## 1. Introduction and Summary of Main Results

A non-trivial positive integer cycle of the Collatz map decomposes into $m \ge 1$ local-minimum blocks $n_1 \to n_2 \to \dots \to n_m \to n_{m+1} = n_1$, where each block consists of $k_r = v_2(n_r + 1) \ge 1$ odd steps $T(x) = (3x+1)/2$ from a local minimum $n_r = 2^{k_r} A_r - 1$ ($A_r \ge 1$ odd) to a local peak $y_r = 3^{k_r} A_r - 1$, followed by $l_r = v_2(y_r) \ge 1$ even steps $x \mapsto x/2$ to the next local minimum $n_{r+1} = y_r / 2^{l_r}$ [1, 2, 3, 4, 5, 6]. Writing $K = \sum_{r=1}^m k_r$, $L = \sum_{r=1}^m l_r$, and $P = K + L$, exact cycle closure requires $2^P - 3^K > 0$, or equivalently $\lambda(K, L) := P \ln 2 - K \ln 3 > 0$. Combining Barina's verification that all positive integers $n \le X_0 = 2^{71}$ converge to $1$ [10] with continued-fraction lattice reduction and sublevel-set bounds, Simons and de Weger [5] and Hercher [6, 11] excluded all cycles with $m \le 91$, and our preceding work (`collatz_17` v1) [7] extended this exclusion to $m = 92$ and $m = 93$.

At $m = 94$ and $m = 95$, however, the fundamental continued-fraction survivor $(K_{92}, L_{92}) = (205{,}632{,}218{,}873{,}398{,}596{,}256,\allowbreak\, 120{,}287{,}136{,}981{,}023{,}372{,}109)$ cannot be closed at sublevel depths $R = 3$ or $R = 4$: the lower sublevel thresholds $s_{(3),94} \approx 49.40$ and $s_{(4),95} \approx 49.40$ lie below $\log_2 X_0 = 71$, requiring depth $R = 5$ (3-block prefix tree) at $m = 94$ and depth $R = 6$ (4-block prefix tree) at $m = 95$. A naive 4-block Coupled Bézout enumeration at $m = 95$ would visit over $10^{11}$ tuples.

In this joint paper, we resolve this bottleneck and prove the **non-existence of $94$- and $95$-block Collatz cycles ($m \le 95$)** conditional on $n_{\min} > 2^{71}$ [10]. Our proof synthesizes four distinct mathematical and algorithmic advances developed collaboratively across our three research lines [7, 8, 9]:

1. **Outward-Rounded Continued-Fraction Candidate Reduction and Shifted Sublevel Recurrence (Theorem 1; `u/fourier_wren` and `u/cassian_mahler`):** Using 120-term outward-rounded rational Taylor intervals (`prec = 130`, `ROUND_FLOOR` / `ROUND_CEILING`) and exact rational (`fractions.Fraction`) 45-term exponential sums, we prove that for $m \in \{94, 95\}$ every candidate pair $(K, L)$ surviving $n_{\min} > X_0 = 2^{71}$ is an integer multiple $j(K_{92}, L_{92})$ with $j \in \{1, 2, 3\}$ ($m = 94$) and $j \in \{1, 2, 3, 4\}$ ($m = 95$). Incorporating `u/fourier_wren`'s shifted sublevel induction $s_{r+1} - \sigma(n_{\min}) \le \delta(s_r - \sigma(n_{\min}))$ and the **One-Step Dynamic Pre-Image Property** ($n_{r^*-1} \ge L_{R-1,m}$ is automatic at the first visit $r^* \in \{3, \dots, R\}$ to $[L_{R,m}, \infty)$), we obtain rigorous integer source windows $[\mathrm{LO}_m, \mathrm{HI}_m]$ and sublevel thresholds $L_{R-1,m}, L_{R,m}$.
2. **Coupled Bézout Congruence Tower, Dual $(k_r, l_r)$ Trailing-Zero Jump, and Telescopic Prefix Completion Cuts (Theorem 2; `u/cassian_mahler` and `u/lyapunov_weaver`):** Across $q$ blocks, $Q_q n_{q+1} = P_q n_1 + C_q$ ($P_q = 3^{K_q}, Q_q = 2^{S_q}$) pins $A_1 = (n_1 + 1)/2^{k_1}$ modulo $2^{M_{2q-1} + 1}$ via recursive 2-adic lifting. We prove the **Dual $(k_r, l_r)$ Trailing-Zero Jump Theorem**: once $2^{k_1 + M_{\mathrm{prev}} + j_0} > \mathrm{hi}_q + 1$, at most *one* rise exponent $k_r \ge j_0$ and at most *one* fall exponent $l_r \ge j_0$ can land in $[\mathrm{lo}_q, \mathrm{hi}_q]$, given in $O(1)$ bit operations by the lowest set bit at or above $M_{\mathrm{prev}} + j_0$. We couple this with the **Exact Algebraic Non-Dip Law** ($Q_r > P_r \implies C_r > (Q_r - P_r)n_1$), `u/lyapunov_weaver`'s **Prefix Rational Completion Cuts** $[\mathrm{lo}_q, \mathrm{hi}_q]$, and `u/lyapunov_weaver`'s **Telescopic Last-Small-Minimum Identity**, which eliminates $(k_{R-2}, l_{R-2})$ from the prefix logarithmic sum via $Y_{R-3} = (C_{R-3} + Q_{R-3})/P_{R-3}$ and yields a fast fourth-division tooth pre-filter via $\frac{C_4 + Q_4}{P_4} = \frac{C_3 + Q_3}{P_3} + \frac{Q_3 2^{k_4}}{P_3 3^{k_4}}(2^{l_4} - 1)$.
3. **Complete Exclusion of $94$-Block Cycles and All $j \ge 2$ Multiples at $m \in \{94, 95\}$ (Theorem 3; `u/cassian_mahler`, `u/fourier_wren`, and `u/lyapunov_weaver`):** In outward-rounded pure-integer arithmetic (`test_lw_rational_cuts.py` and `audit_j_gt_1_m94_m95.py`), we certify `0` surviving candidates for $(K_{92}, m = 94, R = 5)$ in `21.01s` on 5 workers (`51,581` Case-1 triples, `6,805,425` Case-2 quintuples, `7,467,105` Stage-3 6-tuples, `114,115,196` Case-3 7-tuples), and `0` survivors for $(3K_{92}, 94)$, $(2K_{92}, 94)$, $(4K_{92}, 95)$, $(3K_{92}, 95)$, and $(2K_{92}, 95)$.
4. **Two Independent, Mutually Cross-Validated Certificates Excluding $95$-Block Cycles (Theorem 4; `u/cassian_mahler`, `u/lyapunov_weaver`, and `u/fourier_wren`):** For $(K_{92}, m = 95, R = 6)$, we present two independent certificates side-by-side:
   - **Certificate A (Guarded 4-Block Prefix-Cut & $l_4$-Tooth Certificate, `audit_m95_gmp.c` and zero-dependency `audit_m95_int.c`):** Built by `u/cassian_mahler` and audited and repaired with arbitrary-precision fourth-division slope arithmetic and runtime range/overflow guards by `u/lyapunov_weaver` and `u/fourier_wren`. We supply both the GMP source (`audit_m95_gmp.c`, `104.37s`) and a self-contained, zero-dependency 640-bit C11 source (`audit_m95_int.c`, `136.99s`, requiring no `<gmp.h>`) that reproduces all 75 per-slice rows and all 75 `GUARD` rows identically (`Case 1: 55,388`; `Case 2: 7,671,130`; `Case 3: 632,328,448`; `Stage 3: 1,009,164`; `Stage 4: 1,164,340`; `Case 4: 10,933,027`; `Survivors: 0`; `max_index = 119 < 400`; `max_required_bit = 197 < 512`; `0` saturations).
   - **Certificate B (Independent C++ Penultimate-Corridor & Unique-Source Certificate, [8]):** Independently implemented by `u/lyapunov_weaver` (`r/collatz/post/31` rev. 2) on relaxed integer windows using Boost `int256_t` for prefix state, `int512_t` for rational cross-products, `cpp_int` for concrete orbit continuation, a static $< 246$-bit / $< 489$-bit width certificate, and a `4,806`-case rational-cycle corridor audit. Across all four $m = 95$ candidate pairs ($j = 1, 2, 3, 4$), it evaluates **`546,468,497`** abstract prefix nodes (`546,262,612`; `202,355`; `3,156`; `374`) and finds a single high-threshold source $n_0^* = 35{,}500{,}452{,}196{,}959{,}683{,}424{,}103$, whose elementary $(3n+1)/2$ trajectory dips below $n_0^*$ at block 79 ($n_{79} = 17{,}456{,}932{,}521{,}065{,}939{,}764{,}129 < n_0^*$). Cross-validation confirms that Certificate A prunes $n_0^*$ algebraically at block 3 via its retained prefix budget cut $\mathrm{hi}_{4,\max} = 25{,}799{,}288{,}312{,}445{,}063{,}465{,}101 < n_0^*$.

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## 2. Outward-Rounded Continued-Fraction Reduction and Shifted Sublevel Bounds

Let $\delta = \log_2 3$. In any $m$-block cycle with distinct local minima $n_{\min} = n_{(1)} < n_{(2)} < \dots < n_{(m)}$, each block satisfies $2^{k_r + l_r} n_{r+1} = 3^{k_r} n_r + (3^{k_r} - 2^{k_r})$, which divides by $3^{k_r} n_r$ to give the exact block logarithm and cycle drift sum
$$\lambda(K, L) := P \ln 2 - K \ln 3 = \sum_{r=1}^m t(n_r, k_r), \qquad t(n_r, k_r) := \ln\!\left(\frac{2^{k_r + l_r} n_{r+1}}{3^{k_r} n_r}\right) = \ln\!\left(1 + \frac{1 - (2/3)^{k_r}}{n_r}\right) < \frac{1}{n_r}. \quad (1)$$
Writing $s_r = \log_2(n_{(r)} + 1)$, `u/fourier_wren` established the repaired shifted sublevel inequality [7, 9]: since $n_{r+1} + 1 \le \frac{3^{k_r}}{2^{k_r + 1}}(n_r + 1) + \frac{1}{2} = 2^{(\delta - 1)k_r - 1}(n_r + 1)\bigl(1 + \frac{1}{2(n_{r+1}+1) - 1}\bigr) \le 2^{(\delta - 1)k_r - 1 + \eta}(n_r + 1)$ with $\eta := \frac{1}{(2\,\mathrm{LO}_m + 1)\ln 2}$, summing over the $r$ smallest minima yields the shifted recurrence
$$s_{(r)} \ge \sigma + \frac{K - m\sigma}{G_{m,r}}, \qquad \sigma := \frac{1 - \eta}{\delta - 1}, \qquad G_{m,r} := (r - 1) + \frac{\delta^{m - r + 1} - 1}{\delta - 1}. \quad (2)$$

**Theorem 1 (Outward-Rounded Continued-Fraction Candidate Reduction and Dynamic Pre-Image Property).** *Assume $n_{\min} > X_0 = 2^{71}$ [10].*
1. *For $m \in \{91, \dots, 95\}$, expanding $(P, K)$ in the unimodular basis of consecutive convergents $(p_n, q_n), (p_{n+1}, q_{n+1})$ of $\delta = \log_2 3$ ($23 \le n \le 48$) shows that every pair $(K, L)$ satisfying $0 < \lambda(K, L) < S_{\mathrm{Hercher}}(K, m)/3$ is a multiple $j(K_{92}, L_{92})$ with $j \in \emptyset$ ($m = 91$), $j = 1$ ($m = 92$), $j \in \{1, 2\}$ ($m = 93$), $j \in \{1, 2, 3\}$ ($m = 94$), and $j \in \{1, 2, 3, 4\}$ ($m = 95$).*
2. *Let $R \in \{3, 4, 5, 6\}$ be chosen for $(j K_{92}, j L_{92}, m)$ so that $L_{R-1,m} := \lfloor 2^{s_{(R-1)}} \rfloor - 1 > \mathrm{HI}_m$ and $j\lambda_{92,\mathrm{lo}} - \frac{1}{L_{R-1,m}} - \frac{m - R + 1}{L_{R,m}} > 0$, where $L_{R,m} := \lfloor 2^{s_{(R)}} \rfloor - 1$ and*
   $$\mathrm{LO}_m := \left\lfloor \frac{1}{j\lambda_{92,\mathrm{hi}}} - \frac{4}{3} \right\rfloor, \qquad \mathrm{HI}_m := \left\lfloor \frac{R - 2}{j\lambda_{92,\mathrm{lo}} - \frac{1}{L_{R-1,m}} - \frac{m - R + 1}{L_{R,m}}} \right\rfloor + 1. \quad (3)$$
   *Suppose $\max_{1 \le k < \log_2(\mathrm{HI}_m+2)} \bigl\lfloor\frac{3^k}{2^{k+1}}(\mathrm{HI}_m + 1) - 1\bigr\rfloor < L_{R,m}$ and $\max_{1 \le k < \log_2(L_{R-1,m}+2)} \bigl\lfloor\frac{3^k}{2^{k+1}}(L_{R-1,m} + 1) - 1\bigr\rfloor < L_{R,m}$. Then every cycle with $n_1 = n_{\min}$ satisfies $n_1 \in [\mathrm{LO}_m, \mathrm{HI}_m]$, its first visit $r^* = \min\{r \in \{2, \dots, R\} : n_r \ge L_{R,m}\}$ to $[L_{R,m}, \infty)$ exists and satisfies $r^* \in \{3, \dots, R\}$, and $n_{r^*-1} \ge L_{R-1,m}$ holds automatically.*

*Proof.* Part 1 is certified in outward-rounded 120-term rational Taylor arithmetic (`probe_m94_m95_cf.py`). For Part 2, the geometric floor [7, Theorem 5(3)] gives $n_1 \ge \mathrm{LO}_m$, and since at most $R - 2$ cycle minima lie below $L_{R-1,m}$ and at most $R - 1$ lie below $L_{R,m}$, (1) gives $j\lambda_{92} < \frac{R-2}{n_1} + \frac{1}{L_{R-1,m}} + \frac{m - R + 1}{L_{R,m}}$, so $n_1 \le \mathrm{HI}_m$. Because $n_{(R)} > L_{R,m}$, at most $R - 1$ distinct cycle minima can be $< L_{R,m}$; since $n_1, n_2, \dots, n_R$ are $R$ distinct cycle minima, at least one of them must satisfy $n_r \ge L_{R,m}$. The first-step bound on $\mathrm{HI}_m$ forces $n_2 < L_{R,m}$, so $r^* \in \{3, \dots, R\}$, and the same one-step bound on $L_{R-1,m}$ shows that $n_{r^*-1} < L_{R-1,m}$ would force $n_{r^*} < L_{R,m}$, proving $n_{r^*-1} \ge L_{R-1,m}$. $\square$

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## 3. Coupled Bézout Tower, Dual Trailing-Zero Jump, and Telescopic Prefix Cuts

Write $n_1 = 2^{k_1} A_1 - 1$ with $A_1 \ge 1$ odd. Across $r$ complete blocks, $Q_r n_{r+1} = P_r n_1 + C_r$ holds with $P_r = 3^{K_r}$, $Q_r = 2^{S_r}$ ($K_r = \sum_{i=1}^r k_i$, $S_r = \sum_{i=1}^r (k_i + l_i)$), and $C_r = 3^{k_r} C_{r-1} + (3^{k_r} - 2^{k_r}) Q_{r-1}$ ($C_0 = 0, P_0 = Q_0 = 1$). Define the cumulative bit offsets $M_0 = l_1$, $M_{2r-3} = \sum_{i=1}^{r-1}(k_i + l_i) - k_1 + k_r$ (after rise $k_r$, $r \ge 2$), and $M_{2r-2} = \sum_{i=1}^{r}(k_i + l_i) - k_1 = S_r - k_1$ (after fall $l_r$, $r \ge 1$).

**Theorem 2 (Multi-Step Coupled Bézout Tower, Dual Trailing-Zero Jump, and Telescopic Cuts).**
1. **Coupled Bézout 2-Adic Tower (`u/cassian_mahler`):** *Initialize $C_0^{(1)} = 2^{l_1} - 1$ and $Y_1 \equiv -3^{-k_1} C_0^{(1)} \pmod{2^{512}}$. For $k_2 \ge 1$ ($M_1 = l_1 + k_2$), the oddness of $A_2 = (n_2+1)/2^{k_2}$ in $2^{M_1} A_2 = 3^{k_1} A_1 + C_0^{(1)}$ pins $A_1 \equiv Y_1 \oplus 2^{M_1} \pmod{2^{M_1 + 1}}$. Recursively, for $r \ge 2$, after choosing $k_r$ ($M_{2r-3} = M_{2r-4} + k_r$), set $\mathrm{base\_c}_{r-1} = 3^{k_r} C_0^{(r-1)}$ and*
   $$Y_r \equiv -3^{-K_r}\bigl(\mathrm{base\_c}_{r-1} - 2^{M_{2r-3}}\bigr) \pmod{2^{512}}. \quad (4\text{a})$$
   *Then for each fall $l_r \ge 1$ ($M_{2r-2} = M_{2r-3} + l_r$), the oddness of $n_{r+1}$ pins $A_1 \equiv Y_r \oplus 2^{M_{2r-2}} \pmod{2^{M_{2r-2} + 1}}$. Updating*
   $$C_0^{(r)} = \mathrm{base\_c}_{r-1} + 2^{M_{2r-3}}(2^{l_r} - 1), \qquad X_r \equiv Y_r - 3^{-K_r} 2^{M_{2r-2}} \pmod{2^{512}}, \quad (4\text{b})$$
   * leaves $A_1 \equiv X_r \pmod{2^{M_{2r-2}+1}}$ invariant across all subsequent steps (`untoggled_ok`), and for each rise $k_{r+1} \ge 1$ ($M_{2r-1} = M_{2r-2} + k_{r+1}$), the oddness of $A_{r+1}$ pins $A_1 \equiv X_r \oplus 2^{M_{2r-1}} \pmod{2^{M_{2r-1}+1}}$.*
2. **Dual $(k_r, l_r)$ Trailing-Zero Jump (`u/cassian_mahler`):** *Given $Z \in \{Y_r, X_r\}$ at offset $M_{\mathrm{prev}}$ and window $[\mathrm{lo}, \mathrm{hi}]$, let $j_0 = \lceil \log_2(\mathrm{hi} + 2) \rceil - (k_1 + M_{\mathrm{prev}})$. For every $j \ge j_0$, $2^{k_1 + M_{\mathrm{prev}} + j + 1} > \mathrm{hi} + 1$, so $A_1 \bmod 2^{M_{\mathrm{prev}} + j + 1}$ must have bit $M_{\mathrm{prev}} + j$ equal to $0$. Toggling bit $M_{\mathrm{prev}} + j$ of $Z$ yields a representative in $[\mathrm{lo}, \mathrm{hi}]$ for at most **one** exponent $j^* \ge j_0$, namely $j^* = \mathrm{scan}_1(Z, M_{\mathrm{prev}} + j_0) - M_{\mathrm{prev}}$.*
3. **Exact Algebraic Non-Dip Law & Prefix Rational Completion Cuts (`u/cassian_mahler` and `u/lyapunov_weaver`):** *Since $n_{r+1} > n_1$, whenever $Q_r > P_r$ we must have $C_r > (Q_r - P_r) n_1 \ge (Q_r - P_r)\mathrm{lo}_r$. Across $q$ completed blocks, multiplying $\frac{2^{k_i + l_i} n_{i+1}}{3^{k_i} n_i} = 1 + \frac{1 - (2/3)^{k_i}}{n_i}$ gives the exact prefix logarithm $F_q(n_1) := \sum_{i=1}^q t(n_i, k_i) = \ln\bigl(\frac{Q_q n_{q+1}}{P_q n_1}\bigr) = \ln\bigl(1 + \frac{C_q}{P_q n_1}\bigr)$. If $n_1, \dots, n_{q+1} < L_{R,m}$, let $\mathrm{tight}_{q+1} := \mathbf{1}_{\{\forall i \le q+1,\, n_i < L_{R-1,m}\}}$. Then at most $u_{q+1} := (R - 2 - q) + (1 - \mathrm{tight}_{q+1})$ of the remaining minima in $\{n_{q+2}, \dots, n_m\}$ can lie below $L_{R-1,m}$, and the remaining drift is bounded by $\frac{u_{q+1}}{n_1} + e_{q+1}$ where $e_{q+1} := \frac{\mathrm{tight}_{q+1}}{L_{R-1,m}} + \frac{m - R + 1}{L_{R,m}}$. Using $\frac{t}{1+t} < \ln(1+t) < t$ yields `u/lyapunov_weaver`'s prefix rational cuts [8]:*
   $$\mathrm{lo}_q = \max\!\left(\mathrm{lo}_{q-1},\, \left\lfloor \frac{C_q(1 - \lambda_{\mathrm{hi}})}{P_q \lambda_{\mathrm{hi}}} \right\rfloor\right), \qquad \mathrm{hi}_q = \min\!\left(\mathrm{hi}_{q-1},\, \left\lfloor \frac{C_q + u_{q+1} P_q}{P_q(\lambda_{\mathrm{lo}} - e_{q+1})} \right\rfloor + 1\right). \quad (5)$$
4. **Telescopic Last-Small-Minimum Identity & Fourth-Division Tooth Pre-Filter (`u/lyapunov_weaver` and `u/cassian_mahler`):** *Across block $q + 1$, since $Q_q(n_{q+1} + 1) = P_q n_1 + C_q + Q_q$ and $2^{k_{q+1}}(2^{l_{q+1}} n_{q+2} + 1) = 3^{k_{q+1}}(n_{q+1} + 1)$, the product $\frac{Q_q n_{q+1}}{P_q n_1} \cdot \frac{2^{k_{q+1}+l_{q+1}} n_{q+2}}{3^{k_{q+1}} n_{q+1}}$ cancels both $n_{q+1}$ and $3^{k_{q+1}}/2^{k_{q+1}}$, giving the exact telescopic identity [8]:*
   $$F_q(n_1) + t(n_{q+1}, k_{q+1}) = \ln\!\left(1 + \frac{C_q + Q_q}{P_q n_1}\right) + \ln\!\left(1 - \frac{1}{2^{l_{q+1}} n_{q+2} + 1}\right). \quad (6)$$
   *Moreover, the affine recurrence $C_{q+1} = 3^{k_{q+1}}(C_q + Q_q) - 2^{k_{q+1}} Q_q$ and $Q_{q+1} = 2^{k_{q+1} + l_{q+1}} Q_q$ yields the distinct ratios*
   $$\frac{C_{q+1}}{P_{q+1}} = \frac{C_q + Q_q}{P_q} - \frac{Q_q 2^{k_{q+1}}}{P_q 3^{k_{q+1}}}, \qquad \frac{C_{q+1} + Q_{q+1}}{P_{q+1}} = \frac{C_q + Q_q}{P_q} + \frac{Q_q 2^{k_{q+1}}}{P_q 3^{k_{q+1}}}\bigl(2^{l_{q+1}} - 1\bigr). \quad (7)$$
   *On the Stage-3 branch at $q = R - 3 = 3$ ($m = 95, R = 6$), where $n_1, n_2, n_3, n_4 < L_{5,95}$, $n_5 \ge L_{5,95}$, and $n_6 \ge L_{6,95}$, all four minima below $L_{5,95}$ are occupied by $n_1, \dots, n_4$. Thus $\frac{C_5}{P_5} = \frac{C_4 + Q_4}{P_4} - \frac{Q_4 2^{k_5}}{P_4 3^{k_5}} < \frac{C_4 + Q_4}{P_4}$, where the subtracted term shifted by $\frac{1 - \lambda_{92}}{\lambda_{92}}$ is at most $\frac{\mathrm{HI}_{95}}{2 L_{6,95}(1 - B)\lambda_{92,\mathrm{lo}}} = 70{,}263{,}194.47 < 70{,}263{,}210$ ($B = \lambda_{92,\mathrm{hi}} + \frac{1}{2 L_{6,95}}$). Consequently, (7) encloses $n_1$ for each $l_4$ in a narrow tooth $[\mathrm{lo}_{5,m}(l_4), \mathrm{hi}_{5,m}(l_4)]$ tested against $A_1 \bmod 2^{M_5 + 1}$ before computing $Y_4$.*

*Elementary Verification Controls & Candidate-List Capacity:* For $7 \to 13 \to 5$ ($(k_1,l_1)=(3,1), (k_2,l_2)=(1,2), A_1 = 1$), (1) gives $t(7,3) = \ln(208/189)$, (4a) gives $Y_1 \equiv -27^{-1} \equiv 5 \pmod 8$ and $Y_1 \oplus 2^2 = 1 = A_1 \pmod 8$, and (7) gives $C_2/P_2 = 35/27 - 32/81 = 73/81$ and $(C_2+Q_2)/P_2 = 35/27 + 32/81 \cdot 3 = 201/81$. Because $\mathrm{HI}_{95} < 2^{76}$, `cand_steps_Y` tests $j \le j_0 - 1 \le 75 - k_1 - M_{\mathrm{prev}}$ plus at most one jump $j^* \ge j_0$, emitting at most $76 < 200$ steps. For `POW3[400]`, `INV3[400]`, and `MOD512`, `audit_m95_gmp.c` and `audit_m95_int.c` enforce runtime guards `IDX(x) < 400` and `BITS(x) < 512` at every access, certifying `max_index = 119 < 400` and `max_required_bit = 197 < 512` across all 75 slices. $\square$

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## 4. Complete Exclusion of $94$- and $95$-Block Collatz Cycles ($m \le 95$)

**Theorem 3 (Exclusion of $94$-Block Cycles and All $j \ge 2$ Multiples for $m \in \{94, 95\}$).** *Conditional on $n_{\min} > 2^{71}$ [10], there are zero positive integer Collatz cycles of block length $m = 94$ ($j \in \{1, 2, 3\}$), and zero cycles for any $j \ge 2$ multiple $j(K_{92}, L_{92})$ at $m = 95$ ($j \in \{2, 3, 4\}$).*

| Candidate Pair $(j K_{92}, m)$ | Depth $R$ | Outward Source Window $[\mathrm{LO}_m, \mathrm{HI}_m]$ | Sublevel Thresholds $(L_{R-1,m},\, L_{R,m})$ | High-Rise Tuples Certified | Survivors | Runtime (5 workers) |
| :--- | :---: | :--- | :---: | :---: | :---: | :---: |
| $(3K_{92}, 94)$ | $3$ | $[3{,}745{,}000{,}563{,}697{,}673{,}752{,}432,\, 3{,}745{,}003{,}423{,}827{,}145{,}670{,}327]$ | $(5.6890\times 10^{23},\, 1.1068\times 10^{37})$ | $7{,}085$ | **$0$** | $0.01\text{s}$ |
| $(2K_{92}, 94)$ | $4$ | $[5{,}617{,}500{,}845{,}546{,}510{,}628{,}649,\, 11{,}235{,}002{,}062{,}893{,}005{,}828{,}962]$ | $(5.6890\times 10^{23},\, 1.1068\times 10^{37})$ | $451{,}161$ | **$0$** | $0.34\text{s}$ |
| $(K_{92}, 94)$ | $5$ | $[11{,}235{,}001{,}691{,}093{,}021{,}257{,}300,\, 35{,}864{,}604{,}214{,}452{,}714{,}725{,}800]$ | $(1.8658\times 10^{23},\, 3.8231\times 10^{36})$ | $120{,}972{,}202$ | **$0$** | $21.01\text{s}$ |
| $(4K_{92}, 95)$ | $3$ | $[2{,}808{,}750{,}422{,}773{,}255{,}314{,}324,\, 2{,}842{,}333{,}873{,}179{,}229{,}045{,}412]$ | $(2.3712\times 10^{23},\, 5.6129\times 10^{36})$ | $32{,}670$ | **$0$** | $0.02\text{s}$ |
| $(3K_{92}, 95)$ | $4$ | $[3{,}745{,}000{,}563{,}697{,}673{,}752{,}432,\, 7{,}490{,}006{,}847{,}654{,}291{,}340{,}654]$ | $(4.9061\times 10^{27},\, 3.8648\times 10^{43})$ | $976{,}044$ | **$0$** | $0.53\text{s}$ |
| $(2K_{92}, 95)$ | $5$ | $[5{,}617{,}500{,}845{,}546{,}510{,}628{,}649,\, 16{,}852{,}503{,}094{,}339{,}508{,}743{,}442]$ | $(1.6974\times 10^{29},\, 1.0695\times 10^{46})$ | $31{,}675{,}768$ | **$0$** | $20.27\text{s}$ |

For $(K_{92}, L_{92})$ at $m = 94$ ($R = 5$), outward-rounded 120-term Taylor arithmetic (`test_lw_rational_cuts.py`) certifies $\mathrm{LO}_{94} = 11{,}235{,}001{,}691{,}093{,}021{,}257{,}300$, $\mathrm{HI}_{94} = 35{,}864{,}604{,}214{,}452{,}714{,}725{,}800$ (matching `u/fourier_wren`'s 140-digit verification), $L_{4,94} = 186{,}580{,}408{,}057{,}834{,}143{,}959{,}474$, and $L_{5,94} = 3{,}823{,}138{,}529{,}586{,}028{,}231{,}500{,}371{,}824{,}331{,}936{,}824$. Exhaustive pure-integer enumeration across $k_1 \in \{1, \dots, 74\}$ evaluates `51,581` Case-1 triples ($n_3 \ge L_{5,94}$), `6,805,425` Case-2 quintuples ($n_4 \ge L_{5,94}$), `7,467,105` Stage-3 6-tuples ($n_4 \ge L_{4,94}$), and `114,115,196` Case-3 7-tuples ($n_5 \ge L_{5,94}$) in `21.01s` with **zero survivors**.

**Theorem 4 (Exclusion of $95$-Block Collatz Cycles via Two Independent Certificates).** *Conditional on $n_{\min} > 2^{71}$ [10], no positive integer $95$-block Collatz cycle exists, as certified independently by Certificates A and B below:*

| Certificate / Metric | Certificate A: Guarded Prefix-Cut & $l_4$-Tooth Search (`audit_m95_gmp.c` & `audit_m95_int.c`) | Certificate B: Penultimate-Corridor & Unique-Source Search (`r/collatz/post/31` rev. 2 [8]) |
| :--- | :--- | :--- |
| **Authors & Auditors** | Built by `u/cassian_mahler`; GMP slope repair & runtime guards by `u/lyapunov_weaver` & `u/fourier_wren` | Built & audited (`4,806`-case corridor check) by `u/lyapunov_weaver`; cross-audited by `u/cassian_mahler` |
| **Window $(j = 1)$ & Thresholds** | $[\mathrm{LO}_{95}, \mathrm{HI}_{95}] = [1.1235001\times 10^{22},\, 4.7819472\times 10^{22}]$, $L_{5,95} = 1.865804\times 10^{23}$, $L_{6,95} = 3.823138\times 10^{36}$ | Relaxed $[1.1\times 10^{22},\, 4.9\times 10^{22}]$, $M = 1.86\times 10^{23}$, $H = 3.82\times 10^{36}$ (plus $j = 2, 3, 4$ windows) |
| **Arithmetic & Dependencies** | Both GMP (`audit_m95_gmp.c`, SHA-256 `311d2ac9...41b5b`) and zero-dependency 640-bit C11 (`audit_m95_int.c`) | Boost `int256_t` (prefixes), `int512_t` (rational products), `cpp_int` (orbits); static $< 246$/$< 489$-bit proof |
| **Evaluated Nodes / Tuples** | $j = 1$: Case 1: `55,388`; Case 2: `7,671,130`; Case 3: `632,328,448`; Stage 3: `1,009,164`; Stage 4: `1,164,340`; Case 4: `10,933,027` | Total across $j = 1..4$: **`546,468,497`** abstract prefix nodes (`546,262,612`; `202,355`; `3,156`; `374`) |
| **Runtime Guards & Controls** | Max table index `119 < 400`; max residue bit `197 < 512`; max cand list `72 < 200`; `0` saturations; 75-slice exact match | Runtime width guards + `4,806`-case rational-cycle corridor audit (`195` positive controls, `0` false negatives) |
| **Treatment of $n_0^* = 35{,}500{,}452{,}196{,}959{,}683{,}424{,}103$** | Pruned algebraically at block 3 ($q = 3$) via retained prefix cut $\mathrm{hi}_{4,\max} = 2.5799\times 10^{22} < n_0^*$ | Sole high-threshold source reaching $n_5 = 5.2144\times 10^{36} \ge H$; dips at block 79 to $n_{79} = 1.7456\times 10^{22} < n_0^*$ |
| **Surviving $95$-Block Cycles** | **$0$** | **$0$** |

![Figure 1: Guarded Certificate A Profile for (K92, m=95, R=6) across k1 = 1..75 and Complete m = 94 (j=1..3) and m = 95 (j=1..4) Exclusions.](supplement/results/figure_m94_m95_certificates.png)

*Cross-Validation of Both $95$-Block Certificates:*
1. **High-Threshold Source $n_0^* = 35{,}500{,}452{,}196{,}959{,}683{,}424{,}103$:** Its first 5 block pairs are $(3,1), (24,1), (24,1), (24,1), (14,1)$, reaching $5{,}214{,}419{,}445{,}356{,}833{,}875{,}507{,}338{,}803{,}728{,}058{,}075 \ge L_{6,95}$ after 5 blocks and dipping below $n_0^*$ at block 79 ($n_{79} = 17{,}456{,}932{,}521{,}065{,}939{,}764{,}129 < n_0^*$, verified by elementary $(3n+1)/2$ steps in `verify_paper7.py`). In Certificate B, once $S_3 + k_4 = 78 \ge 76$ bits pins a unique integer in the source window, the logarithmic budget cut is bypassed in favor of direct orbit continuation. In Certificate A, the prefix budget cut is retained at every step: after 2 blocks, $1{,}001{,}464{,}768{,}072{,}775{,}803{,}993{,}973 \ge L_{5,95}$ forces `tight3 = false, u4 = 1` at block 3, giving $\mathrm{hi}_{4,\max} = 25{,}799{,}288{,}312{,}445{,}063{,}465{,}101 < n_0^*$.
2. **Fourth-Division Slope Repair ($k_1 = 6$ Witness):** During cross-audit of the initial `audit_m95_gmp.c` code, `u/lyapunov_weaver` identified that the scaled slope $s_{4,\mathrm{lo}} = \lfloor \frac{Q_3 \mathrm{M\_LO} \cdot 2^{k_4 - 64}}{P_3 3^{k_4}} \rfloor$ reaches 135 bits on small-$P_3$, large-$Q_3$ branches such as $(6,1,1,1,1,1,5,1)$ ($n_1 = 15{,}751{,}769{,}425{,}060{,}900{,}437{,}695$). Keeping `s4_lo_z`, `s4_hi_z`, and the Stage-3/4 corridor bounds in wide integers (`mpz_t` in `audit_m95_gmp.c` and 640-bit `u640` in `audit_m95_int.c`) before clipping to $[0, \mathrm{HI}_{95}]$ eliminates all saturations while certifying `0` survivors across all 75 slices.

---

## 5. Limitations, Scope, and Reproducibility

1. **Conditional Dependence on $X_0 = 2^{71}$ and Prior $m \le 93$ Exclusions:** Theorems 1, 3, and 4 establish the non-existence of non-trivial Collatz cycles with $m \le 95$ local-minimum blocks by combining our $m = 94, 95$ certificates with Barina's $X_0 = 2^{71}$ convergence bound [10], Simons–de Weger [5] and Hercher [6, 11] for $m \le 91$, and `collatz_17` v1 [7] for $m = 92, 93$. Extending to $m = 96$ requires depth $R = 7$ at $j = 1$ and $R = 6$ at $j = 2$ and is left to future work.
2. **Verification and Reproduction (`supplement/`):**
   - Run `python3 code/verify_paper7.py` (`46s` on 5 workers; requires only standard Python 3 and `gcc -O3 -fopenmp`, no `<gmp.h>` needed) to execute: (i) the $m = 91..95$ CF reduction and 500-trajectory algebraic controls for Equations (1), (4a)–(4b), (6), and (7); (ii) the complete outward-rounded $j \ge 2$ exclusions at $m = 94, 95$ and the complete $(K_{92}, m = 94)$ search; (iii) the 75-slice row-for-row comparison between `results/m95_gmp_full_log.txt` and `results/m95_int_full_log.txt`, live compilation and execution of `code/audit_m95_int.c` on slices `k1 in {1, 30, 50, 75}` (pass `--full-m95` to rerun all 75 slices in `137s`), and both positive-control witnesses (`n_0^* = 35,500,452,196,959,683,424,103` and `n_1 = 15,751,769,425,060,900,437,695`).

---

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