Astra 15A — dimensional enlargement

An instrument for deciding Euclidean admissibility and transporting orientation from a table of squared distances, in exact rational arithmetic.

Lu Semita, EmergenceByDesign   browser entry B1   exact rationals, no external dependencies

1The result

The metric fixes orientation only up to one global sign.

Everything below is computed in the page from exact rational arithmetic. Nothing is fitted, rounded, or drawn from a stored answer.

For rational squared-distance data the centered Gram criterion decides Euclidean admissibility and the exact embedding rank before any realisation is chosen, and without a tolerance.

For accepted data of affine rank three the metric determines the complete orientation array up to one global sign. That remaining sign is not metric-determined: it must be declared.

B = −½ J D J    χ(t) = χ(r)·sign det C(t,r)

A reflection leaves every distance unchanged and reverses every non-degenerate orientation. So no further computation on the distances can choose between a configuration and its mirror. The instrument therefore refuses to invent the sign, and instead transports one that you supply.

2Figure 1
mirror pair

One metric, two realisations.

Two point sets related by a reflection. Move the controls: the drawings diverge, the distance matrix does not.

Stipulated coordinates +1

Reflected coordinates −1

squared distances agree yes
D(left)
= D(right)
Figure 1The two realisations share one matrix of exact squared distances. Drawings use a screen projection; every decision on this page uses the rational matrix, never the picture.
3Figure 2
transport

One declared sign fixes the whole array.

Each cell is an ordered 4-subset of the configuration in Figure 1. Choose the reference sign and watch every cell resolve at once.

declared sign of reference (0,1,2,3)
resolution policy η²

Figure 2Orientation of every 4-subset, from one supplied sign. Flipping the selector inverts the entire array; no cell resolves independently. Degenerate subsets return zero and consult no selector at all.

Centered Gram matrix B

Classification

Figure 3Double-centering the distance matrix. A positive semidefinite B certifies Euclidean admissibility; its rank is the least affine dimension carrying the data.
4Workbench

Put your own data through it.

Integers and rational strings such as "1/10". Zero-based labels. Decimals, exponents, Booleans and duplicate fields are refused rather than silently coerced. Up to 12 points and 128 digits per scalar in the browser.

WorkbenchThe trace shows the exact intermediates that decided the branch — the centered Gram matrix, the gate comparison, the cross-Gram determinant and the receipt actually consulted.
5Scope

What a chosen frame permits.

A target factors through an observation exactly when it is constant on every fibre of that observation. Each record fixes its own domain, states what is settled there, and names the bridge a stronger claim would need.

Selecting a record exposes its warrant. It does not manufacture a verdict for a question the record does not cover.

6Verification
and sources

Check it rather than trust it.

The conformance cases, the full texts and the original Python release are embedded in this file. Nothing here loads from the network.

Browser conformance

Runs the declared cases against stipulated expected answers, then the exact algebra and receipt guards.

Not run in this session.

Embedded release integrity

Recomputes every byte count and SHA-256 in the embedded release against its manifest, plus the pinned archive digest.

Not checked in this session.

A file cannot authenticate itself. Recomputing a hash stored in the same document shows internal consistency only; an independent trust anchor has to come from outside this page.

Documents

Source packages

The Python audit suite runs after extraction; browser byte-verification does not re-execute it.

Individual release artifacts

Every file in the original release, with the byte count and digest each one is checked against. Saving one here writes the same bytes the integrity check verifies.

Astra 15A — browser entry B1. Lu Semita, EmergenceByDesign.

Exact rational arithmetic throughout; the drawings are illustrations of data that was decided algebraically. Figures, matrices and verdicts on this page are generated by the embedded engine at the moment you read them.

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