Mersenne Block Dynamics: A Framework for the Collatz Conjecture

Mersenne Block Dynamics: A Framework for the Collatz Conjecture Stephen R. Campbell Simon Fraser University [email protected] December 18, 2025 Abstract We introduce Mersenne block dynamics, a bit-level structural decomposition of the accelerated Collatz (Syracuse) map on odd integers based on the 2-adic valuation n(x):=ν2(x+1). This yields the canonical decomposition x=P(x) 2n(x)+(2n(x)−1), where n(x)is the length of the Mersenne tail. Using the associated odd factor a(x), we partition each Syracuse orbit into deterministic Mersenne blocks. Inside a block, the odd values increase strictly while the tail length decreases by one at each step, producing a rigid “wedge” pattern in the binary expansion. The exit from a block is governed by the exponent r(x):=ν2(3n(x)a(x)−1) and induces a coarse-grained block map B(x) := Sn(x)(x). We derive explicit transition identities for Band exact step-count bookkeeping across the time scales C→T→S→B. Using the exact natural-density law for (n(x),r(x)), and assuming an orbit-mixing hypothesis for typical B-orbits, we model successive block parameters (n(xk) and r(xk)) as independent and identically distributed (i.i.d.) geometric(1/2) random variables. This intrinsic statistical model predicts an expected logarithmic drift of 2 log2(3/2) −2≈ −0.83 bits per block step, recovering the classical probabilistic heuristic within a structural coordinate system. 2020 Mathematics Subject Classification: Primary 11B83; Secondary 37P99. Keywords: Collatz conjecture, Syracuse map, Mersenne blocks, 2-adic valuations, probabilistic models. 1 Contents Introduction 3 1 An Overview of Mersenne Block Dynamics 6 1.1 The Collatz, Terras, and Syracuse maps .................... 6 1.2 Notation and conventions ............................ 7 1.3 Bit strings, Mersenne tails, and even prefix segments ............. 7 1.4 Definition of Mersenne blocks and wedges ................... 8 2 Mersenne Inter-Block Dynamics 11 2.1 The Mersenne block map ............................. 11 2.2 Collatz in Mersenne block form ......................... 12 2.3 Total stopping times and time scales ...................... 13 3 Mersenne Intra-Block Dynamics 16 3.1 Stairs inside a block ............................... 18 3.2 Matrix formulation of the intra-block stair dynamics ............. 19 4 Mersenne Trans-Block Dynamics 20 4.1 Block peak and exit exponent .......................... 21 4.2 Direct block transition theorem ......................... 21 4.3 The Mersenne block drop ............................. 22 4.4 Stairs and exits .................................. 24 4.5 Residue-class constraints and the image of B.................. 25 5 Intrinsic Statistics and Logarithmic Drift 28 5.1 Exact distribution of n(x)............................ 29 5.2 Exact joint distribution of (n, r)......................... 29 5.3 The orbit-mixing heuristic ............................ 29 5.4 Expected logarithmic drift ............................ 31 Conclusion, outlook, and prospects 32 Acknowledgements 35 References 35 Appendix: Computational Implementation and Reproducibility 36 2 Introduction The Collatz conjecture (the 3x+ 1 problem) asks whether repeated iteration of C(n) = (3n+ 1 if nis odd, n/2if nis even,n∈N, eventually reaches 1for every starting value n≥1. Despite its elementary definition, the conjecture remains open and is widely regarded as extraordinarily difficult [8]. It has been verified computationally to very large bounds (see, e.g., [13,10] and references therein), and no nontrivial cycles are known; recent work of Hercher rules out all odd m-cycles up to length 91 [5]. On the analytic side, results of Terras and Tao provide rigorous evidence of “typical” descent (in suitable senses) without resolving convergence for all starting values [16,15]. A standard reduction (formalized by Möller [12]) compresses Collatz dynamics to odd integers by folding each odd step together with all subsequent divisions by 2. This yields the accelerated Collatz (or Syracuse) map Son odd integers, S(x) = 3x+ 1 2ν2(3x+1) , so that S(x)is the next odd value after xin the original Collatz trajectory. Between Cand S, one also encounters the intermediate Terras acceleration [16], T(n) = (n/2neven, (3n+ 1)/2nodd, giving a natural ladder of time scales C→T→Sfrom “step-by-step” to “odd-to-odd” dynamics. Mersenne block dynamics as a canonical coarse-graining. This paper introduces a further acceleration of the Syracuse dynamics obtained by grouping the orbit into variablelength blocks determined directly by the binary expansion of the current odd value. For an odd integer x, define n(x) := ν2(x+ 1) ≥1, so n(x)is the length of the trailing run of 1’s in the binary expansion of x(the Mersenne tail). Equivalently, writing x+ 1 = 2n(x)a(x)with a(x)odd yields the canonical decomposition x= (a(x)−1) 2n(x)+2n(x)−1, a(x) := x+ 1 2n(x)odd. This tail decomposition partitions each Syracuse orbit into contiguous Mersenne blocks. Starting from a block start x, the block consists of the n(x)odd values x, S(x), S2(x), . . . , Sn(x)−1(x), 3 whose trailing-1run decreases deterministically in length from n(x)down to 1, producing a rigid right-angled triangular “wedge” pattern in the trailing bits (Figure 2). Inside a block the dynamics is fully explicit (Theorem 3.1) and strictly increasing (Proposition 3.4); in particular, the first n(x)−1Syracuse steps in a nondegenerate block are forced stair steps in the valuation sense s(u) := ν2(3u+ 1) = 1 (Lemma 3.7). Thus the only nontrivial 2-adic division in a block is concentrated in its final exit step. The exit is measured by the exponent r(x):=ν2 3n(x)a(x)−1, and it induces a coarse-grained block map B(x):=Sn(x)(x), which jumps from one block start to the next. On this scale, the Collatz conjecture is equivalent to the pure block statement that every odd x≥1eventually satisfies Bk(x) = 1 (Theorem 2.6). Main results snapshot. Deterministically, the Mersenne Block Dynamics Framework (MBDF) yields an explicit closed form for the Syracuse iterates inside each block and an exact start-to-start transition identity for the induced block map B(Theorems 3.1 and 4.4), together with exact ratio/time bookkeeping and an explicit correction term for the block ratio B(x) x(Corollaries 4.12 and 4.5). We also characterize the arithmetic image and inverse structure of B: one always has 3∤B(x)(Proposition 4.15), and Bis surjective onto the admissible odd set {y:yodd,3∤y}, in fact with infinitely many preimages at every fixed block length n(Theorem 4.17). Finally, the intrinsic joint law of (n(x), r(x)) is exactly independent geometric(1/2) in natural density (Proposition 5.2); assuming orbit-mixing along typical Borbits (Heuristic 5.3), this intrinsic law recovers the classical negative expected logarithmic drift 2 log2(3/2) −2≈ −0.83007 per block step in block coordinates (Proposition 5.6). What the framework delivers (and what it does not). We do not prove the Collatz conjecture. The purpose of the MBDF is to provide a structural coordinate system in which (i) the deterministic part of the dynamics is isolated and solvable in closed form, and (ii) the remaining arithmetic difficulty is localized to the behavior of the exit exponent r(x)along block orbits. Concretely, this manuscript provides: (i) Explicit intra-block mechanics and a rigid bit-level wedge: closed forms for Sj(x)across a block and a deterministic decrement of the tail length at each step (Theorem 3.1, Corollary 3.2). (ii) Exact transition identities and time bookkeeping across scales: a direct startto-start block transition formula (Theorem 4.4) together with exact step-count relations across C→T→S→B(Corollary 4.12); in particular, one block step corresponds to exactly nSyracuse steps, n+rTerras steps, and 2n+rCollatz steps, with naturaldensity expectations E[n]=2,E[n+r]=4,E[2n+r]=6(Proposition 2.12). 4 (iii) A clean dominant factor plus an exact correction term: an exact ratio identity for B(x)/x which isolates the dominant heuristic factor 3n(x)/2n(x)+r(x)and controls the remaining correction (Corollary 4.5). This makes precise what is dropped when one models block ratios by 3n/2n+r. (iv) Residue-class control of exits and a block-level inverse picture: a congruence characterization of r(x)(Lemma 4.13) and the resulting geometric residue-class law for rat fixed n(Corollary 4.14), together with a modular description of the image of B and explicit infinite families of preimages. In particular, B(x)is never divisible by 3 (Proposition 4.15), and Bis surjective onto the admissible odd set {y:yodd,3∤y}; moreover, this surjectivity holds for every fixed block length n(Theorem 4.17). (v) Exact intrinsic statistics and an explicit single heuristic input: the joint law of (n(x), r(x)) is exactly independent geometric(1/2) in natural density (Proposition 5.2). To pass from these static residue-class laws to orbit-level predictions along a deterministic B-orbit, we isolate a single non-rigorous assumption—an orbit-mixing hypothesis for typical block orbits (Heuristic 5.3). Under this hypothesis the classical expected logarithmic drift is recovered in block coordinates (Proposition 5.6). The upshot is a sharp separation: static congruence structure and intrinsic distributions are proved exactly, while dynamical independence/mixing along orbits is explicitly identified as the remaining heuristic gap. Orbit-level diagnostics and reproducibility details are provided in the computational appendix. Relation to classical encodings. Many standard approaches encode accelerated Collatz dynamics by recording parity vectors or, equivalently, the valuation sequence sj:= ν2(3xj+ 1) along a Syracuse orbit xj+1 =S(xj); see, e.g., [8,17,9]. MBDF can be viewed as a coarse-graining of this encoding: if xis a block start, then s(Sj(x)) = 1 for 0≤j≤ n(x)−2(Lemma 3.7), while the unique non-stair valuation completing the block satisfies s(L(x)) = 1 + r(x)(Lemma 4.10). Thus the pair (n(x), r(x)) packages long forced runs of s= 1 into a single block-length parameter and isolates all nontrivial 2-adic division to the exit exponent. This viewpoint is also compatible with 2-adic formulations (e.g. the Bernstein–Lagarias conjugacy map) in which valuations and digit patterns govern time scales [1]. Finally, from the inverse-iteration (preimage tree/graph) perspective [9,17], MBDF supplies a residue-class-friendly organization of backward branches for the coarse-grained map Bvia explicit preimage families (Theorem 4.17). Organization. Section 1introduces the tail decomposition and defines Mersenne blocks and wedges. Section 2defines the block map Band proves the equivalence between block and Syracuse formulations of Collatz, together with time-scale comparisons. Section 3develops the rigid intra-block dynamics, including a matrix formulation. Section 4derives explicit exit and transition formulas for Bin terms of (n(x), r(x)) and records a sufficient condition for a net block contraction. Section 5proves the intrinsic residue-class statistics and derives the expected drift under the orbit-mixing hypothesis. We conclude with a summary, outlook, and concrete prospects for further work. 5 1 An Overview of Mersenne Block Dynamics To understand the long-term behavior of the Collatz map, we introduce a structural framework called Mersenne Block Dynamics. This framework shifts the perspective from the step-by-step iteration of individual integers (be they both odd and even, or just odd) to the dynamics of Mersenne blocks, the deterministic activities within them, transitions between them, and their global orbits. In this overview, we establish the fundamental definitions of the Collatz, Terras, and Syracuse maps, and the canonical decomposition of odd integers that underpins the entire framework. 1.1 The Collatz, Terras, and Syracuse maps For a nonzero integer m, let ν2(m)denote the 2-adic valuation, i.e. the largest k≥0with 2k|m. Definition 1.1 (Collatz, Terras, and Syracuse maps).The Collatz map C:N→Nis C(n) = (3n+ 1 if nis odd, n/2if nis even. The Terras map T:N→Nis T(n) = (n/2if nis even, (3n+ 1)/2if nis odd. Equivalently, T(n)=C(n)for even nand T(n) = C2(n)for odd n. The accelerated Collatz or Syracuse map Sacts on odd integers by S(x) := 3x+ 1 2ν2(3x+1) , x odd, so that S(x)is always odd. Each step of Scorresponds to one application of x7→ 3x+ 1 followed by all possible divisions by 2, i.e. one full Collatz “odd-to-odd” segment. Remark 1.2 (Equivalent formulations and time scales).It is standard (e.g., [8]) that the classical Collatz conjecture ∀n≥1,∃i≥0 : Ci(n) = 1 is equivalent to either of the accelerated formulations ∀n≥1,∃t≥0 : Tt(n) = 1 and ∀x≥1odd,∃j≥0 : Sj(x) = 1. In this manuscript we work primarily on the odd integers with S(and with the further coarse-grained block map Bdefined in §2). The maps Cand Twill be used mainly for time-scale comparisons and for connecting with the existing literature. 6 C T S B merge even steps merge remaining even steps merge Mersenne blocks Figure 1: A hierarchy of time scales. The Collatz map Cis the finest scale; the Terras map Tfolds each odd step together with one division by 2; the Syracuse map Sfolds each odd step together with all divisions by 2up to the next odd integer; and the block map B(§2) folds whole Mersenne blocks into a single step. 1.2 Notation and conventions •Iteration indices. We use ifor Ci(Collatz time), tfor Tt(Terras time), jfor Sj (Syracuse time), and kfor Bk(Mersenne Block time). •Domain. Unless explicitly stated otherwise, symbols such as xdenote positive odd integers. The functions n(x),a(x),P(x),L(x), and r(x)are only defined for odd x. •Blocks vs. exits. AMersenne block rooted at xwill mean the wedge segment x, S(x), . . . , Sn(x)−1(x). Its last value L(x):=Sn(x)−1(x)is the block peak. The next odd iterate B(x):=Sn(x)(x)begins the next block, and the step L(x)7→ B(x)is the block exit. 1.3 Bit strings, Mersenne tails, and even prefix segments We now isolate the maximal trailing run of ones in the binary expansion of an odd integer. Lemma 1.3 (Canonical Mersenne-tail decomposition).Let x≥1be odd and define n(x) := ν2(x+ 1) (≥1). Then there is a unique even integer P(x)≥0such that x=P(x) 2n(x)+2n(x)−1.(1) Equivalently, x+ 1 = 2n(x)a(x)with a(x) := x+ 1 2n(x)odd, P(x)=a(x)−1even. Proof. Since xis odd, x+ 1 is even, so we can write x+ 1 = 2na, with n=ν2(x+ 1) ≥1and aodd. Then x= 2na−1=(a−1)2n+ 2n−1, 7 so setting P(x) := a−1and n(x):=ngives (1) with P(x)even. For uniqueness, suppose x=P2n+ (2n−1)=P′2n′+ (2n′−1) with P, P′even and n, n′≥1. Then x+ 1 = 2n(P+ 1) = 2n′(P′+ 1), and P+ 1,P′+ 1 are odd. Hence n=n′and P+ 1 = P′+ 1, so P=P′. Definition 1.4 (Mersenne tail, even prefix segment, and odd factor).For odd x≥1we define: •the Mersenne-tail length n(x) := ν2(x+ 1) (≥1); •the odd factor a(x) := x+ 1 2n(x)(odd); •the even prefix segment P(x):=a(x)−1(even). Equivalently, n(x)is the length of the trailing run of 1’s in the binary expansion of x. Remark 1.5 (Binary picture and Mersenne tail).Write (y)2for the binary expansion of a nonnegative integer yas a bit string. Since P(x)is even, (P(x))2ends in a 0-bit. The decomposition (1) says that in base 2 we can write xas x= (P(x))2 | {z } ends with a 0-bit 11 ···1 | {z } n(x)ones The trailing block of n(x)ones is the Mersenne tail of x. The special case P(x) = 0 gives the pure Mersenne numbers 2n(x)−1, whose entire binary expansion is the tail. For example, x= 27 has (27)10 = (11011)2with a trailing run of two 1’s, so n(27) = ν2(28) = 2 and (P(27))2= (110)2. 1.4 Definition of Mersenne blocks and wedges Definition 1.6 (Mersenne block and Mersenne wedge).Let x≥1be odd and set n:= n(x) = ν2(x+ 1) ≥1. •The finite Syracuse segment x, S(x), S2(x), . . . , Sn−1(x) is called the Mersenne block rooted at x. It contains nodd terms and n−1Syracuse steps. We refer to xas the block start. 8 •The last value in the block, L(x):=Sn−1(x), is called the block peak. The next odd iterate B(x):=Sn(x) Equivalently, because n:= n(x), the start-to-start block transition consists of n(x) Syracuse steps: B(x) = Sn(x)(x). •In binary, the iterates in the block have Mersenne-tail lengths n, n −1, . . . , 1. If we stack their bit strings vertically (with S0(x)=xon the top row and Sj(x)on row jbelow it, for 0≤j≤n−1), the ones in the tails form a stair-step right-angled triangular pattern. We call this triangle of ones the Mersenne wedge rooted at x(See Corollary 3.2). Remark 1.7 (Degenerate and nondegenerate blocks).If n(x) = 1, then the Mersenne block at xconsists only of the single value x; there is no visible wedge. For a Mersenne tail of length n(x)≥2we obtain a genuine wedge of height n(x)in the trailing bits. For example, if x= 31 then n(x)=ν2(32) = 5 and the wedge has the schematic form 11111 1 1 1 1 1 1 1 1 1 1 The full Syracuse iteration of the block, including even prefixes P(Sn(x)(x)), is as follows: 11111 1 0 1 1 1 1 1000111 1101011 10100001 Remark 1.8 (Bit-string vs block dynamics).We will consistently distinguish: •bit-string dynamics: how the bits of a single integer xevolve under one step x7→ S(x); •block dynamics: how finite segments of the Syracuse orbit, grouped into Mersenne blocks, evolve as we apply a coarse-grained block map B. Mersenne tails and the triple (n(x), a(x), P(x)) are bit-string objects. Mersenne blocks, Mersenne wedges, and the block map B(defined in Section 2below) live at the block level. 9 Map f τf(27) max{ft(27) : 0 ≤t≤τf(27)}τC(27)/τf(27) C111 9232 1 T70 4616 ≈1.59 S41 3077 ≈2.71 B17 911 ≈6.53 Table 1: Total stopping times and maximal values for x0= 27 corresponding to Fig. 3. Here max{ft(27) : 0 ≤t≤τf(27)}denotes the maximum value attained along the orbit under fbefore reaching 1. In particular, one block step represents about 2Syracuse steps, 4Terras steps, and 6Collatz steps on average. Proof. The exact step counts are proved in Corollary 4.12. The expectations follow from the intrinsic geometric laws for n(x)and r(x)(Propositions 5.1 and 5.2), which give E[n] = E[r]=2. Heuristic consequence. If successive block starts sample (n, r)with weak dependence as in Heuristic 5.3, then for typical x0with large τB(x0)one expects τS(x0)≈2τB(x0), τT(x0)≈4τB(x0), and τC(x0)≈6τB(x0)(see 5.7). Remark 2.13 (Degenerate “no-wedge” class).The equality case τS(x0)=τB(x0)in Lemma 2.9 is structurally degenerate and occurs if and only if every Mersenne block along the B-orbit of x0has length n(xr)=1. In this case each block step is a single Syracuse step, so the Syracuse and block time scales coincide. Equivalently, every odd iterate of the Syracuse orbit lies in the residue class 1 (mod 4), so the orbit never visits an odd integer ≡3 (mod 4). In this “non-compressing” class of cases, the Mersenne wedges are degenerate, and the coarse-grained dynamics of Boffers no time compression beyond the original Syracuse dynamics. 3 Mersenne Intra-Block Dynamics We now describe the Syracuse dynamics inside a single Mersenne block in closed form. Write x= (P+ 1)2n−1with P=P(x)even and n=n(x)≥1as in Lemma 1.3 and Definition 1.4. Since Pis even, the shifted factor P+ 1 is odd, so it is natural to set a:= P+ 1 = a(x) = x+ 1 2n(x)(odd). Equivalently, x= 2na−1and P=a−1. We regard P(x)as the canonical prefix parameter, and use a(x) = P(x) + 1 only as a notational convenience when it streamlines formulas. Theorem 3.1 (Local Mersenne block dynamics for S).Let x= (P+ 1)2n−1with Peven and n≥1. Then for 0≤j≤n−1, Sj(x) = 3j(P+ 1) 2n−j−1.(2) 16 Proof. We use induction on j. For j= 0 we have S0(x) = x= (P+ 1)2n−1, so (2) holds. Assume (2) holds for some jwith 0≤j≤n−2, i.e., Sj(x)=3j(P+ 1)2n−j−1. Then 3Sj(x) + 1 = 33j(P+ 1)2n−j−1+ 1 = 3j+1(P+ 1)2n−j−3 + 1 = 3j+1(P+ 1)2n−j−2 = 23j+1(P+ 1)2n−j−1−1. The factor in parentheses is odd, so ν2(3Sj(x) + 1) = 1 and Sj+1(x) = 3Sj(x)+1 2= 3j+1(P+ 1)2n−j−1−1. This is exactly (2) with jreplaced by j+ 1. Corollary 3.2 (Bit-string structure along a wedge).In the setting of Theorem 3.1, for each 0≤j≤n−1we can write Sj(x) = Pj2n−j+2n−j−1, where Pj:= 3j(P+ 1) −1 is an even integer. Thus: •the lower n−jbits of Sj(x)form a Mersenne tail of 1-bits; •the bit immediately above the tail is 0(since Pjis even); •the higher bits encode the evolving even prefix segment Pj. As jincreases from 0to n−1, the tail length decreases from ndown to 1. In particular, Sj(x) + 1 = 2n−j(Pj+ 1), and since Pjis even, Pj+ 1 is odd. Hence n Sj(x)=ν2 Sj(x)+1=n−j(0 ≤j≤n−1), so the Mersenne-tail length decreases deterministically by one at each step inside the block. Remark 3.3 (Explicit evolution of the even prefix).The sequence of even prefixes (Pj)satisfies the linear recurrence Pj+1 = 3Pj+ 2, P0=P, which solves to Pj= 3j(P+ 1) −1as in Corollary 3.2. Thus the entire Mersenne wedge and, equivalently, the entire Mersenne block rooted at xis determined by the pair (P(x), n(x)). 17 Proposition 3.4 (Strict increase inside a Mersenne block).Let x= (P+ 1)2n−1with P even and n≥1. Then S0(x)< S1(x)<··· < Sn−1(x). Proof. From (2), Sj(x)=3j(P+ 1)2n−j−1. For 0≤j≤n−2, Sj+1(x)−Sj(x) = 3j+1(P+ 1)2n−j−1−1−3j(P+ 1)2n−j−1 = 3j(P+ 1)3·2n−j−1−2n−j = 3j(P+ 1)2n−j−1(3 −2) = 3j(P+ 1)2n−j−1>0. Remark 3.5 (Local regularity vs global complexity).Within a single Mersenne block, the dynamics of Sis as regular as one could hope for: •the odd values are given by a simple explicit formula; •they form a strictly increasing sequence; •their tails shrink in a perfectly predictable way, forming a Mersenne wedge. The intra-block evolution is explicit; the global difficulty is pushed into the exit exponent r(x) and its behavior along block orbits. 3.1 Stairs inside a block Definition 3.6 (Step valuations, stairs, and exits).For odd x≥1define s(x):=ν2(3x+ 1) (≥1), x+:= S(x) = 3x+ 1 2s(x). We say: •the step x7→ x+is a stair step if s(x)=1; •the step x7→ x+is an exit step if s(x)≥2. Lemma 3.7 (Stair valuations inside a block).Let x= (P+ 1)2n−1with n=n(x)≥2. Then for 0≤j≤n−2, sSj(x)= 1. That is, the first n(x)−1Syracuse steps starting at xare all stair steps. Proof. In the proof of Theorem 3.1 we saw that for 0≤j≤n−2, 3Sj(x) + 1 = 23j+1(P+ 1)2n−j−1−1, with the factor in parentheses odd. Thus ν2(3Sj(x) + 1) = 1, i.e. s(Sj(x)) = 1. 18 3.2 Matrix formulation of the intra-block stair dynamics In this subsection we recast the rigid intra-block dynamics as the iteration of a single 2×2 integer matrix acting on integer pairs that encode rationals as ratios. This eliminates the need for induction and makes the “3nexpansion vs. 2n+rcontraction” balance completely transparent. The odd-branch Terras map. Recall the Terras map Tfrom Definition 1.1. On odd inputs it acts by Todd(x) := 3x+ 1 2.(3) Whenever s(x)=ν2(3x+ 1) = 1, the Syracuse map agrees with this odd branch: S(x)=T(x) = Todd(x), and the output is again odd. By Lemma 3.7, if xbegins a Mersenne block of length n(x) then sSj(x)= 1 for 0≤j≤n(x)−2, so along the Mersenne wedge we have Sj(x) = Tj odd(x) (0 ≤j≤n(x)−1).(4) A matrix model via integer pairs. We encode a rational number as an integer pair (u, v)with v= 0, interpreted as the ratio u/v. In particular, we represent an integer xby the column vector v(x) := x 1. Then (3) is encoded by the matrix update A:= 3 1 0 2∈M2(Z), A v(x) = 3x+ 1 2.(5) Interpreting u vas the rational number u/v recovers Todd(x) = (3x+ 1)/2. (Scaling u vby a nonzero integer does not change the ratio u/v.) This matrix viewpoint is purely bookkeeping: it just tracks the numerator and denominator of the rational iterate simultaneously, and the underlying map is recovered by taking the ratio of the two entries. Closed form for Ajand for Tj odd.Since Ais triangular, its powers admit a closed form. For all j≥0, Aj=3j3j−2j 0 2j.(6) Applying (6) to v(x)yields Tj odd(x) = 3jx+ (3j−2j) 2j=3j(x+ 1) −2j 2j.(7) 19 Recovery of the intra-block formula. If xbegins a Mersenne block of length n= n(x) = ν2(x+ 1), write x= 2na−1with aodd. (8) Then (7) simplifies for every 0≤j≤nto Tj odd(x)=3ja2n−j−1.(9) In particular, for 0≤j≤n−1we are still within the block and Sj(x)=Tj odd(x)by (4), so (9) reproduces Theorem 3.1 without induction. Exit exponent and residue classes. The nth application of the odd branch produces the raw exit numerator Tn odd(x) = 3na−1,(10) which is even. The true block exit divides by at least an additional power of 2: r(x):=ν2 3na−1, B(x) = Sn(x) = 3na−1 2r(x).(11) Equivalently, if we iterate the Terras map T(Definition 1.1) starting at odd x, then the first niterates apply the odd branch, landing at 3na−1, and the next r(x)iterates simply divide by 2. Thus B(x)=Tn(x)+r(x)(x).(12) Expansion cost vs. total contraction across a block. The representative matrix A in (5) has eigenvalues 3and 2; after nodd-branch steps the numerator scales like 3nwhile the baseline denominator scales like 2n, producing the familiar factor (3/2)n. The additional division by 2r(x)in (11) is what turns a typically-expanding stair regime into an overallcontracting block transition. 4 Mersenne Trans-Block Dynamics The rigid intra-block phase described in §3ends when the Mersenne tail has been stripped down to a single trailing 1. The next Syracuse step then performs a division by a (potentially large) power of 2and lands at the start of the next Mersenne block. This block exit can be viewed at two resolutions: •coarsely, as the direct start-to-start transition x7→ B(x)(one step of the block map); •finely, as the single Syracuse step from the block peak L(x)to the next block start B(x). We treat the fine-grained picture first. 20 4.1 Block peak and exit exponent Definition 4.1 (Block peak and exit exponent).Let xbe odd and write x= 2n(x)a(x)−1 with a(x)odd. The block peak is L(x):=Sn(x)−1(x), and the exit exponent is r(x):=ν2 3n(x)a(x)−1. Lemma 4.2 (From block peak to next block start).Let xbe odd with n=n(x)and a=a(x). Then L(x) = 2 ·3n−1a−1, B(x) = 3na−1 2r(x). Proof. By Theorem 3.1 with j=n−1(equivalently, (9) with j=n−1), we have L(x)=Sn−1(x)=3n−1a2n−(n−1) −1=2·3n−1a−1. Then S(L(x)) = 3L(x)+1 2ν2(3L(x)+1) =3(2 ·3n−1a−1) + 1 2ν2(3na−1) =3na−1 2r(x). But S(L(x)) = Sn(x) = B(x)by Definition 1.6. Example 4.3 (A single block: x= 15).We have 15+1 = 16 = 24, so n(15) = 4 and a(15) = 1. The Mersenne block rooted at 15 is 15 S −→ 23 S −→ 35 S −→ 53, so the block peak is L(15) = 53. The raw exit numerator is 34·1−1 = 80, which has r(15) = ν2(80) = 4, hence the next block start is B(15) = S4(15) = 80 24= 5. In this example the block ratio is B(15)/15 = 1/3, while the dominant factor 3n/2n+r= 34/28= 81/256 ≈0.316 differs from 1/3only by the small correction factor in (14). 4.2 Direct block transition theorem Theorem 4.4 (Direct block transition theorem).Let xk=Bk(x0)denote the sequence of block starts along a block orbit. Write each xk= 2nkak−1with akodd, and set rk:= r(xk) = ν2(3nkak−1). Then xk+1 =B(xk) = 3nk(xk+1)−2nk 2nk+rk=3nkak−1 2rk.(13) 21 Proof. By Lemma 4.2 applied to xk, we have xk+1 =B(xk) = (3nkak−1)/2rk. Using ak= (xk+ 1)/2nkgives the alternative expression xk+1 =3nk(xk+1)−2nk 2nk+rk. Corollary 4.5 (Exact block ratio formula and error bound).Let xbe odd with x= 2na−1 and exit exponent r=r(x). Then the block ratio is exactly B(x) x=3n 2n+r·1−1 3na 1−1 2na .(14) Moreover, the logarithmic deviation from the heuristic model is bounded by the reciprocal of the input size: log 1−1 3na 1−1 2na≤C 2na(15) for an absolute constant C. This confirms that the approximation B(x)/x ≈3n/2n+rbecomes exponentially accurate as xgrows. Proof. The identity (14) follows by dividing B(x) = (3na−1)/2rby x= 2na−1and factoring out the dominant terms. For the bound, let δ1= (3na)−1and δ2= (2na)−1. For large x, these are small, and we apply the estimate |log(1 −y)| ≤ 2|y|for small |y|. The logarithmic correction is ∆err = log(1 −δ1)−log(1 −δ2). Using the mean value theorem or Taylor expansion, |∆err| ≈ |δ2−δ1|< δ2=1 2na. Thus, the error is bounded by C/x for a suitable constant C, vanishing rapidly for large orbits. Remark 4.6 (Cost–revenue viewpoint for a block step).The direct transition formula (13) isolates two exponents associated with a block start xk: the Mersenne-tail length nk= n(xk)and the exit exponent rk=r(xk). These govern the dominant multiplicative factor 3nk/2nk+rkappearing in (14), while the remaining correction factor in (14) tends to 1when ak= (xk+ 1)/2nkis large. Heuristically, one may think of each block step as “paying” nkmultiplications by 3(coming from the stair regime) and then “collecting” nk+rkdivisions by 2at the boundary. The block map tends to contract when the exit exponent rkis frequently large enough to offset the typical 3/2growth inside the block. 4.3 The Mersenne block drop The Collatz conjecture asserts that orbits fall to 1eventually. This is stronger than merely requiring that the orbit’s size decreases at some point. However, any finite total stopping time must include at least one drop below the starting value. The next theorem gives a sufficient criterion for when a drop in size occurs across a block. 22 Theorem 4.7 (Mersenne block drop theorem).Let x>1be odd with Mersenne tail length n(x) = n, odd factor a(x)=a, and exit exponent r(x)=r. If r > n log2(3/2) + 1,(16) then the block map contracts: B(x)< x. Proof. We seek a sufficient condition for the contraction B(x)< x. First, observe the strict upper bound on the block map: B(x) = 3na−1 2r<3na 2r. Second, for x > 1, we have 2na≥2, which implies the strict lower bound on x: x= 2na−1>1 2(2na) = 2n−1a. Combining these, it is sufficient to enforce the condition 3na 2r≤2n−1a, which ensures B(x)<3na 2r≤2n−1a < x. Dividing by aand rearranging 3n 2r≤2n−1yields 3n≤2n+r−1⇐⇒ nlog23≤n+r−1. Solving for rgives r≥n(log23−1) + 1 = nlog2(3/2) + 1. The hypothesis (16) satisfies this inequality strictly. Remark 4.8 (Zig-zag behavior).Within a nondegenerate Mersenne block, the orbit increases monotonically: x<S(x)<··· < Sn(x)−1(x)=L(x). The drop occurs in the exit step from L(x)to B(x)=Sn(x)(x), followed by the start of the next block. Viewed on the Syracuse time scale, this creates a characteristic “zig-zag” pattern: steady growth inside each wedge, followed by a sharp descent at each block boundary. Corollary 4.9 (Downward block-to-block drift condition). r≥ ⌊nlog2(3/2) + 1⌋+ 1 =⇒B(x)< x Proof. Let α:= nlog2(3/2) + 1. Since α < ⌊α⌋+ 1 and r∈Z, the hypothesis r≥ ⌊α⌋+ 1 implies r > α, i.e. r > n log2(3/2) + 1. Therefore B(x)< x by Theorem 4.7. 23 4.4 Stairs and exits We now make the staircase nature of intra-block dynamics precise. Recall the stair/exit terminology from Definition 3.6. Lemma 4.10 (Exit valuations).Let xbe odd with Mersenne tail length n(x) = nand exit exponent r(x) = r. Then the exit step completing the block transition, L(x)=Sn−1(x)7→ B(x) = Sn(x), has valuation s(L(x)) = 1 + r. Proof. By Lemma 4.2, we have 3L(x) + 1 = 2(3na(x)−1). Since r=ν2(3na(x)−1), it follows that ν2(3L(x)+1)=1+r. But s(L(x))=ν2(3L(x) + 1) by definition. Proposition 4.11 (Stairs per block).Let xbe odd with Mersenne tail length n=n(x)≥2, and define J(x):=n(x)−1. Then: •the Mersenne block rooted at xcontains exactly J(x)stairs, namely the steps Sj(x)7→ Sj+1(x)for 0≤j≤n−2; •the block transition from this block to the next is completed by a single exit step L(x)7→ B(x), whose valuation is 1+r(x). Proof. The first claim is exactly Lemma 3.7. The second claim is Lemma 4.10. Corollary 4.12 (Exact time cost of one block transition).Let xbe odd and set n:= n(x) and r:= r(x). Then the start-to-start block transition x7→ B(x)consists of: •exactly nSyracuse steps; •exactly n+rTerras steps; •exactly 2n+rCollatz steps. Equivalently, B(x)=Sn(x) = Tn+r(x)=C2n+r(x). Proof. Write the odd Syracuse segment completing the block transition as x0:= x, xj+1 := S(xj) (0 ≤j≤n−1),so xn=B(x). By Lemma 3.7, the first n−1steps are stairs, so s(xj) = ν2(3xj+ 1) = 1 for 0≤j≤n−2. By Lemma 4.10, the exit step has valuation s(xn−1) = 1 + r. 24 For an odd input u, the relation 3u+ 1 = 2s(u)S(u)shows that the odd-to-odd segment u7→ S(u)consists of s(u)Terras steps (one odd Terras step followed by s(u)−1halving steps) and 1 + s(u)Collatz steps (one 3u+ 1 step followed by s(u)halvings). Summing over the nSyracuse steps in the block gives τT(x→B(x)) = n−1 X j=0 s(xj) = (n−1) ·1+(1+r)=n+r, and τC(x→B(x)) = n−1 X j=0 1+s(xj)= (n−1) ·2+(2+r)=2n+r. Hence B(x) = Tn+r(x)and B(x)=C2n+r(x), while B(x)=Sn(x)holds by definition. 4.5 Residue-class constraints and the image of B The block coordinates (n(x), a(x), r(x)) make several modular features essentially one-line consequences of the transition formula B(x) = (3n(x)a(x)−1)/2r(x)(Lemma 4.2). Lemma 4.13 (Congruence characterization of the exit exponent).Let xbe odd and write x= 2na−1with n=n(x)≥1and a=a(x)odd, and let r=r(x)=ν2(3na−1). Then for any integer t≥1: (i) r≥tif and only if 3na≡1 (mod 2t)(equivalently, a≡3−n(mod 2t)); (ii) r=tif and only if 3na≡1+2t(mod 2t+1)(equivalently, a≡3−n(1+2t) (mod 2t+1)), where 3−ndenotes the (unique) inverse of 3nmodulo 2t(or 2t+1). Proof. By definition, r=ν2(3na−1) means that 2r|(3na−1) but 2r+1 ∤(3na−1). Thus r≥tis equivalent to 2t|(3na−1), i.e. 3na≡1 (mod 2t), proving (i). Similarly, r=t is equivalent to 3na−1≡2t(mod 2t+1), i.e. 3na≡1 + 2t(mod 2t+1), proving (ii). Since gcd(3n,2m) = 1 for all m, the inverse 3−n(mod 2m)exists and is unique. Corollary 4.14 (A geometric residue-class distribution for rat fixed n).Fix n≥1. For each t≥1there is exactly one odd residue class a(mod 2t+1)for which r(2na−1) = t. Equivalently, there is exactly one residue class x(mod 2n+t+1)such that n(x) = nand r(x)=t. Consequently, among the 2todd residue classes modulo 2t+1, exactly one yields r=t. In particular, within the set {x:n(x)=n}the relative frequency of r=t(per full modulus period) is 2−t, and the relative frequency of r≥tis 2−(t−1). Proof. By Lemma 4.13(ii), the condition r=tis the single congruence 3na≡1+2t (mod 2t+1). Since 3nis invertible modulo 2t+1, this congruence has a unique solution a (mod 2t+1). The right-hand side 1+2tis odd, and 3nis odd, so the unique solution class is also odd. 25 Proof. From Corollary 4.12, the number of Collatz steps in a block transition is exactly 2n(x)+r(x). Under the independent geometric model (Definition 5.4), we have E[N]=2 and E[R] = 2. By linearity of expectation, E[2N+R]=2E[N]+E[R] = 2(2) + 2 = 6. Remark 5.8 (Operational utility of coarse-graining).This result quantifies the efficiency of the Mersenne Block framework. On average, iterating the block map Bcompresses the dynamics by a factor of roughly 6compared to the original Collatz map C. This explains the significant reduction in total stopping times observed in Table 1(where τC(27) = 111 and τB(27) = 17, a ratio of ≈6.5) and validates Bas a computationally effective acceleration. Remark 5.9 (Arithmetic vs. logarithmic contraction).This result recovers the classical probabilistic heuristic for the 3x+ 1 map but in block coordinates. Note the dichotomy: •In arithmetic mean, the model is critical: E[3N/2N+R] = 1. •In logarithmic mean, the model is contractive: E[∆] ≈ −0.83. This tension mirrors the phenomenon utilized by Tao [15] in the analysis of typical orbits using logarithmic density; the distribution of these logarithmic densities is visualized in Figure 6. In the Mersenne block framework, this drift arises from simple intrinsic statistics: an average block has length 2(adding ≈1.17 bits of growth via 32) but an average exit adds 2divisions by 2(removing 2bits), resulting in a net loss of ≈0.83 bits. Proposition 5.10 (Frequency of upward steps).The probability that a random block step increases the value (i.e., ∆>0) is Pr(∆ >0) = X n≥1 2−n1−2−⌊nlog2(1.5)⌋≈0.286. Thus, roughly 71% of Mersenne block transitions are contractive. Proof. For each n,∆>0iff R≤ ⌊nlog2(3/2)⌋, so Pr(∆ >0|N=n)=1−2−⌊nlog2(3/2)⌋, then sum over nwith weights 2−n. 32 Figure 6: Histogram of block log-increments ∆ = log2(B(x)/x)for odd x≤200,000. The distribution is centered near the theoretical expected drift of ≈ −0.83 bits per step, illustrating the net contraction of the Mersenne block map. Conclusion, outlook, and prospects Mersenne block dynamics reorganizes the accelerated Collatz map into two sharply separated phases: a completely deterministic intra-block regime and an irregular exit at the block boundary. Every odd integer xadmits the canonical tail decomposition x= (a(x)−1)2n(x)+2n(x)−1, n(x)=ν2(x+ 1), and this partitions the Syracuse orbit into Mersenne blocks of length n(x). Inside a block we have the closed form Sj(x)=3ja(x) 2n(x)−j−1 (0 ≤j≤n(x)−1), so the odd values increase strictly while the Mersenne-tail length decreases by one at each step, producing the rigid “wedge” pattern in the trailing bits. All nontrivial behavior is therefore concentrated in the single exit step, measured by the exponent r(x)=ν2 3n(x)a(x)−1, and encoded by the induced block map B(x) = Sn(x)(x). The explicit transition identity B(x) = 3n(x)a(x)−1 2r(x) 33 (and its equivalent ratio form in Corollary 4.5) isolates the dominant multiplicative factor 3n(x)/2n(x)+r(x)together with an exact correction term. At the level of time scales, Lemma 2.9 relates total stopping times across the hierarchy C→T→S→B, and Theorem 2.6 restates Collatz equivalently as eventual absorption at 1under iteration of B. Finally, the intrinsic statistics analyzed in Section 5demonstrate the predictive power of this framework. We established that the block length n(x)follows an exact geometric distribution, and that a heuristic independence model for the pair (n(x), r(x)) successfully recovers the standard probabilistic prediction of the Collatz map, yielding an expected logarithmic drift of ≈ −0.83 bits per block step. Outlook. From the Mersenne block perspective, the central obstacle is no longer intra-block complexity (which is rigid and explicit), but rather understanding how the exit exponent r(x)behaves along B-orbits xk+1 =B(xk). Concrete directions suggested by the framework include: •Residue-class dynamics and mixing for the induced map B.Both n(x) = ν2(x+ 1) and r(x)=ν2(3n(x)a(x)−1) are defined by 2-adic valuations, so a natural target is to study the induced action of Bon odd residue classes modulo 2m(for example via the evolution of a(x)=(x+1)/2n(x)mod 2m). Establishing equidistribution or mixing statements at fixed 2-adic scales along typical block orbits would provide a rigorous pathway to orbit-level statistics in block coordinates. •From intrinsic (static) statistics to orbit statistics. Section 5proves an exact residueclass fact: for a uniformly random odd integer x(in natural density), the block parameters satisfy Pr(n(x) = n, r(x) = r)=2−(n+r), so n(x)and r(x)are independent geometric(1/2) random variables at a single block start (Proposition 5.2). The remaining heuristic input needed for drift predictions is therefore dynamical: one seeks conditions under which successive block starts along a typical B-orbit sample these residue classes with sufficiently weak dependence (Heuristic 5.3). •Deterministic descent criteria. The sufficient condition in Theorem 4.7 shows how large exits can overwhelm the systematic (3/2)n(x)growth inside a block. Strengthening such criteria—for example, by relating r(x)to constraints on a(x)or on residues modulo 2m—could yield new deterministic mechanisms for forcing net contraction across blocks, complementing probabilistic drift heuristics. •Refining the probabilistic bridge and computational experiments. The exact ratio identity (Corollary 4.5) expresses B(x)/x as a dominant factor 3n(x)/2n(x)+r(x)times an explicit correction term. Future work could incorporate this correction and possible residue-class dependencies into more refined (e.g. Markov) block models, and test such refinements using statistics gathered along long B-orbits rather than only over all odd integers in an interval. 34 Prospects. Problem 5.11 (Finite-time block mixing conditional on long orbits).Fix m≥1. For “typical” starting values x0with large block stopping time τB(x0), the sequence of residues a(xk) mod 2mfor 0≤k≤Kbecomes close to uniform on odd residue classes as K→ ∞ with K≪τB(x0). In particular, the empirical distribution of (n(xk), r(xk)) over 0≤k≤K approaches µ(n, r) = 2−(n+r). Problem 5.12 (Diophantine Constraints on Block Cycles).A non-trivial cycle corresponds to a finite sequence of block parameters ((n0, r0),...,(nk−1, rk−1)) satisfying the exact closure condition xk=x0. While (ni, ri)are the primary discrete parameters, the cycle constraint depends explicitly on the evolving odd factors a(xi). Using the exact ratio identity (Corollary 4.5), the closure condition becomes: k−1 Y i=0 3ni 2ni+ri· k−1 Y i=0 1−1 3nia(xi) 1−1 2nia(xi) = 1. This structurally separates the dominant scaling factor from the small arithmetic corrections. A key prospect is to utilize the modular rigidities of r(x)(Corollary 4.14) to derive strong restrictions on admissible parameter sequences and to obstruct broad families of potential block cycles. Problem 5.13 (Diffusion and Maximum Excursion).While the expected logarithmic drift E[∆] ≈ −0.83 in the heuristic model suggests global descent, the variance of the block transitions determines the rate of diffusion. A further goal is to establish a central limit theorem for τB(x)when xis sampled uniformly from odd integers up to X(or in logarithmic density), as X→ ∞, thereby providing probabilistic bounds on the maximum excursion maxkBk(x)relative to the starting value x. Ultimately, the Mersenne Block Dynamics Framework offers more than a change of variables: it provides a canonical coarse-graining of the dynamics determined directly by the binary structure of the input. By packaging forced “stair” runs into the return-time n(x)and isolating the arithmetic complexity into the single exit exponent r(x), MBDF disentangles the deterministic wedge dynamics from the irregular behavior of the orbit. This separation yields explicit closed forms for block transitions and a rigorous modular description of the image and inverse families (Theorem 4.17), features that are obscured in step-by-step valuation encodings. These coordinates thus supply a natural interface between exact congruence structure and probabilistic heuristics, offering a tractable vantage point for future analytic bounds on the 3x+ 1 problem. Acknowledgements I acknowledge the use of Large Language Models (LLMs) in assisting me in bringing my intuitions, insights, and ideas in this manuscript to fruition. Responsibility for the conceptual and analytical content and for any errors or omissions herein is mine. 35 References [1] Bernstein, D. J., & Lagarias, J. C.,The 3x+ 1 conjugacy map, Canad. J. Math. 48 (1996), no. 6, 1154–1169. [2] Chamberland, M.,A continuous extension of the 3x+1 problem to the real line, Dyn. Contin. Discrete Impuls. Syst. 2(1996), no. 4, 495–509. [3] Cohen, H.,A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer-Verlag, Berlin, 1993. [4] Conway, J. H.,Unpredictable Iterations, in Proc. 1972 Number Theory Conference (Univ. of Colorado, Boulder), 1972, pp. 49–52. [5] Hercher, C.,There are no Collatz m-cycles with m≤91, J. Integer Seq. 26 (2023), Article 23.3.5. [6] Konstadinidis, P. B.,The real 3x+ 1 problem, Acta Arith. 122 (2006), no. 1, 35–44. [7] Krasikov, I., & Lagarias, J. C.,Bounds for the 3x+ 1 problem using difference inequalities, Acta Arith. 109 (2003), no. 3, 237–258. [8] Lagarias, J. C.,The 3x+ 1 problem and its generalizations, Amer. Math. Monthly 92 (1985), no. 1, 3–23. [9] Lagarias, J. C. (ed.), The Ultimate Challenge: The 3x+ 1 Problem, American Mathematical Society, 2010. [10] Leavens, G. T., & Vermeulen, M.,3x+ 1 search programs, Comput. Math. Appl. 24 (1992), no. 11, 79–99. [11] Letherman, S., Schleicher, D., & Wood, R.,The 3n+1-problem and holomorphic dynamics, Experiment. Math. 8(1999), no. 3, 241–251. [12] Möller, H.,Über Hasses Verallgemeinerung des Syracuse-Algorithmus, Arch. Math. 31 (1978), no. 1, 21–33. [13] Oliveira e Silva, T.,Empirical verification of the 3x+ 1 and related conjectures, in The Ultimate Challenge: The 3x+ 1 Problem (J. C. Lagarias, ed.), AMS, 2010, pp. 189–207. [14] Rosen, K. H.,Elementary Number Theory and Its Applications, 6th ed., Pearson, 2011. [15] Tao, T.,Almost all orbits of the Collatz map attain almost bounded values, Forum Math. Pi 10 (2022), e12. [16] Terras, R.,A stopping time problem on the positive integers, Acta Arith. 30 (1976), no. 3, 241–252. [17] Wirsching, G. J.,The Dynamical System Generated by the 3n+ 1 Function, Lecture Notes in Math. 1681, Springer, 1998. [18] Yolcu, E., Aaronson, S., & Heule, M. J. H.,An Automated Approach to the Collatz Conjecture, in Automated Deduction – CADE 28, Lecture Notes in Comput. Sci. 12699, Springer, 2021, pp. 428–445. 36 Appendix: Computational Implementation and Reproducibility A.1 Mersenne Block Algorithm The Mersenne Block framework allows for efficient orbit calculation by skipping the intrablock “stair” steps. A standard Syracuse implementation computes every odd step; a Block implementation computes only the block starts. Algorithm 1 Mersenne Block Step Input: Odd integer x n←count_trailing_zeros(x+ 1) a←(x+ 1) ≫n y←3n·a−1 r←count_trailing_zeros(y) B(x)←y≫r return B(x) For large x, the dominant cost is the multiplication 3n·a. Because n(x)is typically small (mean 2), this step is computationally inexpensive compared to iterating the underlying 2n+rCollatz steps individually. A.2 Orbit-level diagnostics for residue mixing and weak dependence To empirically probe the orbit-level hypotheses central to the probabilistic bridge (Heuristic 5.3 and Problem 5.11), we computed simple diagnostics along a single B-orbit xk+1 = B(xk)of length N= 100,000 block steps. Computational setup. The starting value x0was chosen as a random odd integer of bitlength L≈200,000 (to ensure the orbit did not collapse to 1within the first Nblock steps due to logarithmic drift). We sampled x0uniformly from the odd integers in [2L−1,2L) by choosing a uniform integer u∈ {0,...,2L−1−1}and setting x0= 2L−1+ 2u+ 1. Residue mixing. We tested equidistribution of the odd factor a(x) := (x+1)/2n(x)modulo 2malong the orbit. For m= 6 (so 2m= 64, with 32 odd residue classes), uniformity predicts an expected count E=N/32 = 3125 per bin. Using the observed counts from Figure 7, Pearson’s chi-square statistic is χ2= 32 X i=1 (Oi−E)2 E≈26.64 (df = 31), 37 with p≈0.69 (interpreted here as a descriptive goodness-of-fit measure rather than a formal i.i.d. significance test), consistent with uniform sampling of odd residue classes modulo 64 along this orbit. Lag-1 linear dependence checks. To probe the i.i.d. modeling assumption in Definition 5.4, we computed lag-1 sample correlations for block parameters (nk, rk)=(n(xk), r(xk)): ρ(nk, nk+1)≈ −0.0018, ρ(rk, rk+1)≈0.0023, ρ(nk, rk)≈ −0.0005, ρ(nk, rk+1)≈ ··· As a scale reference, under a simple null of negligible dependence one expects typical fluctuations of order 1/√N≈0.0032 for correlations at this sample size. Thus, we detect no meaningful lag-1 linear dependence in these basic statistics on this trajectory. 1 9 17 25 33 41 49 57 63 2,800 3,000 3,200 3,400 Odd Residue Class a(x) (mod 64) Frequency Count Uniformity of a(xk)along a single B-orbit (N= 105) Expected Uniformity Observed Counts Figure 7: Empirical histogram of a(xk) (mod 64) along a single block orbit of N= 100,000 steps. The counts are consistent with uniformity (χ2≈26.64,p≈0.69,df = 31), and basic lag-1 correlations for (nk, rk)are near zero (see text). Vertical axis truncated for readability. A.3 Data and Code Availability All computations were performed using Python 3.8 with numpy; some figures are rendered directly in LaTeX (PGFPlots) from the computed data. The full reproduction code, including the static distribution analysis (Section 5) and the dynamical orbit validation (Appendix A.2), is provided in the supplementary material file mbd_supplementary.py. 38