Pang Measurement
Vectors

Pang Defines Scale; Canonical Mathematics Defines Precision

Pang is the native canonical unit of linear measure in the Foundational Survey Fabric. Canonical coordinate values are exact reduced rational Pang quantities:

q = n/d Pang

where n ∈ Z, d ∈ N+, and the fraction is reduced exactly.

Coordinate exactness therefore does not depend upon decimal digits or named subunits. Derived measurements may belong to different exact scalar classes; for example, a straight-line Euclidean distance between rational-coordinate Points may be algebraic rather than rational.

One Pang Must Mean the Same Scale Everywhere

Reference Vectors shall demonstrate the uniformity of Pang scale across the Survey frame using exact coordinate differences.

A displacement of exactly one Pang along a canonical axis shall have the same linear scale regardless of where the participating Points lie, provided the test Points are valid within the governing Survey Domain.

Axis-Aligned Separation Components Are Exactly Rational

Given Points A = (x1,y1) and B = (x2,y2), Reference Vectors may directly test the exact coordinate differences:

Δx = x2 - x1 and Δy = y2 - y1.

Because CRPC arithmetic is exact, these component differences are exact rational Pang quantities and require no tolerance or approximation.

Identical Canonical Points Produce Zero Coordinate Difference

Canonically equal Points shall produce exactly Δx = 0 and Δy = 0.

Any nonzero component difference for equal canonical Points constitutes a conformance failure.

Exact Measure Below One Pang Is Already Supported

The corpus shall include exact rational values below one Pang, such as one-half, one-third, or other reduced rational Pang quantities, where the selected fixtures remain valid under the governing Survey Domain.

These cases prove that sub-Pang precision is not a future hierarchical refinement mechanism. It already follows directly from exact rational coordinate representation.

Polygon Area Is Exactly Testable

For a valid normalized SCPE with CRPC vertices, Reference Vectors shall test exact area using the canonical shoelace construction.

Because all vertex coordinates are rational, the resulting signed double-area and absolute area are exact rational square-Pang quantities.

Independent implementations shall produce exactly the same authoritative area result for the same canonical SCPE.

Canonical Normalization Must Preserve Exact Area

Reference Vectors shall demonstrate that permitted normalization steps—such as traversal normalization, unique start-vertex selection, and removal of redundant collinear middle vertices—do not change the exact represented area.

If two valid supported polygon representations are semantically equivalent and normalize to the same SCPE, their exact area shall agree.

Do Not Invent Boolean Area Semantics Ahead of Closure Rules

The original vector plan anticipated unions, differences, intersections, and other composed extents. Those cases remain conditional because the general Boolean/composite output model is not yet fully closed in the Specification.

Reference Vectors may test exact area of canonical SCPE inputs and any composition whose result type and normalization are already explicitly governed. They shall not invent a general composite geometry model merely to exercise area arithmetic.

Coordinate Difference Is Solved; Universal Distance Is Not

Exact coordinate differences are executable now. A general canonical Euclidean spatial-separation operation requires an exact result scalar capable of representing values such as √r when the squared distance is rational but not a rational square.

Until that derived-scalar representation and operation contract are formally governed, Reference Vectors shall not publish arbitrary diagonal-distance results as final canonical outputs.

General Path Length Remains Open

Segment and polygon geometry are defined, but a universal canonical scalar contract for arbitrary path length is not yet fully specified.

Reference Vectors may preserve the proof obligation and may test any special exact case whose result form is already governed. General path-length fixtures remain non-final until the Specification defines the exact output scalar and operation semantics.

Perimeter-Like Measurement Is Not Yet a Closed General Operation

SCPE boundaries are exact geometric objects, but their total Euclidean length may contain irrational segment lengths.

The corpus shall therefore distinguish exact boundary geometry from a still-open general canonical boundary-length scalar representation. No approximate decimal perimeter may be promoted to canonical truth.

Exact Representation Changes Must Preserve the Measurement

Where one exact measurement can be expressed through multiple supported exact forms, a lossless conversion shall preserve identical mathematical meaning.

For CRPC values, normalization to reduced rational form is exact and lossless. Canonicalization shall never change the represented quantity.

Approximation Must Never Masquerade as Canonical Measurement

The corpus shall include derived display or interchange examples in which an exact value is rounded, truncated, decimalized, generalized, or otherwise reduced.

Conforming implementations shall distinguish such approximations from the exact canonical value. A lossy result shall not silently re-enter the canonical layer as though no information were lost.

Different Exact Forms Must Resolve Without Precision Loss

Where supported inputs use mathematically equivalent rational forms, implementations shall normalize and compute without silently discarding exactness.

ECEM does not define coarse and fine semantic states. Mixed-form testing therefore concerns exact representational equivalence, not hierarchical precision inheritance.

Defined Measurements Must Reproduce Exactly

For every measurement operation whose canonical mathematics and result form are actually defined, independent conforming implementations shall return the same exact authoritative result.

A mismatch indicates implementation defect, Specification ambiguity, version incompatibility, fixture defect, or an operation that was tested before its mathematical output contract was sufficiently specified.

Measurement Vectors Must Follow the Exact Result Type

The original page correctly refused to invent formulas while the measurement model was broadly unresolved. The present candidate mathematics now support a narrower but real executable core.

CRPC arithmetic and exact polygon area are ready for Reference Vector construction. General distance and length operations are not yet equally mature because their exact derived-scalar representation remains open. The corpus shall therefore advance by operation, not by pretending that every measurement family has reached the same level of closure.

Referenced Requirements

Pang Measurement Vectors will derive especially from Requirements Findings #10, #24, #34–#37, #41, #53–#55, #66–#67, #69, #72–#74, and #78–#83.