Trials Factors for Closed-Form Relations Among Physical Constants

Proposed closed-form expressions for measured constants—relations such as \sin\theta_W = 3/2\pi or \Omega_\Lambda = 3/(3+\sqrt2)—are routinely assessed by percentage agreement. This paper argues that percentage agreement is the wrong statistic and supplies a cheap replacement. The Problem Percentage agreement is a poor statistic for two reasons. First, it ignores experimental precision: a 0.6\% agreement is impressive for a quantity known to 5\% and catastrophic for one known to 10^{-8}. Second, it ignores the size of the space that was searched. If a hundred equally simple expressions land inside the same interval, exhibiting one of them is not evidence. The second effect is the harder one to see from the inside, because the search is usually conducted informally over months. Candidate expressions that fail are discarded and leave no record; the survivor arrives feeling like a discovery. This paper provides a means of recovering the discarded denominator after the fact. The Method We enumerate expressions built from an alphabet of symbols \Sigma = \{1,\ \sqrt2,\ \sqrt3,\ \sqrt5,\ \sqrt7,\ \pi,\ e,\ \sqrt\pi,\ \varphi\}, \qquad \varphi = \tfrac{1+\sqrt5}{2}, together with integer coefficients a,b,c \in \{1,\dots,5\}, combined in three templates: \frac{a\,s_1}{b\,s_1 + c\,s_2}, \qquad\frac{a\,s_1}{b\,s_2}, \qquad\frac{a\,s_1}{b\,s_2\sqrt{c}},\qquad s_1,s_2 \in \Sigma . This generates 30{,}375 expressions, of which 11{,}416 take distinct values in (0,10). No expression in this family is more complex than the relations we test; several are simpler. For a measurement \mu \pm \sigma we count the distinct enumerated values lying in [\mu-\sigma,\ \mu+\sigma], and report both that count N_{\text{hit}} and the fraction f = N_{\text{hit}} / N_{\text{total}}. The Hit Fraction Is Alphabet-Independent The obvious objection is that N_{\text{hit}} depends on an arbitrary choice of alphabet. It does. The fraction does not, to useful accuracy. Varying the number of symbols from four to nine and the integer range from 3 to 7—a factor of 74 in the number of enumerated expressions—gives fractions that are stable to within tens of percent. We therefore recommend reporting f. Note also that f substantially exceeds the value expected for values distributed uniformly on (0,10). Simple expressions cluster at small values, so the density of candidates in these regions is roughly seven times the uniform expectation. Intuition calibrated on "how unlikely is it that a random number lands here" therefore understates the trials factor, in these cases by about a factor of seven. Four Relations Assessed · \Omega_\Lambda = 3/(3+\sqrt2): Predicted 0.6796 against 0.6847 \pm 0.0073—agreement at 0.7\sigma. But 117 enumerated forms fall inside the same band. A separate objection is more serious: \Omega_\Lambda is not a constant of nature—it evolves with cosmic expansion. A timeless closed form for it implies that we occupy a distinguished epoch (the cosmological coincidence problem). Verdict: weak, and conceptually problematic.· \sin\theta_{13} = 1/(3\sqrt5): Predicted 0.14907 against 0.1491 \pm 0.0023—agreement to about 0.01\%, the tightest of the four. Competing forms inside the band number 76, including 2/(4+3\pi) and 1/(4+e). Verdict: an open coincidence. This is the only one we do not refute, and the only one we would recommend pursuing—neutrino mixing angles are the natural home of discrete flavour symmetries, which supplies a mechanism to look for.· \sin^2\theta_{23} = 1/2: Maximal atmospheric mixing is reproduced by 418 enumerated forms, and is a generic prediction of many flavour-symmetry models. Verdict: no discriminating power.· \lambda_{\text{CKM}} = 1/5: Against the measured Wolfenstein parameter 0.22500 \pm 0.00067, the value 0.200 is discrepant by more than 30\sigma. Verdict: refuted. Two Further Cheap Tests · Measure agreement in standard deviations: A relation should be compared against experimental precision, not against unity. For m_\mu/m_e = 206.7682830(46), an agreement at 0.6\% is a discrepancy of order 10^6\sigma. Quoting percentages on quantities known to eight or more digits systematically misrepresents the evidence.· Check scale consistency: Relations among running couplings must specify the renormalisation scale, and the proposed value must be attainable at some scale. For \sin\theta_W = 3/2\pi, the prediction \sin^2\theta_W = 9/4\pi^2 = 0.22797 lies 81\sigma from the measured \overline{\rm MS} value at M_Z. Integrating the one-loop RGEs, we find \sin^2\theta_W attains 0.22797 at \mu \approx 47.5 GeV. The relation is therefore not excluded—but the scale it selects is unremarkable, lying between the b-quark and Z thresholds. A group-theoretic boundary value that points nowhere is a coincidence with extra steps.· A structural test: can the formula reach the value?: For a mass formula with suppression factor F(n_x,n_y,n_z) = P e^{-\beta \Sigma^2}, enumerating all integer modes up to n = 30 gives a maximum of 1.078. If reproducing some measured mass requires F = 17.9, then no mode numbers can do so—not merely the assigned ones. This test costs a few lines of code and should precede any fit. A Suggested Checklist Before publishing a proposed closed-form relation: 1. Express the discrepancy in standard deviations, not percent.2. Enumerate a comparable expression alphabet and report the hit fraction f.3. If the quantity runs, state the scale and verify the value is attainable.4. If the quantity is epoch-dependent, address why the present epoch.5. Check the attainable range of the formula before fitting.6. Count free integers and discrete choices as parameters.7. Trace each input: verify the output is not the input returned. Why This Matters The trials factor problem is pervasive in fundamental physics. Proposed closed-form relations are routinely assessed by percentage agreement, which systematically overstates the significance of numerical coincidences. This paper provides a cheap, robust replacement: enumerate the space of equally simple expressions and count how many fall inside the experimental band. The hit fraction is stable to variation in the enumeration alphabet, making it a usable alphabet-independent statistic. The method refutes three of four published relations in under an hour of computation. It does not prove that a relation is accidental—Balmer's formula had an enormous trials factor in 1885 and a mechanism in 1913—but it supplies the correct denominator, so that surprise is calibrated rather than assumed. Keywords: trials factor, closed-form relations, physical constants, numerical coincidence, model comparison, enumeration, fine-structure constant, neutrino mixing angles, cosmological constant, CKM matrix