Triangular Root Invariant

$100/\sqrt{21} = 21.821789023599238\ldots$, displayed to 14 decimal places as $21.82178902359924$.

The shared mathematical quantity evaluated across these documents is the algebraic constant $1/\sqrt{21} = 1/\sqrt{T(6)}$, where $T(n) = n(n+1)/2$ is the $n$-th triangular number (triangular number).

Outside of exact analytical identities, domain-specific evaluations represent direct numerical parameter substitutions into standard governing equations rather than universal physical invariants or empirical coincidences. Detailed derivations and proofs are cataloged in Mathematical Derivations.

Abstract

Single final formula: $$\Lambda(n) = \sqrt{\frac{2}{n(n+1)}} = \frac{1}{\sqrt{T(n)}}, \quad \Lambda(6) = \frac{1}{\sqrt{21}} = 0.21821789023599238\ldots,$$ displayed to 16 decimals as $0.2182178902359924$.

$100 \cdot \Lambda(6) = 21.82178902359924$ is the decimal display of $1/\sqrt{21}$. Domain rows below are substitutions of $21$, $42$, $126$, $20/21$, or $1/\sqrt{21}$ into standard formulas, except where an identity is stated.

Core Contributions:

$\Lambda(6) = 1/\sqrt{21}$ up to 15 or more digits of precision.
$n=6$ is the unique positive integer index with $T(n) = 21$. The identity $\Lambda(n) = 1/\sqrt{T(n)}$ holds for every positive integer $n$.
The shared value is the algebraic quantity $1/\sqrt{21}$. Agreement after a parameter is set to $21$, $42$, $126$, $20/21$, or $1/\sqrt{21}$ is a substitution, not evidence of a causal link between domains.
The generic formula $\Lambda(n) = \sqrt{2/(n(n+1))}$ is derived without hardcoded numeric constants such as $21$ or $0.2182$.

1. Single Unified Formula

1.1. Definition and Evaluation

$$\Lambda(n) = \sqrt{\frac{2}{n(n+1)}} = \frac{1}{\sqrt{T(n)}}, \quad n \in \mathbb{N}_{>0}$$

where $T(n) = n(n+1)/2$ is the $n$-th triangular number (triangular number).

Evaluation for $n = 6$: $$\Lambda(6) = \sqrt{\frac{2}{6 \cdot 7}} = \sqrt{\frac{1}{21}} = \frac{1}{\sqrt{21}} = 0.21821789023599238\ldots$$

1.2. Analytical Properties

The function satisfies partial fraction telescoping $\Lambda(n)^{2} = 2(1/n - 1/(n+1))$ (partial fraction decomposition), with $\sum_{n=1}^{\infty} \Lambda(n)^{2} = 2$ and asymptotic linear sum $\sum_{n=1}^{N} \Lambda(n) \sim \sqrt{2}\ln N$ (natural logarithm). Full derivations are in Math-A.md §1.2.

1.3. Uniqueness of $n = 6$

Theorem: $100 \cdot \Lambda(n) = 21.8218\ldots$ for $n \in \mathbb{N}_{>0}$ if and only if $n = 6$. The proof reduces to $n^2 + n - 42 = 0$ with unique positive root $n = 6$ (quadratic formula). Full derivation in Math-A.md §1.1.

Generalization: $\Lambda(n) = 1/\sqrt{k}$ for positive integers $n$ and $k$ if and only if $k = T(n)$. The value $k = 21$ occurs uniquely at $n = 6$.

1.4. Continuous Beta Function Generalization

$$\boxed{\Lambda(x) = \sqrt{2,B(x, 2)} = \sqrt{\frac{2,\Gamma(x)}{\Gamma(x+2)}}, \quad x > 0}$$

where $B(x, 2)$ denotes the Euler Beta function (Beta function). For integer $x = n$, this reduces to $\sqrt{2/(n(n+1))}$. The logarithmic derivative is $\frac{d}{dx}\ln\Lambda(x) = -\frac{1}{2x} - \frac{1}{2(x+1)}$, and the generating function of squares is $G(z) = 2 + 2(1-z)\ln(1-z)/z$. Full derivations are in Math-A.md §1.3 and Math-A.md §1.4.

1.5. Equivalent Representations (Master Summary)

Rows 1--5 are identities for every positive integer $n$. Rows 6--14 equal $\Lambda(6)$ only, and only after the index or parameter named in each row is fixed:

#	Representation	Mathematical Domain
1	Algebraic form $\sqrt{2/(n(n+1))}$	Pure mathematics
2	Triangular form $1/\sqrt{T(n)}$	Number theory
3	Beta function $\sqrt{2,B(n,2)}$	Special functions
4	Gamma function $\sqrt{2,\Gamma(n)/\Gamma(n+2)}$	Special functions
5	Partial fraction $\sqrt{2(1/n - 1/(n+1))}$	Real analysis
6	Polygonal number $1/\sqrt{P(d, n)}$ for $(d, n) \in {(3, 6), (8, 3), (21, 2)}$	Polygonal numbers
7	Fibonacci number $1/\sqrt{F_{8}}$	Recursive sequences
8	$21$ is the second term of OEIS A046183, not a representation of $\Lambda(n)$	Diophantine geometry
9	Coefficient of variation $1/\sqrt{21}$ at the parameter fixed in Section 2.2	Exponential families
10	Stirling, Motzkin, and Standard Young Tableaux counts $1/\sqrt{M_{5}}, 1/\sqrt{c(7,6)}$	Combinatorics
11	Quadratic Gauss sum $1/\lvert g(\chi_{21})\rvert$	Analytic number theory
12	Class number $h(\mathbb{Q}(\sqrt{21})) = 1$, which makes row 13 an identity	Algebraic number theory
13	Dirichlet L-function $L(1, \chi_{21})/(2\log \varepsilon_{21})$	Analytic number theory
14	Cusp forms dimension $1/\sqrt{\dim S_{24}(\Gamma_{0}(6))}$	Modular forms

2. Key Domain Manifestations

2.1. Machine Learning and Artificial Intelligence

Domain	Formula	Instantiation
Transformer Attention (Vaswani et al. 2017)	$1/\sqrt{d_{k}}$	$d_{k} = 21$: Exact $\Lambda(6)$
Xavier Initialization (Glorot and Bengio 2010)	Uniform on $[-\sqrt{6/(n_{in}+n_{out})}, \sqrt{6/(n_{in}+n_{out})}]$	$n_{in}+n_{out} = 126$: the paper's bound endpoint is $\Lambda(6)$. $\sqrt{1/n_{in}}$ is a later shorthand
He Initialization (ReLU) (He et al. 2015)	Gaussian standard deviation $\sqrt{2/n_{in}}$	$n_{in} = 42$: $\Lambda(6)$
Diffusion Noise Schedule (Ho, Jain, and Abbeel 2020)	$\sqrt{1 - \bar{\alpha}_{t}}$	$\bar{\alpha}_{t} = 20/21$ is a chosen schedule value, not a value derived by that paper
Echo State Network (Jaeger 2001 and corrected report)	No required input scale	Jaeger's example sets input weights to $\pm 1$. $1/\sqrt{N}$ is not a formula in that report
Sample-mean standard error	$\text{SE}(\bar{X}) = \sigma/\sqrt{n}$	For independent zero-mean unit-variance variables and $n = 21$: $\Lambda(6)$. This is not the definition of Rademacher complexity

2.2. Statistics and Probability

Coefficient of Variation in Natural Exponential Families with Quadratic Variance Function (Morris 1982): The coefficient of variation equals $1/\sqrt{21}$ for the five parameter choices below. Morris classifies six quadratic-variance families: normal, Poisson, gamma, binomial, negative binomial, and hyperbolic secant. The negative-binomial and hyperbolic-secant families are not included, and the binomial equality also fixes $p = 1/2$:

Distribution	Shape Parameter	Coefficient of Variation
Poisson $\text{Poisson}(\lambda)$	$\lambda = 21$	$1/\sqrt{21}$
Gamma $\text{Gamma}(\alpha, 1)$	$\alpha = 21$	$1/\sqrt{21}$
Chi-squared $\text{Chi-squared}(k)$	$k = 42$	$1/\sqrt{21}$
Binomial $\text{Binomial}(n, p=0.5)$	$n = 21$	$1/\sqrt{21}$
Erlang $\text{Erlang}(k, \lambda)$	$k = 21$	$1/\sqrt{21}$

Skewness of Poisson(21): $\gamma_{1}[\text{Poisson}(21)] = 1/\sqrt{21}$. For every Poisson parameter, coefficient of variation equals skewness. The value $21$ selects the common magnitude, not a unique distribution.

Fisher information: For one observation from $\text{Poisson}(\lambda)$, the Fisher information is $I(\lambda) = 1/\lambda$ (Fisher information). Therefore

$$\sqrt{I(21)} = 1/\sqrt{21} = \Lambda(6).$$

The quantity $\sqrt{I(\lambda)},d\lambda$ is the line element, not the metric component itself.

Jeffreys Prior: $\pi(\lambda) \propto 1/\sqrt{\lambda}$ (Jeffreys prior). The density is fixed only up to normalization, so $\pi(21) \propto \Lambda(6)$. It is not the equality $\pi(21) = \Lambda(6)$.

Standard error of the mean: For $n$ independent observations with population standard deviation $\sigma$, the standard error page gives $\sigma_{\bar x}=\sigma/\sqrt{n}$. Therefore $\text{SE}=1/\sqrt{n}$ requires $\sigma=1$. At $n=21$ that value is $\Lambda(6)$.

2.3. Physics

Domain	Formula	Instantiation
Harmonic Oscillation Period (harmonic oscillator)	$T = 2\pi/\omega$	$\omega=\sqrt{k/m}$. The value $\omega^{2}=21$ is a chosen stiffness-to-mass ratio
Schwarzschild time-dilation factor (gravitational time dilation)	$t_{0}=t_{f}\sqrt{1-r_{s}/r}$	Static exterior clock at $r/r_{s}=21/20$. $\Phi=-10/21$ is algebraic notation, not the Newtonian-limit potential
Reciprocal of the Lorentz factor (Lorentz factor)	$1/\gamma=\sqrt{1 - v^{2}/c^{2}}$	$v/c = \sqrt{20/21}$ is a chosen speed. The page defines $\gamma$ as the reciprocal
Upper-mantle viscosity range (Earth's mantle)	$10^{19}$ to $10^{24}$ Pa·s	$10^{21}$ is one conventional scale inside the reported upper-mantle range, not an exact value
Molecular Vibrational Modes (molecular vibration)	$3N - 6$	Nonlinear molecule with $N = 9$: $21$ vibrational modes
Bernoulli number in the zeta formula (Bernoulli number)	$B_{6} = +1/42$ in both standard sign conventions	$1/42 = 1/(2T(6))$. The two conventions differ at $B_{1}$, not at $B_{6}$

2.4. Engineering and Computer Science

Domain	Formula	Instantiation
Quality Factor (Q factor)	$1/(2\zeta)$	$\zeta = 1/\sqrt{21}$: $Q = \sqrt{21}/2$
Peak Percentage Overshoot (overshoot)	$100\exp(-\pi \zeta/\sqrt{1-\zeta^{2}})$	Underdamped second-order step response. $\zeta = 1/\sqrt{21}$ gives $49.535%$, displayed as $49.54%$
Fixed-length alphabet ceiling (prefix code)	$\lceil \log_{2} N \rceil$	A $k$-bit fixed-length code encodes at most $2^{k}$ symbols. Therefore $N = 21$ needs $5$ bits. This is not the Huffman weighted path length
Hash Table Load Factor (hash table)	$\alpha = n/m$	Choosing $m = 21$ buckets and $n = \sqrt{21}$ stored elements gives $\alpha = \Lambda(6)$. A real table has an integer element count

2.5. Life Sciences and Medicine

Domain	Formula	Instantiation
First-Order Reaction Rate (rate equation)	$k = \ln 2 / t_{1/2}$	Half-life $t_{1/2} = 1/\sqrt{21}$ is a chosen time
Clinical Trial Sample Size (Cohen, Statistical Power Analysis, 2nd ed.)	$n = 2(z_{\alpha}+z_{\beta})^{2}/d^{2}$ per group	For $d = 1/\sqrt{21}$ this is $42(z_{\alpha}+z_{\beta})^{2}$ per group, not a total sample size
Michaelis-Menten Kinetics (Michaelis and Menten 1913, English translation)	$v/V_{\max} = [S]/(K_{m}+[S])$	$[S]/K_{m} = 1/\sqrt{21}$ gives $v/V_{\max} = 1/(\sqrt{21}+1)$, not $\Lambda(6)$
Absolute Bioavailability (bioavailability)	Dose-normalized $\mathrm{AUC}_{\mathrm{extravascular}}/\mathrm{AUC}_{\mathrm{intravenous}}$	Setting that fraction to $1/\sqrt{21}$ gives $21.82%$. It is not a measured drug value
Bazett rate correction (Bazett 1920 and QT interval)	$\text{QTc} = \text{QT}/\sqrt{\text{RR}/1,\text{s}}$	Setting $\text{RR} = 1/\sqrt{21}$ seconds is an input choice, not a measured interval

2.6. Economics, Geology, Astronomy, and Epidemiology

Domain	Formula	Instantiation
Herd Immunity Threshold (herd immunity)	$1 - 1/R_{0}$	Homogeneous mixing, solid immunity, no immune escape, and no nonhuman vector. $R_{0} = 21$ then gives $20/21$
First-Price Auction (first-price sealed-bid auction)	$(n-1)/n \cdot v$	The page states this symmetric BNE for i.i.d. uniform $[0,1]$ valuations. Its general symmetric BNE is $E[y_i \mid y_i<v_i]$, and $n = 21$ gives bid ratio $20/21$
Cournot Oligopoly Equilibrium (Cournot 1838 and linear case Marker)	$(a-c)/((n+1)b)$	Identical firms, inverse demand $p=a-bQ$, and marginal cost $c$. At $n = 20$, the firm output ratio $= 1/21$
Kelly Betting Fraction (Kelly criterion)	$p/l-q/g$	Even money $g=l=1$ and edge $2p-1=1/\sqrt{21}$: the page's fraction is $2p-1$, so $p \approx 0.609109$
Kepler third law in solar units (Kepler's laws)	$T^{2} = a^{3}$ for a solar-mass primary	Period $T = 21$ years gives $a = 21^{2/3} \approx 7.61$ AU. This is not the mass-independent SI form
Gutenberg-Richter law (Gutenberg-Richter law)	$\log_{10} N = a - bM$	$a = 5$, $b = 1$, and $N = 1/\sqrt{21}$ are chosen inputs. They give $M \approx 5.66$
Transmission-line reflection (reflection coefficient)	$\Gamma=(Z_{L}-Z_{0})/(Z_{L}+Z_{0})$	Choosing real $\Gamma=1/\sqrt{21}$ gives $Z_{L}/Z_{0}=(11+\sqrt{21})/10 \approx 1.558258$. The page's acoustic $R$ is an intensity ratio, not this formula
Adiabatic Index (heat capacity ratio)	$1 + 2/f$	Choosing effective degrees of freedom $f = 2\sqrt{21}$ gives $\gamma = 1+\Lambda(6)$. Equipartition uses integer $f$
Effective neutrino number	$N_{\mathrm{eff}} = 3.044$, with numerical and mixing error at most $0.0005$ (Akita and Yamaguchi 2020)	$\ln(21) \approx 3.04452$ is a numerical proximity, not the definition of $N_{\mathrm{eff}}$

3. Scope of the Domain Catalog

Scope: Shared algebraic form does not establish a physical correlation. A row is an identity only when both sides are equal without a free parameter being set to $21$, $42$, $126$, $20/21$, or $1/\sqrt{21}$.

4. Combinatorial and Number-Theoretic Structure

The integer $21$ satisfies ten combinatorial evaluations: $T(6)$, $\binom{7}{2}$, octagonal $P(8,3)$, icosihenagonal $P(21,2)$, Fibonacci $F_8$, Stirling numbers $S(7,6)$ and $c(7,6)$, Motzkin number $M_5$, and two Young tableaux counts $f^{(3,3,1)}$ and $f^{(3,2,2)}$. The quintuple identity (triangular, semiprime, Carmichael $\lambda(21)=6$, multiplicative order $\text{ord}_{21}(2)=6$, and Fibonacci $F_8$) is unique among integers below $5000$. Luo's theorem (1989) establishes that $21 = F_8 = T(6)$ is one of only five Fibonacci-triangular equalities. The continued fraction of $\sqrt{21}$ has period $6$, matching the index $n=6$. Ramsey number $R(3,3)=6$ completes the sixfold self-reference. Detailed derivations and proofs are in Math-A.md §2.

5. Analytic Number Theory

The quadratic Gauss sum satisfies $|g(\chi_{21})|=\sqrt{21}$ (Gauss sum), yielding $\Lambda(6)=1/|g(\chi_{21})|$. The class number $h(\mathbb{Q}(\sqrt{21}))=1$ (class-number-one fields) makes the Dirichlet class number formula an identity (class number formula): $\Lambda(6)=L(1,\chi_{21})/(2\log\varepsilon_{21})$ where $\varepsilon_{21}=(5+\sqrt{21})/2$. The cusp form dimension $\dim S_{24}(\Gamma_0(6))=21$ (Stein, Proposition 6.1) gives $\Lambda(6)=1/\sqrt{\dim S_{24}}$. The Euler zeta ratio $\zeta(6)/\zeta(4)=2\pi^2/21$ (particular values of the Riemann zeta function) yields $\Lambda(6)=\sqrt{\zeta(6)/(2\pi^2\zeta(4))}$. These five analytic formulas (elementary algebra, Gauss sums, Dirichlet class numbers, modular forms, and zeta identities) all evaluate to $\Lambda(6)=1/\sqrt{21}$. Detailed derivations and proofs are in Math-A.md §3.

6. Reproducibility

6.1. Final Expression

$$\boxed{\Lambda(x) = \sqrt{2,B(x,2)} = \sqrt{\frac{2,\Gamma(x)}{\Gamma(x+2)}} = \frac{\sqrt{2}}{\sqrt{x(x+1)}}}, \quad \Lambda(6) = \frac{1}{\sqrt{21}}$$

7. Sources

Each link was checked against the sentence it supports. A substitution row uses the cited formula, not a result discovered by that source.

Number theory

Triangular numbers: Wikipedia
Binomial coefficient: Wikipedia
Polygonal formula $P(s,n)=((s-2)n^{2}-(s-4)n)/2$: Wikipedia
Fibonacci indexing with $F_{0}=0$, including $F_{8}=21$: Wikipedia
One-step Stirling identities $S(n,n-1)=c(n,n-1)=\binom{n}{2}$: first kind, second kind
Motzkin numbers, including $M_{5}=21$: Wikipedia
Hook-length formula $f^{\lambda}=n!/\prod h_{\lambda}(i,j)$: Wikipedia
Semiprime: Wikipedia
Multiplicative order: Wikipedia
Carmichael function, including the table value $\lambda(21)=6$: Wikipedia and definition DOI 10.1090/S0002-9904-1910-01892-9
Continued-fraction periodicity of non-square roots: Wikipedia
Pell fundamental solution and even-period convergent rule: Wikipedia
Euler Beta function: Wikipedia
Fibonacci-triangular theorem: Luo Ming, Fibonacci Quarterly 27.2 (1989), PDF and sequence OEIS A039595
Octagonal-triangular sequence: OEIS A046183
Ramsey existence theorem: DOI 10.1112/plms/s2-30.1.264. Evaluation $R(3,3)=6$: Wikipedia
Real primitive character $\chi(n)=(21/n)$, conductor $|D|=21$: Kronecker symbol. Jacobi factorization: Jacobi symbol
Primitive Gauss-sum modulus and coprime product phase: Gauss sum. Conductor definition: Dirichlet character
Class number one and its implication: Wikipedia
Dirichlet class-number formula: Wikipedia
Fundamental unit: Wikipedia
Cusp-form dimension, Proposition 6.1: Stein, citing Diamond and Shurman
Even zeta values: Wikipedia
Bernoulli sign convention, including $B_{6}=+1/42$: Wikipedia
Zeta denominators: OEIS A002432

Statistics and machine learning

Standard error of the mean, $\sigma/\sqrt{n}$ for independent observations: Wikipedia
Natural exponential families with quadratic variance: Morris, Annals of Statistics 10 (1982)
Distribution moments used above: Poisson, gamma, chi-squared, binomial, Erlang
Jeffreys prior: Wikipedia
Scaled dot-product attention: Vaswani et al., arXiv:1706.03762
Xavier uniform initialization: Glorot and Bengio, PMLR 9
ReLU initialization, standard deviation $\sqrt{2/n_{l}}$: He et al., arXiv:1502.01852
Diffusion forward-process standard deviation: Ho, Jain, and Abbeel, arXiv:2006.11239
Two-group sample-size formula: Cohen, Statistical Power Analysis, 2nd ed., DOI

Physics, biology, and applied formulas

Static exterior Schwarzschild factor $t_{0}=t_{f}\sqrt{1-r_{s}/r}$: gravitational time dilation. Redshift from infinity is the reciprocal: gravitational redshift
Reciprocal of the Lorentz factor, $\gamma=1/\sqrt{1-v^{2}/c^{2}}$: Wikipedia
Harmonic oscillator period: Wikipedia
Nonlinear molecular vibrations: Wikipedia
Mantle viscosity range: Wikipedia
Calculated baseline $N_{\mathrm{eff}}=3.044$, with numerical and mixing error at most $0.0005$: Akita and Yamaguchi, arXiv:2005.07047
Michaelis-Menten equation, English translation of the 1913 paper: DOI 10.1021/bi201284u
Bazett correction: Bazett 1920, DOI 10.1111/j.1542-474X.1997.tb00325.x. Summary Wikipedia
First-order half-life: Wikipedia
Q factor: Wikipedia. Second-order percentage overshoot: overshoot
Fixed-length alphabet ceiling: prefix code. Huffman weighted path length is a separate claim: Huffman coding
Herd-immunity threshold and its mixing assumptions: Wikipedia
First-price auction: $(n-1)/n \cdot v$ is the page's symmetric BNE for i.i.d. uniform $[0,1]$ valuations. Its general symmetric BNE is $E[y_i \mid y_i<v_i]$: Wikipedia
Kelly criterion: the binary formula $f=p/l-q/g$, and $f=2p-1$ for even money $g=l=1$: Wikipedia. Kelly 1956 gives the same even-money fraction with swapped labels: PDF
Cournot oligopoly: Cournot 1838. The linear identical-firm output $(a-c)/((n+1)b)$: Marker
Echo-state networks: Jaeger 2001, Fraunhofer record. The corrected report sets its example input weights to $\pm 1$ and does not prescribe $1/\sqrt{N}$: PDF
Hash-table load factor $\alpha = n/m$: Wikipedia
Fisher information: Wikipedia
Reflection coefficient $(Z_L-Z_0)/(Z_L+Z_0)$: Wikipedia
Ideal-gas adiabatic index $1+2/f$: Wikipedia
Absolute bioavailability: Wikipedia
AR(1) stationary variance: Wikipedia
Kepler's third law: Wikipedia
Gutenberg-Richter relation: Wikipedia

Cite

If you find Triangular Root Invariant helpful in your research cite simply as:

@misc{triangular-root-invariant,
  author = {NeaByteLab},
  title = {Triangular Root Invariant: The Inverse Root of Twenty-One},
  year = {2026},
  publisher = {GitHub},
  url = {https://github.com/NeaByteLab/Triangular-Root-Invariant}
}

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