🤯 Did You Know (click to read)
The proof does not depend on the size of the multiplier, only on its positivity.
One might suspect that very large step sizes, which sample sparsely, could tame discrepancy. Yet the theorem applies uniformly to every multiplier. Even when d is enormous, the subsequence d, 2d, 3d extends infinitely. The partial sums along that sparse path must exceed every fixed bound. Sparse sampling only slows accumulation, never halts it. Infinite length compensates for thin density. Large multipliers still unlock runaway growth.
💥 Impact (click to read)
The scale contrast is dramatic. A step size in the millions still cannot preserve bounded sums. Sparsity does not neutralize structural bias. Given infinite time, even rare sampling accumulates overwhelming deviation. Arithmetic patience guarantees explosion. Infinity defeats dilution.
This universality underscores the theorem’s strength. No region of the multiplier spectrum offers refuge. Every scale, from tiny to astronomical, participates in divergence. The result transforms arithmetic scaling into an unstoppable engine. Infinite sequences remain vulnerable across all magnifications.
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