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Cycle–Tree Potentials for Collatz Residue Graphs Equal-Slack Cycles, ε-Squeezed Trees, and a Conservative Exceptional Edge Aleksandar Perišić Ocotber 2025 Abstract This note explains only how the certificate ϕ for the accelerated Collatz map is constructed, serving as preparation for a code implementation. We work on the odd-residue graph modulo 2 k and define per-edge weights from the 2-adic valuations v2 (3 r + 1). On each directed cycle we impose equal slack, which uniquely fixes the cycle-wide slack by telescoping and determines ϕ along the cycle after anchoring one vertex. For every in-tree feeding a cycle, we assign a tiny uniform negative tree slack (“ ε -squeezing”) and propagate ϕ outward so that all tree inequalities hold strictly. At the unique exceptional residue r† with 3 r† + 1 ≡ 0 ( mod 2 k ), we enforce a conservative slack (slightly below −ln 2) to harden the most delicate inequality. Finally, we normalize ϕ by a single offset (e.g. ϕ (1) = 0) and specify the export shape ( r, ϕ ( r ) , v2 (3 r + 1) ,succ ( r ) , s ( r )) at fixed precision. The aim is expository: to make the cycle–tree assembly of ϕ , the role of equal slacks, the ε -squeezed trees, and the conservative exceptional edge fully explicit so they can be reproduced verbatim in code. 1 Setting and Notation Fix an integer k≥1and let M= 2k. Consider the directed graph on the odd residues Rk={r∈ {1,2,...,M −1}:r≡1 (mod 2) }, with one outgoing edge per node succ(r) = Fk(r) := odd(3r+ 1) mod M, where odd ( n )removes all powers of 2from n . Thus each node has exactly one successor and the graph is a disjoint union of directed cycles with in-arborescences (in-trees) feeding those cycles. For each r∈ Rkdefine w(r) = ln 3 −v2(3r+ 1) ln 2+ρ+δ, ρ := bln 2 −ln(1 −p) + ζ, (1) where v2 is the 2-adic valuation and b, p, ζ, δ are fixed constants. A certificate is a potential function ϕ:Rk→Rsatisfying the difference constraints ϕ(r)≥ϕ(succ(r))+w(r) (∀r∈ Rk).(2) It is often convenient to record the slack of each inequality, s(r) := w(r) + ϕ(succ(r)) −ϕ(r)≤0. Feasibility of ϕis equivalent to s(r)≤0for all r. 1
2 Equal-Slack on Cycles Let C = {r0, r1, . . . , rL−1} be a directed cycle (indices modulo L ), so succ ( ri ) = ri+1 . Suppose we impose equal slack sCon each cycle edge and replace (2) by the equalities ϕ(ri)−ϕ(ri+1) = −w(ri)+sC, i = 0, . . . , L −1.(3) Summing (3) over itelescopes the left-hand side to 0, hence 0 = − L−1 X i=0 w(ri)+L sC⇒sC=1 LX i∈C w(i). Thus the cycle slack is uniquely fixed by the edge weights. Choose an anchor vertex r∗∈C and set ϕ ( r∗ )=0; then all ϕ on C are determined by chaining (3) . Since sC is typically negative, the cycle constraints hold with a uniform negative margin and therefore remain feasible. 3ε-Squeezed Trees Consider an edge u→v in a tree feeding into a cycle (so v = succ ( u )is already assigned). To reproduce a near-equality profile while keeping feasibility strict, assign a small uniform negative tree slack stree <0(e.g. stree =−10−6) and set ϕ(u) = ϕ(v)+w(u)−stree.(4) Then the corresponding slack is exactly s ( u ) = stree ≤ 0. Applying (4) recursively from the cycle outward assigns ϕ to every node in each in-tree, with all tree edges carrying the same tiny negative slack. 4 The Exceptional Residue There is a unique exceptional residue r† such that 3 r† + 1 ≡ 0 ( mod M ). For this edge, a distinguished negative value close to −ln 2naturally appears. To build in a robust safety margin (independent of floating-point roundoff), set a conservative exceptional slack s(r†) = −ln 2 −10−6,(5) and enforce equality ϕ ( r† ) = ϕ ( succ ( r† )) + w ( r† ) −s ( r† ). This makes the exceptional inequality strictly stronger by 10−6than the exact −ln 2 target, ensuring s(r†)≤0with a tiny buffer. 5 Normalization and Formatting Adding a constant to ϕpreserves all slacks. One may therefore normalize by fixing ϕ(1) = 0,(6) without affecting feasibility. For numerical reporting, one can export r, ϕ(r), v2(3r+1),succ(r), s(r) with fixed decimal precision for ϕand s(e.g. 12 places), and integer fields for residues and v2. 2
6 Why Feasibility Is Preserved Feasibility requires s(r)≤0for all r. • On cycles, sC is determined by the exact identity Pi∈Cϕ ( ri ) −ϕ ( ri+1 ) = 0; the induced value is typically negative, hence s(r)=sC≤0on all cycle edges. •On trees, (4) enforces s(u)=stree <0, so each tree inequality holds strictly. •At the exceptional residue, (5) sets s(r†)<0by design, also strict. •The normalization (6) is a global shift of ϕand does not change any slack. Consequently, the constructed ϕ satisfies (2) everywhere. The (deliberately) nonzero negative slacks serve as a conservative buffer, making the certificate robust against finite-precision arithmetic and minor implementation differences. 7 Algorithmic Outline 1. Build the functional graph r7→ succ(r)and the weights w(r)from (1). 2. Decompose the graph into cycles (with standard tortoise–hare or stack-based DFS). 3. For each cycle C, compute sCand solve (3) around Cwith ϕanchored at one node. 4. Propagate (4) outwards along each in-tree with stree =−10−6. 5. At the exceptional residue r†, use (5) instead of stree. 6. Normalize by ϕ(1) = 0 and export (r, ϕ, v2,succ, s)at fixed precision. (see [1]) References [1] A. Perisic. A Lyapunov Certificate for the Accelerated Collatz Map. Zenodo, 2025. 3
