---
id: test-jj-prism-extended-2026-05-14
type: test
title: Test JJ Prism-Extended — Prism-Family Chebyshev Lock
date_published: 2026-05-14
date_updated: 2026-05-14
project: geometric_wave_compute
status: confirmed
log_subtype: experiment_complete
tags: [test-jj, prism-class, chebyshev, drive-invariant, coupled-pair, gwc, fdtd, prism-7fold, prism-11fold, prism-13fold]
author: Jonathan Shelton
predicts:
  - prism-class-spectral-lock
  - chebyshev-attractor-2cos2pi-N
data_supporting:
  - hpc-039-heptagonal-resonance
  - test-jj-joint-emergence-prereg
---

## TL;DR

GWC Test JJ phase-1 advanced from 341 → 606 of 700 configurations
between 2026-05-13 and 2026-05-14, closing the prism_7fold near_field
gap and adding full prism_11fold + partial prism_13fold pair-coverage.
**The single-cavity {7}-uniqueness of [HPC-039](/research/tests/hpc-039-heptagonal-resonance.html)
generalizes to a prism-family Chebyshev attractor at the coupled-pair
scale.** Each prism over-represents its own `2·cos(2π/N)` ratio; each
prism shows a drive-and-decoherence-invariant joint-spectrum envelope
of comparable tightness:

| Pair | n | BW std/mean (pooled) | BW std/mean (near_field) |
|---|---:|---:|---:|
| prism_7fold × prism_7fold | 150 | 4.4 % | **2.8 %** (tightest in dataset) |
| prism_11fold × prism_11fold | 150 | 4.9 % | **2.9 %** |
| prism_13fold × prism_13fold | 56 / 150 | 7.1 % | (partial) |
| octahedron × octahedron | 175 | 75.6 % (compare) | — |
| bicone × bicone | 150 | 59.2 % (compare) | — |

## Headline findings

- **Prism family is a Chebyshev attractor.** The dominant joint-spectrum
  ratio is `2·cos(2π/N)` (second-order Chebyshev), generic across
  N. Earlier "{7}-fold Fibonacci attractor" framing is superseded.
- **prism_11fold near_field is a new compute primitive** — 12 GHz
  wideband locked (2.9 % CV), 25-peak high-density modulator. Distinct
  from the narrowband prism_7fold near_field clock primitive and the
  high-entropy octa near_field modulator.
- **prism_13fold may be entering a Bessel/cylinder asymptotic regime**
  — its first-order Chebyshev `2·cos(2π/13) = 1.942` is *suppressed*
  (0.70 %) below octa baseline 1.41 %. PROVISIONAL pending the
  remaining 94 of 150 configs (Hetzner phase-1 ETA late 2026-05-14).
- **bicone shared_wall hidden BW lock** — pooled 4.0 % CV, comparable
  to a 7-fold prism, masked in pooled-shape views.

## Math correction

The 2026-05-13 comprehensive doc labeled the dominant ratio
`2·cos(π/7) = 1.247`. The correct label is **`2·cos(2π/7)`** —
`2·cos(π/7) ≈ 1.802` is a different number entirely. Minimal
polynomial for `2·cos(2π/7)` is `x³ + x² − 2x − 1 = 0`. The phenomenon
(over-representation of the value 1.247) is real; only the symbolic
identification was wrong. Propagated to the GWC `tests/` corpus and
the GWC bibliography.

## What this means for the framework

- HPC-039's single-cavity {7}-uniformity result still holds as a
  single-cavity property. Coupled-pair behavior generalizes.
- Cipher v3 circuit-language Layer 1 should add a PRISM-CLASS family
  with `bw_lock_cv` as a new readout, rather than proliferating
  N-specific FRUSTRATE archetypes.
- {7} is *first* in the cycle (lowest odd-prime frustration overtone),
  not *unique* across it. Biology's preferential use of {7}-fold
  rotational machines (ATP synthase, GroEL, proteasome) becomes an
  open question for [Paper 9](/research/paper-status/paper-9-status-2026-05.html).

## Source

Findings doc and reproducibility artifacts are in the GWC project
tree:

- Findings: `geometric_wave_compute/tests/test_jj_prism_extended_2026-05-14.md`
- Dataframe: `geometric_wave_compute/data/2026_05_14_pull/jj_fingerprints.pkl`
  (681 records)
- Build/analysis scripts: `geometric_wave_compute/data/2026_05_14_pull/{build_dataframe.py,analyze.py}`

The frequency-ratio analysis is in `analysis/ratios.json`; the
shape × topology cuts are in `analysis/cuts.json`.

## Status

**Confirmed for prism_7fold, prism_11fold.** Provisional for
prism_13fold pending full sample (~94 more configurations finishing
on Hetzner late 2026-05-14). Bicone shared_wall hidden lock confirmed
at the coupled-pair scale.