HPC-024 was an early test of the framework's prediction that angular deficit (the geometric concentration ratio produced by cone-like or pinch geometries) should scale with the local framerate. The sweep ran 7 geometries: bicone, cone, pyramid, double-pyramid, sphere (control), hemisphere, and torus.
\nSetup.
\n(vertex with maximum angular deficit) vs incident plane-wave intensity at the entrance plane.
\nResult.
\n| Geometry | \nPeak concentration | \nAngular deficit | \n
|---|---|---|
| Sphere (control) | \n840\u00d7 | \n0\u00b0 (no apex) | \n
| Hemisphere | \n1,180\u00d7 | \n\u03c0 (flat boundary) | \n
| Cone (single) | \n2,030\u00d7 | \nvaries with half-angle | \n
| Pyramid (4-sided) | \n2,310\u00d7 | \nvaries with apex angle | \n
| Bicone | \n3,428\u00d7 | \n2 \u00d7 half-angle (waist effect) | \n
| Double-pyramid | \n3,150\u00d7 | \n2 \u00d7 apex angle | \n
| Torus | \n980\u00d7 | \n0\u00b0 (no apex) | \n
Headline finding: the angular-deficit hypothesis holds. Geometries with concentrated apex regions (bicone, double-pyramid, pyramid) produce 2\u20134\u00d7 the concentration of geometries without apex regions (sphere, torus). The bicone result at 3,428\u00d7 motivated the HPC-027 follow-up that swept the bicone half-angle in 5\u00b0 increments.
\nThe sphere-control artifact. Initial readings of the sphere control showed *higher* concentration (~1,500\u00d7) than the prediction. Investigation traced this to a grid-alignment artifact: when a spherical boundary is rendered on a cubic grid, the boundary cells have systematically higher \"effective curvature\" than the true spherical case, producing spurious concentration at certain grid resolutions. The artifact disappears at high enough grid density (>128\u00b3) but was present at the initial 64\u00b3 sweep. Re-running the sphere control at 128\u00b3 gave 840\u00d7 \u2014 the value reported in the table.
\nThis artifact is important because it nearly produced a false positive: at 64\u00b3, the sphere \"beat\" several geometries that the framework predicted should outperform it. Only when the grid- resolution issue was tracked down did the result fall into line with the framework prediction. This is the kind of small artifact that, if uncaught, would have falsely refuted the angular-deficit hypothesis. Documented here so future HPC sweeps run sphere controls at adequate grid resolution.
\nWhy this matters for the dark-matter test. HPC-024's results underpin one of the framework's testable claims about dark matter: *geometric amplification* of gravitational coupling could mimic the observed dark-matter-density profile in galaxy halos without exotic particles. If apex-bearing geometries amplify EM concentration, the framework predicts analogous amplification of *gravitational coupling* in galactic-scale geometries with apex structures (galaxy spiral arms, bar structures). HPC-024 doesn't directly test that gravitational claim \u2014 but it does establish the EM-analog mechanism is real.
\nStatus: confirmed (after sphere-artifact resolved). Result robust across 64\u00b3 \u2192 96\u00b3 \u2192 128\u00b3 once the sphere control is run at sufficient grid density.
\nHPC-024 was an early test of the angular-deficit hypothesis: that geometric concentration in cone-like or pinch geometries should scale with the local framerate. Seven geometries swept on 64\u00b3 FDTD: bicone, cone, pyramid, double-pyramid, sphere control, hemisphere, torus.
\nHeadline finding: geometries with concentrated apex regions (bicone 3,428\u00d7, double-pyramid 3,150\u00d7, pyramid 2,310\u00d7) produced 2\u20134\u00d7 the concentration of geometries without apex regions (sphere 840\u00d7, torus 980\u00d7). The bicone result motivated the HPC-027 follow-up sweep that pinpointed 35\u00b0 as the optimal half-angle.
\nThe sphere-control artifact is documented here as a warning. Initial 64\u00b3 readings gave the sphere 1,500\u00d7 \u2014 anomalously high. Tracked down to a grid-alignment artifact at the spherical boundary that disappears at >128\u00b3 resolution. Important to document because the artifact would have falsely refuted the angular-deficit hypothesis if uncaught. Future HPC sweeps run sphere controls at adequate grid density.
\nDark-matter connection. HPC-024 establishes the EM-analog of the gravitational-amplification mechanism that the framework predicts for galactic dark matter. Apex-bearing geometries amplify EM concentration; the framework predicts they should amplify gravitational coupling similarly. HPC-024 doesn't test the gravitational claim directly but validates the underlying mechanism.
\nStatus: confirmed after sphere-artifact resolved. Result robust at 96\u00b3 and 128\u00b3.
", "see_also": [ "hpc-027-bicone-angular-sweep", "hpc-039-heptagonal-resonance" ], "cited_by": [ "dark-matter-geometric-amplification" ], "attachments": [], "schema_version": "1.0", "generated_at": "2026-05-12T03:27:18.533879Z" }