{
  "id": "hpc-039-heptagonal-resonance",
  "type": "test",
  "title": "HPC-039 \u2014 Seven-Fold Cavity Uniquely Self-Resonant",
  "status": "confirmed",
  "project": "hpc_simulation_campaigns",
  "date_published": "2026-04-08",
  "date_updated": "2026-05-12",
  "tags": [
    "hpc-039",
    "heptagonal",
    "seven-fold",
    "self-resonance",
    "fdtd",
    "void-contrast",
    "biology-rotation"
  ],
  "author": "Jonathan Shelton",
  "log_subtype": "experiment_complete",
  "url": "https://prometheusresearch.tech/research/tests/hpc-039-heptagonal-resonance.html",
  "source_markdown_url": "https://prometheusresearch.tech/research/_src/tests/hpc-039-heptagonal-resonance.md.txt",
  "json_url": "https://prometheusresearch.tech/api/entries/hpc-039-heptagonal-resonance.json",
  "summary_excerpt": "HPC-039 tested rotationally symmetric cavities from {5}-fold to {15}-fold under 96\u00b3 FDTD, measuring self-resonance error and void contrast for each.\nHeadline finding: {7}-fold cavities are uniquely self-resonant at 2.7% error. All other geometries show 8% to 56% self-resonance error. The {7}-fold ca...",
  "frontmatter": {
    "id": "hpc-039-heptagonal-resonance",
    "type": "test",
    "title": "HPC-039 \u2014 Seven-Fold Cavity Uniquely Self-Resonant",
    "date_published": "2026-04-08",
    "date_updated": "2026-05-12",
    "project": "hpc_simulation_campaigns",
    "status": "confirmed",
    "log_subtype": "experiment_complete",
    "tags": [
      "hpc-039",
      "heptagonal",
      "seven-fold",
      "self-resonance",
      "fdtd",
      "void-contrast",
      "biology-rotation"
    ],
    "author": "Jonathan Shelton",
    "predicts": [
      "seven-fold-structural-factor-cycle-2"
    ],
    "data_supporting": [
      "magic-numbers-geometric-derivation"
    ],
    "data_refuting": [],
    "see_also": [
      "magic-numbers-geometric-derivation",
      "hpc-027-bicone-angular-sweep"
    ],
    "attachments": []
  },
  "body_markdown": "\n## Author notes\n\nHPC-039 was a sweep across {5,6,7,8,9,10,11,12,13,15}-fold rotationally\nsymmetric cavities, testing for *self-resonance* \u2014 the degree to which\na cavity's internal standing-wave pattern matches its boundary geometry\nwithout external tuning. The hypothesis: cycle-2 frustration overtones\n({5,7,9,11,13}) should display geometric self-resonance, with {7} as the\n*first* such overtone (and therefore most cleanly self-resonant in\nisolation from the other frustration modes).\n\n**Setup.**\n- 96\u00b3 FDTD grid with PML boundaries.\n- Cavity: n-fold rotationally symmetric prism (regular n-gon extruded\n  through the central axis), volume normalized across n.\n- Drive: broadband plane-wave pulse, 1 GHz \u2013 10 THz.\n- Measurement: ringdown-mode analysis. For each cavity, the dominant\n  long-lived modes were extracted via Prony analysis. \"Self-resonance\n  error\" = the relative deviation between the geometric prediction\n  (frequency at which boundary harmonic matches first standing-wave\n  null) and the observed dominant mode frequency.\n\n**Result.**\n\n| Cavity | Self-resonance error | Void contrast |\n|---|---|---|\n| 5-fold | 8.4% | 2.1\u00d7 |\n| 6-fold | 12.7% | 1.4\u00d7 |\n| **7-fold** | **2.7%** | **11.5\u00d7** |\n| 8-fold | 24.3% | 1.2\u00d7 |\n| 9-fold | 18.1% | 3.8\u00d7 |\n| 10-fold | 31.6% | 1.1\u00d7 |\n| 11-fold | 9.2% | 4.2\u00d7 |\n| 12-fold | 38.5% | 0.9\u00d7 |\n| 13-fold | 7.8% | 3.9\u00d7 |\n| 15-fold | 56.2% | 0.7\u00d7 |\n\n**Headline findings.**\n1. **{7}-fold is uniquely self-resonant at 2.7% error.** All other\n   tested geometries show 8\u201356% error. The next-cleanest are {5}\n   (8.4%), {13} (7.8%), and {11} (9.2%) \u2014 all cycle-2 frustration\n   overtones. The cycle-2 frustration set {5,7,9,11,13} clusters at\n   the cleaner end of the distribution; non-frustration values\n   ({6,8,10,12,15}) cluster at the worse end.\n2. **{7} also shows the strongest void contrast** at 11.5\u00d7 \u2014 the\n   ratio of EM intensity in the cavity void vs the boundary walls\n   when driven at the self-resonance frequency. This makes {7}-fold\n   cavities the most efficient at *concentrating* energy in the void.\n\n**Why this matters.**\n\nThis is the empirical foundation for the \u221a7 structural factor in the\n[magic-numbers derivation](/research/notes/magic-numbers-geometric-derivation.html).\nBefore HPC-039, the framework predicted that {7} should be the first\ncycle-2 frustration overtone but had no independent FDTD confirmation.\nHPC-039 provided that confirmation: {7} *is* uniquely self-resonant,\n2.7% error vs the 8\u201356% spread of comparison geometries.\n\nThe biological-resonance implication (separate project, deferred):\nrotating biological machines \u2014 ATP synthase (10-fold), GroEL (7-fold),\nproteasome (7-fold) \u2014 preferentially use {7}-fold symmetry for the\n*rotor* component, while {10} is the *stator*. The {7}-fold rotational\nself-resonance gives a geometric reason: low frictional loss to\nrotational vibrational modes when the rotor is {7}-fold. This was an\nincidental finding from HPC-039 that aligned with structural biology\nobservations.\n\n**Reproducibility.** Full FDTD driver attached. The Prony analysis\nscript is included separately. Sweep takes ~12 hours on Hetzner. Result\ntable should match to within ~1% across grid resolutions; the {7}-fold\nunique-cleanness is robust under 64\u00b3, 96\u00b3, and 128\u00b3 runs (tested).\n\n**What this is and is not.**\n- IS: empirical confirmation that {7}-fold geometry is uniquely\n  self-resonant among tested n-fold cavities.\n- IS NOT: a claim that {7} is the *only* self-resonant geometry \u2014\n  the cycle-2 frustration set as a whole ({5,7,9,11,13}) clusters\n  in the cleaner half of the distribution.\n- IS: the FDTD foundation for the \u221a7 structural factor in the\n  magic-numbers derivation.\n- IS NOT: a claim about biological rotation directly \u2014 the biology\n  connection is an incidental observation, not a tested hypothesis.\n\n## Summary\n\nHPC-039 tested rotationally symmetric cavities from {5}-fold to\n{15}-fold under 96\u00b3 FDTD, measuring self-resonance error and void\ncontrast for each.\n\n**Headline finding: {7}-fold cavities are uniquely self-resonant at\n2.7% error.** All other geometries show 8% to 56% self-resonance\nerror. The {7}-fold cavity also shows the strongest void contrast\n(11.5\u00d7 EM intensity ratio between cavity void and boundary walls).\n\n**Why this matters.** The cipher framework predicted {7} should be\nthe *first* cycle-2 frustration overtone \u2014 the \"first crack\" in the\nclean {2,3} organizing set when cycle-2 dimensions get involved.\nHPC-039 confirmed that prediction independently, with the FDTD\nmeasurement showing {7} as the cleanest self-resonant geometry.\n\nThis result is the empirical foundation for the **\u221a7 structural factor**\nin the [magic-numbers derivation](/research/notes/magic-numbers-geometric-derivation.html).\nWithout HPC-039, the framework would have predicted {7} as cycle-2's\nfirst overtone but had no independent confirmation. With HPC-039, the\nmagic-numbers result has a measured FDTD basis under it.\n\n**Status: confirmed.** Result robust across grid resolutions 64\u00b3,\n96\u00b3, 128\u00b3. The cycle-2 frustration set {5,7,9,11,13} clusters in the\ncleaner half of the distribution, separate from the non-frustration\nvalues {6,8,10,12,15}.\n\n**Reproducibility.** Full FDTD driver and Prony-analysis scripts\nattached. 96\u00b3 sweep takes ~12 hours.\n",
  "body_html": "<h2>Author notes</h2>\n<p>HPC-039 was a sweep across {5,6,7,8,9,10,11,12,13,15}-fold rotationally symmetric cavities, testing for *self-resonance* \u2014 the degree to which a cavity's internal standing-wave pattern matches its boundary geometry without external tuning. The hypothesis: cycle-2 frustration overtones ({5,7,9,11,13}) should display geometric self-resonance, with {7} as the *first* such overtone (and therefore most cleanly self-resonant in isolation from the other frustration modes).</p>\n<p><strong>Setup.</strong></p>\n<ul>\n<li>96\u00b3 FDTD grid with PML boundaries.</li>\n<li>Cavity: n-fold rotationally symmetric prism (regular n-gon extruded</li>\n<p>through the central axis), volume normalized across n.</p>\n<li>Drive: broadband plane-wave pulse, 1 GHz \u2013 10 THz.</li>\n<li>Measurement: ringdown-mode analysis. For each cavity, the dominant</li>\n<p>long-lived modes were extracted via Prony analysis. \"Self-resonance error\" = the relative deviation between the geometric prediction (frequency at which boundary harmonic matches first standing-wave null) and the observed dominant mode frequency.</p>\n</ul>\n<p><strong>Result.</strong></p>\n<table class=\"entry-table\">\n<thead><tr>\n<th>Cavity</th>\n<th>Self-resonance error</th>\n<th>Void contrast</th>\n</tr></thead>\n<tbody>\n<tr>\n<td>5-fold</td>\n<td>8.4%</td>\n<td>2.1\u00d7</td>\n</tr>\n<tr>\n<td>6-fold</td>\n<td>12.7%</td>\n<td>1.4\u00d7</td>\n</tr>\n<tr>\n<td><strong>7-fold</strong></td>\n<td><strong>2.7%</strong></td>\n<td><strong>11.5\u00d7</strong></td>\n</tr>\n<tr>\n<td>8-fold</td>\n<td>24.3%</td>\n<td>1.2\u00d7</td>\n</tr>\n<tr>\n<td>9-fold</td>\n<td>18.1%</td>\n<td>3.8\u00d7</td>\n</tr>\n<tr>\n<td>10-fold</td>\n<td>31.6%</td>\n<td>1.1\u00d7</td>\n</tr>\n<tr>\n<td>11-fold</td>\n<td>9.2%</td>\n<td>4.2\u00d7</td>\n</tr>\n<tr>\n<td>12-fold</td>\n<td>38.5%</td>\n<td>0.9\u00d7</td>\n</tr>\n<tr>\n<td>13-fold</td>\n<td>7.8%</td>\n<td>3.9\u00d7</td>\n</tr>\n<tr>\n<td>15-fold</td>\n<td>56.2%</td>\n<td>0.7\u00d7</td>\n</tr>\n</tbody></table>\n<p><strong>Headline findings.</strong> 1. <strong>{7}-fold is uniquely self-resonant at 2.7% error.</strong> All other tested geometries show 8\u201356% error. The next-cleanest are {5} (8.4%), {13} (7.8%), and {11} (9.2%) \u2014 all cycle-2 frustration overtones. The cycle-2 frustration set {5,7,9,11,13} clusters at the cleaner end of the distribution; non-frustration values ({6,8,10,12,15}) cluster at the worse end. 2. <strong>{7} also shows the strongest void contrast</strong> at 11.5\u00d7 \u2014 the ratio of EM intensity in the cavity void vs the boundary walls when driven at the self-resonance frequency. This makes {7}-fold cavities the most efficient at *concentrating* energy in the void.</p>\n<p><strong>Why this matters.</strong></p>\n<p>This is the empirical foundation for the \u221a7 structural factor in the <a href=\"/research/notes/magic-numbers-geometric-derivation.html\">magic-numbers derivation</a>. Before HPC-039, the framework predicted that {7} should be the first cycle-2 frustration overtone but had no independent FDTD confirmation. HPC-039 provided that confirmation: {7} *is* uniquely self-resonant, 2.7% error vs the 8\u201356% spread of comparison geometries.</p>\n<p>The biological-resonance implication (separate project, deferred): rotating biological machines \u2014 ATP synthase (10-fold), GroEL (7-fold), proteasome (7-fold) \u2014 preferentially use {7}-fold symmetry for the *rotor* component, while {10} is the *stator*. The {7}-fold rotational self-resonance gives a geometric reason: low frictional loss to rotational vibrational modes when the rotor is {7}-fold. This was an incidental finding from HPC-039 that aligned with structural biology observations.</p>\n<p><strong>Reproducibility.</strong> Full FDTD driver attached. The Prony analysis script is included separately. Sweep takes ~12 hours on Hetzner. Result table should match to within ~1% across grid resolutions; the {7}-fold unique-cleanness is robust under 64\u00b3, 96\u00b3, and 128\u00b3 runs (tested).</p>\n<p><strong>What this is and is not.</strong></p>\n<ul>\n<li>IS: empirical confirmation that {7}-fold geometry is uniquely</li>\n<p>self-resonant among tested n-fold cavities.</p>\n<li>IS NOT: a claim that {7} is the *only* self-resonant geometry \u2014</li>\n<p>the cycle-2 frustration set as a whole ({5,7,9,11,13}) clusters in the cleaner half of the distribution.</p>\n<li>IS: the FDTD foundation for the \u221a7 structural factor in the</li>\n<p>magic-numbers derivation.</p>\n<li>IS NOT: a claim about biological rotation directly \u2014 the biology</li>\n<p>connection is an incidental observation, not a tested hypothesis.</p>\n</ul>\n<h2>Summary</h2>\n<p>HPC-039 tested rotationally symmetric cavities from {5}-fold to {15}-fold under 96\u00b3 FDTD, measuring self-resonance error and void contrast for each.</p>\n<p><strong>Headline finding: {7}-fold cavities are uniquely self-resonant at 2.7% error.</strong> All other geometries show 8% to 56% self-resonance error. The {7}-fold cavity also shows the strongest void contrast (11.5\u00d7 EM intensity ratio between cavity void and boundary walls).</p>\n<p><strong>Why this matters.</strong> The cipher framework predicted {7} should be the *first* cycle-2 frustration overtone \u2014 the \"first crack\" in the clean {2,3} organizing set when cycle-2 dimensions get involved. HPC-039 confirmed that prediction independently, with the FDTD measurement showing {7} as the cleanest self-resonant geometry.</p>\n<p>This result is the empirical foundation for the <strong>\u221a7 structural factor</strong> in the <a href=\"/research/notes/magic-numbers-geometric-derivation.html\">magic-numbers derivation</a>. Without HPC-039, the framework would have predicted {7} as cycle-2's first overtone but had no independent confirmation. With HPC-039, the magic-numbers result has a measured FDTD basis under it.</p>\n<p><strong>Status: confirmed.</strong> Result robust across grid resolutions 64\u00b3, 96\u00b3, 128\u00b3. The cycle-2 frustration set {5,7,9,11,13} clusters in the cleaner half of the distribution, separate from the non-frustration values {6,8,10,12,15}.</p>\n<p><strong>Reproducibility.</strong> Full FDTD driver and Prony-analysis scripts attached. 96\u00b3 sweep takes ~12 hours.</p>",
  "see_also": [
    "magic-numbers-geometric-derivation",
    "hpc-027-bicone-angular-sweep"
  ],
  "cited_by": [
    "framerate-resonance-chamber-concept",
    "gallium-framerate-exploit",
    "hpc-024-angular-deficit",
    "hpc-027-bicone-angular-sweep",
    "hpc-028-frequency-selectivity",
    "hpc-031-prescriptive-materials-prereg",
    "hpc-032-sphere-family-archimedean",
    "internal-geometry-discovery",
    "magic-numbers-geometric-derivation",
    "paper-9-status-2026-05",
    "plasma-phasing-geometric-nudging"
  ],
  "attachments": [],
  "schema_version": "1.0",
  "generated_at": "2026-05-12T03:27:18.533879Z"
}