Summary:
The verified literature converges on one central conclusion: Grinold's fundamental law IR = ICsqrt(BR) is an unrealistic limiting case, and naive breadth counting (470 names x 52 weeks = 24,440 bets) systematically overstates achievable IR because breadth means statistically independent forecasts, not separate positions. The working generalizations are (a) Clarke-de Silva-Thorley's IR = TC * IC * sqrt(BR) with BR = i'Pi^-1 i (reducing to N/(1+(N-1)rho) under constant correlation), and (b) the Qian-Hua/Ye/Ding-Martin family IR = IC / sqrt(1/(phiN) + sigma_IC^2), which caps IR at IC-bar/std(IC) as N grows — for realistic sigma_IC ~ 0.1 and N in the hundreds, the IC-volatility term dominates and universe size barely matters. Applied to the context numbers (IC 0.0125, t=2.35 over 758 weeks), the IC t-stat pins std(IC) ~ 0.146 (about 3.2x the 1/sqrt(470) sampling floor), implying a Qian-Hua gross IR ceiling of roughly (0.0125/0.146)*sqrt(52) ~ 0.62 annualized — versus a naive Grinold prediction of ~1.95 — which is fully consistent with the observed low net Sharpe and implies only ~45-50 effective independent bets per week (cross-sectional rho ~ 0.02). For signal combination, the Sorensen-Qian-Hua-Schoen/Kakushadze framework gives IC-weighted (inverse-covariance) optimal combination; adding an uncorrelated IC-0.02 signal lifts the composite IC to ~sqrt(0.0125^2 + 0.02^2) ~ 0.024, nearly twice the 0.015 obtained by improving the existing signal 20%, making the new-signal route strictly dominant. Parts (3)-(5) of the original question (Sharpe deflation, Kelly, cost/turnover math) did not survive into the verified claim set and remain to be verified separately.
Grinold's fundamental law IR = IC*sqrt(N) requires N to be the number of statistically INDEPENDENT bets, holds only 'under a host of assumptions,' and treating breadth as raw universe size (470 names) or names-times-weeks (470x52) produces IRs far above what managers achieve. Buckle identified the lack of a measurable definition of breadth as the law's key weakness, and Polakow-Gebbie state it as a lemma: sqrt(N) scaling requires independence, not mere separateness — N correlated positions do not deliver sqrt(N) diversification.
Confidence: high | Vote: 3-0 (x4 merged claims)
Evidence: Ding-Martin (J. Empirical Finance 2017): 'practitioners have sometimes taken breadth to be the number of assets in the portfolio selection universe, and this leads to IR values that are far too optimistic relative to what active managers can achieve in practice.' Buckle (J. Asset Management 2004): 'Grinold and Kahn's highly acclaimed fundamental law... has as a weak point the lack of a measurable definition of breadth.' Polakow-Gebbie Lemma 1.1: 'The square root of N in mathematical statistics implies independence amongst statistical units (here bets) rather than simply the notion of separate bets.' Merges claims 1, 6, 14, 15 (all 3-0).
The full fundamental law with constraints is IR = TC * IC_adj * sqrt(BR) (Clarke, de Silva & Thorley, Financial Analysts Journal 58(5), 48-66, Sept-Oct 2002): constrained IR scales linearly in the transfer coefficient. TC is formally TC = mu'w / (sqrt(mu'Omega^-1 mu) * sqrt(w'Omega w)) = E(R_A)/E(R_A)* — the fraction of potential active return retained after constraints (no-short, turnover limits, sector/cap neutrality), evaluated at equal active risk. The paper's premise is that such constraints are materially restrictive and prevent full conversion of forecasting skill into positions. A decile-book construction is itself a constraint, so the strategy's realized IR embeds a TC < 1 relative to mean-variance-optimal weights.
Confidence: high | Vote: 3-0 (x5 merged claims)
Evidence: CST 2002 Eq. 1: 'IR = TC x IC * N^.5'; CFA Institute technical appendix Eq. A11 gives the exact TC formula and Eq. A11-A12 the full law E(R_A) = (TC)(IC_Adj)sqrt(BR) sigma_A. Verifier independently re-derived TC = E(R_A)/E(R_A)* under the equal-active-risk condition the source imposes. Note: this is an ex-ante expected-value relation; ex-post attribution adds a realized-noise term (CST 2005). Merges claims 4, 5, 8, 11, 16 (all 3-0).
Effective breadth has a closed form: BR = i' Pi^-1 i (sum of elements of the inverse active-return correlation matrix), which equals N only when active returns are uncorrelated. Under the Elton-Gruber constant-correlation model, BR = N/(1+(N-1)*rho), strictly less than N for any positive rho. Worked magnitude (verifier-checked): N=470 with rho=0.05 gives effective breadth ~19, not 470. This is the formula for mapping '470 names' to independent bets within a period; independence across the 52 weekly rebalances is a separate assumption not covered by this formula.
Confidence: high | Vote: 3-0 (x2 merged claims)
Evidence: CFA/CST technical appendix Eq. A8: 'we define it as the sum of the elements in the inverse correlation matrix, BR = i'Pi^-1 i'; Eq. A14: 'BR = N/(1+(N-1)rho)... for positive values of rho, breadth is lower than the number of assets.' Verifier confirmed the matrix identity numerically (N=470, rho=0.05 gives 19.22 by direct inversion and by formula). Merges claims 9, 10 (both 3-0).
The corrected fundamental law (Ding working paper Eq. 42; Ding & Martin, J. Empirical Finance 2017) is IR = IC / sqrt(1/(phiN) + sigma_IC^2), with phi >= 1 (empirically ~1.5-2 for Russell universes). Special cases: sigma_IC = 0 recovers Grinold IR = ICsqrt(N); Ye (2008) is phi = 1; N -> infinity gives Qian-Hua IR = IC/sigma_IC. Because empirical factor ICs run 0.02-0.05 with sigma_IC ~ 0.1, for N in the hundreds the sigma_IC^2 term dominates 1/(phi*N) (for N=470, phi=1.75: 0.0012 vs 0.01) and IR is capped near IC/sigma_IC — an absolute upper bound no matter how many assets are added. The binding constraint at scale is IC volatility (strategy risk), not breadth.
Confidence: high | Vote: 3-0 (x3 merged claims)
Evidence: Abstract: IR is 'positively related to the mean of ICt and the number of assets N... inversely related to the volatility of ICt, and is an absolute upper bound on IR as N tends to infinity'; 'the G&K formula is an unrealistic limiting case of our single factor fundamental law.' Working paper Eq. 42 verified verbatim including phi range and all three special cases; the paper states sigma_IC is '4 to 10 times more important than 1/sqrt(N*phi)' for the universes studied. Merges claims 0, 2, 3 (all 3-0).
Qian & Hua (J. Investment Management 2(3), 2004) derive: true active risk sigma = std(IC) * sqrt(N) * sigma_model (Eq. 2), so risk-model tracking error equals realized active risk only in the knife-edge case std(IC) = 1/sqrt(N) (the pure sampling error of a correlation coefficient); and IR = IC-bar / std(IC) (Eq. 3). Empirically std(IC) bears little relationship to the 1/sqrt(N) floor — across 60 Russell 3000 strategies kappa = std(IC)sqrt(N) averaged 1.5, and their regression of ex-post active risk on kappa gives R^2 ~ 0.98. Consequence: the observed IR pins down std(IC), not nominal breadth, and realized IR is lower than ICsqrt(N) predicts in most cases studied.
Confidence: high | Vote: 3-0 (x4 merged claims)
Evidence: Verified verbatim against the JOIM PDF: 'the active risk is a product of the strategy risk, the square root of breadth, and the risk-model tracking error'; 'IR = IC-bar/std(IC) (3)'; 'the standard deviation of IC bears little relationship to this theoretical sampling error.' Eq. 3 is derived assuming mean-variance-optimal dollar- and factor-neutral long-short weights; general form Eq. 9 includes a dispersion factor (~1.01 empirically). Merges claims 17, 19, 20, 21 (all 3-0).
Worked application to the context numbers (derivation from the verified formulas, arithmetic partially verifier-checked): the IC t-stat pins std(IC) = 0.0125sqrt(758)/2.35 ~ 0.146, i.e. ~3.2x the 1/sqrt(470) ~ 0.046 sampling floor. Naive Grinold predicts annualized gross IR = 0.0125sqrt(47052) ~ 1.95; the Qian-Hua/Ding-Martin ceiling instead gives weekly IR ~ 0.0125/0.146 ~ 0.085, annualized ~ 0.62 gross — consistent with the observed ~0.1 net Sharpe after 5bps costs and tau=0.25 partial rebalancing. Inverting ICsqrt(BR_weekly) = 0.085 gives effective breadth ~46 independent bets per week (about 10% of 470 names), corresponding to constant cross-sectional correlation rho ~ 0.019 via BR = N/(1+(N-1)rho). Under the sigma_IC-dominated regime, raising N barely helps; raising IC-bar or lowering sigma_IC (signal diversification) is the only lever on gross IR.
Confidence: medium | Vote: derived from 3-0 claims
Evidence: The std(IC) inference from the IC t-stat is construction-independent (verifier for claim 19 reproduced: std(IC) ~ 0.146 vs floor 0.046, implied gross IR ~ 0.62). The effective-breadth back-out and rho estimate are this synthesis's application of the verified BR = N/(1+(N-1)rho) formula, not independently verified; the gross-to-net gap attribution to costs+TC assumes the decile book's TC and the stated cost model, which were not separately measured.
Empirical effective-breadth evidence (Polakow & Gebbie, arXiv physics/0601166, later J. Asset Management 2008): using SVD with the Kaiser-Guttman criterion (retain eigenvalues >= 1) to count the effective dimensionality of the return correlation matrix, 41 liquid JSE equities over 4.3 years of daily data had effective dimensionality <= 8 and effective breadth ~3, versus the conventional sqrt(41) ~ 6 — naive breadth overstated the IR multiplier by ~2x (and nominal N by ~5x) even before considering signal correlation. The eigenvalue count estimates the N in IR = IC*sqrt(N).
Confidence: high | Vote: 3-0 (x2 merged claims)
Evidence: Verbatim from the paper: 'we compute the effective dimensionality of the dataset as no more than 8 dimensions. This translated into an effective breadth of about 3. The conventional use of the fundamental law... would estimate breadth here at sqrt(41) = 6, twice that evidenced here.' Caveats: emerging-market universe (correlations likely higher than S&P 500); Kaiser-Guttman is a heuristic stopping rule the authors themselves hedge; the paper's breadth terminology conflates N with sqrt(N). Merges claims 12, 13 (both 3-0).
Multi-signal combination: the canonical source is Sorensen, Qian, Hua & Schoen (J. Portfolio Management 31(2), 39-45, 2004) and Ch. 7 of Qian-Hua-Sorensen's Quantitative Equity Portfolio Management, which derive the single-period composite IC of a multifactor model and the optimal factor weights maximizing IR. The vanilla Sharpe-maximizing combination is w = eta * C^-1 * E (inverse alpha covariance times expected alphas, normalized sum|w|=1), but with many alphas the sample covariance matrix is singular (rank <= number of observations after demeaning), so shrinkage or factor models are mandatory; Kakushadze-Yu show that in the large-N limit without clustering, mean-variance alpha combination under ANY factor-model covariance reduces to a weighted cross-sectional regression — w_i = eta * residual_i / sigma_i — irrespective of the factor covariance used (explicitly covering Ledoit-Wolf-style shrinkage).
Confidence: high | Vote: 3-0 (x3 merged claims)
Evidence: Book ToC confirms 'Ch. 7: Single-period composite IC of a multifactor model / Optimal alpha model: an analytical derivation.' Kakushadze-Yu verified verbatim: sample covariance 'is badly singular as the number of observations is much smaller than N'; 'irrespective of how a factor model for C_ij is built... the optimization invariably reduces to a (weighted) regression.' Merges claims 18, 22, 23 (all 3-0).
Marginal value of a new signal vs improving the old (worked from the standard composite-IC framework, not an independently verified claim): with optimally (inverse-covariance) weighted signals, two uncorrelated signals combine to IC_combined = sqrt(IC1^2 + IC2^2). Adding an uncorrelated IC-0.02 signal to the existing IC-0.0125 signal gives IC ~ sqrt(0.0125^2 + 0.02^2) ~ 0.0236 (+89%), whereas improving the existing signal's IC by 20% gives only 0.015 (+20%) — the uncorrelated-signal route dominates by roughly 4.5x in IC gain, and additionally tends to reduce sigma_IC (strategy risk), compounding the IR benefit under the Qian-Hua/Ding-Martin law. Rough targets holding std(IC) and the ~0.5-Sharpe cost drag fixed: net Sharpe 0.5 needs composite IC ~ 0.021 (~1.7x current), net Sharpe 1.0 needs ~ 0.031 (~2.4x current).
Confidence: medium | Vote: derived from 3-0 claims
Evidence: The sqrt-sum-of-squares combination for uncorrelated equal-vol signals follows directly from the verified w = C^-1 E optimal weighting (claim 22) and the Sorensen et al. composite-IC framework (claim 18), but the specific two-signal formula and the net-Sharpe IC targets were computed in synthesis, not adversarially verified. The IC targets assume cost drag is invariant to the signal change (turnover-neutral improvement) and std(IC) stays at 0.146 — both optimistic simplifications; a more autocorrelated composite would cut turnover and lower the required IC.
Coverage: the verified claim set covers only parts (1), (2), and the fundamental-law half of part (6) of the original question. Nothing on Sharpe-estimator statistics (Lo 2002, deflated Sharpe, PBO/CSCV, minimum track record, Harvey-Liu), Kelly/position sizing, Ledoit-Wolf closed form, or trading-cost math (turnover vs signal AR(1), Garleanu-Pedersen, no-trade bands, Almgren-Chriss, square-root impact) survived verification — those sections of the formula sheet remain unverified and should be researched separately before use. One claim was refuted 0-3 (the loose statement that TC is 'the correlation between forecast alphas and realized active weights' and equals 1 only when unconstrained); use the exact Eq. A11 definition instead. The worked applications (effective breadth ~46/week, rho ~0.019, IC targets of 0.021/0.031 for net Sharpe 0.5/1.0) are synthesis-level derivations from verified formulas, with arithmetic only partially verifier-checked; they assume std(IC) is stable, cost drag is turnover-invariant, and the decile-book TC is subsumed in the observed Sharpe. The Qian-Hua Eq. 3 assumes mean-variance-optimal factor-neutral long-short weights, so for a decile book it is an upper bound. The pageplace book-preview URL returned 403 on re-fetch; those claims rest on ISBN identification plus corroborating sources. Polakow-Gebbie's numbers are from an emerging market (JSE) and a heuristic eigenvalue rule — directionally instructive for S&P 500 but not transferable magnitudes. All results are model-dependent: the Ding-Martin bound requires the single-factor framework with stationary random IC_t.