The Spacetime Diagonal: Overcoming the Von Neumann Bottleneck in 10-GHz Arbitrary-Precision Integer Analysis
Description
1. The Mathematical Invariant: From Rings to Ceilings
if state >= N then state - N) requires wide multi-word multiplexer trees. These trees stall computation, creating a severe routing bottleneck across the silicon die.2. Silicon Realization: The 2D Spacetime Matrix
Physical Time (Clock Cycles)
│
▼ [Chunk 0 / LSB] [Chunk 1] [Chunk 2] [Chunk 16383 / MSB]
T = 1 ────── W0 ────────── IDLE ────────── IDLE ────────────── IDLE
T = 2 ────── W1 ────────── W0 ─────────── IDLE ────────────── IDLE
T = 3 ────── W2 ────────── W1 ─────────── W0 ──────────────── IDLE
│ . . . .
│ . . . .
T = 16384 ── W16383 ────── W16382 ────── W16381 ────────────── W0 ──► [ORACLE]
T = 16385 ── W16384 ────── W16383 ────── W16382 ────────────── W1 ──► [ORACLE]
Note: TheIDLEstates represent a strictly temporary $1.6 \ \mu\text{s}$ priming phase (for a 1-Megabit state vector). At $T = 16385$ and beyond, matrix utilization is exactly 100%, with no dead cycles, bubbles, or pipeline stalls.
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0ps – 15ps (Register Launch): Global clock edge hits. The local chunk registerstate_reglaunches. Simultaneously, wavefront attributes (carry_in,cmp_gt_in,cmp_eq_in) exit the boundaries of Chunk $i-1$.
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15ps – 55ps (Local Accumulation): A local 64-bit parallel-prefix Kogge-Stone adder computessum = state_reg + q_step[i] + carry_in. Becauseq_stepis a static parameter loaded only once per search execution, intermediate prefix logic levels are deeply optimized during physical synthesis.
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55ps – 75ps (Watermark Comparison): A 64-bit localized magnitude tree evaluates the newly stabilizedsum[63:0]against the static, hardwired watermark slice $N[i]$.
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75ps – 85ps (Wavefront Override Logic): High-speed multiplexers resolve the pipelined magnitude tracking flags. If a higher-order chunk evaluates as strictly greater or less than the local modulus slice, it asserts immediate veto power over the traveling tracking flags. If it evaluates as strictly equal, it acts as a transparent wire, seamlessly passing the lower chunk's verdict forward.
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85ps – 100ps (Setup & Hold Settle): The local state register captures the newly computed sum. The resolved tracking wires arrive at the input boundary of Chunk $i+1$ with ample setup time to spare.
3. Asymptotic Throughput and Scientific Significance
4. Conclusion
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Additional details
Dates
- Copyrighted
-
2026-09-19