The aggregation law: why an index is less intermittent than its stocks, and why it cannot be why the index is rough

Stock indices are less intermittent than their constituent stocks and, at the same time, rougher: the log-volatility of an index has a smaller intermittency coefficient λ² and a larger Hurst exponent H than the log-volatility of the individual names it is built from. Two mechanisms have been proposed for the intermittency half — aggregation destroying log-correlation, or a realized-volatility measurement artifact — and a nested factor model with a rough common factor has been fit to reconcile both halves at once. This note reports a sharper result than a reconciling fit: a closed-form aggregation law with a necessity structure. Analytically, then on synthetic aggregates of known cascades, and then estimator-free on a committed 180-stock S&P-500 panel, aggregating D imperfectly-correlated H = 0 cascades into an equal-weight aggregate variance drives the intermittency down by λ²_agg = ρλ²₀ + (1−ρ)λ²₀/D_eff — so the lower intermittency of an index is aggregation — but cannot raise H: a weighted sum of H = 0 fields is again H = 0, because linear mixing preserves the correlation exponent. The roughness half is therefore not aggregation, not iid microstructure noise (excluded separately), and not constituent-level liquidity (excluded on the own panel): a genuinely rough common volatility factor is necessary, not merely one sufficient modelling choice. The structural argument is first-order in the (weak) intermittency, and the real panel prices its own second-order term: the covariance-mixture law holds estimator-free to [0.958, 1.016] at the six dyadic lags the slopes are read on and [0.894, 1.046] across all integer lags 1…32. The calibration-validated slope-ratio is ρ ≈ 0.58 (calibrated 0.587) with a moving-block-bootstrap 95% interval [0.43, 0.65]; converged to 2,000 replicates at the declared block length the interval excludes the synthetic comparator 0.65, so the honest statement is that 0.65 sits on the boundary of the interval, and the boundary moves ±0.03 with the resampling choices — and the comparator is itself cross-measurement. The single-common-cascade approximation is measurably violated: ρ(τ) falls with lag, the fingerprint of exactly the rough, shorter-scale common factor the necessity argument forces. The delta over the nested-factor fit is thus a necessity argument where the prior work supplied a sufficiency fit — established structurally and on synthetic aggregates, and consistent with, but not powered to exclude, an aggregate as rough as the index on the real panel at its 1,259-day length. Scope stated up front (the 2026-08-30 revision). This is the second version of the note. An adversarial audit of the sealed real-data record found several sentences in the first draft stronger than the measurements support; every one is restated in the direction the measurements point, and nothing is softened by omission. In particular: the real-data half is downgraded from closing the necessity gap to consistent with, but not powered to exclude, an aggregate as rough as the index at this length — because at the panel's own N = 1,259 the programme's own matched-length ladder reads the index cohort at Ĥ = 0.0245, and no H-detection positive control has been run. "Ĥ = 0 at every D (grid max 0.000)" becomes the per-D median Ĥ is 0 at every D; 3 of 41 subset fits are non-zero, max 0.050. The estimator-free band is stated both ways. The ρ interval becomes a boundary statement with its resampling movement, and ρ is reported as band-conditional (0.53–0.65 across reasonable slope bands). The comparator's cross-measurement nature and its two matched alternatives are named. The "predicted by D_eff = 180" explanation of the 0.58-vs-0.65 offset is withdrawn against the measured log-space D_eff (135 / 72.5), which the closed form can credit with at most ≈ 0.02 of the 0.07 gap. ρ is redefined as the noise-excluded ratio of common-to-total log-correlated covariance amplitude (ρ(0) = 0.285). The covariance-mixture identity is marked algebraic by construction, and the necessity argument scoped first-order with its measured second-order bound. The liquidity exclusion carries its rank statistic (Spearman −0.21, p = 0.004 — noisier names read less rough, the opposite of the artifact prediction). Two open cells are named rather than papered over, both pre-deposit requirements for any stronger claim: an H-detection positive control (inject a known-H ≈ 0.11–0.13 common factor at the measured per-stock noise variance and N = 1,259, and report the recovered Ĥ distribution), and a committed cite pack for the four external references, so the novelty delta can be re-checked offline. Files. The note itself, and — included because this is a standalone deposit and the note must be self-supporting — the Aggregation-law audit addendum (2026-08-30), which is the citable basis for every number the revision moved: the ρ lag-band sensitivity table, the 2,000-replicate block bootstrap at four block lengths (raw and calibration-inverted), the per-cell d-sweep, ρ(0), the measured log-space D_eff with the closed form's attribution of the offset, the Spearman rank statistic, the mixture band stated both ways, the factor-loading dispersion, and what the "sealed 0.65" comparator actually is. Its method section carries the seeds and the exact reproduction recipe. Both documents are additionally deposited as typeset PDFs (rendered 2026-08-30 via headless Chromium from the committed Markdown; `tools/build_pdf.py`), included alongside the Markdown as two more files in this same deposit. Supporting sealed record. The real-data measurements are sealed in Fenopan record v17, deposited separately and wired to this note. Evan Tabak Atlas (ORCID 0009-0007-7374-2338). Licensed CC BY 4.0. Deposited as a standalone record: the generating repository is private and remains so by the author's decision of 2026-08-30, so this description carries the record's own regeneration block verbatim and no repository links. Every number in the deposited files is reproducible from the deposited object and the named committed inputs.