Author notes — full detail, auditor-facing

HPC-027 swept the half-angle of a bicone resonant cavity from 10° to 60° in 5° steps under 96³ FDTD simulation. The hypothesis (cipher-predicted) was that bicone geometry should concentrate incident EM via the double-pinch focusing of the apex region, with an optimal angle somewhere in the 30–45° range determined by the standing-wave geometry of the two cones meeting at the central waist.

Setup.

• Grid: 96³ cells, FDTD with PML boundaries.
• Cavity: two cones meeting apex-to-apex at the central plane, each with

half-angle θ ∈ {10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°, 55°, 60°}.

• Source: plane-wave incident on one cone's base, broadband pulse

10 GHz – 8 THz.

• Measurement: peak EM concentration at the apex region (central plane),

normalized to incident plane-wave intensity at the entrance.

Result.

Headline finding: 35° is the optimal half-angle, with peak EM concentration of 3,428× incident intensity. The 30–40° range gives >2,000× — a broad optimum, not a sharp resonance, which is consistent with geometric (not chromatic) concentration.

Why 35° specifically. The bicone's central waist is where the two cones' standing-wave nodes overlap. The optimal half-angle is the one that places the first standing-wave node of each cone exactly at the waist plane — a geometric constraint, not a frequency-tuned one. 35° is roughly the solution of a transcendental equation involving the ratio of cone height to base radius; the FDTD result confirms the geometric prediction.

Sphere control: a spherical cavity of equivalent volume showed peak concentration of ~840×, well below the bicone optimum. The result is geometry-driven, not just cavity-volume-driven.

Patent significance. This result underlies two of the framework's provisional patents (TPU + Generator, filed 2026-03-29). The 35° optimum with broadband response gives a geometric EM concentrator with no chromatic tuning, no metamaterial requirements, no exotic fabrication — just shape. The patent application calls out the 35° half-angle and the 30–40° tolerance range explicitly.

Reproducibility. The full FDTD driver is attached. To reproduce: download HPC-027_bicone_apex_sweep.py.txt, rename to .py, run with NumPy + the project's FDTD engine (engine_4d.py.txt works after renaming). 96³ run takes ~4 hours on a single Hetzner box. The result should match the table above to within ~3% (small grid-resolution variance is expected).

What this is and is not.

• IS: a geometry-only mechanism producing 3,428× broadband EM

concentration with a broad geometric optimum.

• IS NOT: a magic energy source. Energy is concentrated, not created.

The waist sees 3,428× intensity *because* the surrounding cone volume has correspondingly less. Conservation holds.

• IS: directly patentable as a geometric EM concentrator design pattern.
• IS NOT: optimal across all cavity shapes. HPC-032 ran the Archimedean-solid

sweep and HPC-039 ran the {7}-fold-cavity test — both are separate geometries with their own optima.