Exploring Collatz Dynamics with Human-LLM Collaboration

Exploring Collatz Dynamics with Human-LLM Collaboration

Abstract

Figures & tables

Dynamical Systems

Abstract

Dynamical Systems

Explore similar work

Appendix

Adaptive Search in Collatz Exponent-Code Space via 2-adic and 3-adic Constraints

Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective

Stellar Colosseum: A Many-Agent Harness for Long-Horizon Research in Mathematics and Theoretical Computer Science

Adaptive Search in Collatz Exponent-Code Space via 2-adic and 3-adic Constraints

Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective

Stellar Colosseum: A Many-Agent Harness for Long-Horizon Research in Mathematics and Theoretical Computer Science

Edward Y. Chang

math.DSMar 10, 2026

Authors: Edward Y. Chang

Organizations: Computer Science, Stanford University · QuadriumAI

We present a comprehensive structural analysis of the Collatz conjecture through ~1014 computational experiments yielding 630 formal results. By systematically deploying 29 distinct mathematical paradigms--including transfer operator spectral theory, S-unit equations, p-adic interpolation, martingale methods, modular sieving, formal language theory, cascade algebra, discrete logarithm obstruction, and Diophantine approximation--we establish a Paradigm Exhaustion Theorem: every known framework for promoting distributional convergence ("almost all orbits descend") to pointwise convergence ("all orbits descend") encounters an irreducible structural obstruction when applied to the Syracuse map. On the unconditional side, we prove: (i) the Syracuse transfer operator has a uniform spectral gap for all M, implying equidistribution modulo any power of 2; (ii) any nontrivial cycle of length L satisfies D > 2^F for all L >= 3, giving ord_D(2) > F and F+1 distinct residues mod D; (iii) divergent starting points have natural density 0 and Hausdorff dimension ~0.68; (iv) the formal language of divergent-compatible v-sequences is not context-free; (v) cylinder-averaged density-1 convergence is proved unconditionally via spectral contraction on the invariant core I_2; (vi) a discrete logarithm triple filter achieves 100% cycle blockage for all L tested. We identify the Distributional-to-Pointwise Gap as the irreducible core and prove it equivalent to the divergence component. The modular sieve is permanently nonempty via the Mersenne Bypass. The present work is not a proof of the Collatz conjecture; it characterizes why the conjecture resists proof. The 29-paradigm exhaustion constitutes the most comprehensive structural survey of Collatz attack surfaces to date. Produced through human-LLM collaboration; see Section 12.

Panel of Figure 1 — Figure 1: Proof dependency diagram. Green nodes form the Route A core spine (linear chain), terminating at the Robustness Corollary; the red WMH node is the sole open hypothesis. Cyan nodes form the Route B spine (v5): Carry Contamination $\to$ cross-core alphabet $\to$ $2$ -adic expander $\to$ $j{\geq}5$ bound $\to$ anti-correlation $\to$ scaling regime ( $\beta_{w}{\approx}1.80$ , computational evidence for $j{\geq}3$ ); the CIC is the open hypothesis for this route. Violet nodes form Route C (v6): $I_{2}$ spectral contraction yielding unconditional cylinder-averaged density- $1$ convergence. Dark red nodes form the cycle exclusion spine (v7): $D>2^{F}$ $\to$ $\mathrm{ord}_{D}(2)>F$ $\to$ triple filter ( $100\%$ blockage, computational); the open node $D\nmid Q$ for $L>116$ is the remaining gap. Nodes marked “(comp.)” are computational observations, not formal theorems. Amber nodes are supporting results. Orange : Fiber-57 programme. Gray : exploratory.

Panel of Figure 2 — Figure 2: The burst-gap decomposition of a Collatz orbit. Bursts (red) are maximal runs of iterates with $k\geq 2$ ; gaps (green) are runs of downward iterates ( $k=1$ ). The Persistent Exit Lemma (Lemma 4.4 ) shows that when a burst ends at a persistent state, the subsequent gap has length exactly $1$ .

Panel of Figure 3 — Figure 3: Architecture of the burst–gap conditional framework (Sections 4 – 6 ). Green boxes denote results proved unconditionally; red boxes denote open components. The formal conditional proof (Theorem 8.14 ) derives the mean burst and mean gap bounds from the Orbit Equidistribution Conjecture, then applies the Burst–Gap Criterion. The phantom-cycle analysis (Figure 5 ) provides independent quantitative evidence that the equidistribution assumption is not artificially strong.

Panel of Figure 4 — Figure 4: Known-Zone Decay for $M=12$ . The known zone $Z_{k}$ decreases by at least $2$ per odd-to-odd step (generically $3$ when a burst step occurs). The worst-case bound $Z_{k}\leq M-2k$ reaches zero after $\lceil M/2\rceil=6$ odd-to-odd steps. Empirically, typical orbits in the tested residue classes ( $M\leq 18$ ) reach $Z_{k}=0$ in $1$ – $3$ steps.

$K$	$M(K)$	$M^{+}(K)$	$R(K)$	ratio
3	1	1	$1.06\times 10^{-2}$	—
4	1	1	$1.57\times 10^{-2}$	—
5	2	1	$1.05\times 10^{-2}$	$0.67$
6	5	3	$8.67\times 10^{-3}$	$0.83$
7	8	4	$7.60\times 10^{-3}$	$0.88$
8	9	4	$4.68\times 10^{-3}$	$0.61$

Table 1: Phantom family counts by depth

K

. By Theorem 7.15 , every primitive cyclic composition is a phantom family. Here

M(K)

is the total number of primitive families at depth

K

(summed over all

\ell

M^{+}(K)

counts those with expanding drift

\Delta>0

, and

R(K)

is the per-orbit gain contribution at depth

K

. The last column shows

R(K)/R(K{-}1)

, confirming geometric decay. Selected small-

K

families are shown below.

Panel of Figure 5 — Figure 5: The phantom-cycle analysis. The four green boxes are proved unconditionally in Sections 7 – 7.6 . They show that the per-orbit expanding-family gain is well within the contraction budget ( $R<\varepsilon$ , margin $\geq 4.65\times$ ), providing independent structural evidence that the Orbit Equidistribution Conjecture is the sole conceptual bottleneck. The formal conditional proof of convergence remains Theorem 8.14 , which derives the needed burst–gap hypotheses from equidistribution.

$K$	$R(K)$	$K$	$R(K)$	$K$	$R(K)$
3	$1.062\times 10^{-2}$	9	$4.730\times 10^{-3}$	15	$1.548\times 10^{-3}$
4	$1.573\times 10^{-2}$	10	$3.604\times 10^{-3}$	16	$1.218\times 10^{-3}$
5	$1.047\times 10^{-2}$	11	$2.858\times 10^{-3}$	17	$1.128\times 10^{-3}$
6	$8.670\times 10^{-3}$	12	$2.526\times 10^{-3}$	18	$9.702\times 10^{-4}$
7	$7.603\times 10^{-3}$	13	$2.065\times 10^{-3}$	19	$7.819\times 10^{-4}$
8	$4.676\times 10^{-3}$	14	$1.727\times 10^{-3}$	20	$7.495\times 10^{-4}$

Table 2: Exact values of

R(K)

for

K=3,\dots,20

, computed by Möbius inversion of the primitive necklace counts

M(K,\ell)

Result	Status	Role
Shadow sparsity (Prop. 9.1 )	Proved	Expanding shadows exponentially rare
Shadow return time (Thm. 9.4 )	Proved	Re-encounters separated by $\geq\ell+K-1$ steps
Finite-depth reduction (Prop. 9.6 )	Proved	WMH reduces to $K\leq 55$
Hierarchical consistency (Prop. 9.9 )	Proved	$\delta_{K}$ non-decreasing
Monotonicity constraint (Cor. 9.10 )	Proved	$\delta_{3}<0.0105$ necessary
Repulsion trapping (Lem. 9.20 )	Proved	Shadow encounters bounded: $\lfloor m/K\rfloor$ blocks

Table 3: Established results toward the Weak Mixing Hypothesis (Part 1 of 2). Proved : unconditional. Proved (ens.) : proved for the ensemble (uniform random start), not for individual orbits. Numerical : computational observation.

Result	Status	Role
Odd-skeleton reduction (Prop. 9.58 )	Proved	Collatz $\Leftrightarrow$ $x_{t}<0$ for all odd $n_{0}$
Block law (Thm. 9.61 )	Proved	$(a_{0},\ldots,a_{m-1})$ : unique residue class, density $\prod 2^{-b_{j}}$
I.i.d. valuations (Cor. 9.62 )	Proved	$a_{j}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mathrm{Geom}(1/2)$ under natural density
I.i.d. cycle types (Cor. 9.63 )	Proved	$(L_{i},r_{i})$ i.i.d.; Cramér rate unconditional on ensemble
Run-length invariant (Prop. 9.64 )	Proved	$L(n)=v_{2}(n{+}1){-}1$ ; exact burst formula
Single-cycle crossing (Cor. 9.65 )	Proved	$p_{\mathrm{cross}}\approx 0.7137$ (ensemble)

Table 4: Established results toward the Weak Mixing Hypothesis (Part 2 of 2). Proved : unconditional. Proved (ens.) : proved for the ensemble (uniform random start), not for individual orbits. Numerical : computational observation.

Result	Status	Role
Amplification hypothesis (Hyp. 9.37 )	Open	Orbit hits $S^{+}(K)$ at most $A^{\mathrm{cross}}_{K}\times$ uniform
Depthwise recurrence (Conj. 9.12 )	Open	Would imply WMH
Gain-observable recurrence (Conj. 9.18 )	Open	Weaker target; would imply observable WMH
Alignment renewal (Conj. 9.22 )	Open	Missing bridge; would close recurrence
Spectral Diffusion (Conj. 9.50 )	Open	Would imply Amplification Hyp. with margin
Orbit covariance summability (Conj. 9.166 )	Open	Would give CLT $\Rightarrow$ crossing $\Rightarrow$ Collatz

Table 5: Open problems and structural reductions. Open : conjecture. Structural : logical reduction (not a conjecture).

	Tao (2019)	Route A (WMH)	Route C ( $I_{2}$ , v6)	Cycle Excl. (v7)
Conclusion	Almost all $<f(n)$	Every orbit $\to 1$	Density- $1$ conv.	No cycles of length $L$
Quantifier	Log-density	Universal	Cylinder-averaged	Per- $L$
Status	Unconditional	Cond. on WMH	Unconditional	$L\leq 116$ proved; $L>116$ open
Open assumption	None	$\sum\delta_{K}{<}0.557$	None	$D\nmid Q$ for $L>116$
Technique	Entropy / ergodic	Phantom $+$ i.i.d.	$\widetilde{B}_{2}$ spectral	$D>2^{F}$ , discrete log
Unconditional	Almost-all conv.	$e^{-0.1465k}$ decay; $91\%$ at depth $13$	$\rho\leq 5/32$ ; cyl-avg no-escape	$\mathrm{ord}_{D}(2){>}F$ ; triple filter $100\%$ ; $L\leq 116$

Table 6: Structural comparison with Tao [ 14 ] . Three complementary approaches to the same problem: Tao controls the typical orbit unconditionally; Route A controls every orbit conditionally; Route B provides unconditional structural results plus a conditional reduction.

Panel of Figure 6 — Figure 6: Landscape comparison with Tao [ 14 ] . Horizontal axis: structural depth of the analysis. Vertical axis: scope of the conclusion. The dashed line marks the distributional-to-pointwise barrier. Route A crosses it via the WMH; Route B via the CIC. Both share the same barrier. Route C (violet) bridges spectral coupling analysis between Routes A and B. The Cycle Exclusion framework (v7, dark red) sits at the algebraic extreme: $D{>}2^{F}$ , $\mathrm{ord}_{D}(2){>}F$ , and the discrete-log triple filter provide exact obstructions to cycle existence. Items marked “(comp.)” are computational observations from exact enumeration, not yet formal theorems.

Layer	Content	Status
Algebraic	Map Balance, Scrambling, carry-free decomposition	Solved
Combinatorial	Phantom universality, $\|I_{r}\|=5$ , absorption	Solved
Probabilistic (annealed)	i.i.d. block law, path-conditional bijection	Solved
Dynamical (orbitwise)	Anti-concentration on $5$ -element core	Open

Table 7: Separation of concerns in the Collatz reduction. “Solved” indicates that the layer is fully resolved within the reduction framework; the sole remaining input is dynamical (Conjecture 12.2 ).

Appendix figures & tables50 assets

Supplementary material from the paper’s appendix.

$M$	$K=1$		$K=5$		Cycle states
	DAG?	$\max V$	DAG?	$\max V$
$6$	Yes	32	Yes	8	—
$7$	Yes	34	Yes	8	—
$8$	Yes	43	Yes	10	—
$9$	Yes	47	Yes	11	—
$10$	No	—	No	—	26

Appendix

Table 8: Acyclicity of the net-positive state graph

G_{M}

and the maximum ranking value

\max_{s}V_{M}(s)

. “Cycle states” counts vertices on cycles when the graph is not a DAG.

$M$	Core states	Exits	Returns	Outside cycles	Aug. DAG?	$\max V^{+}$
13	127,009	7,280	2,141	4,885	Yes	103
14	254,017	14,234	4,186	9,619	Yes	108
15	508,033	28,074	8,265	18,947	Yes	115
16	1,016,065	55,699	16,441	37,577	Yes	140
17	2,032,129	110,891	32,737	74,737	Yes	147

Appendix

Table 9: Re-entry analysis for the augmented return graph

G_{M}^{+}

$M$	$\|C\|$	$\|E\|$	Returns	$\|H\|$	$\max L$	$\max V$	Compress
13	59,416	5,036	869	869	5	348	1.6%
14	118,832	10,072	1,705	1,705	4	230	1.5%
15	237,664	20,144	3,347	3,347	4	301	1.5%
16	475,328	40,288	6,560	6,560	5	385	1.5%
17	950,656	80,576	12,988	12,988	6	486	1.5%

Appendix

Table 10: Exit-return map

H

in the exact

(r,b)

model (

B=15

g=2

|H|

counts edges of

H

(exits that return). Compression is

|H|/|G_{C}^{+}|

Appendix

Table 11: Budget equation for known net-positive cycles.

Layer	Statement	Status
A	density- $1-n^{-15}$ closure on $\mathbb{N}$	unconditional
B	$\delta(n)\leq C_{0}+0.0566\log_{2}n$	unconditional
B ′	infinite BF staircase realises B	constructive
B ′′	post-escape suffix is generic Collatz	empirical, $8$ BF champions
C-Haar	$\tau\leq C(k+\log_{2}n)$ on Haar-conull $B_{k}$	Lagarias + Birkhoff
C-pointwise	$\tau\leq C(k+\log_{2}n)$ on $\mathbb{N}\cap B_{k}$	open (Haar/pointwise gap)

Appendix

Table 12: The five pointwise layers of v6 and their statuses. Layer C-pointwise is the residual obstruction; everything above the line is established.

Panel of Figure 7 — Figure 7: Touch growth on the bounded window $[1,1024]$ . Left: the number of touched values in $[1,1024]$ as the seed cap increases through $S=2^{k}$ . Right: a heat strip showing the first exponent $k$ for which each value $n\in[1,1024]$ is touched by some seed at most $2^{k}$ . The entire window is saturated by $S=1024$ , so the visually sparse inner region of the onion plot cannot be explained by missing low integers.

Panel of Figure 8 — Figure 8: Exact first-passage diagnostics for the below-start criterion. Left: below-start traces $\Delta_{t}(n)=\log_{2}C^{t}(n)-\log_{2}n$ for every seed $2\leq n\leq 2^{12}$ , truncated at the first negative value. Right: exact first below-start time $\sigma(n)$ for all $1\leq n\leq 2^{16}$ , plotted on a dyadic horizontal scale. The left panel emphasizes the jagged above-start excursions that can precede first descent; the right panel shows that first descent is highly variable across seeds, making it a natural pointwise object of study for any induction-based Collatz strategy.

#	Level	Contribution	Primary source
1	Critical	Fix $\nu_{2}(9m_{t}{+}1)\geq 6$ overclaim $\to$ $\geq 3$ with explicit computation	Moderator
2	Critical	Fix Theorem 8.14 statement/proof mismatch (tail-control gap)	Moderator
3	Critical	Fix Gap Distribution Lemma scope ( $\mathrm{mod}\;2^{m}$ , $m{>}2$ overconstrained)	Validation (Claude)
4	High	Gap Distribution Lemma: $G\sim\mathrm{Geom}(1/2)$ , $E[G]=2$	Moderator (direction) + Claude (proof)
5	High	Valuation Distribution Lemma: $\Pr(k{=}j)=2^{-(j-1)}$ , $E[k]=3$	Claude
6	High	Convergence Corollary: net log-contraction $\approx-1.15$	Claude

Appendix

Table 13: Contribution attribution (Part 1 of 40).

#	Level	Contribution	Primary source
25	Critical	Almost-all crossing theorem: non-crosser density $\leq e^{-0.1465\,k}$ ; ensemble side fully closed; architectural diagnosis (deterministic realization barrier)	Claude
26	High	Exact cycle log correction $C(n)=O(1/n)$ ; sufficient crossing criterion (succeeds for all $n_{0}\leq 5\times 10^{6}$ except $\{27,31,63\}$ )	GPT (theorem) + Claude (verification + integration)
27	High	Multi-cycle threshold analysis: $n^{}/M$ decay, $2^{k}{-}1$ hardest-start family, analytic obstruction diagnosis for $n^{}<M$ proof	Claude
28	High	Universal one-cycle crossing: $n^{*}<1$ iff $r\geq r_{\mathrm{all}}(L)$ ; $P_{\mathrm{all},1\mathrm{cyc}}=0.4194$ ; $41.9\%$ of odd starts in classwise deterministic crossing blocks	GPT (theorem) + Claude (verification + integration)
29	High	Multi-cycle universal crossing: $R_{k}\to 0$ exponentially by Kesten theory ( $\rho\approx 0.84$ ); classwise universal descent for a.e. odd start	Claude
30	Med	Route A closure: adversarial $n^{*}$ grows $\sim e^{0.023k}$ , proving all-blocks-universal approach impossible	Claude

Appendix

Table 14: Contribution attribution (Part 2 of 40).

#	Level	Contribution	Primary source
44	High	Self-reinforcing mixing loop (Thm. 9.131 ): five-step loop closing scrambling $\to$ uniformity $\to$ episode bound $\to$ tail $\to$ TV $\to$ refresh; single remaining open step identified	Claude
45	High	Fiber-averaged transition matrix (Prop. 9.134 ): exact $8\times 8$ stochastic matrix at resolution $R$ , spectral gap $\gamma\geq 0.85$ for $R\geq 10$ , conditional orbit mixing (geometric TV contraction); remaining gap identified as non-divergence (Rem. 9.135 )	Claude
46	High	Corrected cascade PGF verification: independent derivation confirming $G_{\mathrm{cas}}(z)=z/(4{-}2z{-}z^{2})$ , $\mathbb{E}[S]=5$ , $\rho=\sqrt{5}{-}1$ ; coefficient recurrence check through $s=25$	GPT (independent verification)
47	High	Post-burst continuation bridge (Prop. 9.127 ): exact geometric law $\Pr(a_{k}{=}r)=2^{-(r-1)}$ on burst-exit fibers; $q_{\mathrm{ep}}=1/3$ exactly; TV transfer bound $\|q_{\mathrm{ep}}^{\mathrm{orbit}}-1/3\|\leq\Pr(\mathcal{E}_{R})$	GPT (theorem + verification)
48	High	Exact rational spectral gap (Prop. 9.136 ): explicit $T_{10}$ with entries in $\mathbb{Q}$ , $\|\lambda_{2}\|<0.146$ , $\gamma>0.854$ ; verified by exact characteristic polynomial	Claude
49	High	Non-divergence threshold quantification (Rem. 9.137 ): max single-cycle growth $\approx 8.5$ bits, $43$ -bit safety margin for $n_{0}\geq 2^{68}$ , Cramér bound $\Pr(\text{gain}>43)\leq 10^{-12}$	Claude

Appendix

Table 15: Contribution attribution (Part 3 of 40).

#	Level	Contribution	Primary source
56	Critical	Pair-return automaton (Prop. B.1 ): depth- $2$ partial return kernel with spectral radius $\rho(\tilde{B}_{2})=129/1024\approx 0.1260$ ; per-return information cost $c_{0}=\log_{2}(1024/129)\approx 2.989$ bits	GPT (concept) + Claude (formalization + computation)
57	Critical	Bounded invariant core (Prop. B.10 ): $\|I_{r}\|=5$ for all $r\geq 2$ (three fixed points $q=k\cdot 8^{r-1}-1$ , $k\in\{1,4,8\}$ , plus one $2$ -cycle); projective limit (Thm. B.11 ): $\varprojlim I_{r}=\{-1\}$ in $\mathbb{Z}_{8}$	Claude (computation + proof)
58	Critical	Absorption Bottleneck Lemma (Lem. B.15 ): channel capacity $\leq\log_{2}5\approx 2.322<c_{0}$ ; per-return deficit $0.667$ bits; geometric decay at rate $\alpha=645/1024\approx 0.630$ (Cor. B.17 )	GPT (information-theoretic framing) + Claude (proof + computation)
59	High	M-value collapse discovery: at $r\geq 3$ , all five $I_{r}$ elements produce the same post-absorption residue; effective channel capacity drops to zero bits. At $r\geq 4$ , first $n\equiv 1\pmod{8^{r}}$ coincides with $n=1$	Claude (computation + diagnosis)
60	Critical	Orbit autonomy diagnosis: orbit $\bmod\;8^{r}$ is not autonomous because $v_{2}(3n{+}1)$ depends on all digits of $n$ ; absorption for general $r$ cannot be proved by finite mod- $8^{r}$ computation alone. Confirms algebraic reduction has reached its natural fixed point	Claude (analysis) + Moderator (diagnosis)
61	High	Absorption theorem (Thm. B.13 ): verified computationally for $r=2,\ldots,10$ (max 194 steps); all five $I_{r}$ elements reach fixed point $1\pmod{8^{r}}$	Claude (computation)

Appendix

Table 16: Contribution attribution (Part 4 of 40): v3 fiber-57 programme.

#	Level	Contribution	Primary source
68	High	$q\equiv 7$ two-step return theorem (Prop. B.3 ): for $q=8m+7$ , the fiber-57 orbit returns in exactly two Syracuse steps with quotient $q^{\prime}=9m+8$ ; the return is perfectly uniform when $m\bmod 8$ is uniform	Claude (algebraic proof) + GPT (cross-validation)
69	High	$q\equiv 3$ minimum return-gap theorem (Prop. B.4 ): for $q=8m+3$ , the orbit cannot return to fiber 57 in fewer than five Syracuse steps; each intermediate residue $\bmod\;64$ is algebraically determined and $\neq 57$	Claude (algebraic proof + verification)
70	High	Kernel decomposition remark: fiber-57 returns split into a fast uniform branch ( $q\equiv 7$ , 14.4%, 2-step) and a slow sparse branch ( $q\equiv 3$ , 11.0%, $\geq 5$ -step with 9 new low-order bits), explaining why even short orbits mix quickly	GPT (kernel-split concept) + Claude (computation + formalization)
71	High	Invariant core rigidity remark: chain map $q\mapsto 9q+8\bmod 8^{r}$ acts as a permutation on $I_{r}$ at every depth (three fixed points plus one 2-cycle, $H_{AB}=0$ ); known-gap sub-stochastic contraction $\rho(\tilde{B}_{2})=129/1024<1$ is dynamic, not algebraic	Claude (algebraic construction + discovery)
72	Med	Operator identification: three distinct operators disambiguated: $\tilde{B}_{2}$ (sub-stochastic partial kernel, $\rho=129/1024$ ), $P$ (row-stochastic kernel, Perron root $1$ ), $P_{g}$ (gap-conditioned stochastic kernel); resolves conflation present in earlier drafts	GPT (operator taxonomy) + Claude (exact construction + numerical verification)
73	Med	Sub-stochastic $R_{\mathrm{sub}}$ at depth 2: full first-return-to- $I_{2}$ spectral radius $\rho\approx 0.149$ (Monte Carlo); partial kernel $\tilde{B}_{2}$ (gap- $2$ + gap- $5$ only) gives exact $\rho=129/1024\approx 0.1260$ ; the gap- $\geq 6$ tail accounts for the remaining $\approx 0.023$ discrepancy	Claude (computation) + GPT (diagnostic assessment)

Appendix

Table 17: Contribution attribution (Part 5 of 40): v4 return structure and operator identification.

#	Level	Contribution	Primary source
76	High	Sturmian obstruction concept: adversarial approach asking whether any single orbit can avoid class $5$ at every Sturmian depth, yielding the Carry Independence Conjecture framework	Moderator (direction) + Claude (formulation)
77	Critical	Carry Contamination Theorem (Thm. 10.4 ): for any depth- $D$ word, $n_{D}\bmod 8$ is exactly equidistributed over $\{1,3,5,7\}$ ; proof via affine parameterization $n_{D}=R_{w}+2\cdot 3^{D}\cdot m$	Claude (proof)
78	High	Orbit Parameterization Theorem (Thm. 10.3 ): $n_{0}=a_{w}+2^{S_{D}+1}m$ , $n_{D}=R_{w}+2\cdot 3^{D}m$ with $R_{w}$ odd; exact stride formulas	Claude (proof)
79	High	$(3/4)^{D}$ survivor law (Cor. 10.5 ): $\|C_{D}\|=2\cdot 3^{D-1}$ compatible words; each depth prunes exactly $1/4$	Claude (proof)
80	High	Refined mod- $16$ /mod- $32$ transition rules (Thm. 10.10 ): class $3$ bottleneck ( $1/2$ survival), class $7$ safe harbor (never $\to$ class $5$ ), class $1$ ( $3/4$ survival); stationary distribution $\pi=(3/7,2/7,2/7)$	Claude (proof + computation)
81	High	Class- $3$ recurrence theorem (Thm. 10.12 ): class $7$ runs bounded by $K-3$ steps, class- $1$ self-loop only sustains $n_{0}=1$ ; unconditional for any $n_{0}>1$	Claude (proof + computation)

Appendix

Table 18: Contribution attribution (Part 6 of 40): v5 Sturmian obstruction, cross-core dynamics, and carry independence.

#	Level	Contribution	Primary source
88	Critical	Periodic elimination: exhaustive cycle-equation search through period $13$ ( $238{,}811$ contracting words), zero non-trivial natural fixed points; $S$ -unit equation for integer fps; three-layer obstruction (carry contamination $+$ Möbius rigidity $+$ $S$ -unit)	Claude (computation + proof)
89	Critical	$2$ -adic expander property: $\|\Phi_{b}(x){-}\Phi_{b}(y)\|_{2}=2^{V}\|x{-}y\|_{2}$ ; measure shrinkage $\mu_{2}(T_{j})\leq 0.522^{jD}$ ; unconditional $j{\geq}5$ bound: count of $K$ -bit integers surviving $j$ non-descending rounds $\leq 2^{K(0.856-0.202j)}$ , zero for $j\geq 5$	Claude (proof + computation)
90	High	Exhaustion-sequence rigidity: counterexample residues must be aperiodic AND unbounded; periodic $\Rightarrow$ cycle contradiction; bounded $\Rightarrow$ eventually periodic	Claude (proof) + GPT (Pushes 58–62)
91	Critical	Compound carry anti-correlation: exact two-round survival $S_{2}/S_{1}^{2}\approx 0.635$ (constant in $K$ , verified $K=8$ – $20$ ); transport graph edge density $=1.0$ ; actual Collatz orbit independence $P(\text{slow}_{2}\|\text{slow}_{1})/P(\text{slow}_{2})\approx 1.000$ ; compound carries cover all joint residue cells mod $(p_{1},p_{2})$ ; no per-prime-pair obstruction	Claude (computation + analysis)
92	High	Distributional-to-pointwise barrier precisely localized: $\mu_{2}(T_{\infty})=0$ (unconditional) but $T_{\infty}\cap\mathbb{N}=\varnothing$ is the exact remaining content; anti-correlation bounded (constant factor, not exponential in $K$ ); barrier equivalent to Collatz itself	Claude + GPT (independent convergence)

Appendix

Table 19: Contribution attribution (Part 7 of 40): compound carry analysis and scaling regime.

#	Level	Contribution	Primary source
93	High	GPT Pushes 43–87 assessment: transport graph reformulation (Pushes 82–87); exhaustion-sequence rigidity (Pushes 58–62); carry-conspiracy route (Pushes 68–87); honest joint diagnosis that remaining gap IS the Collatz conjecture	GPT (analysis) + Claude (verification) + Moderator (synthesis)
94	High	Scaling regime computation (Pushes 37–39): effective measure $\mu(D)\sim 0.645^{D}$ via exact enumeration to depth 8 (20,951 nd words); overlap correction worsens with $D$ ( $\omega$ drops from $0.64$ to $0.42$ ); $r_{\mathrm{exact}}$ converges from $0.578$ toward $r_{\mathrm{union}}\approx 0.645$	Claude (computation + analysis)
95	Critical	Measure-weighted $\beta$ discovery: $\beta_{w}\approx 1.80$ (vs uniform $\beta_{u}\approx 2.91$ ), improving the $j\geq 5$ bound to $j\geq 3$ ; identifies that the survival probability is dominated by words with small $V$ , making the effective natural depth $D\approx K/1.80$ rather than $K/2.91$	Claude (computation + insight)
96	Critical	Anti-correlation asymptotic irrelevance: proved that $\gamma\approx 0.635$ is a constant factor that vanishes in the $K\to\infty$ limit and cannot change the $j$ threshold; three-round data confirms $S_{3}/S_{1}^{3}\approx 0.316\approx 0.635^{1.9}$	Claude (analysis)
97	High	Three-level nail characterization: (1) $c<1/2$ (technical), (2) density $\neq$ pointwise (structural), (3) $T_{\infty}\cap\mathbb{N}=\varnothing$ is the conjecture (fundamental); no further computational push within the Sturmian framework can close the gap	Claude (synthesis) + GPT (Pushes 108–127 confirmation)

Appendix

Table 20: Contribution attribution (Part 8 of 40): scaling regime, definitive nail, and v6 spectral closure.

#	Level	Contribution	Primary source
98	Critical	Complete cascade theorem (Thm. C.29 ): zero inter-BF generic steps; exact deterministic step count $4k+e_{k}$ with $e_{k}\in\{5,6\}$ period- $4$ ; total cascade growth $=k\cdot\log_{2}(81/64)$ exactly; net gain after full deterministic region bounded by $\log_{2}3$ or $2\log_{2}3-1$	Claude (proof + computation, verified $k=1\ldots 20$ )
99	High	Post-cascade closed form (Prop. C.30 ): $P_{k}=(a_{k}\cdot 81^{k}\cdot 32-65)/17$ with $a_{k}\in\{57,41,45,61\}$ period- $4$ ; iterated exit map formula $\mathrm{exit}^{k}(n)=(81^{k}n+65(81^{k}-64^{k})/17)/64^{k}$ ; AP structure: post-cascade $=P_{k}+m\cdot 81^{k}\cdot 256$ , all $\equiv 47\pmod{64}$	Claude (derivation + verification $k=1\ldots 12$ )
100	High	Post-cascade recurrence: $P_{k+4}=81^{4}P_{k}+164590400$ ; auxiliary recurrence $A_{k+4}=2^{24}A_{k}+64148175$ ; verified $19/19$ cases	Claude (proof + computation)
101	Critical	Carry chain structural theorem (Thm. C.32 ): $v\!=\!1$ streak length $=\mathrm{trailing\_ones}(n)-1$ exactly (100% verified, $10{,}000$ trials); after every streak, $\mathrm{trailing\_ones}(n_{L})=1$ exactly, forcing $v_{\mathrm{rec}}\geq 2$ ; formula $n_{L}=2\cdot 3^{L}\cdot m-1$	Claude (proof + computation)
102	High	Renewal drift computation (Cor. C.33 ): $E[v\mid v\geq 2]=3$ (exact geometric sum); drift $=-0.830$ bits/cycle $=-0.415$ bits/step; variance $=2.69$ ; growth episodes ( $\mathrm{trailing\_ones}\geq 4$ ): $+0.116$ /cycle; descent episodes ( $\mathrm{trailing\_ones}\leq 3$ ): $-0.946$ /cycle	Claude (computation + analysis)
103	High	Syracuse cocycle spectral gap (Rem. C.34 ): transfer operator $L_{M}$ on odd residues mod $2^{M}$ , Perron eigenvalue $\lambda_{1}=4/3$ ; spectral gap exists for $M\leq 12$ : $\mathrm{gap}(8)=1.0$ , $\mathrm{gap}(10)=0.33$ , $\mathrm{gap}(12)=0.39$ ; connection to Kontorovich–Miller programme	Claude (computation + analysis)

Appendix

Table 21: Contribution attribution (Part 9 of 40): v6 update — complete cascade algebra, carry chain, renewal drift, and Syracuse cocycle spectral analysis.

#	Level	Contribution	Primary source
109	Critical	Post-cascade universalization (Rem. C.36 ): BF encounter statistics for post-cascade AP family are indistinguishable from generic random orbits ( $50\,000$ generic $+$ $18\,000$ structured orbits). The AP structure is irrelevant; the problem reduces to generic Collatz dynamics	Claude (computation + analysis)
110	High	$\eta$ approach failure (Rem. C.35 , item 11): post-cascade height loss $\eta=(\log_{2}n_{\text{start}}-\log_{2}n_{\text{re-entry}})/k$ is negative for all observed BF re-entries (min $\eta$ ranges from $-6.28$ to $-0.26$ ). The contraction inequality $\rho<1$ cannot hold. Reframes the target from height loss to modular depth	Claude (computation + analysis)
111	High	BF re-entry depth distribution ( $60\,000$ post-cascade orbits): $92$ BF re-entries observed; $98.9\%$ at $k^{\prime}=1$ , $1.1\%$ at $k^{\prime}=2$ , $0\%$ at $k^{\prime}\geq 3$ ; depth tail $\Pr[k^{\prime}\geq j]\approx 2^{-6(j-1)}$ matches equidistribution prediction	Claude (computation)
112	High	Carry chain bit sensitivity (Rem. C.37 ): each $v\geq 2$ step scrambles $\approx 40\%$ of bottom $M$ bits; combined with $v\geq 2$ density $\geq 1/2$ , gives bit scrambling every $O(1)$ steps; maintaining $6k+8$ specific bits requires exponentially unlikely carry conspiracy	Claude (computation + analysis)
113	High	Conditional BF anti-correlation: $\Pr[\text{B}_{1}\text{ hit within 50 steps}\mid\text{just hit B}_{1}]=0.54\times$ unconditional rate; orbits exiting BF cascade are slightly less likely to re-enter	Claude (computation)
114	Med	Telescoping bound: orbit length $L(n)=O(\log n)$ with ratio $3.1$ – $5.2$ ; combined with per-step BF probability $2^{-(6k+8)}$ , gives $K_{*}(n)\leq O(\log\log n)$ heuristically	Claude (computation + analysis)

Appendix

Table 22: Contribution attribution (Part 10 of 40): v7 — universalization, zero persistence, entropy injection, and Ising formalization.

#	Level	Contribution	Primary source
119	Critical	IFS Fourier mode analysis (Rem. C.40 , Fourier section): $v=1$ IFS noise $\hat{\mu}(\xi)=\mathbf{1}_{\xi\text{ even}}$ : odd modes annihilated, even modes form closed invariant subspace. Mixing requires $v\geq 2$ steps (prob $1/2$ ). Per-step gap $\approx 1/M$ , explaining why gap shrinks while mixing time stays $O(M)$ . Multiplicative order $\mathrm{ord}(3,2^{M-1})=2^{M-3}$ verified for $M\leq 20$	Claude (analysis + computation)
120	Critical	Direct $K_{*}$ bound from zero persistence (Rem. C.41 ): $\mathbb{E}[\text{total BF}]\leq k_{0}/12+L\cdot 2^{-13}/(1-2^{-6})$ ; for $n=10^{20}$ : $\mathbb{E}\leq 0.92$ . Post-cascade values $P_{k}\equiv 55\pmod{64}$ , $P_{k}\notin B_{k^{\prime}}$ for any $k^{\prime}\leq 10$ , all $k=1\ldots 9$ . Zero same- $k$ multiple re-entries in $50{,}000$ orbits. Conditional on orbit finiteness and generic $\approx$ post-cascade equivalence only	Claude (computation + synthesis)
121	High	$10{,}000$ -orbit BF encounter census: average encounters $=0.006$ (for $k=1\ldots 5$ ); max $=2$ ; $0.6\%$ of orbits have any BF encounter; $50{,}000$ -orbit same- $k$ re-entry test: exactly $0$ orbits with $\geq 2$ encounters at same depth	Claude (computation)
122	High	Skew-product formulation and Galperin analogy (Rem. C.42 ): $(\omega,\theta)$ coordinate system where $\omega$ is the $2$ -adic symbolic coordinate (Bernstein–Lagarias conjugacy) and $\theta=\sum(\log_{2}3-v_{i})$ is the Archimedean phase. Deterministic discrepancy theorem identified as the precise form of the distributional-to-pointwise bridge. Native irrational constant is $\log_{2}3$ , not $\pi$	GPT (conceptual framework via Galperin billiards analysis); Claude (formalization and integration)
123	Med	GPT assessment of proof status: the distributional-to-pointwise bridge is correctly identified as the sole remaining barrier; $\pi$ is not the right replacement multiplier ( $\log_{2}\pi-2<0$ ); the critical drift threshold is $a=4$ (not $\pi$ ); Bernstein–Lagarias $2$ -adic conjugacy already provides a partial linearization	GPT (independent analysis of v18–v22 materials)
124	Critical	$v=1$ streak density theorem (Rem. C.43 ): exactly $2^{-s}$ fraction of odd residues mod $2^{s+2}$ allow $v\!=\!1$ streak of length $s$ , for all $s\leq 15$ . Matches independence prediction exactly. All long-streak residues are of the form $n\equiv 2^{s+1}-1\pmod{2^{s+2}}$	Claude (proof + computation)

Appendix

Table 23: Contribution attribution (Part 11 of 40): v7 — IFS Fourier analysis, direct

K_{*}

bound, skew-product formulation, and GPT side notes.

#	Level	Contribution	Primary source
127	Critical	Multi-step carry propagation measurement (Rem. C.46 ): bit-flip spread $\approx 1.5\,t$ bits after $t$ steps (linear). Post-cascade carry isolation: $P_{k}(q)$ values differ only at bits $\geq 6k+7$ , with $2$ – $4$ bits affected. Five-step formal proof structure: union bound gives $\Pr(\text{fresh zone contaminated})\leq 4\cdot 2^{-\ell}$ . Predicted vs. actual $B_{1}$ encounter rates match to ratio $0.98$	Claude (computation + proof structure)
128	Critical	Route A obstruction (Rem. C.47 ): envelope coefficient $1+\alpha=1.0566>1$ means height control is expansive at BF-depth scale. $\ell\leq(1+\alpha)k+O(1)$ : restart depths do not contract. Kills naive Borel–Cantelli from size alone. Conditional Route A theorem: need post-escape loss $\eta>6\alpha=\log_{2}(81/64)=0.3399\ldots$	GPT (diagnosis + conditional theorem); Claude (verification + integration)
129	Critical	Route B pullback residue formula and zero-compatibility census (Rem. C.47 ): $r_{k,\pi}=3^{-t}(2^{S_{\pi}}\,n_{k}^{*}-b_{\pi})\bmod 2^{6k+8+S_{\pi}}$ fully explicit. Among $309{,}136$ cross-depth pairs: zero compatible. Among $308{,}580$ same-depth pairs: zero compatible. Pullback density decays as $\approx 2^{-(j-8)}$ at mod $2^{j}$ . Low-bit residue sets identical across depths — incompatibility is structural at full modulus	GPT (Route B program design); Claude (formulas, computation, census)
130	High	Post-cascade family algebra: $P_{k}=(a_{k}\,81^{k}\cdot 32-65)/17$ , $P_{k}\equiv 47\pmod{64}$ , $P_{k+4}=81^{4}\,P_{k}+164{,}590{,}400$ . All post-cascade orbits ( $k=1\ldots 6$ ) reach $1$ with zero BF re-entries. $\log_{2}P_{k}-\log_{2}n_{k}^{*}=0.3399\ldots$ per BF lap (= Karp surplus $\log_{2}(81/64)$ )	GPT (closed forms); Claude (verification + orbit tracing)
131	High	Target lemma formulation: no infinite compatible chain in BF pullback tower based at $n\in\mathbb{N}$ . $k\to k+4$ periodicity of $a_{k}$ may reduce compatibility to a finite automaton on $(k\bmod 4,\pi\bmod 2^{M})$	GPT (target lemma statement); Claude (periodicity analysis)

Appendix

Table 24: Contribution attribution (Part 12 of 40): v7 — carry chain formalization, Route A/B analysis.

#	Level	Contribution	Primary source
132	Critical	3-bit shortfall theorem (Rem. C.47 ): for same $\pi$ , $v_{2}(r_{k_{1},\pi}-r_{k_{2},\pi})=6k_{1}+5+S_{\pi}$ , shortfall $=3$ bits exactly. Algebraically proved from $n_{k}^{*}=(a_{k}\,2^{6k+5}-65)/17$ with $a_{k}$ odd. For different $\pi$ : min shortfall $=3$ in $400{,}000+$ random trials. The BF cascade word $(1,1,1,3)$ achieves maximum mismatch valuation; shortfall still $3$	Claude (proof + computation)
132b	Critical	Different- $\pi$ shortfall theorem: Case 1 ( $S_{1}\neq S_{2}$ ): shortfall $\geq 14$ . Case 2 ( $S_{1}=S_{2}$ , $t_{1}\neq t_{2}$ ): shortfall $\geq 13$ . Case 3 ( $S_{1}=S_{2}$ , $t_{1}=t_{2}$ , $\pi_{1}\neq\pi_{2}$ ): $v_{2}(b_{1}-b_{2})\leq S-2$ (exhaustive $S\leq 21$ ), shortfall $\geq 16$ . $\|D_{\mathrm{odd}}\|\equiv 3\bmod 4$ (algebraic proof). $1.5\text{M}+$ trials, min shortfall $=13$ . Closes the algebraic gap for non-cascade incompatibility	Claude (proof + computation)
133	High	BF staircase non-nesting: $n_{k_{2}}^{}\notin B_{k_{1}}$ for all $k_{2}>k_{1}$ ( $k\leq 11$ ). $n_{k_{2}}^{}\bmod 2^{6k_{1}+8}=(-65/17)\bmod 2^{6k_{1}+8}$ for $k_{2}>k_{1}$ , while $n_{k_{1}}^{*}$ has a nonzero $a_{k_{1}}\,2^{6k_{1}+5}/17$ correction. The cascade works via orbit passage, not cylinder nesting	Claude (proof + verification)
134	High	Large-scale orbit census: $14/100{,}000$ random orbits hit multiple BF depths—all via cascade mechanism ( $B_{2}\to B_{1}$ in $4$ steps). Zero independent multi-depth re-entries. $0.54\%$ of orbits hit any $B_{k}$ at all	Claude (computation)
135	Critical	Different- $\pi$ 3-case shortfall proof: Case 1 ( $S_{1}\neq S_{2}$ ): shortfall $\geq 14$ . Case 2 ( $S_{1}=S_{2}$ , $t_{1}\neq t_{2}$ ): shortfall $\geq 13$ via $v_{2}(3^{d}-1)\leq v_{2}(d)+1$ . Case 3 ( $S_{1}=S_{2}$ , $t_{1}=t_{2}$ , $\pi_{1}\neq\pi_{2}$ ): $v_{2}(b_{1}-b_{2})\leq S-2$ (exhaustive $S\leq 21$ ), shortfall $\geq 16$ . Min different- $\pi$ shortfall $=13$ in $1.5\text{M}+$ trials	Claude (proof + computation)
136	High	$\|D_{\mathrm{odd}}\|\equiv 3\pmod{4}$ algebraically proved: $(a_{k_{2}}\,2^{6\Delta k}-a_{k_{1}})/17\equiv 3\pmod{4}$ since $a_{k}\equiv 1\pmod{4}$ for all $k$	Claude (proof)

Appendix

Table 25: Contribution attribution (Part 13 of 40): v7 — shortfall theorems, pullback incompatibility.

#	Level	Contribution	Primary source
137	High	Post-cascade convergence verified $k\leq 200$ ( $P_{200}$ : $1275$ bits). Deterministic prefix ( $47\bmod 64$ ) is expansive (period $10$ , ratio $3.604$ ). Convergence requires generic Collatz dynamics, not cascade structure	Claude (computation + analysis)
138	High	$A$ -coordinate framework verified: $x=(32\cdot 64^{k}\cdot A-65)/17$ with $A=a_{k}$ for champions. Key identity $17P_{k}+65=(81/64)^{k}(17n_{k}^{}+65)$ . Each BF restart imposes $A\equiv c\bmod 2^{m}$ with $m\geq 4$ ; shortfall $\geq 3$ makes all non-cascade restarts incompatible. Quantitative bound: $K_{}(x)\leq 1+\lfloor(H-6k-5)/4\rfloor$	Claude (proof + GPT framework)
139	High	$v_{2}(b_{\pi_{1}}-b_{\pi_{2}})\leq S-2$ algebraically proved (two cases): Case A (first difference at $j\leq t{-}2$ ): nonzero term with $v_{2}\leq S{-}2$ . Case B (differ only at last two positions): contradiction via $v_{t}^{1}=v_{t}^{2}=1\Rightarrow v_{t-1}^{1}=v_{t-1}^{2}$ . Bound is tight: achieved by $(S{-}2,2)$ vs $(S{-}1,1)$	Claude (algebraic proof)
140	High	Extended $P_{k}$ structure: $P_{k}\equiv 111\pmod{128}$ for all $k$ (7-bit deterministic prefix). Stopping time $\sigma(P_{k})\in[8,64]$ for $k\leq 500$ , mean $\approx 16.6$ , independent of $k$ . $P_{k}\to 1$ verified for $k\leq 5000$ ( $P_{5000}$ : $31{,}700$ bits). Binary digit density of $P_{k}\to 0.500$ as $k\to\infty$	Claude (computation + analysis)
141	High	Divergence threshold: need $P(v{=}1)>0.7076$ for divergence (from geometric tail $E[v\mid v\geq 2]=3$ ). Measured $P(v{=}1)\approx 0.50$ in all tested orbits. Lag-1 anti-correlation: $P(v{=}1\mid\text{prev }v{=}1)\approx 0.37$	Claude (algebraic + computational)
142	Medium	Correction: Syracuse mod $2^{M}$ does not project real orbits (carry term $3k\cdot 2^{M-v}$ breaks correspondence). Non-trivial cycles of abstract map at $M\in\{10,11,12,20\}$ are sporadic and die within two levels	Claude (analysis + correction)

Appendix

Table 26: Contribution attribution (Part 14 of 40): v7 — post-cascade structure, A-coordinate framework.

#	Level	Contribution	Primary source
143	High	Consecutive $v{=}1$ bound: $k$ consecutive $v{=}1$ steps require $n\equiv-1\pmod{2^{k+1}}$ , density $1/2^{k}$ among odd integers. Algebraic proof by induction on $k$ . Verified $k=1\ldots 7$	Claude (proof + computation)
144	High	$v{=}1$ self-limitation: after $v{=}1$ with $n\equiv 3\pmod{8}$ (half of all $v{=}1$ events), next value $\equiv 1\pmod{4}$ , forcing $v\geq 2$ . Only $n\equiv 15\pmod{16}$ (quarter of $v{=}1$ ) permits $v{=}1\to v{=}1$ . Unconditional algebraic constraint	Claude (proof)
145	High	Mod- $2^{K}$ Markov chain ergodicity: irreducible for $K=3\ldots 8$ , spectral gap $>0.72$ , stationary $P(v{=}1)\to 0.50$ as $K\to\infty$ . Mixing time $O(1)$ steps (independent of $K$ )	Claude (computation + analysis)
146	High	Exactly $2$ ambiguous residues mod $2^{K}$ : those with $v_{2}(3r+1)\geq K{-}1$ . All other transitions are fully deterministic from bottom $K$ bits. The ambiguous residues are $r=(2^{K+1}-1)/3$ and $(2^{K+2}-1)/3\bmod 2^{K}$ (the $\ldots 010101$ patterns)	Claude (proof + computation)
147	High	1-bit-per-step injection: the Collatz map mod $2^{K}$ injects exactly $1$ new bit per step from position $K$ via carry chain. The injected bit is approximately uniform ( $P\approx 0.50$ ) and approximately independent of the current mod- $2^{K}$ state	Claude (proof + computation)
148	High	$E[v\mid v\geq 2]=3$ algebraically for uniform distribution over odd residues mod $2^{K}$ . Proof: $P(v=k)=1/2^{k}$ , so $E[v\mid v\geq 2]=2\sum_{k\geq 2}k/2^{k}=3$ . Verified $E[v\mid v\geq 2]\approx 3.0$ along actual orbits	Claude (proof + verification)

Appendix

Table 27: Contribution attribution (Part 15 of 40): v8 — equidistribution attack, 1-bit injection, consecutive v=1 bound.

#	Level	Contribution	Primary source
150	High	Coupling framework: two orbits from $n$ and $n+2^{K}$ diverge in bottom $K$ bits after $O(2^{K-1})$ steps. Post-divergence mixing in $O(1)$ steps via spectral gap. Transition graph is strongly connected ( $K\leq 6$ ). Every tested orbit hits ambiguous residues within $O(2^{K})$ steps	Claude (computation + analysis)
151	High	Noise independence: MI(state, bit $K$ ) $<0.02$ bits ( $K\leq 8$ ). Ensemble branching balanced (algebraic: 50/50 split from $2$ -residue structure). Successive branch correlations $<0.01$ at lags $1$ – $3$ . Cone of dependence: after $T=5$ steps, all input bits $1$ – $30$ affect lowest output bits equally ( $\approx 2.0$ )	Claude (computation + analysis)
152	Critical	Complete conditional proof architecture: $K_{}(n)<\infty$ (unconditional) $+$ orbit equidistribution () $\Rightarrow$ $E[v]=2>\log_{2}3$ $\Rightarrow$ uniform contraction $\Rightarrow$ Collatz conjecture. (*) framed as noise-plus-mixing (Kifer 1988): deterministic dynamics with rare $1$ -bit injection	Claude (synthesis)

Appendix

Table 28: Contribution attribution (Part 16 of 40): v8 — coupling framework, noise independence, conditional proof completion.

#	Level	Contribution	Primary source
153	High	Disjoint-slice argument: decisive bit position of $n_{0}$ for successive branch choices increases monotonically (avg $10\to 43$ at $K=4$ ); $76\%$ of consecutive branch pairs have zero overlap in decisive bit sets. Carry chain depth $O(\log B)\approx 12$ vs slice separation $O(2^{K})$ : carry cannot bridge disjoint slices for $K\geq 4$	Claude (computation + analysis)
154	High	Higher-order correlations quantified: $\chi^{2}/\mathrm{df}$ on $4$ -tuples drops from $21.8$ ( $K=4$ ) to $3.2$ ( $K=7$ ). Branches are NOT i.i.d. but max absolute deviation $<1\%$ . Structured starting values (all-ones, alternating, Fibonacci-like) still yield $P(1)\approx 0.50$ and $\|\mathrm{lag\text{-}1~corr}\|<0.02$	Claude (computation + analysis)
155	Critical	Branch correlations cannot shift $E[v]$ below threshold: impact bounded by $(2/2^{K-1})\cdot K$ ; for $K\geq 7$ : $\|\Delta E[v]\|<0.22$ vs margin $0.42$ . Proof relaxed from i.i.d. to $\phi$ -mixing: (W1) $E[b_{i}]=1/2$ (proved), (W2) $\phi(L)\to 0$ (verified), (W3) spectral gap $>0$ (verified). Residual gap = formal $\phi$ -mixing bound	Claude (computation + synthesis)
156	High	Empirical spectral gap from actual orbits: $0.91$ ( $K=3$ ), $0.86$ ( $K=4$ ), $0.78$ ( $K=5$ ), $0.67$ ( $K=6$ ). $\mathrm{TV}(\pi,\mathrm{uniform})<0.02$ for all $K$ tested. Mixing time $\leq 3$ steps from any initial state	Claude (computation)
157	Medium	Normalization time for structured $n_{0}$ : $E[v]$ converges to $2.0\pm 0.1$ within $10$ – $70$ steps from pathological binary patterns. Orbits from $2^{63}-1$ , alternating bits, runs-of- $8$ all reach generic regime quickly	Claude (computation)

Appendix

Table 29: Contribution attribution (Part 17 of 40): v8 — disjoint-slice argument and weak-dependence framework.

#	Level	Contribution	Primary source
158	High	Worst-case branching under uniform distribution: $E[v\|\text{uniform, worst}]=2-1/2^{K-1}$ . For $K\geq 4$ : $\geq 1.875>\log_{2}3$ . Even adversarial branch resolution cannot break contraction if distribution is uniform	Claude (computation + algebra)
159	Critical	Renewal argument: $E[v]=3-2p$ where $p=P(v{=}1\text{ at renewal})$ . Contraction iff $p<0.7075$ . Empirical $p\approx 0.496$ (margin $0.21$ ). After $v\geq 2$ : mod- $4$ class is $\approx 50/50$ , giving $P(v{=}1\text{ at renewal})\approx 1/2$	Claude (computation + synthesis)
160	High	No fixed critical block length: $n_{0}=2^{B}-1$ produces $B-1$ consecutive $v=1$ steps. Episode ratio $2(K-1)/K>\log_{2}3$ for $K\geq 5$ (proved). Block-level contraction requires long-run average, not short-block bounds	Claude (computation + analysis)
161	Medium	No orbit sustains $P(v{=}1)>0.60$ over $500$ steps ( $10$ K trials). Divergence threshold $0.7075$ is never approached. Anti-correlation: after $v=1$ from $3\bmod 8$ , forced $v\geq 4$	Claude (computation)

Appendix

Table 30: Contribution attribution (Part 18 of 40): v8 — renewal argument, worst-case analysis, Boolean function complexity.

#	Level	Contribution	Primary source
162	Critical	Adversarial Markov chain: even the adversary minimizing $v$ at every ambiguous step cannot sustain $\bar{v}<\log_{2}3$ for $K\geq 4$ . Worst 100-step avg $\geq 1.90$ (K=8). The deterministic transitions ( $98\%$ +) force sufficient $v\geq 2$ steps	Claude (computation + synthesis)
163	Critical	GPT “Route A” critique: (i) circular (assumes growth bound $=$ conjecture), (ii) wrong ( $6$ bits/step vs. actual $1$ bit/step), (iii) insufficient (finite BF-depth $\not\Rightarrow$ convergence). 2-adic ergodicity on $\mathbb{Z}_{2}$ gives nothing for $\mathbb{N}$ (Haar measure $0$ )	Claude (analysis of external proposal)
164	Critical	Conditional proof via unique non-atomic ergodicity: if Haar is the unique non-atomic $S$ -invariant measure on $\mathbb{Z}_{2}^{\times}$ , then every orbit converges to $1$ . Proof: unbounded $\Rightarrow$ equidistributes $\Rightarrow$ $E[v]=2>\log_{2}3$ $\Rightarrow$ contraction $\Rightarrow$ contradiction	Claude (synthesis)
165	High	Trapped orbit contradiction sketch: non-convergent $\Rightarrow$ unbounded $\Rightarrow$ sustained $P(v{=}1)\geq 0.7075$ $\Rightarrow$ frequent $-1\bmod 2^{K}$ $\Rightarrow$ peeling exhausts $\Rightarrow$ recovery bits fresh $\Rightarrow$ $P(\text{return})$ exponentially small	Claude (synthesis)

Appendix

Table 31: Contribution attribution (Part 19 of 40): v8 — adversarial Markov chain, GPT critique, unique ergodicity.

#	Level	Contribution	Primary source
166	Critical	Spectral gap of Syracuse transfer operator $\approx 0.73$ for $K=3,\ldots,10$ . $\|\lambda_{2}\|$ oscillates around $0.25$ with no trend. Model fits predict gap $\to 0.70$ as $K\to\infty$ . Stationary distribution $=$ exact uniform for all $K$ tested	Claude (exact computation)
167	High	Diameter of Syracuse chain mod $2^{K}$ is exactly $K-1$ : from state $1$ , reachable set grows as $4^{t}$ per step ( $4$ after $1$ , $16$ after $2$ , $64$ after $3$ ). Doeblin condition holds for $K=3,4$ only. Mixing time $=O(K)$ but spectral gap $=O(1)$	Claude (exact computation)
168	High	Multi-step contraction: from ANY initial state, TV $<0.01$ after $5$ steps ( $K\leq 6$ ), $10$ steps ( $K\leq 8$ ). One-step contraction $=1.0$ (no single-step mixing), but bulk mixing is extremely rapid	Claude (computation)
169	Medium	Gap characterization: the proof reduces to one statement in four equivalent forms: (1) uniform spectral gap, (2) unique non-atomic ergodicity, (3) orbit equidistribution, (4) Birkhoff sum convergence. All evidence supports $\|\lambda_{2}\|\leq 0.30$ universally	Claude (synthesis)

Appendix

Table 32: Contribution attribution (Part 20 of 40): v8 — uniform spectral gap, transfer operator analysis.

#	Level	Contribution	Primary source
170	Critical	Universal no-target-sharing theorem: for each $v=1,\ldots,K-1$ , the $v$ -class Syracuse map mod $2^{K}$ is a bijection of all $n$ odd residues. Proved via coset disjointness: $C_{r}=(3r+1)/2^{v}+2^{K-v}\mathbb{Z}$ are pairwise disjoint with total $n$ . Verified $K=3,\ldots,10$	Claude (discovery + proof)
171	Critical	Row-sum decomposition: $(M\mathbf{1})_{i}=1-1/n+\mathrm{overflow}_{i}$ where overflow comes from the unique $v\geq K$ source. Each $v$ -class contributes exactly $1/2^{v}$ to every row sum	Claude (algebraic proof)
172	Critical	Exact closed-form: $\\|M\mathbf{1}-\mathbf{1}\\|_{1}=5/12-(2/3)(-1/2)^{K-1}$ . Verified by exact rational arithmetic $K=3,\ldots,15$ . Bound: $C(K)\leq 1/2$ for all $K$ , equality iff $K=4$	Claude (formula + verification)
173	Critical	Uniform spectral gap proved: $\|\lambda_{2}(M_{K})\|\leq 1/2$ for all $K\geq 3$ . Six-step proof: NTS $\to$ row decomposition $\to$ $L^{1}$ bound $\to$ primitivity $\to$ Dobrushin $\to$ Gelfand. All three formal gaps (G1: row sums, G2: column support, G3: primitivity) closed	Claude (complete proof)
174	High	Eigenvalue recalibration: actual $\|\lambda_{2}\|\approx 0.25$ (not $1/\sqrt{2}$ ). The $(1/\sqrt{2})^{K-2}$ operator norm at $T=K-2$ is a transient artifact of $M_{1}$ ’s nilpotency (index $K-1$ exactly)	Claude (computation + analysis)
175	High	Layer NTS in overflow lifts: each layer of the overflow source’s lift average satisfies its own no-target-sharing property (“each_once” verified for all layers, all $K$ tested)	Claude (discovery)

Appendix

Table 33: Contribution attribution (Part 21 of 40): v8 — universal no-target-sharing, closed-form

L^{1}

, spectral gap proved.

#	Level	Contribution	Primary source
178	Critical	Recurrence proved from first principles: $b_{K+1}=2b_{K}-4(-1)^{K}$ via Ansatz $b_{K}=(5/12)\cdot 2^{K}+(4/3)\cdot(-1)^{K}$ . Base case $b_{3}=2$ , closed-form verified $K=3,\ldots,15$ by exact rational arithmetic	Claude (algebraic proof)
179	Critical	Non-atomic unique ergodicity (Proposition C.48 ): uniform spectral gap implies Haar measure on $\mathbb{Z}_{2}^{\times}$ is the unique non-atomic $S$ -invariant measure. Standard projective-limit argument	Claude (synthesis)
180	High	Distributional-to-pointwise barrier analysis: identified two precise obstructions (atomic invariant measures from cycles; $\mu(\mathbb{N})=0$ for Birkhoff). Catalogued three candidate strategies: Diophantine cycle exclusion, carry-propagation bounds, Tao-style density strengthening	Claude (analysis)
181	Critical	Symmetric State Identity: closed-form expression $x_{i}=(3^{i}R_{\mathrm{suff}}+2^{S_{\mathrm{suff}}}R_{\mathrm{pre}})/(2^{S}-3^{L})$ for intermediate cycle states, verified over $123{,}792$ instances	Gemini (derived), Claude (verified)
182	Critical	Algebraic Parity Lock: if $(2^{S}{-}3^{L})\mid R(W)$ then every intermediate state $x_{i}$ is odd. $R_{\mathrm{suff}}$ always odd (leading term $3^{L-i-1}$ ); numerator odd; denominator odd; quotient odd	Gemini (proved), Claude (verified)
183	High	Domain Void (computational): for $L\leq 10$ , all non-trivial $D\mid R$ solutions have $2^{S}<3^{L}$ (negative domain). Solutions correspond to known negative cycles $\{-1\}$ , $\{-5,-7\}$ , $\{-17,\ldots\}$ . Zero positive-domain solutions found	Claude (exhaustive computation)

Appendix

Table 34: Contribution attribution (Part 22 of 40): v8 — recurrence proof, 2-adic non-atomic unique ergodicity, barrier analysis.

#	Level	Contribution	Primary source
197	Critical	Complete $L=3$ cycle exclusion (proved). For $S\geq 7$ : $R_{\max}/D<2$ forces $m=1$ . Equation $3\cdot 2^{a}+2^{b}=2^{S}-36$ has $v_{2}=2$ , forcing $a\leq 2$ ; both branches contradict	Claude (proved)
200	Critical	Even- $m$ parity obstruction (proved, all $L$ ): if $m$ is even, $N_{m}=m\cdot 2^{S}-3^{L-1}(3m{+}1)$ is odd, but $\mathrm{LHS}$ has $v_{2}\geq 1$	Claude (proved)
201	Critical	Complete $L=4$ cycle exclusion (proved). $M_{0}<3$ for $S\geq 9$ : $m=2$ killed by parity, $m=1$ killed by 2-adic cascade ( $a=2,b=4\Rightarrow S=8$ )	Claude (proved)
202	Critical	Complete $L=5$ cycle exclusion (proved). Even $m$ killed by parity; $m=1$ cascade gives trivial; $m=3$ : $2^{d}=3(2^{S}{-}512)$ never a power of 2	Claude (proved)
205	Critical	$m=1$ cascade $=$ trivial cycle (proved, all $L$ ): $q_{0}=1$ , $\sigma_{k}=2k$ , residual $=2^{S}-4^{L}=0$ iff $S=2L$	Claude (proved)
206	Critical	$3\mid m$ divisibility obstruction (proved, all $L$ ): $R_{\det}\equiv 2^{\sigma_{L-1}}\pmod{3}\in\{1,2\}$ , so $3\nmid R_{\det}$ hence $m\nmid R_{\det}$	Claude (proved)

Appendix

Table 35: Contribution attribution (Part 23 of 40): v10 — 2-adic cascade, fundamental identity, cycle exclusion

L\leq 5

#	Level	Contribution	Primary source
212	Critical	Three-case cycle exclusion (proved, Cases A&B): $R\equiv q_{L-1}\cdot 2^{\sigma_{L}}\pmod{m}$ . Case A ( $q_{L-1}=1$ ): $R\equiv 2^{\sigma_{L}}\neq 0$ . Case B ( $m\nmid q_{L-1}$ ): $R\neq 0$ . Both algebraically proved	Claude (proved)
213	Critical	Unconditional cycle exclusion for $L\leq 111$ (proved): $q_{0}(m)\leq 6\cdot(3/2)^{L}<2^{68}$ for $L\leq 111$ , within Barina (2020) verified range. Combined with even- $m$ parity: no non-trivial $L$ -cycle for any $L\leq 111$	Claude (proved)
214	Critical	Descent property (proved): Collatz verified to $2^{N}$ implies no $L$ -cycle for $L\leq N/\log_{2}(3/2)-O(1)$ . Any $L$ -cycle requires non-convergent $q_{0}\leq 6\cdot(3/2)^{L}$	Claude (proved)

Appendix

Table 36: Contribution attribution (Part 24 of 40): v10 — General Cycle Exclusion Theorem, unconditional

L\leq 111

#	Level	Contribution	Primary source
208c	Critical	Fundamental Identity $\sigma$ -convention correction: $\sigma_{1}=v_{2}(3m{+}1)$ (not $v_{2}(3q_{0}{+}1)$ ). Identity verified correct with corrected convention. Conclusions (Results 212–214) unaffected	Claude (corrected)
215	High	Case C power-of-2 obstruction (verified $m\leq 50{,}000$ ): only 6 instances with $m\mid q_{L-1}$ and $q_{L-1}\neq 1$ . All have $R>m(2^{\sigma_{L}}-3^{L})$ ; walk sum overshoots. No cycle	Claude (computation)
216	Critical	Complete Algebraic Reduction (proved): $\Delta=(m-T^{L}(m))\cdot 2^{\sigma_{L}}$ . Cycle exists iff $T^{L}(m)=m$ . All other cases algebraically excluded	Claude (proved)
217	Critical	Case C with $r\geq 2$ excluded (proved): if $q_{L-1}/m\geq 2$ then $\Delta=(1{-}r)\,m\,2^{\sigma_{L}}<0$ . Walk sum overshoots; no cycle	Claude (proved)

Appendix

Table 37: Contribution attribution (Part 25 of 40): v10 —

\sigma

convention correction, Case C power-of-2 obstruction.

#	Level	Contribution	Primary source
218	Critical	Pure Expansion Obstruction (proved): all $v_{i}=1$ requires $2^{L}\mid(m{+}1)$ . Maximum pure expansion $L_{\max}=v_{2}(m{+}1)$ . Every orbit must inject $v\geq 2$ contracting steps	Claude (proved)
219	Critical	Affine Fractional Identity (proved): $q_{L}=((m{+}1)\cdot 3^{L}-2^{L}+\mathcal{P}_{L})/2^{S_{L}}$ where $\mathcal{P}_{L}\geq 0$ is the perturbation sum from $v\geq 2$ steps	Claude (proved)
220	High	Perturbation Recurrence (proved): $\mathcal{P}_{L+1}=3\mathcal{P}_{L}+2^{S_{L}}-2^{L}$ with $\mathcal{P}_{0}=0$ . Balanced at divergence threshold ( $3\approx 2^{\log_{2}3}$ )	Claude (proved)
221	Critical	$S$ -unit Reformulation (proved): $(m{+}1)\cdot 3^{L}-T^{L}(m)\cdot 2^{S_{L}}=2^{L}-\mathcal{P}_{L}$ . A family of $S$ -unit equations over $\mathbb{Z}[1/6]$ , one per step $L$ . Divergence requires simultaneous solutions with $S_{L}<1.585L$	Claude (proved)
222	High	Bit Growth Bound (proved): $\log_{2}T^{L}(m)\leq\log_{2}m+L\log_{2}3-S_{L}+1$ . Pure $v{=}1$ expansion grows as $(3/2)^{L}$ , fails at $L=v_{2}(m{+}1)$	Claude (proved)
—	Medium	Three paradigm bridges evaluated and ruled out: Boolean Fourier (distributional only), Arithmetic Automata (pumping fails), Pointwise Martingale (inapplicable to deterministic orbits). Distributional-to-pointwise gap confirmed robust across all paradigms	Claude / Gemini / User

Appendix

Table 38: Contribution attribution (Part 26 of 40): v10 — Divergence attack, fractional identity, paradigm bridges.

#	Level	Contribution	Primary source
223	Critical	Episode Transition Identity (proved): $v_{k+1}=1+v_{2}(2a\cdot 3^{k+1}+3^{k+1}-1)$ for odd $x$ with $k{+}1$ trailing $1$ -bits. Exact formula linking episode parameters to valuation drops	User (proposed) / Claude (proved, verified)
224	High	Intrinsic Valuation Pattern (proved): $v_{2}(3^{n}-1)=1$ if $n$ odd, $v_{2}(n)+2$ if $n$ even. Consequence: $v_{k+1}\geq 2$ always	Claude (proved)
225	High	Episode Density Mean (proved for uniform $a$ ): $E[\rho_{E}]=2>\log_{2}3$ . Verified computationally for $k=1,\ldots,50$	Claude (proved, verified)
—	Medium	S-unit equation structure analysis: $v_{2}(\text{LHS})=v_{2}(m{+}1)$ constant for all $L$ ; Subspace Theorem blocked (coefficients depend on $L$ ); congruence tower verified	Claude (computed)

Appendix

Table 39: Contribution attribution (Part 27 of 40): v10 — Episode decomposition, S-unit analysis, congruence tower.

#	Level	Contribution	Primary source
226	Critical	$v^{}=2$ Parity Condition (proved): minimum valuation $v^{}=2$ iff $a\equiv k\bmod 2$ . Density of minimum-density episodes is exactly $1/2$ for all $k$	Claude (proved)
227	High	Episode Transition Formula (proved): $x^{\prime}=a\cdot 3^{k+1}+(3^{k+1}-1)/2$ gives the complete transition $(k,a)\to(k^{\prime},a^{\prime})$	Claude (proved)
228	High	Half-Propagation Bound (verified): $P(k^{\prime}\geq 1\mid v^{*}=2)=1/2$ . Consecutive low-density episodes thin by $1/2$ per step	Claude (verified)
229	Medium	Congruence Tower Consistency (verified): $m{+}1\equiv(2^{L}-\mathcal{P}_{L})\cdot 3^{-L}\bmod 2^{S_{L}}$ for all $L$ ; tower refines at each step	Claude (verified)
—	Medium	Distributional-to-pointwise barrier confirmed robust across all paradigms: divergent v-sequences exponentially rare ( $\leq 2^{-0.35K}$ density), but pointwise exclusion remains open	Claude / User / Gemini

Appendix

Table 40: Contribution attribution (Part 28 of 40): v10 — Coefficient discrepancy attack.

#	Level	Contribution	Primary source
230	Critical	Periodic $v$ -Sequence Exclusion (proved): a divergent orbit cannot have a periodic $v$ -sequence. Periodic $\Rightarrow$ cycle $\Rightarrow$ excluded by Result 216	Claude (proved)
231	Critical	Ultimately Periodic $v$ -Sequence Exclusion (proved): a divergent orbit cannot have an ultimately periodic $v$ -sequence. Eventually periodic orbit $\Rightarrow$ cycle $\Rightarrow$ excluded	Claude (proved)
232	Critical	Aperiodic $v$ -Sequence Necessity (proved): any divergent orbit must have an aperiodic $v$ -sequence. Corollary: not generable by any finite automaton	Claude (proved)
233	High	Reconstruction Consistency (verified): all consistent divergent $v$ -sequences of length $15$ crossed $S_{L}/L=1.585$ within 48 additional steps (median $12.7$ )	Claude (verified)

Appendix

Table 41: Contribution attribution (Part 29 of 40): v10.1 — Carry accumulation and

2

-adic reconstruction.

#	Level	Contribution	Primary source
—	High	Cancellation phenomenon: $(m{+}1)\cdot 3^{L}+\mathcal{P}_{L}$ has its first $S_{L}$ binary digits matching $2^{L}$ exactly. Congruence tower stabilizes once $S_{L}>\log_{2}(m{+}1)$	Claude (verified)
—	Medium	Density-zero strengthening: at most $2^{0.98K}$ starting points $m<2^{K}$ can have divergent $L$ -step prefixes. $\mathcal{P}_{L}/3^{L}\to r_{\infty}$ (constant) in divergent regime	Claude (computed)
—	Medium	Complete paradigm exhaustion: nine distinct paradigms tested (Fourier, automata, martingale, Baker, Subspace, episode, $2$ -adic, carry, counting). All produce distributional results only. Distributional-to-pointwise gap confirmed as the exact remaining content of the Collatz conjecture (divergence component)	Claude / User / Gemini

Appendix

Table 42: Contribution attribution (Part 30 of 40): v10.1 — Carry and cancellation analysis.

#	Level	Contribution	Primary source
—	High	$\mathcal{P}_{L}$ Non- $S$ -unit (Result 235, proved): perturbation sum has prime factors outside $\{2,3\}$ , blocking the Subspace Theorem. Measure Rigidity Non-Applicability (Result 237, proved): $\times 2,\times 3$ theorems do not apply to Collatz ( $\times 3{+}1$ , variable exponent, $\mathbb{Z}$ )	Claude (proved) / User (Subspace strategy)
—	Medium	CF Structure of $\log_{2}3$ (Result 234): large partial quotient $a_{9}=23$ . Baker’s theorem yields $\|S_{L}/L-\alpha\|>\exp(-C(\log L)^{2})/L$ , too weak. Inverse tree covers all odd $\leq 10{,}000$ at depth $60$ (Result 236). Entropy ordering: $h_{\text{div}}<h_{\text{conv}}$ (Result 239)	Claude (computed/proved)
—	Medium	$v$ -value bit determinism: density of $v=k$ is exactly $2^{-k}$ (Result 240). Consecutive $v$ independence verified at $K\geq 12$ (Result 241). Max consecutive $v{=}1$ run $=16$ for $m\leq 10^{5}$ (Result 242)	Claude (proved/verified)

Appendix

Table 43: Contribution attribution (Part 31 of 40): v10.2 — Number-theoretic and dynamical paradigms.

#	Level	Contribution	Primary source
—	High	Paradigm Exhaustion Theorem (Result 249, proved): 18 paradigms systematically applied, all encounter the distributional-to-pointwise barrier. Barrier Characterization (Result 250, proved): the gap is equivalent to the Collatz conjecture (divergence component)	Claude / User
—	Medium	Exception Set Invariance (Result 246): $E$ is $T$ -invariant. Density Amplification Barrier (Result 247): density arguments cannot reach “all $n$ ”. Divergent-compatible $v$ -sequences decay as $\sim 0.43^{L}$ . $v$ -sequence state complexity is infinite (Result 243); entropy rate $\to 0$ (Result 244). Three remaining viable routes: Tao amplification, $2$ -adic potential theory, $\mathcal{P}_{L}$ growth contradiction	Claude (proved/computed)
—	Low	$3$ -adic constraint is automatic (orbit coprime to $3$ ). Multi-prime constraints mod $p$ for $p\geq 5$ add nothing. Orbit coupling: consecutive odd orbits merge (verified). 250 cumulative results across 181 scripts	Claude (verified)

Appendix

Table 44: Contribution attribution (Part 32 of 40): v10.2 — Paradigm exhaustion and barrier characterization.

#	Level	Contribution	Primary source
—	High	$2$ -Adic Mahler Identity (Result 252, proved): $(m{+}1)+\Sigma_{\infty}=0$ in $\mathbb{Z}_{2}$ for ALL orbits. $2$ -adic convergence automatic: $v_{2}(\text{term}_{j})=j$ (Result 253). No Hasse-type conflict (Result 254). Cantor Rigidity proof refuted (Result 251): nonlinear maps generate aperiodic sequences from rational seeds	User (Mahler proposal) / Claude (verification, refutation)
—	Medium	Crossing time analysis (Result 255): max $=85$ for $m\leq 50{,}000$ . Sieve coverage $>99.99\%$ at $L=20$ (Result 272). Random walk model matches to $5\%$ (Result 258). Drift-variance exp $(-0.086L)$ matches late-crossers (Result 273)	Claude (computed/verified)

Appendix

Table 45: Contribution attribution (Part 33 of 40): v10.3 —

2

-adic Mahler Identity and proof attempt refutations.

#	Level	Contribution	Primary source
—	Medium	Renormalization convergence $B\geq 8$ (Result 264). Moment match Var $[S_{L}]\approx 2L$ to $3$ sig figs (Result 265). Renewal gaps geometric $(0.46)$ (Result 266). Martingale $M_{L}\approx(m{+}1)$ with drift (Result 268). Descent window: all $m\leq 50{,}000$ descend within $L=100$ (Result 270). Orbit coalescence median $56$ steps (Result 271). Digit sum $E[\Delta\mathrm{hw}]\approx-0.1$ (Result 274). 275 cumulative results across 203 scripts	Claude (computed/verified)

Appendix

Table 46: Contribution attribution (Part 34 of 40): v10.3 — Comprehensive probabilistic and dynamical survey.

#	Level	Contribution	Primary source
—	Medium	Sieve completeness: survivor decay $\lambda\approx 0.90$ per step (Result 276). Tao amplification limits: $5\%$ non-escapers at $K=12$ (Result 277). Running condition entrapment: $P(v{=}1)\approx 68\%$ among survivors (Result 278). Survivor branching factor $\approx 0.876$ (Result 279). Super-exponential decay fit $R^{2}=0.997$ (Result 280)	Claude (computed/verified)

Appendix

Table 47: Contribution attribution (Part 35 of 40): v10.4 — Sieve completeness and survivor decay analysis.

#	Level	Contribution	Primary source
—	Medium	Compactness: $3$ persistent survivors mod $16$ through $L{=}14$ (Result 281). $2$ -adic Cantor structure with $\dim_{H}\approx 0.68$ (Result 282). Borel–Cantelli: $\mu(C_{\mathrm{div}})=0$ proved (Result 283). Effective Tao: $C\approx 2$ – $3$ , $A=2$ (Result 284). Sieve synthesis: three closure routes ranked (Result 285). 285 cumulative results across 213 scripts	Claude (proved/computed)

Appendix

Table 48: Contribution attribution (Part 36 of 40): v10.4 — Topological closure and sieve synthesis.

#	Level	Contribution	Primary source
286	Critical	Post-cascade recovery analysis: $\eta<0$ for all observed re-entries, disproving linear height-loss per BF depth. BF re-entry depth $k^{\prime}\leq 2$ in all 92 observed re-entries across $60{,}000$ post-cascade orbits	Claude (computation + diagnosis)
287	Critical	Complete cascade algebra: zero inter-BF generic steps, exact step count $4k+e_{k}$ with period- $4$ correction, post-cascade closed form $P_{k}=(a_{k}\cdot 81^{k}\cdot 32-65)/17$	Claude (proved)
288	Critical	Carry chain structural theorem: every $v{=}1$ streak $=\mathrm{trailing\_ones}-1$ , followed by forced $v\geq 2$ recovery. Renewal drift $-0.415$ bits/step exact	Claude (proved)
289	High	Syracuse cocycle spectral gap: max modular cycle gain $=+\log_{2}3$ for all $M\leq 14$ . Eleven approaches to universal descent ruled out	Claude (computation + synthesis)
290	High	Hybrid sieve measure-preservation: $\nu(E_{a}^{-1}(C_{r,M}))=\nu(C_{r,M})$ , killing the hybrid sieve approach definitively	GPT (formal proof), instructed by Moderator
291	High	Route A/B strategic bifurcation: Route A (prefix-growth bound $\delta(n)\leq C_{0}+0.0566\log_{2}n$ ) expansive ( $1+\alpha>1$ ); Route B (pullback tower with 4 concrete lemmas) proposed as alternative	GPT (framework), instructed by Moderator

Appendix

Table 49: Contribution attribution (Part 37 of 40): v10.5–v11 — paradigm exhaustion expansion, BF re-entry, cascade algebra.

#	Level	Contribution	Primary source
308–590	High	Deep cycle exclusion campaign across 28 paradigms: algebraic zero-avoidance, Horner chain, bitset DP, CRT hierarchy, greedy optimization, polynomial hash, spectral methods, exponential sums, lattice reduction, transfer matrix eigenvalues, entropy/information-theoretic, algebraic geometry. All 283 results produced by Claude under Moderator direction	Claude (proved/computed/verified)
—	Critical	Paradigm count expanded from 18 to 28 through systematic deployment of new attack frameworks. Each paradigm encounters the distributional-to-pointwise barrier or a structural limitation	Claude (execution), Moderator (direction)

Appendix

Table 50: Contribution attribution (Part 38 of 40): v12–v15 — deep cycle exclusion campaign (Results 308–590).

#	Level	Contribution	Primary source
591	Critical	$D>2^{F}$ for all $L\geq 3$ unconditionally proved : computation for $L\leq 2500$ , Laurent (2008) bound for $L>2494$ . $D/2^{F}=2^{L}(1-2^{-\beta})$ where $\beta=\lceil L\log_{2}3\rceil-L\log_{2}3$	Claude (proved)
592–594	Critical	Unconditional corollaries: $\mathrm{ord}_{D}(2)>F$ (hence $\mathrm{ord}_{D}(2)\nmid F$ ), $\gamma^{L}\not\equiv 1\bmod D$ , $G(L)\neq 0\bmod D$ (no single-block zero), $Q_{\mathrm{const}}\not\equiv 0\bmod D$ . The powers $\{2^{0},2^{1},\ldots,2^{F}\}$ are $F{+}1$ distinct residues in $\mathbb{Z}/D\mathbb{Z}$	Claude (proved)
595	Critical	Geometric sum formulation: $S_{\mathrm{mono}}=\sum_{j}2^{v_{j}}\gamma^{a_{j}}G(m_{j})$ for $k$ -block decompositions. Zero condition $\Leftrightarrow$ discrete log constraint $2^{v_{j}}\equiv T_{j}\bmod D$ with $v_{j}\in[0,F]$	Claude (formulated)
596–605	High	No $k$ -block zeros for $k=1..4$ at all tested $L$ . Progressive CRT saturates every prime factor for $L\geq 21$ . Baker/LMN crossover at $L\approx 2{,}494$ . For $L=5$ : residue $0$ is the only avoided residue	Claude (computed/verified)
606–618	High	Two-block Diophantine analysis, $p$ -adic valuations ( $v_{2}(3^{n}{-}2^{n})=v_{3}(3^{n}{-}2^{n})=0$ proved), 2-block zero requires $\gcd(m,6)=1$ , Baker crossover computation	Claude (proved/computed)

Appendix

Table 51: Contribution attribution (Part 39 of 40): v16 — discrete log obstruction,

D>2^{F}

unconditional proof.

#	Level	Contribution	Primary source
619–621	Critical	Integrality obstruction census: for $L=3..500$ (2-block), exactly one algebraic hit ( $L=7$ , $T=2^{4}$ ), blocked by $bw>F$ . For $L=5..30$ (3-block): zero integral solutions. Triple filter achieves $100\%$ blockage	Claude (computed/proved)
622–623	High	Expected $k$ -block zeros for fixed $k$ : $O(L^{2k-2}/3^{L})\to 0$ . Total $\|A\|/D\to 0$ at rate $\approx 10^{-0.026L}$ ; no $k$ -value has $E[\text{hits}]>1$ for $L\geq 20$	Claude (proved/computed)
624–627	High	Triple filter theorem formalized. $\|A\|/D$ crosses below $1$ at $L=4$ . Two-block target formula derived	Claude (formulated)
—	High	Three-party contribution analysis (April 18–19): seven categories, revised aggregate Claude $37\%$ , GPT $30\%$ , Moderator $33\%$ . Dead-end identification and strategic direction reclassified as moderator-instructed	Claude (analysis), Moderator (direction + correction)

Appendix

Table 52: Contribution attribution (Part 40 of 40): v17 — integrality obstruction, contribution analysis.

Figure 9: Syracuse chord diagram at depth

m=12

(

2^{11}=2048

odd residue classes). Gray chords show the full map

r\mapsto T(r)

; coloured overlays highlight five phantom family shadows (Theorem 7.15 ). Each family

\sigma=(k_{1},\dots,k_{\ell})

occupies

2^{m-K}

residue classes; families with positive log-drift

\Delta=\ell\log_{2}3-K>0

(e.g.

\sigma=(1,1,1)

\Delta\approx+1.75

) are the expanding compositions whose phantom shadow gain is bounded by Theorem 7.19 .

Figure 10: Orbit equidistribution evidence (Conjecture 8.2 ). Four Syracuse orbits with long trajectories (195–357 odd steps) are traced mod

2^{8}

(128 odd residue classes). Chord color encodes time (dark = early, light = late). Right panels: angular visitation histograms with uniform expectation (dashed red). The coefficients of variation (CV

\approx 0.76

–

0.83

) and reduced chi-squared (

\chi^{2}/\nu\approx 2.8

–

4.9

) are consistent with moderate sampling noise over 32 angular bins, supporting the hypothesis that orbit residues are approximately equidistributed.

Figure 11: Syracuse chord structure at scale $2^{32}$ and $2^{64}$ . Red chords: expanding steps ( $v_{2}(3r{+}1)=1$ , per-step gain $\log_{2}3-1\approx 0.585$ bits); blue chords: contracting steps ( $v_{2}\geq 2$ ). The near 50–50 split is consistent with the identity $\mathbb{E}[k]=2$ per Syracuse step (Section 4 ), which underpins the contraction budget $\varepsilon=2-\log_{2}3$ . The inner void corresponds to the known zone of small residues; the uniform angular filling of the outer annulus at both scales provides visual support for the Orbit Equidistribution Conjecture.

Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

See all 474 papers

Jul 10, 2026

We study a symbolic search space for the Collatz conjecture based on finite exponent codes of the accelerated map. Each code records the number of divisions by two after every 3n + 1 step and determines three quantities: real drift, a 2-adic start representative, and a 3-adic endpoint representative. Their combination defines the 2-3-infinity diagnostic. Counterexample-like codes should exhibit near-critical drift, small 2-adic start representatives, and endpoints compatible with growth on the scale of (3/2)^k. We prove that every infinite code generated by a fixed positive integer has asymptotically vanishing 2-adic and 3-adic residue rates. Experiments with random critical codes, mechanical critical codes, and adaptive evolutionary search at lengths 100, 200, and 400 show that adaptive search improves finite-length trade-offs, while all methods retain clearly positive residue rates. The proposed framework is not a verification method for the Collatz conjecture, but a symbolic diagnostic approach for investigating obstruction structures in exponent-code space.

Oliver Kramer

Computational Intelligence Department of Computer Science University of Oldenburg

Topic pending

Ask PDF

Mar 4, 2026

We study the Collatz total stopping time

τ(n)

over

n\le 10^7

from a probabilistic machine learning viewpoint. Empirically,

τ(n)

is a skewed and heavily overdispersed count with pronounced arithmetic heterogeneity. We develop two complementary models. First, a Bayesian hierarchical Negative Binomial regression (NB2-GLM) predicts

τ(n)

from simple covariates (

\log n

and residue class

n \bmod 8

), quantifying uncertainty via posterior and posterior predictive distributions. Second, we propose a mechanistic generative approximation based on the odd-block decomposition: for odd

m

, write

3m+1=2^{K(m)}m'

with

m'

odd and

K(m)=v_2(3m+1)\ge 1

; randomizing these block lengths yields a stochastic approximation calibrated via a Dirichlet-multinomial update. On held-out data, the NB2-GLM achieves substantially higher predictive likelihood than the odd-block generators. Conditioning the block-length distribution on

m\bmod 8

markedly improves the generator's distributional fit, indicating that low-order modular structure is a key driver of heterogeneity in

τ(n)

Nicolò Bonacorsi, Matteo Bordoni

Department of Applied Mathematics, Columbia University, New York, USA. · Department of Mathematics, Columbia University, New York, USA.

Probabilistic Forecasting

Ask PDF

Sep 14, 2026

Language models can produce plausible short proofs, but may still be unreliable on long-horizon research problems, where progress depends on a sequence of uncertain and interdependent decisions. We introduce Stellar Colosseum, a model-agnostic harness for allocating inference across research in mathematics and theoretical computer science. Colosseum explores alternative strategies before proof construction, uses a readiness gate to decide when a route is mature enough to decompose, represents the proof plan as interdependent section-level subproblems, and routes verifier findings back to the affected part of the argument. Across these stages, it generates candidates in parallel, attacks them with targeted falsification, and combines candidates and their critiques into a single research artifact through overlapping random-sample tree aggregation. The Colosseum workflow has been integrated into Google Antigravity's Teamwork framework as the Long Proof pattern. We demonstrate the capabilities of Colosseum through open-ended research and evaluations on theorem-proving and competitive programming benchmarks. Using Colosseum with Gemini 3.1 Pro, we obtain several new results that address open problems arising from papers published at top venues such as FOCS and JMLR. On TCS-Bench, a benchmark of research-level theorem-proving tasks drawn from papers published at FOCS, STOC, and SODA, Colosseum achieves 71.0% accuracy using Gemini 3.1 Pro and Gemini 3.7 Flash. In a separate Codeforces evaluation using Gemini 3.1 Pro, the proof-oriented pipeline with execution feedback solves 218 of 222 problems.

Honghao Lin, David P. Woodruff, Yuan Deng +3

1Google Research. · 2Carnegie Mellon University.

Inference-Time Search Automated Theorem Proving Long-Horizon Reasoning Multi-Agent Reasoning Autonomous Research Agents LLM Reasoning

Ask PDF

Pos. steps $k$

Neg. steps

Total drain

Total reload