Research Note: Resonant Bit-Serialized HDC (RBS-HDC) Evaluation¶
Date: June 2026
Project: Ephemerides Mechanism (ephemerides-spectral)
Objective: Evaluate the feasibility of a "Resonant Bit-Serialized" approach to eliminate FPU dependency while preserving the phase-space architecture.
1. Executive Summary¶
The transition from complex-valued hypervectors to a Resonant Bit-Serialized (RBS) representation is not only feasible but aligns perfectly with the "Gear-Ratio" philosophy of the Antikythera mechanism. By mapping continuous phases to discrete cyclic groups (\(\mathbb{Z}_{2^K}\)), we replace floating-point complex arithmetic with bit-serial integer addition, enabling extreme hardware efficiency (e.g., FPGA/ASIC implementation without FPUs).
2. Proposed Data Format: Bit-Interleaved Phases (BIP)¶
Instead of 128-bit complex floats (2x f64), we represent the \(D=65536\) state vector as an array of \(K\)-bit integers.
- State Vector \(H\): \(H \in (\mathbb{Z}_{2^K})^D\).
- Recommended \(K\): 16 bits (65,536 sectors) for high-precision or 32 bits for astronomical grade precision.
- Bit-Serialized Layout: In hardware, these \(D\) components are processed as bit-serial streams. In software, this is a
uint16_t[65536]oruint32_t[65536]array.
3. Algebraic Rigor & Mapping¶
3.1 Binding Operator (\(\otimes\))¶
- Complex HDC: \(z_1 \otimes z_2 = z_1 \cdot z_2\) (complex multiply).
- RBS-HDC: \(\phi_1 \otimes \phi_2 = (\phi_1 + \phi_2) \pmod{2^K}\) (modular addition).
- Rigor: This is a group isomorphism. The unitary property (Norm=1.0) is preserved because every element in \(\mathbb{Z}_{2^K}\) corresponds to a unit-magnitude phasor.
3.2 The Propagator (\(e^{-i L t}\))¶
The evolution of the system Laplacian \(L\) is mapped from the continuous domain to the discrete bit-serial domain.
- Diagonal Evolution (Mean Motion): $\(\phi_j(t) = (\phi_j(0) + \Omega_j \cdot t) \pmod{2^K}\)$ where \(\Omega_j\) is the fixed-point frequency signature of the \(j\)-th dimension.
- Implementation: This is a Numerically Controlled Oscillator (NCO) at each dimension. Bit-serial adders can update all 65,536 dimensions in parallel without an FPU.
3.3 Off-Diagonal Couplings (Perturbations)¶
Gravitational perturbations (the "fiber couplings") are modeled as phase-dependent nudges. - Interaction: \(\Delta \phi_i = W_{ij} \cdot \text{LUT\_Sin}(\phi_j - \phi_i)\). - FPU-less: Uses a small Look-Up Table (LUT) or CORDIC bit-serial rotation to compute the interaction term.
4. Micro-Architecture Benefits¶
- Ditch the FPU: All operations are integer addition and bit-shifts.
- Energy Recovery: The "Resonant" aspect can be physically realized in hardware using Resonant SRAM (rCiM) or LC-tank oscillators, potentially reducing power by 10-100x for "Always-On" ephemeris tracking.
- Deterministic Latency: Bit-serial processing provides fixed, predictable timing for N-body evolution, critical for real-time haptic or spectral feedback.
5. Logic Sketch for FPU-less Evolution¶
# Fixed-point Spectral Evolution (Conceptual)
def evolve_rbs_hdc(state_vector_bip, frequencies_fixed, delta_t):
"""
Evolve the Bit-Interleaved Phase (BIP) vector.
state_vector_bip: uint32[65536] - The phase of each dimension.
frequencies_fixed: uint32[65536] - Pre-calculated fixed-point frequencies.
"""
# Bit-serial update (integer addition)
# Equivalent to psi_t = exp(-iLt) @ psi_0
for j in range(65536):
state_vector_bip[j] = (state_vector_bip[j] + frequencies_fixed[j] * delta_t) % (2**32)
return state_vector_bip
6. Benchmark Results (Prototype)¶
Benchmarked on a modern workstation using the EphemerisBIPInstrument prototype against the floating-point EphemerisHDCInstrument.
| Metric | FPU (Complex64) | RBS-HDC (UInt32) | Benefit |
|---|---|---|---|
| Execution Time | 1379.0 ms | 4.5 ms | ~305x Speedup |
| Memory (State) | 1024 KB | 256 KB | 4x Compression |
| Terra Phase Err (20yr) | Reference | 0.0002 rad | High Fidelity |
Conclusion: The integer-only approach provides massive speedups and significant memory savings while maintaining astronomical precision. The 0.0002 rad error (approx. 0.01 degrees) is well within the tolerance for edge mapping applications.
7. Phase 8: Dimensional Expansion & SNR Scaling¶
We explored the effect of increasing the hypervector dimension \(D\) on resonance quality and performance.
| D (log2) | Size (MB) | Time (ms) | SNR | Terra Err (rad) |
|---|---|---|---|---|
| 16 | 0.25 | 2.4 | 2,621 | 0.000205 |
| 17 | 0.50 | 5.7 | 5,243 | 0.000205 |
| 18 | 1.00 | 25.7 | 10,486 | 0.000205 |
| 20 | 4.00 | 106.1 | 41,943 | 0.000205 |
Key Findings¶
- Linear SNR Gain: Resonance Sharpness (SNR) scales perfectly linearly with \(D\). Expanding to the 1MB Hypervector (\(D=2^{18}\)) provides a 4x increase in signal strength for planetary extraction compared to the standard \(2^{16}\) baseline.
- ALU-Native Resilience: The system maintains machine-precision 32-bit phase discretization across all dimensions. The 0.0002 rad error is the structural limit of our current static Laplacian propagator, not the bit-serialized format.
- Kernel Synthesis: DE441 and DE442 synthesis remains coherent within the 1MB lattice; the differences between kernels are currently sub-threshold relative to the propagator drift.
7.1 The Structural Limit (Newtonian/LTI)¶
The current 0.0002 rad (~0.01°) residual error represents the Structural Limit of the Phase 8 model. This limit arises from two primary factors:
1. LTI Assumption: The SolarSystemLaplacian is currently a Linear Time-Invariant (LTI) system. It uses a static snapshot of gravitational couplings, ignoring the fact that off-diagonal interaction strengths "breathe" with the epochs.
2. Newtonian Limit: The model follows purely Newtonian mean motions. At this level of precision, General Relativistic effects (time dilation, frame-dragging) start to bleed through as phase drift.
8. Phase 9: Dynamic Perturbations & Relativistic Friction¶
Phase 9 attacks the LTI limit (§7.1) by making the SolarSystemLaplacian breathe — off-diagonal couplings now modulate with the resonant phase difference between the bodies they connect, and a small Post-Newtonian correction is applied to the diagonal (Mercury's 43"/century relativistic precession is the canonical entry).
8.1 Breathing Couplings (Algebraic Form)¶
The static fiber coupling \(W_{ij}\) between two bodies is replaced by a phase-dependent form:
where \((n_{ij}, m_{ij})\) is the resonance ratio (e.g. \(5{:}2\) for the Jupiter–Saturn Great Conjunction) and \(\alpha\) is the modulation depth (10% in the prototype). This is the "fiber bundle" framing applied at the level of the propagator: the connection form depends on where in the bundle you are, not just on the static graph topology.
Mathematical positioning. The Phase-9 phrase "breathing Laplacian" is a project codename. The formal name is state-dependent (non-autonomous) graph Laplacian: the matrix \(L = L(\phi(t))\) depends on the current state, so \(\dot\psi = -i L(\phi)\,\psi\) is no longer a single matrix exponential but a chunk-wise integration. Equivalently, in the dynamical-systems literature this is an adaptive Kuramoto-family network with phase-difference-dependent coupling (PDDP): standard Kuramoto fixes \(K_{ij}\) and varies \(\phi\); we let \(K_{ij}\) be a smooth function of \((\phi_i, \phi_j)\). The vibrating-lattice intuition (the instantaneous spectrum of \(L(\phi)\) defines phonon-like normal modes at each instant) is correct as a static-snapshot picture; the dynamics, however, is 1st-order phase rotation rather than 2nd-order Newtonian, so "vibrating lattice" describes the snapshot but not the flow. See the ephemerides notebook §1.4 for the full positioning across the three vocabularies (spectral graph theory / dynamical systems / DNLS-on-a-graph) and the connection to nonlinear-mode bookkeeping a future Hamilton/Delaunay derivation would unlock.
8.2 ALU-Native Implementation¶
The breathing term is implemented without exiting the integer ALU. The continuous \(\cos(\cdot)\) is replaced by a precomputed integer cosine LUT (1024 × int32, Q1.14 amplitude) keyed on the top 10 bits of the residue \((n_{ij} \phi_i - m_{ij} \phi_j) \bmod 2^{32}\). The modulation is then applied as integer scaling:
— all integer ops, zero FPU calls in the inner loop. The LUT itself fits in 4 KB, three orders of magnitude smaller than the 256 KB hypervector it modulates.
8.3 Fixed-Point Frequency Discipline¶
Mean motions are stored as signed int64 in residues per day (Q-format: MODULO = 2^32 residues per revolution). The conversion is:
Range/precision properties:
- Earth's mean motion → ~11.76 M residues/day; Phobos (smallest period) → ~13.5 G residues/day.
- The pre-flight bounds check rejects |delta_t| > 6.8 × 10^8 days (~1.86 Myr) so omega * delta_t cannot saturate int64.
- Q-format underflow (frequencies that round to zero residues/day) emits a RuntimeWarning at construction time. The floor is ~13 Gyr period — never trips for real bodies, but the guard exists so the assumption is checkable.
8.4 Overflow Discipline¶
The encode path distinguishes two kinds of overflow:
- Signed int64 saturation in omega * step: a real bug. Wrapped in np.errstate(over='raise'), raised as OverflowError with a useful message.
- Unsigned uint64 wraparound on the phase accumulator: intentional — the cyclic-group reduction we want. Run under np.errstate(over='ignore') plus warnings.filterwarnings(..., 'overflow encountered') so callers who promote RuntimeWarning to error don't see spurious noise.
The combination is: pre-flight bounds check (primary defense) + scoped errstate (catches genuine overflow) + lenient cyclic-group context (preserves the modular-arithmetic semantics).
9. Conclusion¶
The RBS-HDC approach is highly feasible and provides a rigorous path to FPU-less celestial mechanics. It transforms the \(D=65536\) complex phase space into a high-dimensional "Gear Train" where each bit-serial dimension acts as a microscopic gear, maintaining the "Antikythera Spirit" in a modern silicon context. Phase 9 extends that gear train with phase-dependent fiber couplings: the breathing Laplacian is the modular-arithmetic version of a connection form, integrated as integer chunks plus a single fractional-day remainder.
Cross-pollination note. The chess-spectral Z_{640} phase-operator engine (§20.13–§20.17 of the chess notebook) is algebraically isomorphic to this BIP design at the group level. The two projects differ only in the modulus: chess uses a non-power-of-2 (\(640 = 2^7 \cdot 5\)) and pays an explicit % 640 per op; ephemerides uses \(2^{32}\) and gets cyclic-group reduction for free as uint32 overflow. The off-diagonal LUT pattern documented here transfers directly.
Next Steps: - Re-measure the \(0.0002\) rad floor with breathing couplings active across the full 26-body Laplacian (Phase 9 currently exercises only the J–S 5:2 term). - Add resonance entries for Neptune–Pluto \(3{:}2\), Io–Europa \(1{:}2\), Earth–Moon (precession), and the Trojan asteroids. - Benchmark 16-bit vs 32-bit phase discretization against JPL DE442 across a 200-year window. - Explore CORDIC-based topocentric rendering for mobile/edge displays (the cosine LUT is the first step; CORDIC handles the trickier topocentric transforms).