Canonical
Operations
The specification boundary for the exact operations by which canonical BitPangea space is measured, composed, compared, normalized, ordered, and converted without changing its underlying truth.
This section translates accepted Foundational Survey Fabric requirements into the specification rules for canonical spatial operations. It now incorporates the candidate mathematical design selected through Formal Mathematical Decisions FMD-01 through FMD-09 wherever those decisions directly resolve exact comparison, geometric predicates, normalization, ordering, validity, precision conversion, and transformation behavior.
The candidate core now defines a deterministic exact operational profile for Point, Segment, and SCPE geometry. General Boolean composition remains intentionally incomplete: union, intersection-result geometry, and difference-result geometry still require formal closure rules for arbitrary outputs, and the known closed-set / difference compatibility issue remains open. This section therefore integrates what is solved without pretending that the complete operations algebra is finished.
Specification Principle At the Survey layer, equal means equal. Tolerance belongs above.
Exact Core Operations Selected · General Composition Still Open
FMD-01 supplies exact CRPC arithmetic. FMD-05 defines the current primitive geometry. FMD-06 supplies exact predicates and validity rules. FMD-07 supplies deterministic normalization and canonical ordering. FMD-09 supplies exact precision-conversion semantics and prohibits silent snapping.
FMD-02 supplies orientation sign conventions, FMD-03 and FMD-04 provide the inherited frame and Domain validity envelope, and FMD-08 supplies canonical machine serialization. None of FMD-01 through FMD-09 establishes a complete closed algebra for arbitrary union, intersection, or difference outputs, so those composition obligations remain open.
Exact Measure Where the Scalar Model Is Closed
The Foundational Survey Fabric shall support exact canonical measurement of valid spatial expressions.
Exact polygonal area is currently closed for valid CRPC SCPE geometry through rational shoelace mathematics. Straight separation, Segment length, path length, and boundary length may require exact non-rational scalar forms that are not yet fully specified.
Measurement shall remain geometric. Functional meanings such as travel cost, accessibility, convenience, or legal significance belong to higher Architecture domains.
Required Operation · General Canonical Output Form Still Open
The specification shall support exact union of valid canonical spatial expressions where mathematically defined.
The current Point / Segment / SCPE primitive set does not yet guarantee that every union result can be represented as one canonical SCPE. A union may produce a disconnected set, a more complex boundary, or geometry requiring a composite canonical form not yet selected.
Therefore union is a required operation whose complete canonical result algebra remains open.
Exact Relationship Known · General Result Geometry Still Open
The current core can determine exact geometric intersection relationships among supported primitives, including Segment intersection classes and Point/SCPE containment.
Where an operation is required to return the exact common geometry itself, the canonical result form remains open when the result is not directly representable as one current primitive.
Intersection remains geometric only and creates no ownership, access, governance, or institutional meaning.
Exact Spatial Subtraction · Closure Issue Explicitly Preserved
The Foundational Survey Fabric shall support exact difference operations sufficient to derive the portion of one valid canonical extent not shared by another.
General canonical closure is not yet established. In particular, subtracting one closed set from another may yield a result that is not closed, which conflicts with any universal assumption that all canonical extent results remain closed.
The Specification shall not silently replace ordinary set difference with regularized, closure-forcing, or application-specific semantics merely to force a closed result. The operation must remain mathematically explicit.
Canonical Predicates Use Exact Truth
Equality, ordering, orientation, incidence, touch, crossing, containment, boundary contact, separation, and related Survey-layer comparisons shall be evaluated exactly.
Approximate comparison, engineering margins, fuzzy matching, rendering thresholds, and epsilon-based coincidence may exist above the Foundational Survey Fabric, but shall not redefine canonical Survey truth.
Equivalent Space, One Canonical Meaning
Two supported spatial expressions are canonically equivalent when they denote the same exact spatial meaning under the governing Specification.
For Point, Segment, and SCPE geometry, equivalent valid inputs normalize to the same canonical representation. Equivalent space does not imply equivalent higher-layer identity, ownership, rights, history, function, or authority.
Deterministic Order Without Semantic Rank
Canonical ordering shall be deterministic wherever needed for normalization, serialization, verification, exact reproducibility, or conformance.
The current core uses lexicographic Point ordering by (x,y) for Segment endpoint order and SCPE start-vertex selection.
Canonical ordering remains semantically neutral and implies no importance, priority, ownership rank, cultural hierarchy, or geographic significance.
Canonicalize Equivalence Without Moving Space
Point coordinates normalize to reduced CRPC form. Standalone Segment endpoints normalize to lexicographic order. SCPE normalization removes exact redundant collinear middle vertices, enforces counterclockwise traversal, selects the lexicographically least start vertex, and omits a repeated terminal closure vertex.
Normalization shall satisfy: N(N(G)) = N(G) and geom(N(G)) = geom(G).
Invalid geometry shall not be silently repaired into valid geometry through normalization.
Canonical, Valid Noncanonical, or Invalid
Conforming implementations shall distinguish at least: canonical valid, valid noncanonical, and invalid.
Invalid conditions include malformed exact values, coordinates outside the valid Survey Domain, impossible or unsupported geometry, self-intersecting SCPEs, zero-area SCPEs, prohibited duplicate vertices, and other failures defined by the governing primitive validators.
Invalid or ambiguous input shall not be assigned canonical spatial meaning through guesswork.
Lossless Means Same Exact Meaning
A precision conversion is lossless only when the exact mathematical object before and after conversion is identical.
For example, 2/4 → 1/2 is lossless CRPC normalization. By contrast, converting exact 1/3 to exact decimal rational 0.333 changes the value and is therefore lossy.
Lossy reduction may be useful for rendering, display, storage, or convenience, but shall never be represented as complete canonical truth.
Separate Truth From Representation
Canonical precision is the exact mathematical value. Storage precision is an implementation choice. Rendering precision is a presentation choice.
Implementations may store or display less detail where useful, provided any reduction is distinguishable from the complete canonical expression and does not alter canonical meaning.
Operations Must Not Move Place
Canonical operations shall not silently snap references to grid lines, precision boundaries, subdivision boundaries, vertices, preferred increments, nearby rational values, or other derived structures.
If an input is not valid under the governing canonical mathematics, the result is invalid rather than a nearby replacement.
Reveal Spatial Conflict Without Adjudication
Exact predicates shall provide sufficient geometric truth for higher Architecture domains to detect overlap, boundary conflict, invalid containment, incompatible extents, or other spatial contradictions.
The Foundational Survey Fabric does not decide the institutional, legal, ownership, access, or governance consequences of those facts.
Derived Operations Must Preserve Canonical Meaning
Exact translation by CRPC offsets and uniform scaling by positive rational factors preserve the current CRPC coordinate domain when outputs remain inside the valid Survey Domain.
Arbitrary rotation and more general transformations may produce coordinates outside the current rational scalar domain. Their complete canonical output representation therefore remains open.
Derived transformations shall not redefine Pang scale, permanent orientation, or authoritative Survey position.
Exact Threshold Tests Without Semantic Meaning
Threshold-based proximity may be evaluated exactly where the required scalar comparison is exactly representable under the governing measurement rules.
Terms such as near, local, neighboring, remote, or convenient remain higher-layer interpretations and shall not become Survey-layer semantics.
An Operation Is Canonical Only If Its Result Type Is Defined
A mathematically meaningful operation does not automatically have a canonical FSF result merely because the set-theoretic answer exists.
Canonical operation closure requires that the Specification define a valid result type, exact normalization rules, deterministic serialization, and conformance behavior for the output.
Where those are not yet defined, the operation remains mathematically required but Specification-open rather than implementation-defined.
Integrated Candidate Mathematics
Exact comparison and predicates; lexicographic ordering; deterministic Point / Segment / SCPE normalization; validity classes; lossless versus lossy precision conversion; no silent snapping; exact polygon area; exact CRPC-preserving translation and positive rational scaling; neutral conflict detection.
Still Open in the Specification
General canonical output forms for union, intersection, and difference; closed-set / difference compatibility; exact general distance and path-length scalar closure; arbitrary rotation/transformation closure; complete operation syntax; complete error taxonomy; and final complexity/resource limits.
Core Predicate and Normalization Operations Ready · Composition Algebra Not Yet Closed
No contradiction was found between Specification 05 and FMD-01 through FMD-09. The solved operational core is sufficiently exact for prototype validators, geometry normalization, Point/Segment/SCPE comparison, Spatial Ground membership support, and first-wave Conformance and Reference Vectors.
Full Specification 05 adoption remains blocked by unresolved general composition/output closure, the difference-closure issue, full measurement scalar closure, arbitrary transformation closure, and the usual Specification-wide Conformance, independent implementation, adversarial, traceability, and adoption gates.
Primary Derived Findings
This specification section is derived especially from Requirements Findings #55–#58 and #65–#75.
Architectural Boundary Canonical operations may calculate, compare, compose, and normalize space. They may not invent meaning, tolerate ambiguity, silently move the place being described, or prescribe how higher Architecture domains must use the result.