Normalization
Vectors

Normalization Is Now a Deterministic Candidate Function

The candidate mathematics define normalization over supported canonical objects rather than treating normalization as an open future concept.

At minimum, the executable normalization surface includes CRPC values, Points through normalized components, Segments, SCPEs, and FSF-CJSON-1.0 canonical serialization.

Normalization may change representation. It may not change exact spatial meaning.

Equivalent Rational Values Must Reduce Identically

Reference Vectors shall include valid mathematically equivalent rational inputs whose canonical result is one reduced CRPC value.

The normalization rules include greatest-common-divisor reduction, positive denominator, normalized sign, and canonical zero as 0/1.

Point Normalization Is Component Normalization

A Point is canonical when its x and y CRPC components are canonical.

Differently expressed valid rational components that denote the same exact values shall normalize to the same canonical Point.

Segment Endpoint Order Must Be Deterministic

Reference Vectors shall include the same valid Segment supplied with reversed endpoint order.

Canonical Segment normalization orders the two endpoints lexicographically so that equivalent endpoint permutations produce one canonical Segment representation.

Equivalent Polygon Traversals Must Converge

Reference Vectors shall include valid SCPE representations differing only by traversal direction or starting vertex.

Canonical normalization selects counterclockwise traversal and then selects the unique lexicographically least vertex as the canonical start vertex.

Redundant Middle Vertices Must Disappear Deterministically

Where a valid SCPE contains a vertex lying exactly between adjacent collinear vertices, normalization removes that redundant middle vertex.

The reduction shall repeat to a fixed point so that equivalent polygonal boundaries do not acquire distinct canonical forms merely because one input contains unnecessary collinear subdivision.

Repeated Terminal Closure Must Not Create Alternate Geometry

Where an accepted input form repeats the first SCPE vertex as a terminal closure marker, Reference Vectors shall prove that canonical normalization removes the repeated terminal Point.

The canonical SCPE representation stores the cycle without duplicating the first vertex at the end.

Equivalent Supported Geometry Must Normalize to Equality

Reference Vectors shall demonstrate that valid supported geometry representing the same canonical spatial set produces identical normalized geometry.

Semantic equivalence shall therefore be reproducible through exact normalization rather than visual similarity or implementation-specific simplification.

Canonical Objects Must Produce Canonical Bytes

Reference Vectors shall include canonical FSF-CJSON-1.0 objects and exact expected serialization where the profile requires one canonical byte sequence.

The corpus shall prove canonical integer-string representation, required object structure, deterministic key ordering, prohibited unknown fields where applicable, UTF-8 encoding, and exact canonical bytes.

Canonical Round-Trip Must Converge

Where a valid supported noncanonical input form is permitted, vectors shall demonstrate the deterministic path:

parse → validate → normalize → canonical object → canonical serialize.

Repeating the process on the canonical output shall produce the same canonical object and the same canonical bytes.

Normalizing Canonical Output Again Must Change Nothing

For every supported normalized object G, Reference Vectors shall prove:

N(N(G)) = N(G).

Idempotence applies to the mathematical canonical object and, where canonical serialization is defined, to the resulting canonical representation.

Normalization Must Never Move Place

Reference Vectors shall include exact Points and geometry near other representable values to prove that normalization performs only permitted exact canonicalization.

It shall not round, snap, approximate, quantize, migrate, or otherwise change the represented position or extent.

Normalization Cannot Become Lossy Conversion

Exact CRPC values and geometry shall retain all canonical mathematical information through normalization.

Any operation that truncates, rounds, decimalizes, generalizes, or otherwise loses information belongs to a lossy conversion pathway and shall not be reported as canonical normalization.

Invalid Input Must Remain Invalid

Reference Vectors shall include malformed CRPC, malformed geometry, invalid SCPEs, prohibited serialization forms, duplicate or unknown fields where forbidden, and other invalid cases that superficially resemble normalizable input.

A conforming implementation shall reject those cases rather than infer intent or silently repair them into canonical validity.

Only Defined Alternate Forms May Participate in Normalization

The current candidate profile supports normalization and canonical serialization for the representations explicitly defined by the Specification.

Future aliases, human-readable syntaxes, alternate interchange formats, or geometry families shall not be treated as valid normalization inputs until the Specification formally defines their semantics and conversion rules.

Every Implementation Must Converge to the Same Canonical Result

Independent conforming implementations shall produce the same normalized CRPC, Point, Segment, SCPE, and canonical FSF-CJSON result for identical supported inputs.

A mismatch indicates implementation defect, Specification ambiguity, version incompatibility, fixture defect, or an input form tested beyond the current normalization contract.

Normalization Is Now One of the Strongest Executable FSF Proof Surfaces

The original page correctly deferred the corpus while addressing, geometry, ordering, serialization, and normalization algorithms were unresolved. Those dependencies are now substantially defined for the candidate core.

Section 06 can therefore support exact executable Reference Vectors across CRPC, Segment, SCPE, and FSF-CJSON normalization. The remaining work is concentrated in future representations and geometry families that the Specification has not yet admitted.

Referenced Requirements

Normalization Vectors will derive especially from Requirements Findings #8, #10, #21, #30, #57–#59, #64, #66, #70–#75, and #78–#83.