# φ Substrate — Hardware Designs (Three Paths) Status gate: no further behavioral simulation is required. The primitive set (P1–P11), the three architecture rules, and the two-mode operating principle are measured. The remaining open items — √(Fₙ·Pₙ·2ⁿ) origin (a Bessel/basin geometry question) and trained-readout capacity (post-build characterization) — do not gate hardware. What follows are buildable designs, with component values computed from the validated SPICE decks. Every design implements the same three rules learned from real failures: **(1) scale C, not the coupling/rotation element**, for frequency ladders; **(2) energy injection must saturate** (tanh-class); **(3) clamp the state nodes and power up near the operating point.** And the same two-mode principle: write mode = self-oscillating (memory in phase); read mode = sub-threshold + pilot comb (coherent forced response). --- ## Path 1 — Silicon (bench build now; IC later) ### 1a. Bench strand array — OTA + BJT-array, audio band The discovery that de-risks this build: the mandatory tanh injector **is an OTA**. The LM13700's diff-pair transfer I = Iabc·tanh(Vin/2VT) is exactly the saturating element the array requires; its Iabc pin is the write/read mode switch. **Frequency plan (C₀ = 10 nF → audio band, f₀ = 477 Hz):** | n | Cₙ = 10n·φ^(−n/2) | E24 combo | error | fₙ (Hz) | |---|------|------------|-------|--------| | 0 | 10.000 nF | 10n | 0.00% | 477.5 | | 1 | 7.862 nF | 7.5n + 360p | 0.02% | 607.3 | | 2 | 6.180 nF | 5.1n + 1.1n | 0.32% | 772.6 | | 3 | 4.859 nF | 4.3n + 560p | 0.03% | 982.7 | | 4 | 3.820 nF | 3n + 820p | 0.01% | 1250.0 | | 5 | 3.003 nF | 3n | 0.09% | 1590.0 | | 6 | 2.361 nF | 2.2n + 160p | 0.03% | 2022.6 | | 7 | 1.856 nF | 1.3n + 560p | 0.22% | 2572.8 | Use C0G/NP0 or polypropylene. The audio band means the **pilot comb, probe tones, and readout are all a computer sound card** (or any 96–192 kS/s ADC/ DAC): the entire instrumentation rack is a laptop. **Per-strand circuit (two state nodes x1, x2):** - **Gyrator (rotation, k = 30 µ℧):** LM13700 A→B and B→A cross-connection. 30:1 resistive input dividers (linear to ±0.9 V), **Iabc = 47 µA** (set by R from V+ to pin: R ≈ (V+ − 1.2V)/47µ; at ±12 V rails, 220 kΩ). One LM13700 (dual) = one gyrator. - **Saturating injector (write mode):** second LM13700 section per node, output tied back to its own node, referenced to the array's xr rail. **10.3:1 input divider** (10 kΩ / 1.0 kΩ), **Iabc = 3 µA** → small-signal 5.6 µ℧, saturation ±3 µA — the SPICE deck's exact numbers. **Mode switch: Iabc(write) = 3 µA, Iabc(read) = 1 µA** — one resistor switched per strand (or one shared DAC line). - **φ-cell (operating-point reference + restoring):** the translinear core needs matched NPNs: THAT300P (quad, current production) or CA3046-class arrays; hand-matched BC547B (ΔVbe < 1 mV) is acceptable per the √N result — the lattice averages mismatch. One shared reference cell for the array (the P10 architecture), buffered by a TL072 follower to drive all divider references. - **Clamps:** BAT54S dual Schottky per state node to the rails. - **Coupling:** 2 MΩ metal-film between adjacent x1 nodes (chain), exactly as simulated. - **Supply:** ±12 V linear; currents are µA-scale — battery operation is trivial and kills mains hum at these frequencies. **BOM per strand:** 2× LM13700 (~$2 ea), 1× THAT300P or 5× BC547B, 1× TL072, 4× BAT54S, C ladder pair, ~20 resistors. **8-strand array ≈ $60–90** plus a sound card. Perfboard is fine at 0.5–2.6 kHz; star-ground the xr rail. **Bring-up = the validated sequence:** free-run check (8 lines, √φ adjacent ratios, φ spiral ratios) → pilot comb lock check → read-mode ladder (the 1–3% product test, now against real-device theory) → genome write/read (phase demod against the comb). ### 1b. IC path (tapeout track) Port the cells to subthreshold CMOS (translinear law holds in weak inversion) or BiCMOS; MIM-cap φ ladder (capacitor matching is the process's best asset — rule 1 was chosen for this); Monte-Carlo with foundry Pelgrom data; cascode the mirrors to kill the +0.5% systematic. The bench array 1a is the direct de-risk of every block. --- ## Path 2 — Vacuum tubes (the native phase medium) Per-node: **Colpitts triode oscillator**, one half of a 12AU7 (or 6J1/6AK5 pentode) per strand. Van der Pol limiting is native (grid conduction); Adler injection locking is native (this physics was discovered here). **Tank ladder (L = 2.5 mH fixed on every strand — rule 1 in tube form: Cₙ = C₀·φ^(−n), since f ∝ 1/√C gives the √φ frequency steps):** | n | Cₙ | fₙ | |---|-----|-----| | 0 | 1013 pF | 100.0 kHz | | 1 | 626 pF | 127.2 kHz | | 2 | 387 pF | 161.8 kHz | | 3 | 239 pF | 205.8 kHz | | 4 | 148 pF | 261.8 kHz | | 5 | 91 pF | 333.0 kHz | | 6 | 57 pF | 423.6 kHz | | 7 | 35 pF | 538.8 kHz | Silver-mica caps; split each Cₙ 2:1 for the Colpitts divider. Grid leak 100 kΩ + 100 pF (the saturation mechanism — rule 2 is free). Cathode bias 1 kΩ bypassed. B+ 150–200 V, heaters 6.3 VAC. Coupling: 2–5 pF gimmick caps between adjacent tanks (weak, per the quenching lesson — start weak). Clamps/rule 3: grid conduction self-limits; power-up is inherently gentle (heater warm-up ≈ soft start). **Role:** the phase/Kuramoto medium. Verified prediction to demonstrate on the bench: the φ-detuned pair is the **last to injection-lock** as coupling increases (Hurwitz; measured in the van der Pol sims). Drift is irrelevant — sync tolerates drift; that is what sync is for. The x²=x+1 amplitude identity does not port (3/2-power law, no translinear loop) — tubes carry the phase identity only. **Scale:** 4 nodes = 2× 12AU7 + PSU ≈ shoebox. Readout: any oscilloscope or an RTL-SDR at 100–540 kHz. --- ## Path 3 — Motors, lamps, gears (the exact-ℕ medium) - **Nodes:** DC motor + flywheel units (swing equation = second-order Kuramoto; verified locking at K=0.15 with 20% drive mismatch). - **Coupling/damping:** #47 incandescent pilot lamps (6.3 V, 150 mA) wired between generator windings — the lamp's thermal R(T) is the adaptive damper (HP200A mechanism), and **filament temperature is the sync readout**: the verified signature is that locked machines' lamps equalize brightness (380 K/380 K vs 363/402 K unlocked). Photodiodes on each lamp → the order parameter as a light level. - **φ itself:** Fibonacci gear pairs, 3D-printed — 34:21 (626 ppm), 89:55 (91 ppm), **233:144 (13 ppm)**. Integer teeth make mechanical φ *cheaper and more precise than the analog silicon lattice* (which needs ~5,900 cells for ~1,000 ppm). This is LAYER 1 — Fibonacci as the axiom iterated in ℕ — in brass (or PLA). - **Rules:** rule 1 = inertia (flywheel mass) scales the frequency, not drive torque; rule 2 = the lamp IS the saturating damper; rule 3 = soft-start the drive current. **Role:** room-scale exhibit and the highest-precision φ artifact of the three paths. Speed: Hz. --- ## The split, restated as a build plan The axiom has two physical identities and the media divide them: **amplitude identity x²=x+1** → silicon translinear loop (path 1) and Fibonacci teeth (path 3); **phase identity φ = most-irrational winding number** → tube injection locking (path 2) and swing-equation machines (path 3). Build order by information gained per dollar: 1. **Path 1a bench array (~$90, sound-card instrumented)** — every P1–P11 result becomes physically checkable, and it is the direct de-risk of the IC. 2. **Path 2 four-node tube rack** — the Hurwitz lock-resistance demo in the medium that invented the physics. 3. **Path 3 gear-and-lamp machine** — 13 ppm mechanical φ and sync you can see, for the bench that visitors understand in ten seconds.