The Banach-Tarski theorem concerns finite equidecomposability under Euclidean isometries. For a solid ball, a paradoxical decomposition cannot consist entirely of Lebesgue-measurable pieces, because volume is invariant under those motions and finitely additive on measurable sets. For the sphere, the corresponding obstruction uses rotation-invariant surface-area measure. The acting group explains the difference between dimensions: the isometry groups of the line and plane, regarded as discrete groups, are amenable, whereas the rotation group in three dimensions contains a non-amenable free subgroup. Here amenability means the existence of a left-invariant finitely additive probability measure on all subsets of the group.

The historical development began with Vitali’s 1905 construction of a non-measurable subset of the real line [1]. Hausdorff then obtained a paradoxical decomposition of the sphere after removing a countable exceptional set [2]. Banach and Tarski converted this group-theoretic phenomenon into the 1924 theorem that any two bounded subsets of ℝ n , n ≥ 3 , with nonempty interiors are finitely equidecomposable [3]. Von Neumann introduced the group-theoretic concept now called amenability to explain why the phenomenon depends on the acting group [4]. Tarski subsequently gave an algebraic formulation relating paradoxicality to the nonexistence of invariant finitely additive measures [5]. Later work sharpened both the geometry and the algebra. Robinson reduced ball-doubling to five pieces and proved that five is minimal [6]. De Groot and Dekker developed explicit free subgroups of orthogonal groups [7], while Świerczkowski supplied a particularly economical free rotation construction [8]. Dougherty and Foreman showed that the pieces can be chosen with the property of Baire, first in a short announcement [9] and then in a full treatment [10]. Świerczkowski’s later classification proved that two equal-angle rotations about perpendicular axes generate a free subgroup whenever their rational cosine avoids a short exceptional list [11]. Tomkowicz and Wagon give a modern synthesis of the subject, including its links with amenability, set theory, and geometric decomposition [12].

The standard constructions raise a specific question about representation. Once a free rotation pair and an orbit representative are fixed, each point of the free orbit has a unique reduced-word label. We ask how integer quaternions can record both that label and its depth in the Cayley graph, while keeping the arithmetic code separate from the measure of a geometric set.

In this work, we combine a fixed system of unnormalized integer-quaternion lifts, a prime-valuation depth register, an exact cancellation rule, and a transversal-relative orbit address. Quaternion parametrization of rotations, Świerczkowski’s freeness criterion, reduced-word normal forms, and transfer of a paradoxical partition through a free action are standard ingredients. The norm identity follows directly from multiplicativity, and the depth register is deliberately redundant. We use these ingredients to distinguish a word, its quaternion lift, an orbit point, and a paradoxical piece within one explicit construction. Sections 3 - 5 carry out this construction for the norm-five generators. We derive the rational rotations, apply the freeness criterion, prove the coding identities, and transfer the standard four-piece partition to sphere orbits.

We set a group G act on a set X . Two subsets E , F ⊆ X are called G -equidecomposable if there are finite partitions E =   ⊔ r = 1 m E r and F =   ⊔ r = 1 m F r , together with elements g 1 , ⋯ , g m ∈ G , such that

Equation (1) means that corresponding pieces differ only by allowed group motions. The symbol ⊔ denotes disjoint union.

A subset E ⊆ X is G -paradoxical if it has a finite partition

and group elements g r , h s ∈ G for which both ⊔ r   g r A r = E and ⊔ s   h s B s = E . Thus the same original pieces can be rearranged into two disjoint copies of E .

For a ball B ⊂ ℝ 3 , the classical theorem asserts paradoxicality under the Euclidean isometry group. If every piece in such a decomposition were Lebesgue measurable, invariance and finite additivity would give μ ( B ) = 2 μ ( B ) , which contradicts 0 < μ ( B ) < ∞ . At least one piece must therefore lie outside the domain of Lebesgue measure. The conclusion is non-measurability, not failure of finite additivity on measurable sets.

We choose F 2 = 〈 a , b 〉 be the free group on two generators. Its elements are reduced words in the alphabet Σ = { a , a − 1 , b , b − 1 } . A word w = s 1 ⋯ s n is reduced when s r + 1 ≠ s r − 1 for every r . Its reduced length is denoted by ℓ ( w ) = n , and the identity element e has length zero.

The Cayley graph Γ ( F 2 , Σ ) is a 4-regular tree. Its vertices are reduced words, and an edge joins w to the reduction of w s for each s ∈ Σ . Word length is the graph distance from e .

Suppose that a faithful representation ρ : F 2 → S O ( 3 ) has been fixed. The explicit representation used in this paper is constructed in Section 3. Every nonidentity rotation of the unit sphere S 2 fixes exactly two antipodal points. So by defining the exceptional set D and its complement X by

Since F 2 is countable and each set Fix ( ρ ( g ) ) contains two points, D is countable. It is also invariant under the action: if x is fixed by ρ ( g ) , then ρ ( h ) x is fixed by ρ ( h g h − 1 ) . Thus X is invariant. Its action is free, meaning that ρ ( g ) x = x with x ∈ X forces g = e .

Throughout the spherical construction, σ denotes completed rotation-invariant surface-area measure on the unit sphere, normalised by σ ( S 2 ) = 4 π . Since D is countable, σ ( D ) = 0 and σ ( X ) = 4 π . Measurability of subsets of X will always refer to this measure.

The orbits of F 2 partition X . The Axiom of Choice selects a transversal M ⊂ X , meaning that M contains exactly one point from each orbit. The orbit map

is then bijective. Surjectivity follows from the definition of a transversal. For injectivity, suppose ρ ( g ) m = ρ ( h ) n . The points m , n ∈ M belong to the same orbit, hence m = n . Freeness then gives h − 1 g = e , so g = h .

Equation (4) is the precise location of word addresses in the Banach-Tarski argument. A point of X is not intrinsically a word. It becomes a pair ( g , m ) only after the representation ρ , the generators a , b , and a transversal M have been chosen.

The free action gives a paradoxical decomposition of X , the complement of D in S 2 . The standard Banach-Tarski proof then absorbs the countable set D , extends the sphere decomposition radially to the punctured ball, and absorbs the centre. These steps are classical and are treated in detail in [6][12]. They are logically separate from the arithmetic code developed below. The code applies directly to the free orbit structure on X ; the auxiliary absorption pieces do not acquire unique reduced-word labels from Equation (4).

We define the lift of each letter in Σ by the four-entry map

Each lift has norm five. For s ∈ Σ , the inverse-letter lift is the conjugate, q s − 1 = q ¯ s = 5 q s − 1 . Thus conjugate lifts represent inverse rotations only projectively: R ( q ¯ s ) = R ( q s ) − 1 since a nonzero real scalar does not change the induced rotation. They are not multiplicative inverses in the quaternion algebra.

For a reduced word w = s 1 s 2 ⋯ s n , define its unnormalised quaternion lift by

The lift uses integer arithmetic. Since R ( q r ) = R ( q ) R ( r ) , Equations (6), (11), and (12) imply

Thus the projective quaternion in Equation (12) and the rational rotation matrix in Equation (11) carry the same group element.

We set v 5 ( m ) denote the 5-adic valuation of a positive integer m , namely the largest r ≥ 0 for which 5 r divides m .

Theorem 2 (prime-depth identity). For every reduced word w ∈ F 2 , put n = ℓ ( w ) . Then

Proof. For w = s 1 ⋯ s n , where n = ℓ ( w ) , by Equation (7) and the definition of the inverse lifts, N ( q s r ) = 5 for every r . Multiplicativity of the quaternion norm gives N ( Q 5 ( w ) ) = ∏ r = 1 n N ( q s r ) = ∏ r = 1 n 5 = 5 n . Taking v 5 gives v 5 ( N ( Q 5 ( w ) ) ) = n = ℓ ( w ) .

Theorem 2 grades the address by depth. It does not assign volume, mass, or probability to a subset of the sphere.

Prime-quaternion code can be defined as:

Proposition 1 (injectivity). The map C 5 is injective on reduced words.

Proof. If C 5 ( u ) = C 5 ( v ) , then Q 5 ( u ) = Q 5 ( v ) . Equation (13) gives ρ 5 ( u ) = ρ 5 ( v ) . The representation ρ 5 is faithful by Theorem 1, so u = v . □

The second component of Equation (15) is intentionally redundant: Theorem 2 shows that it is exactly the word length. Its purpose is to expose the depth as a prime valuation, not to create a stronger invariant. In particular, word length depends on the chosen generating set and is not invariant under arbitrary changes of generators or conjugation.

The reduction step must precede depth assignment. Let u = t 1 ⋯ t m be an arbitrary word, and let red ( u ) = w have length n . We define the raw quaternion product Q 5 raw ( u ) = q t 1 ⋯ q t m .

Proposition 2 (exact cancellation factor). If c = ( m − n ) / 2 is the number of inverse-pair cancellations required to reduce u , then

Proof. Each elementary cancellation removes an adjacent pair s s − 1 or s − 1 s . Its quaternion product is q s q ¯ s = N ( q s ) = 5 , a real scalar that commutes with every quaternion. Removing c such pairs therefore removes the scalar factor 5 c , leaving the product associated with the reduced word.

Equation (16) explains why the valuation of a raw product records raw symbolic history rather than reduced group depth. For example, a a − 1 has raw product 5 and raw norm 25, although its reduced word is e and its reduced depth is zero.

We take w = a b a − 1 . Quaternion multiplication gives

Equation (17) agrees with ℓ ( w ) = 3 . The associated rotation, obtained either from Equation (6) or by multiplying A 5 B 5 A 5 − 1 , is

Exact calculation gives ρ 5 ( w ) T ρ 5 ( w ) = I 3 and det ρ 5 ( w ) = 1 .

For s ∈ Σ , we let H s be the set of nonempty reduced words whose first letter is s . Let L = { a − r : r ≥ 1 } be the negative a -ray. By defining

the four sets in Equation (22) are disjoint and cover F 2 . Left multiplication and free reduction give

To verify the first identity, write a word of P 2 as a − r v , where r ≥ 1 and v begins with b or b − 1 . Multiplication by a either decreases r by one or, when r = 1 , exposes the first letter of v . This produces exactly P 2 ⊔ P 3 ⊔ P 4 . The second identity follows because b H b − 1 is the complement of H b in F 2 .

Equations (22) and (23) imply two disjoint reconstructions:

Thus the four original pieces P 1 , P 2 , P 3 , P 4 give two reconstructions of F 2 . Figure 2 shows the partition through depth three. Colours indicate membership in Equation (22), and black outlines mark the negative a -ray assigned to P 1 .

Figure 2. The prefix partition on the Cayley tree through depth three. The identity and the negative a-ray belong to P 1 ; all other vertices are coloured by their first letter.

By retaining the free action of F 2 on X , the complement of D in S 2 , and the transversal M from Section 2, we define four subsets of X by

The bijection in Equation (4) shows that the E r are disjoint and cover X . Transferring Equation (24) through the action gives

Equation (26) is the four-piece paradoxical decomposition of the free part of the sphere. It uses the same rotations that define the quaternion code.

Suppose all four E r were measurable with respect to σ . Rotation invariance and finite additivity applied to Equation (26) would give

Since the four pieces also partition X , adding these equalities would imply σ ( X ) = 2 σ ( X ) , contrary to σ ( X ) = 4 π . Hence the pieces cannot all be surface-measurable. This argument alone does not determine the measurability of any specified E r .

For each x ∈ X , Equation (4) gives a unique pair ( g , m ) ∈ F 2 × M with x = ρ 5 ( g ) m . We can define the M -relative address of x by

The representative m identifies the orbit and C 5 ( g ) identifies the point within that orbit. Proposition 1 and the bijectivity of Θ make Equation (27) injective. The dependence on M is unavoidable. Selecting M uses choice, and a different transversal changes the address of every point in the affected orbits. The code is therefore exact after the choices have been fixed, but it is not canonical. Equation (25) distinguishes the finite partition from the word labels. Each piece E r is indexed by the entire language P r and all representatives in M . A single word g instead specifies one point ρ 5 ( g ) m in each orbit. Thus C 5 ( g ) is a code within an orbit, P r is a set of words, and E r = P r M is a subset of the sphere. The equality v 5 ( N ( Q 5 ( g ) ) ) = ℓ ( g ) concerns a word and its chosen lift. It does not describe the surface-area measure of E r . The argument above shows that the four pieces cannot all be surface-measurable; it does not establish non-measurability of each individual piece.

Equation (26) treats X , not the complete sphere or solid ball. The countable set D can be absorbed by a separate equidecomposition, and the resulting sphere decomposition can be extended along radial half-open segments. A further absorption handles the centre. Robinson’s refinement gives five pieces for the ball and proves that four cannot suffice [6]. These auxiliary steps do not alter Theorems 1 and 2, but they do show why the present address should be described as a code for the free-group skeleton rather than as a complete constructive listing of the five ball pieces.

The integer quaternions 1 + 2 i and 1 + 2 j give an exact arithmetic representation of the free-group action considered here. Their rational rotations have perpendicular axes and common-angle cosine −3/5, so Świerczkowski’s criterion proves freeness for this pair. For a reduced word w , the unnormalised quaternion product Q 5 ( w ) projects to the same rotation as the matrix word and satisfies N ( Q 5 ( w ) ) = 5 ℓ ( w ) . The 5-adic valuation of the norm is exactly the Cayley-graph depth. For unreduced words, every inverse-pair cancellation contributes an explicit scalar factor 5. These identities give a reproducible, depth-graded code and explain its algebraic content without attaching physical meaning to the quaternion norm. After removing the countable fixed-point set and choosing a transversal, each point of the remaining sphere has a unique orbit representative and reduced-word label. The four word sets then give four spherical subsets satisfying Equation (26). These subsets cannot all be measurable with respect to rotation-invariant surface-area measure. Each piece is a union indexed by a set of words and the transversal, rather than a single word. The resulting distinction is exact. Addressability concerns unique labels inside a chosen free orbit structure. Measurability concerns membership in the domain of an additive measure. The prime-quaternion code supplies the former and leaves the latter unchanged. Future work could compare other prime-norm quaternion pairs, study the dependence of depth statistics on the generating set, or formalise coding on Baire-property decompositions. Any physical or tensorial extension would require separate axioms and cannot be inferred from the set-theoretic decomposition alone.

Payam Danesh and Raoul Bianchetti jointly developed the mathematical framework, carried out the analysis, and checked the proofs. Both authors contributed to drafting and revising the manuscript and approved the final version.