Protected-State-Correction-Theory

Descriptor-Fiber Anti-Classifier Bounds and Compatibility Lift on Supported Witness Classes

Steven Reid
Independent Researcher
ORCID: 0009-0003-9132-3410
sreid1118@gmail.com
April 2026

Abstract

This paper formalizes a branch-limited quantitative layer for the OCP anti-classifier program. On finite witness classes with binary exactness verdicts, we define descriptor fibers and prove two finite-class results: a purity criterion for perfect descriptor-only classification and an irreducible descriptor-only error lower bound. We then define computable invariants: descriptor-fiber mixedness (DFMI), irreducible descriptor error lower bound (IDELB), and compatibility lift (CL). On current OCP witness artifacts, amount-only descriptors are mixed (DFMI=1) with nonzero irreducible error, while adding compatibility information collapses the lower bound to zero on the tested rank witness class. Scope is intentionally restricted to supported finite witness families.

Keywords: anti-classifier, descriptor fibers, recoverability, incompatibility, finite witness classes, branch-limited invariants

1. Scope and Placement

This branch is a quantitative extraction layer under the restricted-linear and fiber no-go program. It is not a universal theory and is not a replacement for core projector/recoverability branches.

2. Setup

Let W be a finite witness set with exactness verdict v: W -> {0,1} and descriptor map a: W -> A. For each descriptor value alpha in im(a), define:

• E_alpha = |{w in W : a(w)=alpha, v(w)=1}|
• F_alpha = |{w in W : a(w)=alpha, v(w)=0}|

The preimage a^{-1}(alpha) is a descriptor fiber.

3. Theorem 1 (Descriptor-Fiber Purity)

A perfect deterministic descriptor-only classifier h: A -> {0,1} satisfying h(a(w))=v(w) for all w in W exists iff every descriptor fiber is verdict-pure:

• for each alpha, either E_alpha=0 or F_alpha=0.

Proof sketch

Necessity: if h is exact, all witnesses with same descriptor must share a label.
Sufficiency: if every fiber is pure, assign the unique fiber label to h(alpha).

4. Theorem 2 (Irreducible Descriptor-Only Error)

Define:

• IDELB(a;W) = (sum_alpha min(E_alpha,F_alpha))/|W|

For any deterministic descriptor-only classifier h: A -> {0,1}:

• Err_W(h) >= IDELB(a;W).

Majority vote on each fiber attains this bound.

Corollary

If any descriptor fiber is mixed (E_alpha>0 and F_alpha>0), perfect descriptor-only classification is impossible on W.

5. Quantitative Invariants

Define:

• DFMI(a;W) = |{alpha : E_alpha>0 and F_alpha>0}| / |im(a)|
• CL(b|a;W) = IDELB(a;W) - IDELB((a,b);W)

CL measures reduction in irreducible descriptor-only error after adding compatibility descriptor b.

6. Supported Witness Results

Computed from:

• data/generated/unified-recoverability/rank_only_classifier_witnesses.csv
• data/generated/unified-recoverability/candidate_library_budget_witnesses.csv

Summaries:

• data/generated/descriptor-fiber-anti-classifier/meta_classifier_invariants.csv
• data/generated/descriptor-fiber-anti-classifier/meta_classifier_invariants.json

Observed on current witness suites:

1. Rank tuple descriptor: DFMI=1.0, IDELB=0.5.
2. Budget tuple descriptor: DFMI=1.0, IDELB≈0.2857.
3. Rank + row-space proxy: IDELB=0 on tested rank witness class, giving positive CL.

7. Status

• Mathematical status: PROVED ON SUPPORTED FAMILY (finite witness classes).
• Computational status: VALIDATED on generated artifacts.
• Promotion status: branch-limited quantitative layer, not universal branch core.

8. Limits

1. Finite witness classes only.
2. No unrestricted continuous theorem is claimed here.
3. No universal branch transfer law is claimed.
4. The layer quantifies known anti-classifier structure; it does not replace branch-level exactness theorems.

9. Relation to the OCP Spine

This paper strengthens anti-classifier diagnostics in a scoped way:

• consistent with OCP-049, OCP-050, OCP-052, OCP-053;
• useful for classifying descriptor insufficiency and compatibility gain;
• kept branch-limited by design.

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