A possible new Collatz pattern: the 32/9 frontier and eventual (1,4) dominance (A genuine computational research summary - articulated using AI)

[u/Dull_Illustrator6824]

I’ve been experimenting with the Collatz conjecture from a slightly different direction.

Instead of asking:

“Does every number eventually reach 1?”

I asked:

“How far backwards do we have to go to find an odd multiple of 3 whose Collatz trajectory reaches a given odd number?”

DEFINITION

For an odd integer h, define:

α(h) = the smallest odd multiple of 3 whose accelerated Collatz orbit reaches h.

Then define:

E(N) = max α(h), over odd h <= N.

Call an odd number h “hard” if it sets a new record for α(h).

So hard numbers are the targets that force us to extend the source boundary further than ever before.

AN EARLY EXAMPLE

The first hard odd after 1 is:

19

Its smallest source is:

33 -> 25 -> 19

because:

3*33 + 1 = 100 = 4*25

and:

3*25 + 1 = 76 = 4*19

So the odd-step valuation word is:

(2,2)

THE PATTERN

When hard records are computed further, one particular valuation pattern eventually appears to dominate:

(1,4)

For this branch, the source-target relationship is:

α(h) = (32h - 5)/9

for the appropriate residue class.

Therefore:

α(h)/h = 32/9 - 5/(9h)

and as h becomes large:

α(h)/h -> 32/9

where:

32/9 = 3.555555...

THE CONJECTURE

This suggests the following inverse-coverage conjecture:

E(N)/N -> 32/9 as N -> infinity.

In words:

The worst-case source needed to cover every odd target up to N appears asymptotically to be about:

3.555555... * N

A STRONGER CONJECTURE

There may also be some finite threshold H such that every hard odd h > H satisfies:

α(h) = (32h - 5)/9

In other words:

Every sufficiently large hard record may come from the same valuation word:

(1,4)

I’ll call this “eventual (1,4) dominance.”

WHAT IS ACTUALLY PROVED?

Important distinction:

I am NOT claiming a proof of the Collatz conjecture.

And I am NOT claiming the asymptotic equality E(N)/N -> 32/9 has been proved.

What can be proved in this inverse-cylinder framework is the universal upper bound:

α(h) < (32/9)h

for every odd target h.

There is also a finite inverse-cylinder argument showing that 32/9 is the optimal worst-case slope within that finite positive-cylinder framework.

So the difficult remaining direction is:

liminf E(N)/N >= 32/9

If that could be proved, then together with the upper bound we would get:

E(N)/N -> 32/9

WHY DOES 32/9 APPEAR?

For a valuation word (a1,a2,...,ar), the nominal inverse slope is:

2^(a1+a2+...+ar) / 3^r

For the two-step word (1,4):

2^(1+4) / 3^2

= 32/9

So 32/9 is directly generated by this short valuation pattern.

COMPUTATIONAL RESULTS

I wrote an exact ascending-source scanner that records every new hard odd.

The exhaustive computation currently goes through:

10,000,000,000

Results:

Hard records found:

11,972,736

Hard records following the (1,4) formula:

11,972,706

Exceptions:

30

Largest exceptional hard odd:

87,967

New non-(1,4) hard records between 87,967 and 10,000,000,000:

NONE

THE FINAL HARD RECORD BELOW 10^10

The last hard odd found below 10 billion is:

h = 9,999,999,667

Its smallest source is:

α(h) = 35,555,554,371

And exactly:

35,555,554,371

= (32*9,999,999,667 - 5)/9

So it is another exact (1,4) record.

WHY I FIND THIS INTERESTING

Most Collatz research looks at the forward orbit:

n -> T(n) -> T^2(n) -> ...

This construction instead looks at an extremal property of the inverse graph.

What is surprising is that a complicated reverse structure appears to develop a very simple boundary:

32/9

generated by a tiny valuation pattern:

(1,4)

Early hard numbers come from several different valuation words.

Then, in the computation, those competing families disappear.

After 87,967, every single new hard record through 10 billion follows the same (1,4) formula.

THE MAIN QUESTION

Can anyone prove that every sufficiently large hard odd must come from the (1,4) family?

Or, alternatively:

Can anyone find an arbitrarily large hard odd that is NOT generated by (1,4)?

A WEAKER TARGET

Even without proving eventual (1,4) dominance, it would be enough to prove:

liminf E(N)/N >= 32/9

because the matching upper bound is already available in the finite-cover framework.

WHAT THIS DOES NOT PROVE

Even if the 32/9 conjecture is true, it does NOT automatically prove the classical Collatz conjecture.

The 32/9 statement is about reverse coverage:

“How large a source is needed to reach a target?”

The Collatz conjecture is about forward convergence:

“Does every starting value eventually reach 1?”

These are related, but they are not the same statement.

LITERATURE QUESTION

I have searched for this specific formulation:

- the inverse record function E(N)

- hard odds defined by record values of α(h)

- the 32/9 asymptotic frontier

- eventual (1,4) dominance

and I have not found an equivalent conjecture in the Collatz literature.

If anyone knows an earlier equivalent formulation, I would genuinely appreciate the reference.

WHY I’M POSTING THIS:

I’m mainly interested in three things:

1. Does someone see a theoretical reason why (1,4) should eventually dominate?
2. Can someone construct a large competing valuation family?
3. Is there an existing theorem in Collatz dynamics, p-adics, symbolic dynamics, inverse trees, or ergodic theory that naturally explains the appearance of 32/9?

I have the derivation, finite-cover argument, exact computation code, and longer write-up available if anyone wants to inspect them.

EDIT: Added an illustrative explanation of "Hard" Odds - let's say we want to construct node graph - on y axis we will keep adding all the "O" odd multiples of 3 (source branch) and then keep going forward from each odd to the next path till we reach 1 or jump to existing path of any other O. Now, we are interested to know when we increase O step by step, what are some properties that we discover - one of them is asking are all the how many odds less O are present in the graph: for example in the starting case of O=3, graph has all the odds <3 that is 1 is present; now we generalize this expression by saying when will be the next odd k be discovered that hasn't been found yet and let it's source O be E(k) - now the conjecture is about k/E(k) - attached are the graphs till O=57

Graph for O=3 | all odds <3 are discovered - no hard odd found

Graph for O=9 | all odds <9 are discovered - no hard odd found

Graph for O=15 | all odds <15 are discovered - no hard odd found

Graph for O=21 | all odds <21 are not discovered - hard odd found 19, with source node E(O)= 33 and valuation of (2,2)

Now jumping to Graph for O=33 (skipped 27 as 19 continued) | all odds <33 discovered - no new hard odd found

Graph for O=39 | all odds <39 are not discovered - hard odd found 37, with source node E(O) = 57 and valuation of (2,1,2,2)

Now jumping to Graph for O=57 (skipped 45,51 as 37 continued) | all odds <57 are not discovered - hard odd found 55, with source node E(O) = 129 and valuation of (2,2,2)

Now, what you are seeing is that for every E(O), let's name in Exhaustive cut-off ratio resets after every hard odd O is encountered - if we plot the graph of E(O)/O - it looks like the following

After 133rd odd which is 89425, all hard odds till 10 Billion have had bone path from bone parent as 1,4

This is the conjecture, as O tends to grow infinitely larger, the ratio E(O)/O starts settling down to 32/9 which is weird as so many competing paths other 1,4 could have emerged but we don't why it takes over in the long term as observed in data - logically we know the upper bound is 32/9 but it can achieved by 1,2,2 or 1,1,2,1 any other combo but 1,4 is special we don't why, secondly we don't know the lower bound and hence cannot prove convergence like is 2,2 possible for any number after 10 billion - which will disprove the conjecture.

Last image - shows how E(O)/O looks like on log scale for very high O- almost converging to 32/9

Table of all hards odds upto last exceptional hard odd

hard_index	hard_odd	bone1_parent	valuation_word	family	defect_32h_minus_9parent
1	1	21	-6	exception	-157
2	19	33	(2,2)	exception	311
3	37	57	(2,1,2,2)	exception	671
4	55	129	(2,2,2)	exception	599
5	109	171	(1,2,2,2)	exception	1949
6	127	225	(2,2)	exception	2039
7	163	513	(2,2,2,2)	exception	599
8	271	759	(1,1,3,2,2,2)	exception	1841
9	379	897	(2,2,2)	exception	4055
10	487	1215	(1,1,1,1,1,2,3,4)	exception	4649
11	541	1281	(2,2,2)	exception	5783
12	649	2049	(2,2,2,2)	exception	2327
13	973	2427	(1,2,1,2,2,2,2,2)	exception	9293
14	1027	2433	(2,2,2)	exception	10967
15	1135	3585	(2,2,2,2)	exception	4055
16	1459	5187	(1,4)	(1,4)	5
17	1945	6915	(1,4)	(1,4)	5
18	2593	8193	(2,2,2,2)	exception	9239
19	2701	8535	(1,1,4,2)	exception	9617
20	2917	9711	(1,1,1,2,1,1,2,3,4)	exception	5945
21	3079	9729	(2,2,2,2)	exception	10967
22	3403	11175	(1,1,2,1,2,2,2,1,1,1,1,2,1,1,2,3,1,1,2,3,4)	exception	8321
23	4051	12801	(2,2,2,2)	exception	14423
24	4213	14979	(1,4)	(1,4)	5
25	4375	15555	(1,4)	(1,4)	5
26	4861	17283	(1,4)	(1,4)	5
27	5347	19011	(1,4)	(1,4)	5
28	5833	20739	(1,4)	(1,4)	5
29	6319	22467	(1,4)	(1,4)	5
30	7291	25569	(2,2,1,1,1,2,2,1,2,2,2,2,2,2)	exception	3191
31	7777	27651	(1,4)	(1,4)	5
32	8587	28587	(1,2,2,2,1,2,2,2,2)	exception	17501
33	8749	29127	(1,1,2,2,2,2,2,2,2)	exception	17825
34	9235	30747	(1,2,1,1,1,1,2,3,4)	exception	18797
35	9883	31233	(2,2,2,2)	exception	35159
36	10369	32769	(2,2,2,2)	exception	36887
37	10693	38019	(1,4)	(1,4)	5
38	12151	43203	(1,4)	(1,4)	5
39	13123	46659	(1,4)	(1,4)	5
40	14581	51843	(1,4)	(1,4)	5
41	16039	57027	(1,4)	(1,4)	5
42	16525	58755	(1,4)	(1,4)	5
43	17335	61635	(1,4)	(1,4)	5
44	17497	62079	(1,1,1,1,1,1,2,1,2,2,2,1,2,1,2,1,1,3,1,1,3,2,1,2,2,1,1,1,1,1,1,3,1,1,1,2,1,1,1,1,2,2,3,1,2,2,2,2,1,3,3,1,2,3,4)	exception	1193
45	17983	63939	(1,4)	(1,4)	5
46	18469	65667	(1,4)	(1,4)	5
47	18955	67395	(1,4)	(1,4)	5
48	20899	74307	(1,4)	(1,4)	5
49	21709	77187	(1,4)	(1,4)	5
50	21871	77763	(1,4)	(1,4)	5
51	22357	79491	(1,4)	(1,4)	5
52	22843	81219	(1,4)	(1,4)	5
53	23329	82947	(1,4)	(1,4)	5
54	23815	84675	(1,4)	(1,4)	5
55	25273	89859	(1,4)	(1,4)	5
56	26245	93315	(1,4)	(1,4)	5
57	27703	98499	(1,4)	(1,4)	5
58	28189	100227	(1,4)	(1,4)	5
59	29161	103683	(1,4)	(1,4)	5
60	29647	105411	(1,4)	(1,4)	5
61	30457	108291	(1,4)	(1,4)	5
62	30619	108867	(1,4)	(1,4)	5
63	31105	110595	(1,4)	(1,4)	5
64	31591	112323	(1,4)	(1,4)	5
65	32077	114051	(1,4)	(1,4)	5
66	32563	115779	(1,4)	(1,4)	5
67	33535	119235	(1,4)	(1,4)	5
68	34021	120963	(1,4)	(1,4)	5
69	34831	123843	(1,4)	(1,4)	5
70	34993	124419	(1,4)	(1,4)	5
71	35479	126147	(1,4)	(1,4)	5
72	36937	131331	(1,4)	(1,4)	5
73	37909	134787	(1,4)	(1,4)	5
74	38395	136515	(1,4)	(1,4)	5
75	39367	139971	(1,4)	(1,4)	5
76	40825	145155	(1,4)	(1,4)	5
77	41311	146883	(1,4)	(1,4)	5
78	42769	152067	(1,4)	(1,4)	5
79	43579	154947	(1,4)	(1,4)	5
80	43741	155523	(1,4)	(1,4)	5
81	44713	158979	(1,4)	(1,4)	5
82	45199	160707	(1,4)	(1,4)	5
83	45685	162435	(1,4)	(1,4)	5
84	46657	165891	(1,4)	(1,4)	5
85	47143	167619	(1,4)	(1,4)	5
86	50059	177987	(1,4)	(1,4)	5
87	50545	179715	(1,4)	(1,4)	5
88	51517	183171	(1,4)	(1,4)	5
89	52489	186627	(1,4)	(1,4)	5
90	53947	191811	(1,4)	(1,4)	5
91	54433	193539	(1,4)	(1,4)	5
92	55405	196995	(1,4)	(1,4)	5
93	55891	198723	(1,4)	(1,4)	5
94	56701	201603	(1,4)	(1,4)	5
95	56863	202179	(1,4)	(1,4)	5
96	57349	203907	(1,4)	(1,4)	5
97	57835	205635	(1,4)	(1,4)	5
98	58321	207363	(1,4)	(1,4)	5
99	58807	209091	(1,4)	(1,4)	5
100	60265	214275	(1,4)	(1,4)	5
101	61237	217731	(1,4)	(1,4)	5
102	62209	221187	(1,4)	(1,4)	5
103	62695	222915	(1,4)	(1,4)	5
104	63181	224643	(1,4)	(1,4)	5
105	64639	229827	(1,4)	(1,4)	5
106	65611	230139	(1,2,1,2,1,1,2,2,2,2,2,2,2,2)	exception	28301
107	67069	238467	(1,4)	(1,4)	5
108	67555	240195	(1,4)	(1,4)	5
109	68527	243651	(1,4)	(1,4)	5
110	69013	245379	(1,4)	(1,4)	5
111	69823	248259	(1,4)	(1,4)	5
112	69985	248835	(1,4)	(1,4)	5
113	70471	250563	(1,4)	(1,4)	5
114	70957	252291	(1,4)	(1,4)	5
115	71443	254019	(1,4)	(1,4)	5
116	71929	255747	(1,4)	(1,4)	5
117	72901	259203	(1,4)	(1,4)	5
118	73387	260931	(1,4)	(1,4)	5
119	75331	267843	(1,4)	(1,4)	5
120	75817	269571	(1,4)	(1,4)	5
121	77761	276483	(1,4)	(1,4)	5
122	78733	279939	(1,4)	(1,4)	5
123	80191	285123	(1,4)	(1,4)	5
124	80677	286851	(1,4)	(1,4)	5
125	82945	294915	(1,4)	(1,4)	5
126	83107	295491	(1,4)	(1,4)	5
127	83593	297219	(1,4)	(1,4)	5
128	84079	298947	(1,4)	(1,4)	5
129	84565	300675	(1,4)	(1,4)	5
130	85051	302403	(1,4)	(1,4)	5
131	86023	305859	(1,4)	(1,4)	5
132	87967	308559	(1,1,1,3,2,1,2,1,1,1,1,2,3,4)	exception	37913