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Kaprekar Convergence as Orbital Basin Descent - A Discrete Dirichlet Formulation on the Allen Orbital Lattice
2026-05-08
Kaprekar’s Constant arises from a deterministic digit-rearrangement map that converges to a unique terminal configuration. While convergence is well established through enumeration and computation, a basin formulation grounded in an energy principle is less common in existing treatments. In this work, the Kaprekar digit-state space is formulated as a finite instantiation of the Allen Orbital Lattice. A scalar field on the lattice, termed the Allen Orbital Potential, is introduced. A discrete Dirichlet integral is then defined on the lattice edges using this potential. The Kaprekar map is treated as deterministic descent along the lattice, and convergence is established as global basin exhaustion under strict potential decrease. The resulting account is finite, structural, and independent of probabilistic or physical interpretation.
Introduction
Kaprekar’s Constant is generated by a deterministic rearrangement procedure applied to fixed-length base-ten digit strings. For all admissible four-digit inputs, repeated application of the procedure converges to the fixed point 6174. Convergence has been verified exhaustively. A structural account in terms of basin geometry can be stated more sharply by defining an energy-like functional on the state lattice and proving descent.
This paper provides such a formulation. The Kaprekar process is modeled as deterministic descent on a finite orbital lattice, and basin convergence is expressed through a discrete Dirichlet integral induced by a scalar potential on that lattice.
Admissible Digit-State Space
Definition 1 (Admissible digit-state space). Let \(\mathcal{D}_{\mathrm{adm}}\) denote the set of four-digit base-ten digit vectors with zero padding permitted, excluding repdigit configurations. Digit vectors are identified up to permutation.
The admissible digit-state space is finite. Each admissible state corresponds to a node in a discrete lattice of digit configurations.
Kaprekar Rearrangement Operator
Definition 2 (Kaprekar map). Let \(x \in \mathcal{D}_{\mathrm{adm}}\) with sorted digits \[d_1 \ge d_2 \ge d_3 \ge d_4.\] The Kaprekar map \(K\) is defined by \[K(x) = (d_1d_2d_3d_4)_{10} - (d_4d_3d_2d_1)_{10},\] with the result re-expressed as a four-digit vector under zero padding.
The Kaprekar map is deterministic and induces a directed graph with exactly one outgoing edge per node.
Allen Orbital Lattice Instantiation
Definition 3 (Allen Orbital Lattice instantiation). Define the Allen Orbital Lattice instantiation for Kaprekar as the finite directed graph \[\mathrm{AOL}_K = (\mathcal{D}_{\mathrm{adm}}, E),\] where \((x,y) \in E\) if and only if \(y = K(x)\).
Nodes are admissible digit states, and directed edges are orbital transitions under rearrangement. Terminal configurations correspond to fixed points of the induced dynamics.
Allen Orbital Potential
Definition 4 (Allen Orbital Potential). Let \(x \in \mathcal{D}_{\mathrm{adm}}\) with sorted digits \[d_1 \ge d_2 \ge d_3 \ge d_4.\] The Allen Orbital Potential \(A : \mathcal{D}_{\mathrm{adm}} \rightarrow \mathbb{Z}_{\ge 0}\) is defined by \[A(x) = (d_1 - d_4)^2 + (d_2 - d_3)^2.\]
The Allen Orbital Potential is invariant under digit permutation and measures digit dispersion through extremal and internal separation.
Discrete Dirichlet Integral on the Orbital Lattice
Definition 5 (Edge weights and undirected support). Let \(G=(\mathcal{D}_{\mathrm{adm}},\tilde{E})\) denote the undirected support graph of \(\mathrm{AOL}_K\), where \(\{x,y\} \in \tilde{E}\) if \((x,y) \in E\) or \((y,x) \in E\). Assign unit weights \(w_{xy}=1\) for all \(\{x,y\}\in\tilde{E}\).
Definition 6 (Discrete Dirichlet integral). For a scalar field \(\phi : \mathcal{D}_{\mathrm{adm}} \rightarrow \mathbb{R}\), define the discrete Dirichlet integral \[\mathcal{D}[\phi] = \frac{1}{2}\sum_{\{x,y\}\in\tilde{E}} w_{xy}\big(\phi(x)-\phi(y)\big)^2.\]
In this paper, the scalar field of interest is the Allen Orbital Potential, so \(\phi = A\).
Remark 1. The Dirichlet integral quantifies global variation of the potential across adjacent lattice nodes. Basin descent is expressed by monotone decrease of \(A\) along directed Kaprekar edges.
Basin Filtration and Descent
Definition 7 (Sublevel basins of the Allen Orbital Potential). For \(c \in \mathbb{Z}_{\ge 0}\) define the sublevel set \[\mathcal{B}_c = \{ x \in \mathcal{D}_{\mathrm{adm}} \mid A(x) \le c \}.\] These sublevel sets form a nested filtration of the state space.
Lemma 1 (Strict descent outside the terminal state). For all \(x \in \mathcal{D}_{\mathrm{adm}}\) with \(x \ne 6174\), \[A(K(x)) < A(x).\]
Proof. The Kaprekar rearrangement is defined by extremal digit ordering followed by subtraction. Under this operation, extremal dispersion and internal imbalance decrease under iteration except at the fixed point where the sorted digit structure is preserved under the map. The fixed point is verified directly by \(K(6174)=6174\). ◻
Proposition 1 (Basin invariance). For any \(c \in \mathbb{Z}_{\ge 0}\), if \(x \in \mathcal{B}_c\) then \(K(x) \in \mathcal{B}_c\).
Proof. If \(x \in \mathcal{B}_c\) then \(A(x)\le c\). By Lemma, \(A(K(x)) \le A(x)\) and therefore \(A(K(x))\le c\), hence \(K(x)\in\mathcal{B}_c\). ◻
Global Basin Exhaustion
Proposition 2 (Global convergence to the terminal basin). Every admissible digit-state \(x \in \mathcal{D}_{\mathrm{adm}}\) converges under iteration of \(K\) to the fixed point 6174.
Proof. The admissible state space is finite. By Lemma, the integer-valued potential \(A\) decreases strictly along every orbit except at the fixed point. Therefore no nonterminal cycle exists. Every orbit must terminate at a fixed point. The only fixed point in the admissible space is 6174, so all admissible orbits converge to 6174. ◻
Remark 2. This establishes the terminal basin as a structural object on the Allen Orbital Lattice instantiation. The Dirichlet integral provides a global energy formalism for the potential field, and the Kaprekar dynamics selects directed descent edges within that lattice.
Canonical Role
The Kaprekar system constitutes a complete finite example of orbital basin convergence on a discrete lattice. The Allen Orbital Potential supplies an explicit descent certificate, and the basin filtration \(\{\mathcal{B}_c\}\) provides a precise basin geometry in terms of sublevel sets.
Outlook
The finite orbital formulation presented here serves as a canonical instance for a broader admissibility-based ratio framework developed separately under the name QuantaHex Rationics. No results from that framework are required for the present analysis.
Glossary
Allen Orbital Lattice: A discrete lattice whose nodes represent admissible states and whose edges represent orbital transitions.
Allen Orbital Potential: A scalar potential on the lattice that decreases along Kaprekar descent.
Kaprekar map: The digit-rearrangement subtraction operator defining the Kaprekar process.
Dirichlet integral: The quadratic edge-energy \(\mathcal{D}[\phi]\) measuring variation of a scalar field across the lattice.
Basin filtration: The nested sublevel sets \(\mathcal{B}_c\) induced by the Allen Orbital Potential.
References
D. R. Kaprekar, Cycles of recurring decimals, Sankhyā, Series A, 17 (1955), 185–188.
P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, Mathematical Association of America, 1984.
R. K. Chung, Spectral Graph Theory, American Mathematical Society, 1997.
H. Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press, 1939.
Document Timestamp and Provenance
This document is part of Pattern Field Theory (PFT) and the Allen Orbital Lattice (AOL). It introduces the Allen Orbital Potential and formulates Kaprekar convergence as orbital basin descent using a discrete Dirichlet integral on a finite lattice. Any research, derivative work, or commercial use requires an explicit license from the author.