# Q1027 Arc Unit Scale Separation — Deterministic Research Record

A public research route for Q1027’s deterministic Arc Unit sweep, exposing normalized-radius rows, equivalent/unresolved/divergent statuses, and first meaningful threshold crossings.

Canonical URL: https://mcbcore.net/q1027-arc-unit-scale-separation

## Claim

The declared Arc Unit matches the Euclidean reference under its fixed conversion, while spherical and hyperbolic references separate by a deterministic normalized-radius threshold in the formal comparison sweep.

This discovery record points to Q1027’s full first-party research surface. The linked Q page publishes the deterministic sweep summary in HTML, Markdown, JSON, and structured data so readers can inspect the rows without reading the validator source.

The sweep compares the converted Arc Unit prediction with Euclidean, spherical, and hyperbolic references at fixed geodesic and curvature radii. It preserves equivalent, unresolved, and divergent statuses instead of collapsing the scale transition into one score.

Q1027 deterministic Arc Unit scale sweep (status=accepted; formal/computational record; not physical validation).
Physical validation status: not-validated; boundary=Formal and computational results do not by themselves validate a physical, quantum, or cosmological model.
Status thresholds: equivalent ≤ 1.000000e-12 relative error; unresolved below 1.000000e-3; divergent ≥ 1.000000e-3.
Reporting-threshold sensitivity: circumference 5.000000e-4, 1.000000e-3, 2.000000e-3; area 5.000000e-4, 1.000000e-3, 2.000000e-3. Statuses distinguish numerical equivalence, unresolved intermediate residuals, stable divergence, and threshold-dependent divergence.
Normalized-radius rows: ρ = R / curvature radius; curved residuals are repeated across curvature radii 5.000000e-1, 1.000000e+0, 2.000000e+0, 1.000000e+1. Each row reports circumference (C), area (A), combined status, and relative errors εC/εA.
Rows:
Euclidean plane | R=1.000000e-8 | ρ=— | C=equivalent | A=equivalent | combined=equivalent | εC=0 | εA=0
Euclidean plane | R=1.000000e-4 | ρ=— | C=equivalent | A=equivalent | combined=equivalent | εC=0 | εA=0
Euclidean plane | R=1.000000e-2 | ρ=— | C=equivalent | A=equivalent | combined=equivalent | εC=0 | εA=1.725561e-16
Euclidean plane | R=1.000000e+0 | ρ=— | C=equivalent | A=equivalent | combined=equivalent | εC=0 | εA=0
Euclidean plane | R=1.000000e+2 | ρ=— | C=equivalent | A=equivalent | combined=equivalent | εC=0 | εA=0
Sphere | R=2.000000e-8 | ρ=1.000000e-8 | C=equivalent | A=equivalent | combined=equivalent | εC=0 | εA=0
Sphere | R=2.000000e-6 | ρ=1.000000e-6 | C=equivalent | A=equivalent | combined=equivalent | εC=1.666245e-13 | εA=8.343826e-14
Sphere | R=2.000000e-4 | ρ=1.000000e-4 | C=unresolved | A=unresolved | combined=unresolved | εC=1.666667e-9 | εA=8.333332e-10
Sphere | R=2.000000e-2 | ρ=1.000000e-2 | C=unresolved | A=unresolved | combined=unresolved | εC=1.666686e-5 | εA=8.333375e-6
Sphere | R=1.000000e-1 | ρ=5.000000e-2 | C=unresolved | A=unresolved | combined=unresolved | εC=4.167882e-4 | εA=2.083594e-4
Sphere | R=2.000000e-1 | ρ=1.000000e-1 | C=divergent | A=unresolved | combined=divergent | εC=1.668613e-3 | εA=8.337502e-4
Sphere | R=5.000000e-1 | ρ=2.500000e-1 | C=divergent | A=divergent | combined=divergent | εC=1.049313e-2 | εA=5.224650e-3
Sphere | R=1.000000e+0 | ρ=5.000000e-1 | C=divergent | A=divergent | combined=divergent | εC=4.291482e-2 | εA=2.109636e-2
Sphere | R=2.000000e+0 | ρ=1.000000e+0 | C=divergent | A=divergent | combined=divergent | εC=1.883951e-1 | εA=8.767132e-2
Hyperbolic plane | R=2.000000e-8 | ρ=1.000000e-8 | C=equivalent | A=equivalent | combined=equivalent | εC=2.106398e-16 | εA=0
Hyperbolic plane | R=2.000000e-6 | ρ=1.000000e-6 | C=equivalent | A=equivalent | combined=equivalent | εC=1.666245e-13 | εA=8.305256e-14
Hyperbolic plane | R=2.000000e-4 | ρ=1.000000e-4 | C=unresolved | A=unresolved | combined=unresolved | εC=1.666667e-9 | εA=8.333334e-10
Hyperbolic plane | R=2.000000e-2 | ρ=1.000000e-2 | C=unresolved | A=unresolved | combined=unresolved | εC=1.666647e-5 | εA=8.333292e-6
Hyperbolic plane | R=1.000000e-1 | ρ=5.000000e-2 | C=unresolved | A=unresolved | combined=unresolved | εC=4.165452e-4 | εA=2.083073e-4
Hyperbolic plane | R=2.000000e-1 | ρ=1.000000e-1 | C=divergent | A=unresolved | combined=divergent | εC=1.664724e-3 | εA=8.329168e-4
Hyperbolic plane | R=5.000000e-1 | ρ=2.500000e-1 | C=divergent | A=divergent | combined=divergent | εC=1.034121e-2 | εA=5.192098e-3
Hyperbolic plane | R=1.000000e+0 | ρ=5.000000e-1 | C=divergent | A=divergent | combined=divergent | εC=4.048262e-2 | εA=2.057548e-2
Hyperbolic plane | R=2.000000e+0 | ρ=1.000000e+0 | C=divergent | A=divergent | combined=divergent | εC=1.490819e-1 | εA=7.932641e-2
First meaningful threshold crossings:
Sphere | circumference | ρ=1.000000e-1 | R=5.000000e-2 | curvature radius=5.000000e-1 | threshold=1.000000e-3 | ε=1.668613e-3 | status=divergent
Sphere | area | ρ=2.500000e-1 | R=1.250000e-1 | curvature radius=5.000000e-1 | threshold=1.000000e-3 | ε=5.224650e-3 | status=divergent
Hyperbolic plane | circumference | ρ=1.000000e-1 | R=5.000000e-2 | curvature radius=5.000000e-1 | threshold=1.000000e-3 | ε=1.664724e-3 | status=divergent
Hyperbolic plane | area | ρ=2.500000e-1 | R=1.250000e-1 | curvature radius=5.000000e-1 | threshold=1.000000e-3 | ε=5.192098e-3 | status=divergent
Threshold sensitivity:
Sphere | circumference | status=threshold-dependent-divergence | threshold=5.000000e-4, firstρ=1.000000e-1, firstStatus=divergence-found, equivalent=8, unresolved=12, divergent=16 | threshold=1.000000e-3, firstρ=1.000000e-1, firstStatus=divergence-found, equivalent=8, unresolved=12, divergent=16 | threshold=2.000000e-3, firstρ=2.500000e-1, firstStatus=divergence-found, equivalent=8, unresolved=16, divergent=12
Sphere | area | status=threshold-dependent-divergence | threshold=5.000000e-4, firstρ=1.000000e-1, firstStatus=divergence-found, equivalent=8, unresolved=12, divergent=16 | threshold=1.000000e-3, firstρ=2.500000e-1, firstStatus=divergence-found, equivalent=8, unresolved=16, divergent=12 | threshold=2.000000e-3, firstρ=2.500000e-1, firstStatus=divergence-found, equivalent=8, unresolved=16, divergent=12
Hyperbolic plane | circumference | status=threshold-dependent-divergence | threshold=5.000000e-4, firstρ=1.000000e-1, firstStatus=divergence-found, equivalent=8, unresolved=12, divergent=16 | threshold=1.000000e-3, firstρ=1.000000e-1, firstStatus=divergence-found, equivalent=8, unresolved=12, divergent=16 | threshold=2.000000e-3, firstρ=2.500000e-1, firstStatus=divergence-found, equivalent=8, unresolved=16, divergent=12
Hyperbolic plane | area | status=threshold-dependent-divergence | threshold=5.000000e-4, firstρ=1.000000e-1, firstStatus=divergence-found, equivalent=8, unresolved=12, divergent=16 | threshold=1.000000e-3, firstρ=2.500000e-1, firstStatus=divergence-found, equivalent=8, unresolved=16, divergent=12 | threshold=2.000000e-3, firstρ=2.500000e-1, firstStatus=divergence-found, equivalent=8, unresolved=16, divergent=12
Null first-crossing status: numerically-equivalent means every tested row stayed within numerical equivalence; unresolved-intermediate-residuals means residuals remained above equivalence but below the documented divergence threshold.
Interpretation boundary: these are deterministic formal/computational comparisons of the declared Arc model with Euclidean, spherical, and hyperbolic references. They do not report physical measurements or validate a physical, quantum, or cosmological model. A rejected formal report is not a physical conclusion and must not reuse successful report data.

A rejected formal sweep is serialized as a formal-validation failure with its error code, model, observable, and adjacent normalized radii; it is not a physical conclusion and cannot reuse a prior successful sweep.

Formal and computational results do not by themselves validate a physical, quantum, or cosmological model.

## Resources

- [Read the Q1027 research record](/q/Q1027) — Rendered first-party Q1027 record with the published deterministic sweep summary.
- [Open the machine-readable Q1027 record](/q/Q1027.json) — JSON record containing the scale-sweep summary as a typed item property and structured data.
- [Open the Markdown Q1027 record](/q/Q1027.md) — Plain-text Q1027 record for direct inspection and retrieval.