:

x = Fz,
y = OFz,
p(x) = LFz.

In coefficient space, equal-record fibers are affine slices of the form z + ker(OF).

Exactness criterion

Exact target recovery is possible exactly when the target is constant on each record fiber. In the restricted linear class this is equivalent to:

ker(OF) ⊆ ker(LF)

and equivalently:

row(LF) ⊆ row(OF).

This criterion is adopted backbone, not a new universal claim.

Why fiber language helps here

The fiber view makes failure mode geometry visible. Two systems can share observation rank but have different target behavior inside ker(OF). That is the core mechanism behind opposite exactness verdicts.

How key theorems fit this view

OCP-045 (minimal augmentation): add rows until target variation inside the record kernel is removed.

OCP-047, OCP-049, OCP-050: amount-only descriptors can match while target behavior on fibers differs.

OCP-051: a weaker target can be stable under noise while a stronger target remains impossible on the same record map.

OCP-052: exactness on a narrow family can disappear after admissible-family enlargement.

OCP-053: exactness on the true family does not guarantee robustness to model mismatch.

Concrete example

Take two observation operators with the same rank. If the first one hides only target-neutral directions, exact recovery can hold. If the second hides a direction that changes the target, exact recovery fails. Rank did not change, but fiber alignment did.

Artifact-backed evidence

Implementation and generated artifacts:

• src/ocp/fiber_limits.py
• restricted_linear_fiber_geometry.csv
• family_enlargement_false_positive.csv
• model_mismatch_stress.csv

These artifacts are branch-limited. They support declared families only.

Citation anchors for overlap-critical statements

• Factorization/sufficiency backbone:
  • Doob-Dynkin style factorization overview: https://arxiv.org/abs/1801.00974
  • Blackwell comparison context: https://digicoll.lib.berkeley.edu/record/112749/files/math_s2_article-08.pdf
  • Coarse-graining and Blackwell order: https://arxiv.org/abs/1701.07602
• Identifiability/observability overlap:
  • Structural identifiability: https://www.sciencedirect.com/science/article/abs/pii/002555647090132X
  • Nonlinear observability: https://doi.org/10.1109/TAC.1977.1101601
  • Kalman filtering baseline: https://www.cs.unc.edu/~welch/kalman/media/pdf/Kalman1960.pdf

This site is open source. Improve this page.