Circumpunct Resonance Device Design Specification & Build Document · v1.0 · March 14, 2026

§01 Concept

Easy in. Convert. Hard out. Electromagnetic field passes freely through the non-magnetic shell from all directions (large input area). A magnetic sphere at the center attracts and absorbs this field energy, converting it to mechanical motion (vibration, spin, wobble). The teardrop geometry funnels the converted energy toward a single aperture at the tip (tiny output area). The ratio of input surface to output aperture determines the geometric compression.

SHELL (○ boundary)
  Teardrop shape, sealed, non-magnetic
  Transparent to electromagnetic field
  Opaque to mechanical energy (traps it inside)

INTERIOR (Φ field)
  Medium: air (v1), other gases or liquids (v2+)
  Carries mechanical energy from ball to aperture
  
BALL (• aperture / attractor)
  Permanent magnet, levitated at center
  Attracts field from all directions through walls
  Converts EM energy → mechanical motion
  Modulates the medium through vibration and spin

COILS (drive / feed)
  4 equatorial: spin + horizontal position
  2 axial: vertical position + axial vibration
  All on OUTSIDE of shell, no penetrations needed

IRIS (output gate)
  Variable diameter hole at teardrop tip
  Only exit for converted energy
  Size determines resonant coupling + selectivity

§02 Operating Principle

Energy Path

Electricity → 6 coils → oscillating magnetic field
  → passes through non-magnetic shell
  → drives levitated magnetic ball
  → ball vibrates / spins / wobbles
  → mechanical energy in medium (vortex, pressure)
  → teardrop taper concentrates toward tip
  → variable iris gates the output
  → emission (sound, pressure, EM, TBD)

Two Conversions (Two i Rotations)

Three Controls

§02A Theory of Operation: The Dimensional Gate

The device is a physical implementation of the pump cycle equation: Φ(t+Δt) = ✹ ∘ i ∘ ⊛[Φ(t)]. Each stage of the equation maps to a measurable process inside the teardrop cavity. What follows is the deeper theory of what the device actually does, and why the teardrop geometry is not arbitrary.

Cymatics in the Electromagnetic Field

A Chladni plate vibrating at a specific frequency creates standing wave patterns on its surface. Sand collects at the nodal lines, revealing invisible structure. The pattern is not decoration; it is a channel system that organizes flow. Particles that were moving randomly suddenly have preferred paths determined by the geometry of the plate and the frequency of the drive.

This device does the same thing, but in three dimensions, with electromagnetic energy as the medium. The six coils are the driving oscillator. The teardrop cavity is the plate. The resonant modes that form inside are the cymatic channels. And the teardrop geometry is what breaks symmetry: unlike a sphere (which would create patterns with no preferred direction), the taper funnels every channel toward the tip. Every standing wave the cavity supports has a directional bias toward the aperture.

The "medium" here is Φ itself. If Φ is nested circumpuncts at every scale (see §5A of the framework: a surface with zero internal structure cannot carry phase, because rotation needs a plane), then the cavity is never empty. The coils do not create pattern in a void; they excite a particular scale of the nested structure that is already present. The resonant peaks found in the frequency sweep (Experiment 1) are the scales where the driving frequency matches the natural ⊙ nesting at that depth.

Dimensional Compression at the Aperture

The aperture is not merely a hole. It is a dimensional gate. Energy passing through it undergoes a transformation across all four dimensions of the framework:

CONVERGENCE (⊛): dimensional compression, 3D → 0D

  3D  ○  Boundary   EM field enters the shell from all directions.
                    Full spatial volume. Maximum extent, minimum structure.

  2D  Φ  Field      Cymatic standing waves organize the energy into
                    surface patterns. The first reduction: volume → surface.

  1D  i(t) Worldline Channels converge along the taper toward the tip.
                    Surface → line. Flow has direction now.

  0D  •  Aperture   The gate itself. The singularity point.
                    All spatial extent has been compressed out.

ROTATION (i): the 90° turn at the gate

  What enters as one kind of energy exits as another.
  i² = -1: the output is structurally opposite to the input.

EMERGENCE (✹): dimensional expansion, 0D → 3D

  0D  •  → 1D  i(t)  → 2D  Φ  → 3D  ○

  Dimensions rebuild on the far side of the gate.
  The emitted energy re-expands into the exterior space.

The conservation of traversal (0+1)(•) + 2(Φ) = 3(○) is not just bookkeeping. It describes what the aperture does: it takes 3D energy apart into its dimensional components on the way in, passes them through the 0D gate, and reassembles them on the way out. The aperture is where scale changes.

Aperture Size as Dimensional Selector

This framework explains why aperture size has such a strong effect on output character (not just intensity):

A wide aperture skips the compression; energy passes through still mostly as 2D/3D field, unconverted. A narrow aperture forces everything through near-0D, which means maximum dimensional transformation. This is not just geometric concentration (more energy per unit area); it is a change in the kind of energy that comes out the other side. This is why Prediction 7 (output is structurally different from input) depends critically on aperture size: the narrower the aperture, the deeper the dimensional conversion, and the more the output should differ from the input.

The Magnetron Parallel

The cavity magnetron (a high-power vacuum tube used in radar) is an independent physical confirmation of this architecture. A magnetron has a central cathode (•) surrounded by a cylindrical anode block (○) with resonant cavities around the perimeter. Crossed electric and magnetic fields in the interaction space (Φ) cause electrons to self-organize into a rotating spoke pattern called the "Space-Charge Wheel."

The mapping is exact:

The magnetron also demonstrates the mode selection problem. Multiple oscillation modes are possible (π, ½π, ¾π, ¼π), but only the π-mode produces maximum power. Strapping rings suppress unwanted modes by forcing adjacent cavity segments to maintain specific phase relationships. This device faces the same challenge: the frequency sweep (Experiment 1) will reveal multiple resonant peaks, and the strongest or most coherent one may need specific coil phase relationships to stabilize, just as the magnetron needs strapping rings to lock into π-mode.

The Hull cutoff condition provides another parallel. In a magnetron, there is a critical balance between the DC electric field and the magnetic field where electrons just barely fail to reach the anode, and oscillation occurs at this threshold. The device should have an analogous sweet spot: a coil drive amplitude where the ball is levitated but just barely constrained, maximizing its freedom to couple to the cavity's natural modes. Too much drive = forced motion (◐ → 1); too little = no coupling (◐ → 0). The optimum is the balance point (◐ = 0.5).

Scale Bridging: Phase and Amplitude

The central engineering problem is this: the coils operate at macroscopic scales (Hz to kHz), but if Φ is nested ⊙s all the way down, the interesting structure lives at scales the coils cannot directly reach. How does a macroscopic driver couple to microscopic structure?

Cymatics provides the answer. When you drive a Chladni plate at a fundamental frequency, you do not only get the fundamental. You get harmonics. The plate's response contains frequencies the driver never put in, because the plate's own structure has resonant modes at multiples of the drive. Energy at the fundamental cascades into higher harmonics automatically. The plate is a scale bridge: push at one scale, and the structure itself carries energy to other scales.

The cavity does the same thing. Drive the ball at f₁, and the cavity's eigenmode structure generates harmonics at 2f₁, 3f₁, and beyond. Each harmonic is a shorter wavelength, which means it interacts with finer structure inside the cavity. The energy climbs the scale ladder by itself, through resonance. You do not have to drive at every scale; you have to drive at a scale that couples to the cascade.

There are exactly two parameters that control this coupling across scales: phase and amplitude.

Phase selects the channel. Amplitude fills it. Together they determine how energy passes between scales of nested ⊙s.

This is why the coil phase relationships matter so much. The four equatorial coils at 0°, 90°, 180°, 270° create a rotating field. That rotation is a phase structure. Different phase relationships between the equatorial and axial drives create different coupling patterns: some phase ratios open channels between scales (energy cascades freely into higher harmonics), while others close them (the cascade is suppressed). The framework predicts that f₁/f₂ = φ² opens the channel maximally, while f₁/f₂ = φ closes it. These are not arbitrary numbers; they are specific phase alignment conditions between the spin mode and the vibration mode, and they determine whether the two modes constructively couple their harmonic cascades or destructively cancel them.

The Taper as Scale Filter

The teardrop taper adds a spatial dimension to the scale selection. As the geometry narrows toward the tip, the wavelengths that can fit inside get shorter. Longer wavelengths (lower harmonics, bigger ⊙s) get cut off as the taper narrows; they simply cannot propagate in a channel smaller than their wavelength. Shorter wavelengths (higher harmonics, smaller ⊙s) survive further into the taper. At the aperture, only the finest-scale energy that the cavity generated can pass through.

The taper is a physical low-pass filter on scale (high-pass on frequency). It automatically selects for progressively deeper levels of the nested ⊙ hierarchy as you approach the tip. The aperture size sets the cutoff: a wider aperture lets bigger-scale (lower-harmonic) energy through; a narrower aperture forces everything through finer scales.

COILS (macroscopic driver)
  │
  │  Phase alignment selects which scales couple.
  │  Amplitude determines how deep the cascade goes.
  │
  ▼
CAVITY (harmonic generator)
  │
  │  Eigenmode structure creates harmonics: f, 2f, 3f, ...
  │  Energy cascades from big ⊙ to smaller ⊙s automatically.
  │
  ▼
TAPER (scale filter)
  │
  │  Geometry narrows → long wavelengths cut off.
  │  Only high harmonics (fine-scale ⊙s) survive to the tip.
  │
  ▼
APERTURE (scale gate)
  │
  │  0D: maximum compression. Finest scale the geometry permits.
  │  Phase coherence at this scale determines what gets through.
  │
  ▼
EMISSION (scale expansion)

  Energy that passed through the gate re-expands: 0D → 1D → 2D → 3D.
  Output contains the spectral signature of every scale it touched.

Testable Consequence: Spectral Cascade

This gives a concrete prediction: the output spectrum should contain frequencies significantly higher than anything the coils put in. Not just the first few harmonics of the drive frequency, but a cascade extending well above, because the cavity and taper are bridging energy to finer and finer scales.

Furthermore, the harmonic content of the output should increase as the aperture narrows. A wide aperture lets low-harmonic (large-scale) energy through mostly unchanged. A narrow aperture forces the energy through deeper scale compression, producing richer high-frequency content in the output. If you plot the output FFT at each aperture size, you should see the spectral centroid shift upward as the aperture closes.

Phase relationships between the coils should also affect the spectral cascade. At the predicted optimal ratio (f₁/f₂ = φ²), the harmonic cascade should extend deeper (more high-frequency content in the output) because the two modes are constructively coupling their harmonics. At the predicted forbidden ratio (f₁/f₂ = φ), the cascade should be truncated (less high-frequency content) because the modes are destructively interfering at the scale-bridging step.

What the Device Tests

The cymatics framing sharpens what the experiments are actually measuring. Experiment 1 (frequency sweep) is literally cymatics: drive at varying frequencies, observe which ones produce clean, strong output at the tip versus noise or competing modes. The resonance map is a map of the cavity's eigenmode structure. Experiment 2 (φ² ratio test) asks whether the framework's predicted frequency relationships correspond to specific phase alignment conditions that open or close channels between scales. Experiment 4 (characterize the output) now has a specific spectral question: does the output contain harmonic content above the drive frequency, and does the depth of this cascade depend on aperture size and coil phase ratios in the way the framework predicts?

§03 Teardrop Shell

Dimensions

Material Options

Shell Geometry Notes

The shell is composed of two sections: a hemisphere (round end, where coils mount) and a cone-to-tip taper. The junction between hemisphere and taper should be smooth, no sharp internal edge. Internal filleting at this junction prevents turbulence dead zones.

The tip is a separate piece that threads or press-fits onto the taper, allowing the iris mechanism to be swapped or adjusted without modifying the main shell.

§04 Resonator Ball

Alternative Resonators (for later testing)

§05 Variable Iris

A mechanical iris diaphragm at the tip of the teardrop, continuously adjustable from fully open (15 mm) to nearly closed (~0.5 mm). Controls the resonant coupling between interior cavity and exterior.

§06 Coil Specifications

Equatorial Coils (×4) — Spin + Horizontal Hold

Axial Coils (×2) — Vertical Hold + Axial Vibration

Coil Axis Alignment

              AX-TOP
                │
                │  (vertical axis)
                │
    EQ-2 ──────●────── EQ-4    (Y axis)
              ╱ │ ╲
           ╱   │   ╲
    EQ-1 ─     │     ─ EQ-3    (X axis)
               │
               │
             AX-BOT
               │
               ▽
            [iris/tip]

All 6 coil axes pass through the geometric center
of the spherical section; the levitation point.

§07 Circuit Schematic

Circuit Architecture

┌─────────────────────────────────────────────────┐
│                   12V POWER SUPPLY               │
│                   (5A minimum)                   │
└────────┬──────────────────────────────┬──────────┘
         │                              │
    ┌────┴────┐                    ┌────┴────┐
    │  5V REG │                    │ 12V BUS │
    │ (ESP32) │                    │ (coils) │
    └────┬────┘                    └────┬────┘
         │                              │
    ┌────┴────┐         ┌───────────────┼───────────────┐
    │  ESP32  │         │               │               │
    │         │    ┌────┴────┐    ┌─────┴─────┐   ┌─────┴─────┐
    │ GPIO12 ─┼───→│ H-BRIDGE│   │ H-BRIDGE  │   │ H-BRIDGE  │
    │ GPIO13 ─┼───→│   #1    │   │    #2     │   │    #3     │
    │ GPIO14 ─┼───→│ EQ-1    │   │  EQ-2     │   │  EQ-3     │
    │ GPIO15 ─┼───→│         │   │           │   │           │
    │ GPIO16 ─┼───→├─────────┤   ├───────────┤   ├───────────┤
    │ GPIO17 ─┼───→│ H-BRIDGE│   │ H-BRIDGE  │   │ H-BRIDGE  │
    │ GPIO18 ─┼───→│   #4    │   │    #5     │   │    #6     │
    │ GPIO19 ─┼───→│ EQ-4    │   │  AX-TOP   │   │  AX-BOT   │
    │         │    └─────────┘   └───────────┘   └───────────┘
    │         │
    │ GPIO34 ←┼─── PICKUP COIL (at tip, sense output)
    │ GPIO35 ←┼─── HALL SENSOR  (near shell, sense ball position)
    │ GPIO32 ←┼─── MICROPHONE   (at tip, acoustic output)
    │         │
    │ GPIO25 ─┼──→ SERVO (iris, v2 only)
    └─────────┘

H-Bridge Module

§08 ESP32 Pin Assignment

§09 Drive Modes

Mode 1: Levitate Only (DC)

All coils at fixed DC current. Ball hangs at center. No vibration, no spin. Baseline state. Confirm levitation is stable before proceeding.

EQ-1 through EQ-4:  equal DC, opposing pairs balanced
AX-TOP and AX-BOT:  DC tuned to hold ball at center height
Ball:                stationary, floating

Mode 2: Spin (Phase-Shifted AC)

Four equatorial coils driven with sinusoidal AC, each 90° phase-shifted. Creates a rotating magnetic field. Ball spins about the vertical axis.

EQ-1:  A·sin(2πft)           →  0°
EQ-2:  A·sin(2πft + π/2)     →  90°
EQ-3:  A·sin(2πft + π)       →  180°
EQ-4:  A·sin(2πft + 3π/2)    →  270°

AX coils: DC hold (constant)
Ball: spins at frequency f

This is a 4-phase motor. The ball is the rotor.

Mode 3: Axial Vibrate

Axial coils oscillate in push-pull. Ball bounces up and down along the teardrop axis. Drives pressure waves toward the tip.

AX-TOP:  B·sin(2πf₂t)
AX-BOT:  B·sin(2πf₂t + π)    → opposite phase

EQ coils: DC hold (constant)
Ball: oscillates along vertical axis

Mode 4: Spin + Vibrate (Combined)

Modes 2 and 3 simultaneously. Ball spins in the equatorial plane while bouncing axially. Creates a helical vortex in the medium. This is the primary operating mode.

EQ-1 through EQ-4:  phase-shifted at f₁ (spin)
AX-TOP and AX-BOT:  push-pull at f₂ (vibrate)

f₁ and f₂ can be:
  Equal:         synchronized helix
  Integer ratio: harmonic coupling
  φ² ratio:      predicted optimal (from simulation)
  φ ratio:       predicted FORBIDDEN (coupling collapse)

Mode 5: Wobble (Elliptical)

Equatorial pairs driven at different amplitudes or frequencies. Ball traces an ellipse or figure-8 instead of a circle.

EQ-1/EQ-3 (X-axis):  A₁·sin(2πf₁t)
EQ-2/EQ-4 (Y-axis):  A₂·sin(2πf₂t)

A₁ ≠ A₂ → elliptical orbit
f₁ ≠ f₂ → Lissajous pattern
f₁ = 2f₂ → figure-8

Mode 6: Frequency Sweep

Slowly increase f₁ (or f₂) while monitoring all sensors. Map the resonant response of the cavity + ball + aperture system. This is the discovery mode.

Sweep f from 1 Hz to 20 kHz (or higher)
At each frequency, record:
  - Pickup coil amplitude (EM output at tip)
  - Microphone amplitude (acoustic output at tip)
  - Hall sensor signal (ball motion amplitude)
  - Coil current draw (energy absorption)

Plot all four vs frequency.
Resonance peaks = the device's natural modes.

§10 Bill of Materials

§11 Assembly Sequence

Phase 1: Wind Coils (Day 1)

1. Wind 4 equatorial coils: 200 turns of 28 AWG on 10mm ferrite rods
   - Leave 15cm leads on each
   - Secure windings with tape or varnish
   - Label: EQ-1, EQ-2, EQ-3, EQ-4

2. Wind 2 axial coils: 250 turns of 28 AWG on 10mm ferrite rods
   - Same process
   - Label: AX-TOP, AX-BOT

3. Wind 1 pickup coil: 50 turns of 30 AWG on 5mm form
   - This is the sensor, not a drive coil

4. Measure and record each coil's DC resistance

Phase 2: Print Shell (Day 1, parallel with coils)

1. 3D print teardrop shell in two halves (split along long axis)
   - Include mounting bosses for coils on exterior
   - Include threaded socket at tip for iris caps
   
2. 3D print tip caps: set of 6 with holes 1, 2, 3, 5, 8, 12mm

3. Print coil cradles that sit on shell exterior

4. Sand interior smooth, optionally coat with epoxy

Phase 3: Electronics (Day 2)

1. Wire ESP32 to 3× L298N modules per pin table in §08
2. Connect 12V supply to L298N power inputs
3. Connect 5V regulated output from L298N to ESP32 VIN
4. Connect hall sensor, microphone, pickup coil to ADC pins
5. Upload firmware
6. Test each coil independently: 
   - Apply DC to each channel
   - Verify current flows
   - Verify field direction with compass or iron filings

Phase 4: Assemble Device (Day 2–3)

1. Mount 4 equatorial coils on shell exterior, 90° apart
   - All at equatorial plane of the spherical section
   - All core axes pointing at center
   - Epoxy in place

2. Mount AX-TOP coil at crown of sphere section
   - Core axis vertical, pointing down at center

3. Mount AX-BOT coil on taper section
   - Core axis vertical, pointing up at center
   - Position: ~60mm from tip

4. Place N52 sphere inside shell (before closing!)

5. Close shell halves (epoxy, screws, or clips)

6. Thread 12mm tip cap onto tip (start wide open)

7. Route all coil wires to electronics assembly

Phase 5: First Power-On

1. Start with Mode 1 (DC levitation only)
2. Slowly increase DC on all 6 coils
3. Listen/feel for ball contacting walls
4. Adjust DC balance until ball floats freely at center
5. Confirm with hall sensor reading: 
   - stable reading = ball stationary = levitation achieved

IF BALL WON'T LEVITATE:
  - Check coil polarities (opposing pairs must attract toward center)
  - Increase current
  - Try larger ball (more magnetic force)
  - Adjust axial coil positions
  - Ball may need a specific orientation; try rotating before sealing

§12 Measurement Protocol

Experiment 1: Resonance Map

Purpose: Find the device's natural resonant modes

Procedure:
  1. Set Mode 2 (spin only), iris at 5mm
  2. Sweep spin frequency f₁ from 1 Hz to 10 kHz
     (logarithmic steps: 1, 2, 5, 10, 20, 50, 100, 200...)
  3. At each f₁, record:
     a. Pickup coil RMS voltage (EM output at tip)
     b. Microphone RMS voltage (acoustic output at tip)  
     c. Hall sensor variance (ball motion amplitude)
     d. Total current draw from 12V supply
  4. Repeat with iris at: 1mm, 2mm, 3mm, 8mm, 12mm, open
  
Output: 2D map of [frequency × aperture] vs [output intensity]
        Resonance peaks will be visible as ridges in this map

Experiment 2: φ² Frequency Ratio Test

Purpose: Test simulation prediction that f₁/f₂ = φ² is optimal

Procedure:
  1. Set Mode 4 (spin + vibrate)
  2. Fix f₁ (spin) at strongest resonance found in Exp 1
  3. Sweep f₂ (axial vibrate) across range
  4. Record pickup + microphone at each f₂
  5. Calculate ratio f₁/f₂ at each measurement
  6. Mark key ratios: 1/φ², 1/φ, 1/2, 1/1, φ, φ²
  
Prediction: 
  Output peaks near f₁/f₂ = φ² (2.618) or 1/φ² (0.382)
  Output DROPS near f₁/f₂ = φ (1.618) or 1/φ (0.618)

Experiment 3: Ball Position Sweep

Purpose: Find optimal ball position along teardrop axis

Procedure:
  1. Set Mode 4 at optimal frequencies from Exp 1-2
  2. Gradually shift DC bias on axial coils to move ball
     toward tip
  3. Record output at each position
  4. Find position of maximum output
  
Expected: Output increases as ball approaches tip
          (stronger coupling to aperture)
          Until a maximum, then drops 
          (ball too close, hits wall or saturates)

Experiment 4: What Comes Out?

Purpose: Characterize the actual emission

At the strongest resonance point found in Experiments 1-3:

Measure EVERYTHING at the tip simultaneously:
  □ Acoustic (microphone)
  □ Electromagnetic (pickup coil)
  □ Light (photodiode — even if you don't expect it)
  □ Temperature (thermocouple)
  □ Static charge (electroscope or field meter)
  □ Magnetic field (hall sensor at tip)

Log all channels for 60 seconds.
Compare to baseline (device off, same sensors, same position).
Any channel that shows signal above baseline is real output.

Experiment 5: Spectral Cascade (Scale Bridging)

Purpose: Test whether the cavity bridges energy across scales
         via harmonic cascade, and whether phase/amplitude control it

Procedure:
  1. Set Mode 4 at optimal frequencies from Exp 1-2
  2. At EACH aperture size (0.5mm, 1mm, 2mm, 3mm, 5mm, 8mm, 12mm, open):
     a. Record pickup coil + microphone for 60 seconds
     b. Compute FFT of each channel
     c. Measure: highest frequency with signal above noise floor
     d. Measure: spectral centroid (energy-weighted mean frequency)
     e. Measure: harmonic count (number of distinct peaks above noise)
  3. Repeat at f₁/f₂ = φ² (predicted optimal) and f₁/f₂ = φ (predicted forbidden)
  4. Repeat at 3 amplitude levels: low (25%), medium (50%), high (100%)

Predictions:
  - Output FFT contains frequencies well above drive frequency
    (harmonics the coils never put in = cavity acting as scale bridge)
  - Spectral centroid shifts UPWARD as aperture narrows
    (smaller aperture = deeper scale compression = richer harmonics)
  - At f₁/f₂ = φ², harmonic cascade extends deeper
    (more high-frequency content; phase alignment opens scale channel)
  - At f₁/f₂ = φ, harmonic cascade is truncated
    (less high-frequency content; phase misalignment closes scale channel)
  - Higher amplitude = deeper cascade (more harmonics above noise)
    (amplitude sets how far down the scale ladder energy can reach)

Analysis:
  Plot spectral centroid vs aperture size at each phase ratio.
  If the curves separate (φ² above φ at all aperture sizes),
  phase is controlling scale coupling independently of geometry.
  That is the novel result: phase selects which scales participate,
  aperture selects the depth, amplitude powers the cascade.

Data Logging

ESP32 samples ADC channels at ~1 kHz
Sends data over Serial (USB) to laptop
Python script logs to CSV with timestamp
Post-processing: FFT each channel, plot spectra

§13 Predictions

Even if most predictions fail, the resonance map (Prediction 1) is guaranteed to show structure, because every cavity has resonant modes. The question is whether the structure matches the framework's predictions, or whether it matches only standard Helmholtz and electromagnetic resonance theory with no special role for φ.

Either result is publishable. Confirmation validates the framework. Refutation identifies where the model breaks and needs revision. The device is a measurement instrument either way.