Pathological
Edge Cases

Stress Defined Mathematics; Expose Undefined Mathematics

Pathological testing has two legitimate outcomes:

defined behavior survives the hard case exactly
or
the case reveals a genuine Specification gap, unsupported capability, or unresolved dependency.

It shall never force an implementation to fabricate canonical truth merely because a stress case demands an answer.

Touching at the Limit Must Not Become Overlap or Separation

Reference Vectors shall include exact endpoint contact, vertex contact, Point-on-Segment cases, SCPE boundary contact, and shared-edge conditions supported by the current geometry model.

Exact predicates shall determine contact. No epsilon, rendering tolerance, or floating proximity may alter the classification.

Different Syntax Must Not Defeat Exact Equality

The corpus shall include mathematically equal CRPC coordinates supplied through different permitted rational forms and visually similar but mathematically distinct Points.

Equal Points shall normalize to equality. Near-but-distinct Points shall remain distinct.

Near-Degenerate Intersections Must Still Classify Exactly

Reference Vectors shall stress proper crossing, endpoint contact, collinear overlap, collinear disjointness, nearly parallel Segments, and Points lying extremely close to but not on another Segment.

Exact orientation and range predicates shall control the result rather than numerical tolerance.

Complex Polygon Boundaries Must Still Produce One Validity Judgment

The corpus shall include high-vertex-count SCPE candidates, near-self-intersections, exact vertex contacts, redundant collinear runs, reversed traversal, alternate starting vertices, and invalid nonadjacent crossings.

Conforming implementations shall reach the same SCPE validity and normalization result from the same exact input.

Repeated Canonicalization Must Converge and Stop

Reference Vectors shall include SCPEs with multiple removable collinear middle vertices, alternate traversal, repeated accepted closure markers, reducible CRPC components, and other permitted representational redundancy.

Normalization shall converge deterministically to a fixed point and satisfy N(N(G)) = N(G).

Large and Fine Rational Values Must Remain Exact

The corpus shall include valid CRPC values with large numerators or denominators and exact sub-Pang quantities chosen to stress parsing, reduction, comparison, and arithmetic without leaving the supported mathematical model.

Finite exact representation shall remain finite, deterministic, and free from floating substitution.

Nearness Must Not Become Identity

Reference Vectors shall place exact values on opposite sides of equality, containment, intersection, or boundary decisions while remaining arbitrarily close in display terms.

Distinct exact values shall remain distinct regardless of apparent visual coincidence.

Boundary Classification Must Survive the Hardest Parameterized Cases

Reference Vectors shall stress symbolic or parameterized Points at x = ±H, y = ±H, the four corners, strict interior, and exact outside conditions.

FMD-04A already resolves the closed-square geometry and boundary semantics. Final concrete numerical edge fixtures remain blocked until FMD-04B resolves the numerical value of H and that decision is incorporated into the governing Specification. Pathological testing shall not invent that number.

The earlier ±1,000,000 Pang half-span is a superseded working candidate and shall not be used as a canonical stress fixture.

Vertex and Horizontal-Edge Cases Must Not Break Containment

The corpus shall include Points whose horizontal test ray passes through vertices, aligns with horizontal edges, meets exact boundary Points, or encounters multiple candidate crossings.

The half-open y convention and exact boundary test shall produce one deterministic INTERIOR, BOUNDARY, or EXTERIOR result.

Large Coordinate Arithmetic Must Preserve Exact SCPE Area

Reference Vectors shall include valid SCPEs whose shoelace arithmetic stresses numerator size, cancellation, orientation reversal, and normalization.

Exact rational square-Pang area shall remain identical for semantically equivalent normalized geometry.

Supported Exact Transformations Must Not Accumulate Drift

The corpus shall include long but finite chains of supported exact translations and positive uniform scales, including exact inverses and identity-producing compositions.

Arbitrary rotation, reflection as canonical placement, nonuniform scaling, and shear shall not be smuggled into the stress corpus as though they were already closed operations.

Large Valid Objects Must Still Produce Canonical Bytes

Reference Vectors shall include large but valid canonical Points, Segments, and SCPEs serialized through FSF-CJSON-1.0.

Canonical integer strings, object-key ordering, UTF-8 serialization, field rules, and canonical byte output shall remain deterministic under increased object size and representational complexity.

Repeated Interchange Must Not Drift

The corpus shall repeatedly apply:

parse → validate → normalize → canonical serialize → parse

to supported complex fixtures and prove that the canonical object and canonical bytes stabilize rather than mutate across cycles.

Execution History May Affect Performance, Never Truth

Reference Vectors shall repeat identical cases under varied cache state, execution order, process lifetime, test ordering, and independent implementation history.

Canonical results, validity judgments, and required canonical bytes shall remain unchanged.

Difficult Input Must Not Collapse Failure Taxonomy

The corpus shall include hard cases that are mathematically valid but unsupported, genuinely malformed or invalid, and supported but computationally demanding.

Conforming implementations shall preserve the distinction among VALID AND SUPPORTED, VALID BUT UNSUPPORTED, and INVALID.

The Stress Corpus Must Reveal Rather Than Conceal Missing Closure

Reference Vectors shall deliberately probe operations that remain open, including arbitrary exact rotation, non-rational derived-scalar measurement, general Boolean result geometry, and concrete numerical Domain-edge behavior requiring closure of FMD-04B.

The expected outcome is not an invented canonical result. It is a documented open dependency or unsupported state until the governing Specification closes the mathematics.

Difficult Finite Input Must Still Terminate Deterministically

The corpus shall include finite but demanding inputs designed to stress parsing, CRPC reduction, normalization, exact predicates, SCPE validation, canonical serialization, and Conformance evaluation.

Where the profile requires support, the implementation shall terminate deterministically. Final quantitative complexity ceilings remain subject to the governing Specification and Conformance profile.

The Corpus Must Attack the Specification as Well as the Implementation

Adversarial vectors shall include cases specifically designed to expose underspecified predicate behavior, unclear profile obligations, contradictory normalization rules, representation ambiguity, hidden dependency, or higher-layer leakage.

If competent independent implementations disagree because the Specification itself does not determine one result, the vector has successfully exposed a Specification defect rather than an implementation variation.

FSF Stress Must Not Pull Parcel or World Meaning Into Geometry

Reference Vectors shall include geometrically difficult cases near World membership, Parcel-like boundaries, or other higher-layer concepts to prove that the FSF returns only canonical reference and mathematical truth.

Spatial Ground, GSI, Parcel Cadastre, governance, ownership, rights, and civilization semantics remain outside the FSF result.

Independent Implementations Must Survive the Same Hard Cases

For every adversarial case whose governing mathematics is defined, independent conforming implementations shall return the same exact result, validity judgment, normalization result, canonical serialization, or governed unsupported classification.

Divergence under stress indicates implementation defect, Specification ambiguity, version/profile incompatibility, fixture defect, hidden state, unresolved mathematics, or higher-layer leakage requiring correction before adoption readiness can be argued.

The Adversarial Corpus Can Now Be Built Around the Solved Core

The original page correctly reserved pathological testing until enough of the mathematics existed to make hard cases meaningful. That threshold has now been crossed for the candidate core.

Section 12 can therefore support adversarial fixtures across CRPC, exact predicates, SCPE validity and normalization, Domain classification, exact area, restricted transformations, FSF-CJSON, hidden-state independence, termination, and Conformance classification. Open mathematics remains visible as open rather than being papered over by stress-test expectations.

Referenced Requirements

Pathological Edge Cases will derive especially from Requirements Findings #10, #24, #30, #32, #44–#50, #56–#59, #63–#66, #70–#75, and #78–#85.