In standard inflationary theory, the total number of e-folds of accelerated expansion in the early universe is usually freely adjusted by the shape of the inflatonpotential, lacking a rigid lock at the fundamental level. Starting from the hierarchical theory of time structure, this paper demonstrates that there exists a dimensionless scale factor in the transitions between time levels, whose numerical value isuniquely determined as r = 2 by the dimension-doubling law of the Cayley-Dicksonconstruction for Hurwitz algebras, without introducing any free parameters. Byestablishing a “causal-metric isomorphism” correspondence principle, the algebraicdoubling law of information capacity is mapped to the discrete growth of the inflationary scale factor, leading to the strict scaling relation Ne = nc ln2 betweenthe total number of e-folds Ne and the number of causal cycle closures nc. Theatomicity and integrality of causal cycle closures require nc to be a positive integer, while the number of macroscopic level transitions k = 3 in the time hierarchyfurther demands that nc be an integer multiple of 3. By combining this doublearithmetic condition with the phenomenological constraints of standard cosmologyon the horizon problem, the allowed values of nc are compressed to three discretecandidates, yielding a rigid prediction interval Ne = 54.1±2.0 centered on nc = 78.This paper further reveals that the inherent incomplete closure effect during thecritical stage when the time hierarchy approaches the classical time level necessarily leaves a characteristic quasi-sinusoidal modulation imprint in the temperatureangular power spectrum of the cosmic microwave background. The multipole position of this modulation is uniquely determined as l∗ ≈ 26 ± 2 by the numberof microscopic closures per macroscopic transition ns = 26, with a relative amplitude of about 1–2%. A statistical comparison with Planck 2018 data shows thatthe predicted interval is highly consistent with observational constraints, and themodulation position overlaps with known low-l anomaly regions. This modulationprediction, together with the scaling constraint on Ne, constitutes a dual-channelindependent test, providing a clear falsifiable path for the time hierarchy theory.