Constants From Balanced Ternary

Abstract

The companion paper Balanced Ternary by Necessity established that \(\{-1,\, 0,\, +1\}\) is the unique minimal integer-valued state space capable of intrinsically representing a directed transition. This paper asks the next question: what mathematical structure appears when that alphabet is subjected to successive completion demands beyond its primitive one-step content?

Starting from \(\{-1, 0, +1\}\), we construct a derivation ladder. Some constants arise once independent generators, metric comparison and symmetry-preserving operators are admitted. Others appear only when the substrate is examined through increasingly strong analytical completions, including recurrence, refinement, rotation and summation. The claim of this paper is therefore ordered rather than absolute: the constants are not all primitive contents of the ternary alphabet, but they do appear in a strict sequence once each additional structural demand is made explicit.

No physical assumption is made at any step. Mathematical structure is introduced only when required, and each constant enters at the earliest stage permitted by the available machinery. The result is a disciplined map from a minimal discrete substrate to the hierarchy of irrational and transcendental constants naturally exposed by its admissible completions.

Reading Guide

This paper is the third in a sequence.

The philosophical argument for why anything exists at all is developed in On the Necessity of Existence. The formal proof that \(\{-1, 0, +1\}\) is the unique minimal state space for directed transitions is given in Balanced Ternary by Necessity.

This paper begins where that proof ends. It takes the established alphabet and asks what mathematical structure appears when the substrate is carried through successive structural and analytical completion demands.

Each section derives one or more constants. The derivation ladder is sequential: each result depends only on structure or completion machinery already established.

This paper is deliberately silent on physical interpretation. While the reader may notice correspondences, the paper does not comment on them.

Part I: The Geometry of the Alphabet

The Second Axis And The Quarter-Turn

Starting inventory: \(\{-1, 0, +1\}\) on a single axis with generator \(e\) and transition operators \(\tau_e\) and \(\tau_e^{-1}\).

The question: Can a second directed transition exist that is independent of the first?

The requirement: Let \(f\) denote a second generator with the same single-step contract as \(e\), but not reducible to \(\pm e\) by rescaling or relabelling. Independence alone yields a copy of \(\mathbb{Z}^2\), but this by itself does not distinguish a plane from an arbitrary product of two lines.

To obtain an intrinsic notion of right angle without importing physical geometry, require a structure-preserving quarter-turn operator \(J\) acting on the span of the generators such that

\[ J(e) = f, \quad J(f) = -e \]

This implies \(J^2(e) = -e\) and \(J^2(f) = -f\), hence \(J^2 = -1\) on the generator subspace.

The result: A second independent axis by itself does not force an operator whose square is negation. What forces it is the additional demand that the two-generator extension admit an intrinsic, structure-preserving quarter-turn. Under that demand, \(\sqrt{-1}\) enters as a purely structural object: not as a number imported from elsewhere, but as the unique algebraic witness of a quarter-turn that cannot be realised on a single signed line. With this operator, the natural integer-coordinate plane is \(\mathbb{Z}[i]\), where \(i\) denotes the action of \(J\) and \(i^2 = -1\).

\[ \boxed{\quad i = \sqrt{-1} \quad} \]

The Diagonal And \(\sqrt{2}\)

The question: What is the distance of the diagonal step \(e + f\) on the plane?

The requirement: To answer this, impose the minimal structural requirement of an intrinsic quadratic invariant preserved by the quarter-turn \(J\). Assume there exists a function \(\|\cdot\|\) on the integer span of \(\{e, f\}\) such that axis symmetry, quarter-turn invariance, and compatibility with additivity and negation hold.

These conditions determine, up to scale, the Euclidean form

\[ \|ae + bf\|^2 \;\propto\; a^2 + b^2 \]

Fix units such that \(\|e\| = \|f\| = 1\). Then

\[ \begin{aligned} \|e + f\|^2 &= \|e\|^2 + \|f\|^2 \\ &= 1 + 1 = 2 \\ \|e + f\| &= \sqrt{2}\,\|e\| \end{aligned} \]

The result: \(\sqrt{2}\) is the first unavoidable irrational that appears once a plane supports two independent unit steps together with an intrinsic quarter-turn preserving a quadratic invariant.

\[ \boxed{\quad \sqrt{2} = 1.41421356\ldots \quad} \]

The Third Axis And \(\sqrt{3}\)

The question: Can a third directed transition exist that is independent of both \(e\) and \(f\)?

The requirement: Introduce a third generator \(g\), independent of both \(e\) and \(f\), with the same single-step contract. Require that each coordinate plane admits a structure-preserving quarter-turn symmetry, mirroring the two-dimensional case.

This extends the reachability set from \(\mathbb{Z}^2\) to \(\mathbb{Z}^3\). The quarter-turn requirement in each coordinate plane yields three operators:

\[ J_{ef}^2 = J_{fg}^2 = J_{ge}^2 = -1 \]

The unit-cube diagonal step \(e + f + g\) now has

\[ \begin{aligned} \|e + f + g\|^2 &= \|e\|^2 + \|f\|^2 + \|g\|^2 \\ &= 3 \\ \|e + f + g\| &= \sqrt{3}\,\|e\| \end{aligned} \]

The result: The unit-cube diagonal of the minimal three-axis integer-coordinate space has length \(\sqrt{3}\). The paper also notes that the quarter-turn operators do not commute under composition in three dimensions; the smallest consistent algebra accommodating the three pairwise plane-rotation quarter-turn generators is quaternionic in character.

\[ \boxed{\quad \sqrt{3} = 1.73205080\ldots \quad} \]

The First Non-Trivial Integer-Coordinate Distance And \(\sqrt{5}\)

The question: What integer-coordinate distances exist beyond the axis, the unit-square diagonal, and the unit-cube diagonal?

The requirement: The integer-coordinate space \(\mathbb{Z}^3\) contains all integer-coordinate vectors. The squared distances from the origin are of the form \(a^2 + b^2 + c^2\) for integers \(a, b, c\).

The first few distinct squared distances are

\[ 1, \quad 2, \quad 3, \quad 4, \quad 5, \quad \ldots \]

Squared distance \(4\) is simply \(2^2\): a two-step axis move. It is the first composite. Squared distance \(5\) is the first that introduces a genuinely new irrational. It is realised by any integer vector of the form \((1, 2, 0)\) or its permutations and sign changes.

\[ \|(e + 2f)\|^2 = 1 + 4 = 5 \quad\Rightarrow\quad \|e + 2f\| = \sqrt{5}\,\|e\| \]

The result: \(\sqrt{5}\) is the shortest integer-coordinate distance that cannot be reduced to a multiple of the axis length, the unit-square diagonal, or the unit-cube diagonal.

\[ \boxed{\quad \sqrt{5} = 2.23606797\ldots \quad} \]

Part II: The Self-Referential Constants

The Golden Ratio \(\varphi\)

The question: What appears when the integer sequence is subjected to a simple, self-referential additive recurrence?

The derivation: The integers \(\mathbb{Z}\) support integer sequences and therefore many possible recurrences. Not all of them are equally informative. Constant, periodic or purely sign-flipping recurrences remain too degenerate to drive a genuine growth law. To obtain a non-trivial self-referential rule using only addition of previously available terms, we choose the recurrence

\[ a(n) = a(n-1) + a(n-2) \]

This is the Fibonacci recurrence. It is not claimed here that this choice is the only recurrence the integers permit. It is claimed that it is a highly natural and minimal additive recurrence once one asks for non-degenerate growth built from prior terms alone.

Its characteristic equation is

\[ x^2 = x + 1 \quad\Rightarrow\quad x^2 - x - 1 = 0 \quad\Rightarrow\quad x = \frac{1 \pm \sqrt{5}}{2} \]

The positive root is the golden ratio:

\[ \varphi = \frac{1 + \sqrt{5}}{2} \]

The result: \(\varphi\) is not imported. It appears as the growth rate of a chosen but highly natural non-trivial additive recurrence on the integers, and it is built from \(\sqrt{5}\), which was itself exposed by the integer-coordinate geometry.

\[ \boxed{\quad \varphi = \frac{1 + \sqrt{5}}{2} = 1.61803398\ldots \quad} \]

The Natural Exponential \(e\)

The question: What appears once the unit step is examined through arbitrary refinement and compounded growth?

The derivation: The substrate has a unit transition of magnitude \(1\). By itself, this does not yet contain continuous growth. To reach that regime, an additional completion demand must be admitted: the unit step is allowed to be refined into \(n\) equal sub-steps, each of magnitude \(1/n\), while preserving the same total action.

If growth compounds multiplicatively at each refined sub-step, the accumulated factor after \(n\) such steps is

\[ \left(1 + \frac{1}{n}\right)^n \]

As \(n \to \infty\), this converges:

\[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \]

The point is not that the primitive ternary alphabet is already “doing” infinite subdivision. The point is that, once refinement is admitted as an analytical completion of the unit step, this limit is no longer optional. It is the unique invariant associated with maximal compounding under arbitrary equal subdivision.

The result: \(e\) is the natural base of continuous growth exposed when a discrete unit transition is carried through the completion demand of arbitrary refinement.

\[ \boxed{\quad e = 2.71828182\ldots \quad} \]

The Half-Period \(\pi\)

The question: What constant appears once the previously established objects \(i\) and \(e\) are carried through a rotation-based analytical completion?

The derivation: We have established the complex plane \(\mathbb{Z}[i]\) and the exponential function \(e^x\). By themselves, these do not yet compel a circle. To reach that regime, an additional completion demand must be admitted: imaginary exponents are allowed, so that the exponential is examined along the purely imaginary direction \(it\).

Under that demand, the standard power-series expansion gives

\[ e^{it} = \sum_{k=0}^{\infty} \frac{(it)^k}{k!} = \cos t + i\sin t \]

where \(\cos t\) and \(\sin t\) are defined by the even and odd parts of the same series. This turns the exponential into a rotation-valued object on the complex plane.

The function \(e^{it}\) starts at \(1\) when \(t = 0\). It first returns to the real axis on the negative side when \(e^{it} = -1\). The value of \(t\) at which this occurs is \(\pi\).

\(\pi\) is therefore the half-period of unit rotation once the earlier algebraic machinery is carried through this rotational completion. It is not being claimed that raw ternary syntax already contains a physical circle. It is being claimed that, once the complex exponential is admitted, the half-period is no longer optional. It is the unique real number satisfying

\[ e^{i\pi} = -1 \]

There is a second, corroborating route to the same constant. On the integer grid \(\mathbb{Z}^2\), count the number of integer points inside a circle of radius \(r\). In the limit of large \(r\), the ratio of this count to \(r^2\) converges to \(\pi\). This does not make the circle primitive; it shows that once rotation-invariant comparison is admitted, the same constant reappears through lattice counting.

The result: \(\pi\) is the half-period of the unit rotation exposed when the already established objects \(i\) and \(e\) are carried through a rotational analytical completion.

\[ \boxed{\quad \pi = 3.14159265\ldots \quad} \]

Part III: The Convergence

Euler's Identity

The five quantities \(0\), \(1\), \(e\), \(i\), and \(\pi\) have now been derived in sequence, each under its own explicit structural or analytical demand:

• \(0\): the ground state.
• \(1\): the unit excitation.
• \(i\): the quarter-turn witness.
• \(e\): the invariant of arbitrary refinement under multiplicative compounding.
• \(\pi\): the half-period of the rotational completion of the complex exponential.

They satisfy

\[ e^{i\pi} + 1 = 0 \]

This is not presented here as a mystical coincidence. It is a structural closure relation: objects that entered the ladder by apparently different routes are found to be algebraically locked together once the exponential is carried through the rotational regime.

\[ \boxed{\quad e^{i\pi} + 1 = 0 \quad} \]

Part IV: The Information-Theoretic Constants

The Natural Logarithms: \(\ln 2\) and \(\ln 3\)

The question: Given the natural base \(e\), what constants appear when binary and ternary distinctions are measured in that base?

The derivation: The natural logarithm \(\ln x\) is the inverse of the exponential \(e^x\). Once \(e\) has entered the ladder, logarithmic comparison is no longer optional wherever multiplicative distinctions are being measured additively.

The most primitive distinction the substrate supports is binary: ground versus not-ground, \(0\) versus \(\pm 1\). Measured in natural units, the information content of this distinction is

\[ \ln 2 = 0.69314718\ldots \]

The full state space \(\{-1, 0, +1\}\) is ternary. The information content of a single ternary choice is

\[ \ln 3 = 1.09861228\ldots \]

The point is not that the alphabet magically contains logarithms in primitive form. The point is that, once the exponential scale has been admitted, binary and ternary distinctions acquire definite logarithmic measures in that same base.

A further constant follows by the product rule. Since \(10 = 2 \times 5\),

\[ \ln 10 = \ln 2 + \ln 5 = 2.30258509\ldots \]

The result: \(\ln 2\) and \(\ln 3\) are the logarithmic measures of the simplest binary and ternary distinctions available once the ladder has admitted the base \(e\).

\[ \boxed{\quad \ln 2 = 0.69314718\ldots \qquad \ln 3 = 1.09861228\ldots \quad} \]

Part V: The Summation Constants

\(\zeta(2)\): The Basel Sum

The question: What appears once the counting sequence is carried through inverse-square summation?

The derivation: By repeated composition of the unit step, the substrate supports the positive integers \(1,2,3,\ldots\). By themselves, they do not yet compel a global convergent sum. To reach that regime, an analytical completion must be admitted: the counting sequence is treated as the domain of an infinite inverse-power summation operator.

The simplest convergent case is

\[ \zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} \]

Its value, first computed by Euler, is

\[ \zeta(2) = \frac{\pi^2}{6} \]

This does not mean the substrate primitively “contains” the Basel sum. It means that, once inverse-square summation is admitted, the resulting constant closes back onto \(\pi\), already established through rotational completion.

The result: \(\zeta(2)\) is the first convergent inverse-power sum on the integer sequence, and its evaluation returns the previously established constant \(\pi\) in squared form.

\[ \boxed{\quad \zeta(2) = \frac{\pi^2}{6} = 1.64493406\ldots \quad} \]

\(\zeta(3)\): Apéry's Constant

The question: What appears once summation over the positive integers is extended from inverse squares to inverse cubes?

The derivation: By repeated composition of the unit step, the substrate supports the counting sequence \(1,2,3,\ldots\). By itself, this does not yet compel a global sum over all inverse cubes. To reach that regime, an additional analytical completion must be admitted: the integer sequence is now treated as the domain of an infinite summation operator.

Under that demand,

\[ \zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \]

Unlike \(\zeta(2)\), this sum has no known closed form in terms of \(\pi\) or any other previously derived constant. Apéry proved in 1978 that \(\zeta(3)\) is irrational.

The point is not that raw ternary syntax is already evaluating global series in the dark. The point is that, once summation over the integer sequence is admitted as a completion demand, the inverse-cube series yields a constant that appears algebraically independent of the preceding ones and resists reduction to the earlier ladder.

The result: \(\zeta(3)\) is the constant exposed when the already available counting sequence is carried through the stronger analytical completion of inverse-cube summation.

\[ \boxed{\quad \zeta(3) = 1.20205690\ldots \quad} \]

The Euler–Mascheroni Constant \(\gamma\)

The question: What constant measures the stable gap between discrete harmonic summation and its logarithmic continuous comparison?

The derivation: The unit step generates the counting sequence \(1,2,3,\ldots\), and therefore the harmonic partial sums

\[ H_n = \sum_{k=1}^{n} \frac{1}{k} \]

Once the natural logarithm has entered the ladder, this discrete accumulation can be compared with the continuous integral

\[ \int_1^n \frac{dx}{x} = \ln n \]

The difference

\[ H_n - \ln n \]

converges as \(n \to \infty\) to a constant. This limiting gap is the Euler–Mascheroni constant:

\[ \gamma = \lim_{n\to\infty}\left(H_n - \ln n\right) \]

Equivalently,

\[ \gamma = \int_1^{\infty}\left(\frac{1}{\lfloor x\rfloor} - \frac{1}{x}\right)dx \]

The point is not that the substrate is primitively aware of this gap. The point is that, once harmonic summation and logarithmic comparison are both admitted, their mismatch stabilises to a definite residual constant.

The result: \(\gamma\) is the constant exposed by comparing discrete harmonic accumulation with its smooth logarithmic counterpart.

\[ \boxed{\quad \gamma = 0.57721566\ldots \quad} \]

Catalan's Constant \(G\)

The question: What appears once alternating sign is applied to inverse-square summation over the odd integers?

The derivation: The balanced-ternary substrate has sign as an intrinsic feature. Once inverse-square summation is admitted, one may also admit alternating summation over the odd subsequence. This yields

\[ G = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \]

This is Catalan's constant. Like \(\zeta(3)\), it has no known closed form in terms of elementary constants, and its irrationality remains unproven. It arises naturally once the already available sign structure is carried through an alternating inverse-square completion.

The result: \(G\) is the constant exposed when intrinsic sign is applied to the odd inverse-square ladder under alternating summation.

\[ \boxed{\quad G = 0.91596559\ldots \quad} \]

The Lemniscate Constant \(\varpi\)

The question: What constant appears once arc-length measurement is extended beyond the circular regime to a simple self-crossing algebraic curve?

The derivation: The lemniscate of Bernoulli, defined in polar coordinates by \(r^2 = \cos 2\theta\), is the simplest algebraic curve that closes on itself with a crossing at the origin. By itself, the previously established ladder does not yet compel elliptic arc-length measurement. To reach that regime, an additional analytical completion must be admitted: the length of a non-circular smooth curve is now treated as a legitimate global invariant.

Under that demand, the total arc length of the lemniscate is

\[ L = 2\varpi \]

where the lemniscate constant is

\[ \varpi = 2\int_0^1 \frac{dt}{\sqrt{1 - t^4}} \]

This is a complete elliptic integral. The point is not that raw ternary syntax is already measuring arc lengths of exotic curves. The point is that, once arc-length comparison is extended beyond the circle, the next natural class of invariants is elliptic rather than circular, and the corresponding constant is no longer \(\pi\) but \(\varpi\).

The two constants are related by

\[ \varpi = \frac{\Gamma(1/4)^2}{2\sqrt{2\pi}} \]

The result: \(\varpi\) is the constant exposed when arc-length measurement is carried from the circular case into the stronger analytical completion of elliptic geometry.

\[ \boxed{\quad \varpi = 2.62205755\ldots \quad} \]

Summary Of Derived Constants

The following constants have been derived, in order, from the substrate \(\{-1, 0, +1\}\) and the successive structural or analytical demands admitted in this paper.

Every entry in this table appears only once the available inventory is carried through an additional explicit demand. In the earlier sections, those demands were structural: independence, quadratic comparison, symmetry-preserving operators. In the later sections, they were analytical: recurrence, refinement, rotation, summation, and arc-length comparison.

The table is not claimed to be exhaustive. It is claimed to be ordered: each constant depends only on structure or completion machinery already on the page, and no constant could have appeared earlier in the ladder than the stage at which it is introduced.

Discussion

Scope

This paper derives a sequence of mathematical constants from the structural operations available to the balanced-ternary state space, but it now makes an explicit distinction between primitive substrate content and later analytical completion demands. Some constants follow once additional structural commitments such as independent generators, quadratic comparison and symmetry-preserving operators are admitted. Others appear only when the substrate is examined through stronger completion principles such as recurrence, refinement, rotation and summation.

The claim is therefore ordered rather than absolute. The paper does not claim that every later constant is already present in raw ternary syntax. It claims that, once each additional demand is stated openly, the resulting constants enter in a strict derivation ladder and do so at the earliest stage permitted by the available machinery. No physical interpretation is assumed or required.

What Is New

The individual constants are, obviously, not new. What is new is the sequential derivation from a single starting point, together with a clear bookkeeping of ontological status. The sequence is not presented as a flat list of consequences all carrying the same force. It is presented as an ordered map from a minimal discrete substrate to the hierarchy of constants revealed when increasingly strong completion demands are imposed.

What Is Not Claimed

The paper does not claim that all mathematically significant constants can be derived in this way. It does not claim that the derivation chain is the unique such chain. It does not claim that the balanced-ternary substrate, in isolation, is spontaneously “doing calculus” or evaluating global sums without further analytical structure being admitted.

What it does claim is narrower and stronger: given the stated succession of structural and analytical demands, this particular ladder is ordered, coherent, and non-arbitrary. Each link follows from machinery already on the page, and the starting point was itself established as a logical necessity in the companion paper.

Open Question

Whether the constants derived here, taken together, suffice to reconstruct a broader mathematical framework beyond the starting alphabet remains open. The present paper does not resolve that question. It establishes only the ladder itself, the order in which its constants appear, and the completion demands under which they do so.

Conclusion

The balanced-ternary state space \(\{-1, 0, +1\}\) is not inert. Once it is subjected to successive structural and analytical completion demands, it exposes a determinate ladder of constants whose order is not arbitrary. Some arise from comparatively primitive extensions of the substrate, such as independent generators, quadratic comparison and symmetry-preserving operators. Others appear only under stronger analytical completions, including recurrence, refinement, rotation and summation.

The sequence begins with the structure of independent axes and proceeds, in order, through the fundamental irrationals, self-referential growth constants, rotation-based invariants, information-theoretic logarithms, and the simplest convergent integer sums. The paper does not claim that all of these are primitive contents of raw ternary syntax. It claims that, once each added demand is made explicit, the resulting constants enter in a strict sequence at the earliest stage permitted by the available machinery.

No numerical value was tuned. Where a choice of scale is required, it is fixed by an explicit normalisation. No constant is inserted ad hoc. The claim is therefore not that the substrate contains the whole edifice in finished form, but that it supports a disciplined, non-arbitrary derivation ladder whose constants appear in order under its admissible completions.