Canonical Result
Equivalence

Canonical Equivalence Is Now Executable for the Solved Core

Specification Sections 01–07 now provide deterministic candidate rules sufficient to compare independent implementations across the current Mandatory Core.

Conformance may therefore require exact agreement in validity, normalization, predicate outcomes, canonical geometry, and canonical FSF-CJSON-1.0 bytes wherever the governing Specification defines one unique result.

Identical Canonical Inputs, One Authoritative Meaning

Any two conforming implementations shall produce the same authoritative canonical meaning when given identical valid canonical inputs under the same applicable Specification profile.

Where the governing Specification requires one unique canonical representation, normalized form, or canonical byte sequence for that result, conforming implementations shall produce that result identically.

A disagreement indicates a conformance failure, Specification ambiguity, implementation defect, incompatible profile/version, or an attempt to evaluate mathematics that remain outside the defined canonical scope.

Equivalence Applies Wherever the Specification Defines Canonical Truth

Canonical Result Equivalence applies to every Mandatory Core capability that produces or classifies authoritative Survey truth.

The current solved profile includes interpretation of CRPC values and canonical frame meaning; validation of Point, Segment, and SCPE geometry; exact predicate results; canonical normalization; ECEM precision behavior; and FSF-CJSON-1.0 machine interchange.

Open capabilities are outside canonical comparison until the Specification defines their exact semantics and result forms.

Internal Method Does Not Define the Answer

A conforming implementation may use any internal algorithm, programming language, storage engine, indexing strategy, cache, or optimization that preserves the authoritative result.

Internal method is implementation freedom. Canonical output is not.

Canonical Equality Is Exact

Authoritative result equivalence shall be determined by the exact canonical rules of the Specification rather than tolerance bands, approximate numeric agreement, implementation-specific rounding, device precision, or “close enough” comparison.

Where a result is canonical, equivalent means mathematically equivalent under the governing Survey rules.

The Same Input Must Receive the Same Validity Judgment

Independent implementations shall classify the same input identically under the same Specification profile.

For the current core, the required validity classes include: canonical valid, valid noncanonical, and invalid.

An input shall not be canonical in one conforming implementation, valid noncanonical in another, and invalid in a third when all claim the same profile.

Equivalent Meaning Must Normalize to One Form

Valid equivalent representations shall normalize according to the deterministic rules of the governing Specification before authoritative result comparison.

For the current core, this includes reduced CRPC values, lexicographically ordered Segment endpoints, and canonical SCPE normalization through exact redundant-collinear removal, counterclockwise traversal, lexicographically least start vertex, and omission of repeated terminal closure.

Conforming implementations shall satisfy: N(N(G)) = N(G) and geom(N(G)) = geom(G) for supported normalized geometry.

Exact Geometry Questions Must Have the Same Answer

Independent implementations shall return identical authoritative outcomes for exact predicates within the current core.

These include Point equality and ordering, orientation, Point-on-Segment, Segment intersection classification, Point-on-boundary, Point-in-SCPE, polygon validity, connectedness, containment, and geometric equivalence where defined by the Specification.

No implementation may substitute epsilon geometry or tolerance-based coincidence for those exact outcomes.

ECEM Must Mean the Same Thing Everywhere

Independent implementations shall interpret a canonical Point under the Exact Coordinate Extension Model as one exact Survey position.

No conforming implementation may reinterpret the same Point as a parent cell, coarse reference, uncertainty region, rounded location, or collection of finer descendants.

Compatible representational extension shall preserve previously valid Point meaning exactly.

FSF-CJSON-1.0 Must Produce the Same Canonical Bytes

For supported normalized Point, Segment, and SCPE geometry, independent implementations claiming FSF-CJSON-1.0 conformance shall produce identical canonical UTF-8 bytes from the same canonical geometry.

Deterministic comparison therefore includes structured CRPC encoding, required object-key order, semantic array order, prohibited duplicate keys, governed unknown-field behavior, NFC string normalization, and the other byte-level rules defined by Specification 06.

Internal or presentation formats may differ. Canonical FSF-CJSON-1.0 output may not.

Invalid Inputs Must Fail Consistently

Where the Specification defines an input or result as invalid, conforming implementations shall reject it consistently.

Human-readable diagnostics may differ unless the governing test requires a specific canonical failure identifier. Canonical treatment shall not differ.

An implementation shall not invent canonical meaning for input another conforming implementation correctly rejects under the same profile.

Equivalent Results Must Not Depend on Hidden Context

Cache state, database sequence, randomization, user identity, ownership state, session context, wall-clock time, storage ordering, machine locale, or other mutable implementation conditions shall not cause canonical results to differ.

Hidden state may improve performance. It may not participate in foundational spatial meaning.

Equivalence Must Be Demonstrated, Not Assumed

Canonical Result Equivalence shall be demonstrated through shared Reference Vectors or equivalent executable fixtures containing defined inputs and required authoritative outcomes.

The current solved core is sufficiently deterministic for such vectors to test validity classification, normalization, exact predicates, precision semantics, and canonical FSF-CJSON-1.0 output.

A conforming implementation shall reproduce every applicable Mandatory Core vector exactly where the Specification defines an exact result.

Independent Implementations Must Be Compared Directly

Conformance testing shall permit outputs from independent implementations to be compared directly after applying the governing validation, normalization, semantic-equivalence, and canonical serialization rules.

Cross-implementation disagreement is evidence to investigate. It shall not be concealed through tolerance, majority vote, implementation preference, or post-hoc output adjustment.

Compare the Right Thing at the Right Stage

Result comparison should proceed in the governing semantic order:

validate → normalize where permitted → evaluate canonical meaning → compare canonical result → compare canonical bytes where uniquely governed.

Byte comparison shall not replace semantic validation, and semantic equivalence shall not excuse a byte mismatch where FSF-CJSON-1.0 requires one exact canonical byte sequence.

No Equivalence Claim Without Defined Canonical Semantics

Conformance shall not require or accept canonical equivalence claims for mathematics that remain open in the Specification.

Current examples include general Boolean/composite output closure, arbitrary exact rotation, unresolved general distance and path-length scalar forms, and future geometry or operation result types not yet governed.

Two implementations agreeing on an experimental result does not make that result canonical.

Conflict Must Be Resolved at the Correct Layer

If independent implementations produce different authoritative results from the same canonical inputs, the disagreement shall be treated as a Conformance or Specification issue.

The investigation shall determine whether the cause is an implementation defect, Specification ambiguity, incompatible profile/version, invalid test fixture, or evaluation of a capability whose canonical semantics remain open.

Consensus, majority output, implementation popularity, or institutional preference shall not simply choose among conflicting mathematical answers.

Result Equivalence Has Moved From Principle to Executable Proof

The original section correctly required independent implementations to produce one authoritative result. The integrated Specification now provides enough exact candidate behavior to make that requirement executable across the solved Mandatory Core.

This section does not declare equivalence rules for open mathematics. It establishes exact comparison only where the Specification already defines canonical truth.

Primary Referenced Findings

This Conformance section is derived primarily from Requirements Finding #10 and is reinforced by Findings #24, #30, #57–#60, #63–#66, #74, and #78–#83.