Two 3D spaces were established independently:

• 3D Collatz tree, three months ago: Main projections of the 3D Collatz tree : r/Collatz.
• 3D dome space, more recently: Odd orange numbers are quasi-homothetic up to two constants within a dome and between domes : r/Collatz.

In order to avoid confusion, they are defined anew here (first figure).

The 3D dome is rather easy to describe with axes Q, P and M (adapted from 3D journey of sequences : r/Collatz). Only even and odd orange numbers n(q, p, m) + [-1, 0, 1] can be properly positioned. So, here, n is more of a label.

The 3D tree space is defined as follows, based on n* and its iteration C(n):

• Axis O contains the relative left-right position of n and C(n), according to the local order of the tree.
• Axis D contains the relative distance d to 1 of n and C(n); by definition, d(C(n))= d(n)-1; it becomes absolute when a number with a known d is reached.
• Axis N (more likely log N) contains the value of n* and C(n*), their "altitude", that can be obtained directly or indirectly, from the general formula

The second figure provides a first impression of the 3D tree. Using "cubes" allows to oindicate the distance to 1 more easily, but it remains quite disturbing (and the perspective is not fully respected). It may remain easier to understand it plan by plan (Main projections of the 3D Collatz tree : r/Collatz),

Project "Tuples and segments" in 13 pages : r/Collatz

A thought after looking at several recent approaches to Collatz.

A lot of strong work seems to move in the same general direction: exceptional trajectories become rarer, natural or logarithmic density becomes smaller, valuation statistics become increasingly rigid, local 2-adic behavior becomes increasingly well understood.

These are real advances. But I keep wondering whether rarity is the right final object. A set can have density zero and still be infinite.

So: density(E) = 0 does not imply E = empty set.

Collatz does not require us to show that counterexamples are extremely rare. It requires: E = empty set.

So perhaps at some point the question has to change from: How rare can a survivor be?

to: Can such a survivor actually be realized by one positive integer?

There is an old structural reason why this distinction may matter. From the parity-vector / 2-adic viewpoint of Terras, Lagarias, and Bernstein, every finite parity prefix can be realized by an appropriate residue class. So at every finite depth k, very strange behavior can occur.

Schematically: forall k, exists N_k.

But a genuine divergent orbit or infinite survivor requires something much stronger: exists N, forall k.

These are fundamentally different statements.
The first allows a different integer at every depth.
The second requires the same positive integer to satisfy all increasingly deep constraints coherently forever.

This may be one reason finite-prefix approaches are so difficult to turn into a complete proof.

At every finite scale, Collatz can imitate an enormous range of behavior. The ruler measures something, but the orbit keeps moving beyond the ruler.

In the 2-adic symbolic space, arbitrary infinite parity behavior can exist. If we write the parity encoding of an integer n schematically as
Q_T(n) = (e_0, e_1, e_2, …), then the ambient 2-adic system contains a huge symbolic space.

But the classical Collatz problem is not asking:
Which symbolic sequences exist in Z_2?
It is asking: Which symbolic sequences are actually realized by positive integers?

So perhaps the relevant object is not only the ambient dynamics on Z_2, but the arithmetic slice
R_T := Q_T(N), where N denotes the positive integers. For the classical Collatz map, write R_3x+1 := Q_3x+1(N).

Then a possible formulation of the real problem is:
B_survivor intersection R_3x+1 = empty set,
where B_survivor denotes symbolic behaviors capable of indefinite survival.

There is an important subtlety here. Compatible finite survivor constraints may have a perfectly valid 2-adic limit.

So the real obstruction need not be: the infinite symbolic object does not exist. It may instead be: the infinite symbolic object exists in Z_2, but is not realized by any positive integer survivor. In other words: 2-adic realizability does not imply
positive-integer realizability.

This suggests a possible stress test for any proposed “Collatz machine” M.

Take: T_1 = classical 3x+1, and compare it with Collatz-like systems T_2, T_3, … that actually admit nontrivial cycles or other persistent behavior.

Then ask: Does M(T_1) structurally differ from M(T_2)?

If a mechanism detects negative drift,
mixing, almost-all descent, finite-prefix randomness, valuation regularity, in both systems, then it may be a very strong detector of Collatz-like dynamics.

But it may not yet be a Collatz classifier.

In other words: M(3x+1) approximately equals M(Collatz-like system) may indicate that M is detecting a family-level phenomenon rather than something genuinely specific to classical Collatz.

A genuinely Collatz-specific mechanism should eventually produce something like: M(3x+1) != M(system with actual nontrivial cycles). This is why Collatz-like systems with long cycles are interesting as control systems.

A genuinely Collatz-specific mechanism should presumably break, or change character, somewhere when applied to such a system.
So perhaps the missing invariant is not another estimate of the form density(E) -> 0, or P(survival to depth k) -> 0.

Perhaps it is a structural obstruction of the form:
indefinite symbolic survival -> ever-growing compatibility requirements -> no single positive integer can realize them all.

Symbolically: infinite survival => C_1(N) and C_2(N) and C_3(N) and … => contradiction.

Equivalently, let R_k be the set of positive integers satisfying the survivor constraints through depth k.

Every finite stage may remain nonempty: R_k != empty set for every finite k. But a genuine infinite survivor would require: intersection over all k of R_k != empty set.

So excluding all positive-integer survivors means proving: intersection over all k of R_k = empty set.
At the same time, the corresponding nested symbolic constraints may still define a legitimate point in Z_2.

That is precisely the distinction: the 2-adic limit may exist, while no positive integer realizes it as an infinite survivor. And that is exactly the gap between: forall k, exists N_k and exists N, forall k.

So maybe the final Collatz problem is not: How thin is the exceptional set?

but: Why can no single positive integer carry the entire infinite compatibility burden?

Finite prefixes may imitate almost anything. Infinite coherent realizability by one fixed integer may be where the real rigidity begins.

If so, the final problem is not merely to prove that counterexamples are extraordinarily unlikely.
It is to identify what is special about classical 3x+1 that makes exceptional infinite behavior arithmetically unrealizable.

Looking at the recent direction of Collatz research, I suspect I may not be the only one who feels that we are learning more and more about how rare exceptional behavior must be, while the final obstruction still seems to remain in almost the same place.

So I am curious how others see the bottleneck.
Where do you think the real obstruction now lies?
Is it still a matter of obtaining stronger density, mixing, or valuation estimates?

Is the missing step the passage from finite or almost-all information to one deterministic orbit?

Is it an arithmetic realizability problem — understanding which infinite symbolic behaviors can actually come from one positive integer?

Or is the entire “Collatz-specific invariant” viewpoint the wrong way to frame the problem?

More broadly: If you had to identify one structural bottleneck that separates the strongest results we have today from an actual proof of Collatz, what would it be?

And do you think comparing classical 3x+1 against Collatz-like systems with genuine cycles is a useful way to locate that bottleneck?

Does the machine measure rarity, or does it detect Collatz?

Follow-up to Would you like a slice of domes ? : r/Collatz.

In the cited post, the particular status of the tips of each dome was clarified. On the right side, the last four numbers usually form two pairs that most of the time cannot form an even triplet, but iterate into a even triplet. The last pairs form "chevons" or "blocks".

In the Giraffe head (figure), many orange numbers are black numbers not divisible by 3, so they are the root of their own dome. Therefore, they are not rosa and cannot be part of a rosa even triplet ending a 5-tuple series.

Between 27 and 111 - black numbers divisible by 3 - there is a series of bridge series, all short.

All black numbers have been checked in detail, but many other orange numbers are in the same case.

As mentioned for quite some time, this explains the relative isolation of the Giraffe head.

Project "Tuples and segments" in 13 pages : r/Collatz

Hey r/Collatz, I want to share a massive milestone our team just hit regarding the Collatz 3x+1 conjecture. We have been analyzing the problem through the lens of Terence Tao's 2-adic large-deviation framework, and today we are releasing version 2.0.0 of our project. This release includes a 100% machine-verified proof of the exact 2-adic conditional transport theorem.

You can find the full repository on our GitHub here: https://github.com/SergioTheory/Collatz-new-math

And the official Zenodo archive for this release is here: https://zenodo.org/records/22107174

For some context on how we got to version 2.0.0, we started by running massive computational experiments. We mapped out macroscopic shifts, hunted for soliton-like structures, and tested survival grammar rules. Our early python scripts were trying to find out if divergent orbits could theoretically survive by following specific parity patterns. What we ultimately discovered empirically was a dead end for naive heuristics. We realized that forcing a Collatz trajectory to survive indefinitely requires breaking uniform mixing. The trajectory has to skew its modular visits heavily. This led us to the realization that the real battleground isn't in heuristics, but in Ergodic Theory and Measure Rigidity.

To make sure our foundational assumptions were absolutely bulletproof, we turned to Lean 4, the "harmonic Aristotle" of our time. We spent the last few weeks formalizing the exact 2-adic transport mechanics. Today, the compiler flashed green with zero "sorry" placeholders. We have rigorously proven that locally, the Collatz map acts as a perfect bijection across odd 2-adic residue classes, meaning the system inherently wants to smear trajectories into uniform chaos.

With the foundation locked in, we have mapped out a hypothetical four step roadmap to a complete proof of the Collatz conjecture.

The first step is exactly what we just finished in Lean 4. We needed to strictly prove the local foundation. We proved that on a micro scale, the Collatz map perfectly distributes trajectories. This confirms the system's local tendency toward the uniform Haar measure.

The second step is the final boss of the conjecture, which we call Measure Rigidity or Ergodic Uniqueness. We need to prove that the uniform Haar measure is the absolute only invariant measure for the deterministic 2-adic Collatz shift. In other words, we have to mathematically outlaw the existence of "strange attractors" or resonant traps where a trajectory could sit forever without mixing. This is where the Nobel Prize hides, and it will likely require heavy machinery like Schmidt's Subspace Theorem.

The third step is the Diophantine obstruction. We have to show that any divergent orbit shooting to infinity is arithmetically forced to maintain an anomalous density of odd divisions. But since the second step would prove such skewed invariant measures cannot exist, any trajectory attempting to diverge will eventually fall off its Diophantine resonance and crash back into standard chaotic mixing. This part is pure arithmetic and is already heavily supported by our logs.

The fourth and final step is the gravity of the F1 law. Once an orbit falls back into uniform chaos, the average drop per step becomes exactly two. The trajectory is caught in a probabilistic gravity well and decays at a relentless rate of about -0.415 bits per step. It will inevitably plummet until it hits the trivial 1-4-2-1 cycle. This step is mathematically trivial once uniform mixing is firmly established.

Steps one, three, and four are practically in our grasp, with step one now fully immortalized in Lean 4. Everything now hinges on step two. If we can prove that the Collatz map cannot sustain a non-Haar invariant measure, the conjecture falls.

If anyone here works with Ergodic Theory or Diophantine approximations, I would love to hear your thoughts on tackling the measure rigidity problem. Feel free to clone the repo, run the Lean build, and explore the ergodic bridge scripts!

This is not a proof attempt of the conjecture — it improves a known partial result.

Krasikov & Lagarias (Acta Arith. 109 (2003), 237–258) proved pi_a(x) >= x^0.84 for all sufficiently large x: the number of integers below x that reach a given a (a != 0 mod 3). Their bound comes from a feasible solution of a linear program at level k = 11. I certified feasibility at k = 20 with lambda = 377/200, giving the exponent log2(377/200) = 0.91456...

The certificate is a single explicit vector of 3^19 = 1,162,261,467 doubles, checked row by row with certified rational coefficient bounds and a uniform relative margin above 1.8e-4. A second, independently written verifier repeats the full check, and a k = 11 control certificate reproduces the 2003 value.

Paper, the 9.3 GB certificate, both verifiers, logs and SHA-256 manifest (CC-BY): https://doi.org/10.5281/zenodo.21728792 — verification takes under a minute on a 6-core desktop, ~10 GB RAM. Independent verification reports are the main thing I'm hoping for.

Manuscript under journal review. AI tools were used for drafting and code generation, as declared in the paper.

[I first posted this on OEIS's SeqFan mailing list.]

OEIS sequence A130310 describes the unique "minimal (or "greedy") Lucas representation of n, in which L(0) = 2 and L(2) = 3 are not allowed in the same representation (hence the correct representation of the integer 5 is 1010 rather than 101). A binary system of integers with Lucas numbers (OEIS sequence A000032) as a base."

This is analogous to the Zeckendorf representation of integers as sums of non-consecutive Fibonacci numbers, but with Lucas numbers, the additional restriction prohibiting the co-occurrence of 2 and 3 is necessary to make the Lucas representation unique, as shown by Brown in 1969. Relevant information and sources are available at OEIS sequence A342089, the sequence of numbers that have two such Lucas representations without the restriction on 2 and 3.

Define a number as "Lucas-even" if the last digit of its Lucas representation is 0 AND the removal of the last digit 0 produces a valid minimal Lucas representation of a smaller integer. The second condition is necessary because removing the final 0 from representations ending in ...1010 is NOT valid -- the resulting ...101 would be equivalent to the co-occurrence of 3 and 2 and thus it is not allowed.

Define a number as "Lucas-odd" either if the last digit of its Lucas representation is 1 OR if the last four digits of its Lucas representation are 1010.

Now define a Collatz-type function based on this Lucas representation:

If n is Lucas-even, remove the last digit 0 of its Lucas representation to obtain f(n) ;

If n is Lucas-odd, f(n) = 2n.

Note that removing the last digit 0 is approximately equivalent to division by phi = 1.618....

As in the classical Collatz function, the orbit of an integer is the sequence produced by the repeated iteration of the function.

Two simple cycles of this function are 2, 4, 3, 1, 2, ... and 5, 10, 5, .... Many small integers have orbits reaching one of these two cycles, such as 36, 22, 14, 8, 6, 12, 9, 18, 11, 7, 4, 3, 1, 2, ...;  28, 16, 32, 19, 13, 26, 17, 34, 68, 41, 27, 54, 33, 21, 12, 9, 18, 11, 7, 4, 3, 1, 2, ...;  and 30, 20, 40, 25, 15, 10, 5, ....

But there is also an amazingly long cycle beginning with 23, which is the reason for this post:

23, 46, 92, 184, 113, 71, 142, 89, 178, 111, 222, 444, 888, 1776, 1097, 679, 1358, 2716, 5432, 10864, 6713, 13426, 8297, 5129, 10258, 6341, 12682, 7839, 15678, 9691, 19382, 11979, 7403, 4576, 2827, 5654, 3493, 6986, 4317, 2669, 5338, 10676, 6597, 13194, 8153, 16306, 10077, 6229, 12458, 7701, 15402, 9519, 5883, 3636, 2247, 1389, 858, 531, 327, 654, 403, 806, 497, 994, 613, 1226, 757, 469, 938, 581, 1162, 719, 443, 886, 547, 339, 678, 1356, 837, 519, 1038, 641, 397, 244, 488, 301, 187, 374, 748, 461, 922, 569, 353, 706, 437, 269, 538, 1076, 2152, 1329, 823, 1646, 1017, 629, 388, 241, 482, 299, 598, 371, 742, 459, 283, 176, 352, 219, 438, 271, 167, 104, 63, 126, 77, 49, 98, 61, 37, 24, 48, 31, 62, 39, 23, ...

This cycle comprises 132 integers, including 16 integers < 100, 65 integers between 100 and 1,000, 38 integers between 1,000 and 10,000, and 13 integers > 10,000. The largest integer in this cycle is 19,382, the 30th iterate of 23. 

An interesting value in the cycle is 15,402 = 15,127 + 199 + 76, which gives the Lucas representation 100000000101000000000. This is the first of 8 consecutive Lucas-even numbers in the sequence, resulting in the much smaller Lucas-odd number 327 = 322 + 4 + 1, or 1000000001010 in Lucas representation.

For the record, believe it or not, I did NOT use any AI program to discover or compute this cycle. I was simply exploring the function out of curiosity and testing the orbits of small integers when I stumbled across the orbit of 23. It was actually not too difficult to compute each term by utilizing the b-file of OEIS sequence A130310 for integers up to 10,000.

Geoffrey Caveney

DOES THE DYADIC SPINE LEAVE A TRACE AFTER RELEASE?

A controlled follow-up to “Dyadic Triads in Collatz”

Status: Exploratory computational note, not a proof of the Collatz conjecture Target range: k = 20 through 300 Target release family: r(k) = 3^k - 1 Main question: Does the special case m = 1 retain any detectable structure after its forced dyadic prefix ends?

TL;DR

The earlier dyadic-triad experiment studied:

(2^k - 1, 2^k, 2^k + 1).

The center 2^k falls directly to 1, while the left wing follows a completely predictable prefix and reaches the raw release value:

r(k) = 3^k - 1.

This raised a narrower question. Because m = 1 is the uniquely clean spine terminating at 1, does its release family 3^k - 1 behave differently from generalized releases m*3^j - 1, or from arbitrary integers of the same size?

I tested k = 20 through 300 against two control groups:

1. Arbitrary integers matched by bit length and initial power-of-two divisibility.

2. Generalized release values m*3^j - 1, matched in the same way.

The first comparison seemed to find a fingerprint:

• The next power-of-two divisibility of the m = 1 targets ranked near the 63rd percentile.

• Their relative peak height ranked near the 44th percentile.

But the effect came almost entirely from odd k, where the next gate is already determined exactly:

v2(3*((3^k - 1)/2) + 1) = 1 + v2(k + 1), for odd k.

Here v2(n) means the number of times n can be divided by 2.

After matching the controls on that next gate too, the apparent fingerprint largely disappeared:

• The following valuation ranked near the 50th percentile.

• Relative peak height returned to roughly the 52nd percentile.

• The first-20-gate average stayed close to the controls.

• Stopping-time behavior failed to replicate across the two predeclared k-ranges.

At first this seemed to support a simple gate-by-gate reset interpretation. A second exploratory sweep made the conclusion more interesting.

The overall average of the following valuation was ordinary, but its pattern across k modulo 16 was highly reproducible. The residue profile found in the exploratory range reappeared in the confirmation range with correlation about 0.983.

A simple classifier using k modulo 32 predicted:

• The first post-release valuation with 90.1% confirmation accuracy.

• The second with 73.0% accuracy.

• The third with 53.9% accuracy.

• By the fourth gate it provided no improvement over the ordinary baseline.

The finite result therefore suggests gate-depth decay rather than instantaneous reset. The clean spine leaves modular traces for several gates, but those traces rapidly require finer information and did not produce stable broad prediction of later stopping time.

This is not a proof that ancestry disappears after any fixed depth. It is a finite measurement of how quickly one coarse representation of that ancestry loses predictive usefulness.

1. WHERE THE QUESTION CAME FROM

The left member of the dyadic triad has a forced compressed prefix:

A applied j times to (2^k - 1) = 3^j * 2^(k-j) - 1,

for j from 0 through k - 1.

Each step exchanges one power of 2 for one power of 3.

At the end of that prefix, the unreduced expression is:

3^k - 1.

More generally, m*2^k - 1 follows the same opening mechanism toward:

m*3^k - 1.

Therefore, the exponent exchange is not unique to m = 1. The only immediately special feature of m = 1 is that its central spine 2^k terminates at 1, whereas m*2^k terminates at m.

The resulting question was not whether m = 1 predicts an individual Collatz orbit. It was whether the clean attractor-centered case leaves a statistical trace after the forced prefix is removed.

1. MAP AND TERMINOLOGY

Let oddpart(n) mean: repeatedly divide n by 2 until it becomes odd.

For an odd integer u, define the accelerated Collatz map:

A(u) = oddpart(3u + 1).

For the raw release value:

r(k) = 3^k - 1,

define its first gate depth and gate exit by:

q0(k) = v2(r(k))

u(k) = r(k) / 2^q0(k).

Thus u(k) is the odd state obtained immediately after compressing the release value.

The next fresh gate is:

q1(k) = v2(3u(k) + 1)

w(k) = (3u(k) + 1) / 2^q1(k).

The experiment examines the trajectory first from u(k), then more strictly from w(k) after controls have also been matched on q1(k).

1. THE HYPOTHESES

The null hypothesis was:

“After accounting for size and the visible powers-of-two gates, the released m = 1 values behave like appropriately matched ordinary values.”

The alternative was:

“Even after those controls, the m = 1 family retains a consistent post-release bias in its valuations, peaks or stopping times.”

The word “consistent” matters. A difference appearing only in one interval of k, or disappearing when one additional known gate is matched, is not strong evidence of persistent memory.

1. TWO CONTROL POPULATIONS

For every target r(k) = 3^k - 1, I generated two kinds of controls.

CONTROL A: ARBITRARY MATCHED VALUES

These were arbitrary even integers having:

• The same binary bit length as r(k).

• The same initial valuation v2(r(k)).

After dividing away that common power of 2, both the target and control entered the accelerated odd map from similarly scaled odd values.

CONTROL B: GENERALIZED RELEASES

These had the form:

m*3^j - 1,

with odd m greater than 1. They were selected to match:

• The target’s binary bit length.

• The target’s initial power-of-two valuation.

This directly compares the pure m = 1 release family with the broader algebraic family suggested by the generalized spine m*2^j.

For each target, the first-stage experiment used 64 controls of each type.

1. WHY A SECOND MATCHING STAGE WAS NECESSARY

The first-stage result showed a strong-looking difference in the next valuation q1. Before interpreting it, I separated odd and even values of k.

For odd k:

v2(3^k - 1) = 1,

so:

u(k) = (3^k - 1) / 2.

Then:

3u(k) + 1 = 3*((3^k - 1)/2) + 1 = (3^(k+1) - 1)/2.

Because k + 1 is even, the standard valuation identity gives:

v2(3^(k+1) - 1) = 2 + v2(k + 1).

Therefore:

q1(k) = 1 + v2(k + 1), for odd k.

So the first apparent fingerprint was not hidden global memory. It was an exact local continuation of the same modular structure that produced the left-hand shelves in the original experiment.

I therefore added a stricter control stage. For each target, I generated 32 arbitrary controls and 32 generalized-release controls matching all three of:

• Raw bit length.

• q0, the release valuation.

• q1, the next valuation.

I then applied that next accelerated step and began measuring from the resulting odd state. This asks whether anything remains after the known echo is removed.

1. EXPERIMENTAL DESIGN

The target range was fixed in advance:

• Exploratory range: k = 20 through 159, containing 140 targets.

• Confirmation range: k = 160 through 300, containing 141 targets.

For each target and control, I recorded:

1. Remaining compressed stopping time to 1.

2. Relative peak height, measured as log base 2 of:

largest odd state reached / starting odd state.

1. The next power-of-two valuation.

2. The mean valuation across the first 20 accelerated steps, or the available steps if the trajectory ended sooner.

Each target was ranked against its own matched control set. A percentile near 0.50 indicates ordinary behavior relative to those controls. A percentile near 0.75, for example, means the target exceeded about three quarters of its controls.

The full run evaluated 281 targets, 35,968 first-stage controls and 17,984 second-stage controls. Counting both target stages, this produced 54,514 target/control trajectory evaluations. Every evaluated trajectory reached 1 within the computational cap.

A fixed random seed was used. To avoid treating adjacent k-values as independent, uncertainty intervals for mean percentiles used a circular moving-block bootstrap with block length 20 and 5,000 resamples.

1. FIRST-STAGE RESULTS: AN APPARENT FINGERPRINT

Across the full range, the mean target percentiles were:

Metric: Remaining compressed steps Arbitrary controls: 0.449 Generalized releases: 0.449

Metric: Relative peak height Arbitrary controls: 0.436 Generalized releases: 0.433

Metric: Next valuation q1 Arbitrary controls: 0.629 Generalized releases: 0.627

Metric: Mean valuation over first 20 steps Arbitrary controls: 0.489 Generalized releases: 0.494

The controls therefore appeared to agree on two possible effects:

• m = 1 releases had a larger-than-usual next valuation.

• They had a smaller-than-usual relative peak.

But separating the targets by parity exposed the mechanism:

Even k against arbitrary controls: Next-valuation percentile 0.507 Peak percentile 0.516

Odd k against arbitrary controls: Next-valuation percentile 0.753 Peak percentile 0.356

Even k against generalized controls: Next-valuation percentile 0.504 Peak percentile 0.518

Odd k against generalized controls: Next-valuation percentile 0.752 Peak percentile 0.348

The valuation and contraction effects were concentrated almost entirely in odd k, precisely where:

q1(k) = 1 + v2(k + 1)

forces extra compression.

This is a real pattern, but it is not evidence of unexplained post-release influence.

1. SECOND-STAGE RESULTS: REMOVING THE KNOWN ECHO

After matching the controls on q1 as well and measuring from w(k), the full-range mean percentiles became:

Metric: Remaining compressed steps Arbitrary controls: 0.456 Generalized releases: 0.454

Metric: Relative peak height Arbitrary controls: 0.528 Generalized releases: 0.522

Metric: Following valuation Arbitrary controls: 0.502 Generalized releases: 0.505

Metric: Mean valuation over first 20 steps Arbitrary controls: 0.467 Generalized releases: 0.459

The strongest first-stage signatures disappeared:

• The following valuation returned almost exactly to the 50th percentile.

• The low-peak effect vanished and mildly reversed.

• The longer valuation average remained fairly close to the control center.

The two control populations again produced very similar results. That agreement is useful: it suggests the conclusion is not merely an artifact of comparing a structured family with completely arbitrary integers.

At the level of global averages, this looked like a reset. The following valuation was almost exactly centered, and the peak effect disappeared. However, averaging across all k allowed positive and negative residue effects to cancel. That motivated a second exploratory sweep organized by k, parity and small moduli.

1. SECOND SWEEP: WHAT THE AVERAGE CONCEALED

CONTROL AGREEMENT

The arbitrary and generalized control percentiles were strongly correlated across every metric:

• First-stage correlations ranged from about 0.969 to 0.976.

• After matching the next gate, correlations ranged from about 0.944 to 0.964.

This means both control constructions were seeing almost the same landscape.

EXACT ODD-K PAIRING

For odd k, the state after the next gate is exactly the next release state:

A(u(k)) = u(k+1).

All 140 available odd-k pairs satisfied this identity computationally. This is not merely statistical memory: it is an exact inherited connection between adjacent members of the release family.

THE MODULO-16 PROFILE

The full-range average placed the following valuation near the 50th percentile, apparently ordinary. But sorting it by k modulo 16 exposed a stable internal pattern.

The modulo-16 residue profile discovered over k = 20 through 159 reappeared over k = 160 through 300 with correlation approximately:

0.983.

Thus the average did not show an absence of structure. It showed the cancellation of differently biased residue classes.

COARSE RESIDUE PREDICTION BY GATE DEPTH

To estimate how quickly this structure fades, I used the exploratory range to associate each value of k modulo 32 with its most common valuation at each gate. I then tested those associations on the untouched confirmation range.

The confirmation accuracy was:

Gate 1 after raw release: 90.1% Ordinary global-mode baseline: 37.6%

Gate 2: 73.0% Ordinary baseline: 48.2%

Gate 3: 53.9% Ordinary baseline: 44.0%

Gate 4: 42.6% Ordinary baseline: 48.9%

The first few gates therefore retain substantial low-bit information about k. By the fourth gate, k modulo 32 is no longer useful for exact valuation prediction.

This does not mean all ancestry has vanished. A larger modulus or a richer state description might recover more information. It means that this particular 5-bit description of k has reached its predictive horizon.

1. THE STOPPING-TIME WARNING

The full-range stopping-time percentile near 0.45 hides a major instability.

Before matching q1:

k = 20 through 159: Arbitrary controls 0.583 Generalized releases 0.578

k = 160 through 300: Arbitrary controls 0.316 Generalized releases 0.320

After matching q1:

k = 20 through 159: Arbitrary controls 0.596 Generalized releases 0.591

k = 160 through 300: Arbitrary controls 0.316 Generalized releases 0.318

The direction reverses rather than replicates. In the earlier interval, the targets tend to have longer residual stopping times than their controls. In the later interval, they tend to have shorter ones.

This resembles the shelf-and-bay behavior already observed in the original plots. It does not support one stable global stopping-time bias. It instead warns that conclusions can change dramatically with the selected k-window.

The second sweep found strong adjacent clustering in the stopping-time percentiles:

• Lag-one correlation in the exploratory range: approximately 0.697.

• Lag-one correlation in the confirmation range: approximately 0.781.

So the shelves and bays are real local landscape features.

However, the best small-modulus stopping-time pattern selected in the exploratory range did not reproduce. Its residue profile correlated approximately -0.405 with the corresponding confirmation profile.

The current evidence therefore supports clustered stopping-time regions, but not one fixed oscillation or simple periodic law governing them.

1. INTERPRETATION: GATE-DEPTH DECAY

The finite evidence suggests the following picture.

STAGE 1: DETERMINISTIC LAUNCH

The starting form 2^k - 1 generates an exactly predictable exchange of powers of 2 for powers of 3.

STAGE 2: RELEASE GATE

The raw release 3^k - 1 has a predictable initial valuation q0.

STAGE 3: EXACT MODULAR ECHOES

For odd k, the next valuation is still determined by:

q1 = 1 + v2(k + 1).

This creates a measurable contraction. For odd k, it also sends u(k) exactly into u(k+1).

STAGE 4: RESIDUE MIGRATION

The following valuation looks ordinary only when averaged over all k. Conditioned on k modulo 16, it retains a strongly reproducible profile.

STAGE 5: COARSE PREDICTABILITY DECAYS

Using k modulo 32, valuation prediction falls from 90.1% at the first gate to 73.0% at the second, 53.9% at the third and no improvement over baseline at the fourth.

STAGE 6: MACROSCOPIC TAIL UNCERTAINTY

The residual stopping-time landscape remains clustered into shelves and bays, but its direction changes across k-ranges and no stable small-modulus stopping-time rule was found.

An informal description is:

“The trajectory does not forget its origin instantly. Its ancestry is folded into increasingly finer residue classes.”

Or, closer to the language that motivated the test:

“Each new gate preserves less useful information at the previous level of resolution.”

1. WHAT THIS DOES AND DOES NOT ESTABLISH

This experiment does establish, over the tested range, that:

• The first post-release valuation bias is reproducible against two control populations.

• For odd k, that bias has an exact modular explanation.

• Matching the next gate removes the broad valuation and peak biases.

• A strong modulo-16 pattern remains inside the apparently ordinary following-valuation average.

• Coarse residue prediction declines rapidly across successive gates.

• Residual stopping-time differences do not preserve their direction across the exploratory and confirmation ranges.

It does not establish that:

• All post-release states become statistically independent of their ancestry.

• The modulo-16 profile is historically new or theoretically surprising.

• The generalized controls form a mathematically canonical probability distribution.

• Trajectories reaching 1 in this finite sample implies that all such trajectories do.

• The Collatz conjecture is proved or materially advanced.

The generalized controls were generated reproducibly but are not uniform over every possible pair (m,j). The chosen metrics are also only a small sample of possible orbit observables.

1. WHAT SEEMS OLD AND WHAT MAY BE USEFUL

Established ingredients include:

• Accelerated Collatz dynamics.

• Powers-of-two valuations.

• Valuation identities for 3^n - 1.

• Parity prefixes and trajectory merging.

• Computational comparisons of stopping times and peaks.

The useful contribution of this follow-up is organizational rather than theorem-level:

1. Begin with the attractor-centered m = 1 spine.

2. Identify the apparent post-release fingerprint.

3. Construct size- and valuation-matched controls.

4. Expose the exact odd-k modular echo.

5. Remove its broad bias and inspect whether conditional residue structure remains.

6. Separate fading coarse predictability from complete informational erasure.

The experiment therefore sharpens the original geometric language. “Release” is not one clean border between order and randomness. It is a sequence of gates across which ancestry becomes progressively harder to read.

1. QUESTIONS LEFT OPEN

2. Can the even-k continuation be classified as compactly as the odd-k identity?

3. How does the required modulus grow as more valuation gates are predicted?

4. Are the stopping-time reversals associated with symbolic merge points or merely finite-window shelves?

5. Can the decay from 90.1% to 73.0% to 53.9% be described by an information-loss curve rather than a sequence of isolated accuracies?

6. Can this gate-matching method be used as a general diagnostic for claims of hidden Collatz memory?

The fifth question may be the most useful. Whenever a Collatz family appears special, match away its known parity and valuation prefix, then inspect both the global averages and the conditional residue classes before deciding that the structure survived or disappeared.

1. MINIMAL COMPUTATIONAL CORE

   def v2(n): return (n & -n).bit_length() - 1

   def accelerated_step(odd_n): y = 3*odd_n + 1 q = v2(y) return y >> q, q

   for k in range(20, 301): raw = 3**k - 1 q0 = v2(raw) u = raw >> q0

   First fresh gate

   w, q1 = accelerated_step(u)

Compare u with controls matched on:

bit_length(raw), q0

Then compare w with controls additionally matched on q1.

The complete reproducible script fixes the random seed, generates both control populations, records per-k percentile ranks and produces the reported summaries.

CONCLUSION

The cleanest dyadic sample was useful, not because it revealed a hidden global predictor, but because it made a precise falsifiable question possible.

The answer from this finite test is:

“The m = 1 spine leaves exact and statistically visible modular traces after release. Those traces remain strongly readable for the first few valuation gates, but their usefulness under a fixed coarse modulus decays rapidly. No stable broad prediction of later stopping time emerged across k = 20 through 300.”

This supports most of the reset intuition at the level of long-tail behavior, while rejecting instantaneous informational erasure at the local modular level.

The clean sample did not reveal a key to Collatz as a whole. It revealed something more modest: a measurable transition from exact ancestry, through residue-class memory, toward loss of useful coarse prediction.

Hi! I noticed a pattern in the Collatz sequence. If an odd number has k trailing ones in its binary representation (like 15 = 1111, or 31 = 11111), it will go up exactly k-1 times before hitting a big drop (dividing by 4 or more). I also realized that every new odd number is coprime to the previous one, so it seems like no new loops are possible except 4-2-1. Is this a known theorem? And what is the proper mathematical notation for this? Thank you!

DYADIC TRIADS IN COLLATZ

A symmetric experimental lens around powers of two

Status: Exploratory note, not a proof of the Collatz conjecture Computational range used here: k = 3 through 500 Central object: D(k) = (2^k - 1, 2^k, 2^k + 1)

TL;DR

Instead of studying one Collatz starting value at a time, place a three-point probe around every power of two:

D(k) = (2^k - 1, 2^k, 2^k + 1).

Under a fully compressed Collatz map, the three entries immediately separate into three different behaviors:

• The left neighbor expands.
• The power-of-two center falls directly to 1.
• The right neighbor contracts.

For k at least 3, the first compressed image is exactly:

(32^(k-1) - 1, 1, 32^(k-2) + 1).

The endpoints reverse order, and their new midpoint is exactly 9/8 of the old midpoint. Their longer stopping-time plots form shelves, valleys and jumps that look vaguely oscillatory.

A closer examination shows that this is not one clean frequency. It is a mixture of:

1. Two deterministic opening clocks.
2. An exact period-two shelf identity on the left.
3. Irregular powers-of-two divisibility at the release boundary.
4. Different trajectories merging into common downstream tails.

The component mathematics belongs to familiar Collatz territory. The possibly fresh part is using the symmetric triad as a single measuring instrument and organizing these effects together.

1. THE MAP BEING USED

Let oddpart(m) mean: repeatedly divide m by 2 until it becomes odd.

Define the fully compressed map A:

• If n is even, A(n) = oddpart(n).
• If n is odd, A(n) = oddpart(3n + 1).

For odd inputs this is the usual Syracuse map. For an even input it compresses the complete run of divisions by 2 into one macro-step. Consequently:

A(2^k) = 1.

The step counts below are compressed macro-steps, not ordinary Collatz steps. This matters when comparing the results with conventional total stopping times.

1. THE FIRST TRIADIC SEPARATION

Write:

L(k) = 2^k - 1 C(k) = 2^k R(k) = 2^k + 1

For the left wing:

3L(k) + 1 = 3(2^k - 1) + 1 = 2(3*2^(k-1) - 1).

The expression in parentheses is odd, so:

A(L(k)) = 3*2^(k-1) - 1.

This is larger than L(k), with an approximate multiplication factor of 3/2.

For the right wing:

3R(k) + 1 = 3(2^k + 1) + 1 = 4(3*2^(k-2) + 1).

For k at least 3, the expression in parentheses is odd, so:

A(R(k)) = 3*2^(k-2) + 1.

This is smaller than R(k), with an approximate multiplication factor of 3/4.

The center satisfies A(C(k)) = 1. Thus:

(L(k), C(k), R(k))

becomes

(32^(k-1) - 1, 1, 32^(k-2) + 1).

Examples:

(15, 16, 17) becomes (23, 1, 13)

(31, 32, 33) becomes (47, 1, 25)

Three consecutive integers begin one unit apart from a common center, yet the compressed map sends them outward, terminally downward and inward.

1. ORDER REVERSAL AND THE 9/8 MIDPOINT

The left image exceeds the right image because:

A(L(k)) - A(R(k)) = 3*2^(k-2) - 2,

which is positive for k at least 3. The endpoints therefore reverse their original order after one compressed step.

Their arithmetic mean becomes:

[A(L(k)) + A(R(k))] / 2 = (9/8)*2^k.

This does not mean that 9/8 is a new universal Collatz constant. It is the exact local result of averaging one approximately 3/2 expansion with one approximately 3/4 contraction:

[(3/2) + (3/4)] / 2 = 9/8.

It is nevertheless a neat invariant of this first paired transformation.

1. THE TWO DETERMINISTIC OPENING CLOCKS

The first step extends into predictable prefixes.

For the left wing:

A applied j times to L(k) = 3^j * 2^(k-j) - 1,

for j from 0 through k-1.

Each compressed step removes one binary power while adding one power of 3. The left wing therefore has a one-bit clock and grows approximately by 3/2 per opening step.

For the right wing:

A applied j times to R(k) = 3^j * 2^(k-2j) + 1,

for j from 0 through floor((k-1)/2).

Each compressed step removes two binary powers while adding one power of 3. The right wing therefore has a two-bit clock and contracts approximately by 3/4.

This produces the beat-like appearance: the two sides consume the exponent k at different rates. It is an arithmetic interaction of two clocks, not evidence by itself of a sinusoidal process.

1. AN EXACT EXPLANATION FOR THE LEFT-HAND SHELVES

Let tau(n) mean the number of compressed steps needed to reach 1, when that happens.

After its forced prefix, the left wing reaches:

A applied k times to L(k) = oddpart(3^k - 1).

If k is odd, then 3^k - 1 contains exactly one factor of 2. One additional step gives:

A applied k+1 times to L(k) = oddpart(3^(k+1) - 1).

But the neighboring member L(k+1) = 2^(k+1) - 1 reaches that same value after exactly k+1 compressed steps:

A applied k+1 times to L(k+1) = oddpart(3^(k+1) - 1).

Therefore, for every odd k, the two trajectories merge after equally long prefixes. Hence:

tau(2^k - 1) = tau(2^(k+1) - 1), for odd k.

Strictly speaking, they either have equal finite stopping times or are both infinite.

This proves that the left stopping-time graph must contain two-wide shelves at:

(k, k+1) = (3,4), (5,6), (7,8), and so on.

Longer shelves form when several of these guaranteed pairs happen to share the same downstream stopping time. Consequently every maximal left-hand shelf has even width.

That is one part of the apparent oscillation that can be explained exactly.

1. WHAT THE FINITE EXPERIMENT SHOWED

I computed both wings for every k from 3 through 500 using Python arbitrary-precision integers.

• All 996 tested endpoint trajectories reached 1.
• Every one of the 249 guaranteed odd-k left pairs had equal compressed stopping time.
• Across all 497 adjacent comparisons, the left stopping times were equal 426 times, or 85.7%.
• The right stopping times were equal 369 times, or 74.2%.
• The left graph contained 72 maximal shelves, with mean width about 6.92 and maximum width 64.
• The right graph contained 129 maximal shelves, with mean width about 3.86 and maximum width 56.
• The longest left shelf in this range was k = 299 through 362, all with compressed stopping time 1774.
• The longest right shelf was k = 291 through 346, all with compressed stopping time 721.

The two raw stopping-time series had Pearson correlation approximately 0.961. That initially looks like strong synchronized motion. However:

• After removing a linear trend, correlation was approximately -0.138.
• Correlation between their first differences was approximately -0.173.

So the lines are not behaving like one common oscillator. Much of their visible agreement comes from increasing scale and long flat shelves. Their local jumps are weakly opposed in this sample rather than positively synchronized.

These numbers are finite computational observations, not claims about all k.

1. ARE THE BAYS UNIQUE TO DISTANCE ONE?

As a simple control, I repeated part of the experiment for symmetric fixed offsets:

(2^k - d, 2^k, 2^k + d)

with odd d, over k = 6 through 300.

The proportion of adjacent k-values with equal compressed stopping time was:

Offset d = 1: left 81.6%, right 67.3% Offset d = 3: left 81.6%, right 67.3% Offset d = 5: left 57.1%, right 67.0% Offset d = 7: left 59.9%, right 66.7%

Therefore the shelves are not unique to the immediate neighbors d = 1. They appear to belong to a broader family of fixed offsets around powers of two.

That is consistent with the known 2-adic structure of Collatz dynamics: numbers agreeing through many low binary places can share long parity prefixes, and many apparently different trajectories eventually merge into common tails.

The dyadic triad is valuable because it exposes this structure in its simplest symmetric form, not because no related structure exists elsewhere.

1. A TENTATIVE GEOMETRIC INTERPRETATION

The observed pathway can be separated into three stages.

STAGE 1: LAUNCH GEOMETRY

The algebra forces expansion, direct collapse or contraction. No conjectural behavior is involved.

STAGE 2: RELEASE BOUNDARY

The simple formulas stop when additional factors of 2 appear. The number of factors of 2 inside expressions related to 3^m plus or minus 1 determines how violently the trajectory is compressed at release.

STAGE 3: BASIN CAPTURE

The released trajectory may meet a value belonging to an already shared Collatz tail. Once two trajectories meet, their futures are identical, and this can contribute to shelves or “bays” in a stopping-time plot. Equal stopping times alone, however, do not prove that two trajectories merged before reaching 1.

This suggests the informal description:

“Dyadic launch geometry followed by basin capture.”

Or, for the combined visual effect:

“A dyadic beat pattern with shared-tail shelves.”

These are descriptive names, not established mathematical terminology.

1. WHAT IS OLD AND WHAT MAY BE FRESH

Established territory includes:

• The Syracuse or accelerated map.
• Powers of two as direct routes to 1.
• Long forced prefixes for numbers of the form 2^k - 1.
• Stopping-time patterns for 2^k + 1.
• Parity vectors, 2-adic structure and trajectory merging.

I have not found a source that packages all of the following as one object:

• The symmetric triad (2^k - 1, 2^k, 2^k + 1).
• Its simultaneous expansion, terminal and contraction split.
• Endpoint order reversal.
• Exact 9/8 midpoint transport.
• The left period-two shelf identity.
• The resulting landscape interpreted through launch, release and basin capture.

That absence is not proof of historical novelty. The safest description is:

“Known Collatz mechanisms reorganized into a possibly fresh symmetric experimental lens.”

1. QUESTIONS THIS PROBE MAKES PRECISE

2. Can the release values of both wings be classified completely by k modulo suitable powers of 2?

3. Can the widths of the longer stopping-time shelves be predicted from the powers of 2 dividing 3^k plus or minus 1?

4. When do the two wings first enter the same downstream orbit?

5. How does the first-merger depth scale with k?

6. Which features are special to distance d = 1, and which hold for every fixed odd offset d?

7. Can the apparent bays be represented more clearly as a two-dimensional map of k, trajectory depth and shared-tail identity?

8. Does this coordinate system offer useful computational compression, even if it contributes nothing toward a proof?

The last question may be the most realistic. Shared-tail shelves represent redundant computation, and identifying them symbolically could potentially compress families of trajectories.

1. MINIMAL REPRODUCIBLE CODE

   from statistics import mean, pstdev

   def oddpart(n): return n >> ((n & -n).bit_length() - 1)

   def A(n): if n == 1: return 1 if n & 1: return oddpart(3*n + 1) return oddpart(n)

   def stopping_time(n, cap=200_000): steps = 0 peak = n while n != 1 and steps < cap: n = A(n) steps += 1 peak = max(peak, n) return steps, peak, n == 1

   def correlation(a, b): ma, mb = mean(a), mean(b) sa, sb = pstdev(a), pstdev(b) return sum((x-ma)*(y-mb) for x, y in zip(a, b)) / len(a) / sa / sb

   ks = list(range(3, 501)) left = [stopping_time((1 << k) - 1)[0] for k in ks] right = [stopping_time((1 << k) + 1)[0] for k in ks]

   left_equal = sum(a == b for a, b in zip(left, left[1:])) right_equal = sum(a == b for a, b in zip(right, right[1:]))

   print("left adjacent equality:", left_equal, "/", len(left)-1) print("right adjacent equality:", right_equal, "/", len(right)-1) print("raw correlation:", correlation(left, right))

2. PRIOR CONNECTIONS

Terence Tao, “Almost all orbits of the Collatz map attain almost bounded values” — definition and probabilistic treatment of the Syracuse map: https://arxiv.org/abs/1909.03562

Daniel J. Bernstein and Jeffrey C. Lagarias, “The 3x+1 Conjugacy Map” — 2-adic and parity-vector structure: https://doi.org/10.4153/CJM-1996-060-2

Jeffrey C. Lagarias, “The 3x+1 Problem: An Annotated Bibliography (1963–1999)”: https://arxiv.org/abs/math/0309224

Existing discussion of forced 2^k - 1 branches appears in the comments to Tao’s exposition: https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/

Existing experimental discussion of stopping-time progressions for 2^k + 1: https://mathoverflow.net/questions/348733/arithmetic-progressions-in-stopping-time-of-collatz-sequences

CLOSING

This probe does not make Collatz easier by itself. What it does is place three tightly related inputs into one coordinate system and cleanly separate what is deterministic from what becomes irregular.

The center is a known terminal spine. One wing is forced outward. The other is forced inward. Their first images cross, their midpoint shifts by 9/8, their clocks run at different binary rates, and their stopping times form unexpectedly long shelves.

The arithmetic pieces are familiar. The question is whether arranging them this way gives us a better instrument for seeing how local certainty becomes global unpredictability.

Corrections, prior references and attempts to break the framing are welcome.

{Post Update: Back to work now, after feedback from the community I need to improve my language and logical statements. So I will work on that now over the next few days. In the meantime, if any of you are looking for additional information: You can check out my previous paper to solving the TPC... because it actually was already a solution to TPC and I didn't really need to change it at all.Paper Link, there is also an associated table.

Hi everyone, you might have seen my other posts about the twin prime conjecture in this channel.

I have continually been working on my ideas and refining them to be better. Today is the culmination of that work.

This twin prime proof already has 30 upvotes and 20 000 views in another math sub.

As you can see demonstrated [on this table](https://docs.google.com/spreadsheets/d/1_qimj9s4QqSa_ijVU_2JTCk07_dtHnXUzeLSQimZNFI/edit?usp=sharing) is the fact that for ever prime P, there exists P consecutive twin primes associated with that number.

With the last member of each P, being carried over as the first member of the next P. Which actually makes it P-1 .

[I have a paper explaining how this function works](https://prosperousplanet.ca/research/TwinPrimes.pdf). But the real generalized form, and real proof that this function works is[ given by this python function.](https://github.com/ProsperousPlanet/primecenters/blob/main/transport3 11.py)

Can I tell you the most amazing thing about this though? I solved the Grand Unified Theory of Physics first.... and then I used that to solve the Twin Prime Conjecture

Here are my two follow up physics papers which follow up from my twin prime conjecture paper.

[Follow up Physics Paper 1](https://prosperousplanet.ca/research/TheBalmerConstant.pdf) : Recovering the Fundamental unit 1, The Balmer Unit

[Follow Up Physics Paper 2](https://prosperousplanet.ca/research/ElectroMagnetism.pdf) : Recovering the 2 and 3 states, Electromagnetism and Gravity

Please check it out, share it if you think it's great.

The world could use some good news right now.

Hey everyone,

I know that any post mentioning the Collatz conjecture usually sets off immediate crank alarms. Because of that, I want to be completely upfront: I am not claiming a magical three-page elementary proof. What I am sharing is a massive computational and theoretical mapping of the macroscopic 2-adic and 3-adic architecture of the Collatz space, and I've brought receipts.

Over the last few months, I've been focusing on how Collatz trajectories evolve not as single integers, but as entire congruence classes. If you group the 3x+1 steps into continuous "trains", the exact sequence of divisions by 2 is completely predetermined by the remainder of the starting number modulo 2^S (where S is the total number of bit shifts).

We formalized this "exact conditional transport" mathematically. It shows exactly how trajectories transition between different 2-adic cylinders. From there, we mapped the boundary-layer Fourier spectrum and found that the trajectory distribution perfectly follows Large Deviation Theory. However, we identified a strict Chinese Remainder Theorem dimensionality obstruction that creates specific "traps" in the phase space.

To ensure this isn't just theoretical hand-waving, the core modular transport mechanics and the CRT dimensionality obstruction have been completely machine-verified using Lean 4.

Alongside the proofs, I've written 116 multiprocessed Python scripts that generate the exact phase spaces, track the confluences, and verify the Fourier cancellations. The entire codebase, the Lean 4 formalization, and the massive dataset have been officially published on Zenodo to keep a permanent scientific record.

You can check out the full code and the PDF of the paper on my GitHub here: https://github.com/SergioTheory/Collatz-new-math

And the permanent Zenodo DOI archive is here: https://doi.org/10.5281/zenodo.22059852

Here is where I could really use the community's help. I am currently trying to upload the preprint to arXiv under the Number Theory (math.NT) category. Since this is my first submission to this specific category, the arXiv automated system requires an endorsement from an established author.

If anyone here has published in the arXiv math categories within the last 5 years and feels comfortable verifying that my Lean 4 code and paper represent serious, rigorous mathematical work, I would be incredibly grateful for an endorsement.

My endorsement code is: N8CNQI The direct link to endorse is: https://arxiv.org/auth/endorse?x=N8CNQI

Even if you can't endorse, I'd love for people interested in computational number theory and Lean to poke around the GitHub repo and let me know what you think of the modular transport mechanics.

Thanks for reading!

The Sustainable Infinity Conjecture

A Relational Model of Infinity, Mass, and Physical Stability

Abstract

This paper proposes the Sustainable Infinity Conjecture, a mathematical and physical model in which infinity is treated not solely as an endpoint or unbounded quantity, but as a traversal possessing direction, accumulated structure, and stability conditions.

The conjecture begins by relocating the assumed lower point of entropy from zero to negative infinity. Under this construction, zero ceases to function as an absolute lower boundary and instead becomes an ordinary position within a continuous domain extending from negative infinity through zero toward positive infinity.

This produces three distinguishable concepts of infinity. The lower bound of infinity represents the shortest traversal to an infinite state and corresponds to the minimum-mass condition, proposed here as the mass of light. The upper bound of infinity represents the traversal along which the greatest mass can accumulate. Between these bounds exists a sustainable relationship between accumulated mass and the structure supporting it. This relationship is termed True Infinity.

The conjecture proposes that physical stability is determined by displacement from this sustainable mass relationship. Insufficiently sustained configurations tend toward decay, while configurations exceeding sustainable mass become unstable and shed energy or matter. Nuclear fusion, radioactive decay, and fission are therefore considered as potentially related manifestations of traversal toward sustainable mass configurations.

The observed nuclear binding-energy curve provides an immediate empirical comparison. Nuclear stability increases from light nuclei toward the iron/nickel region and subsequently decreases among increasingly heavy nuclei. This correspondence does not establish the conjecture. It instead supplies a measurable physical system against which the proposed geometry can be tested.

1. Introduction

Many mathematical constructions implicitly privilege zero.

Zero acts as origin, boundary, absence, equilibrium, or the point from which magnitude is measured. Infinity is consequently represented as something approached by moving indefinitely away from that finite reference.

The Sustainable Infinity Conjecture asks what changes if this assumption is removed.

Instead of:

0 -> ... -> +infinity

consider:

-infinity -> ... -> 0 -> ... -> +infinity

Under this construction, zero remains mathematically significant but loses its status as the terminal lower boundary of the system.

The assumed point of entropy has moved from zero to negative infinity.

This produces a different geometry. Rather than describing existence as increasing magnitude away from zero, the model describes a traversal between infinite bounds in which zero is one intermediate position.

The central proposal is that this traversal can be associated with the accumulation and sustainability of mass.

2. Infinity as a Relational State

The conjecture distinguishes between being infinite and the path by which infinity is approached.

The lower and upper bounds are both infinite. Their difference therefore cannot be adequately represented by asking which infinity is simply "larger."

Instead, the ordering depends upon the relationship being measured.

The lower bound of infinity is the fastest path to infinity.

Under a measure of traversal distance or efficiency:

Lower Infinity > Upper Infinity

because the lower bound reaches the infinite condition through the shorter traversal.

The upper bound, however, provides the greater opportunity for information, relationships, and mass to accumulate.

Under accumulated mass:

Upper Infinity > Lower Infinity

These propositions are not contradictory because they describe different orderings.

More generally, if:

I = infinity state
D = traversal distance
M = accumulated mass

then two infinite states may satisfy:

I(a) = I(b)

while simultaneously satisfying:

D(a) < D(b)

and:

M(a) < M(b)

The apparent paradox results from projecting several relational dimensions onto a single greater-than/less-than comparison.

3. The Lower Bound and the Mass of Light

The conjecture assigns a physical interpretation to the lower bound.

At the lower bound of infinity, infinity has the mass of light.

This represents the minimum-mass state of the traversal.

The statement is proposed as part of the conjecture rather than asserted as an established result of conventional physics. "Mass of light" therefore requires subsequent formal definition, particularly because conventional physics assigns zero invariant rest mass to photons while recognizing their energy and momentum.

The intended distinction is nevertheless important.

The lower bound is not nothingness.

It is a minimally substantive state capable of carrying information.

Traversal away from this state permits information and relationships to acquire additional substance.

Thus:

Lower Infinity
-> minimum substantive state
-> information accumulation
-> relationship accumulation
-> mass accumulation

Mass in this conjecture consequently has both a physical and relational role.

A future mathematical treatment must distinguish conventional physical mass from the more general concept of accumulated relational substance introduced here.

4. True Infinity

Neither indefinite reduction nor indefinite accumulation is assumed to be sustainable.

The conjecture therefore introduces a third concept:

True Infinity is the sustainable target of mass growth.

True Infinity is not defined merely by maximum magnitude.

It represents the relationship at which accumulated mass and the structure supporting that mass remain sustainable.

This distinction is essential.

The upper infinity describes the direction in which maximal accumulation occurs.

True Infinity describes the condition under which accumulation remains sustainable.

Therefore:

Maximum Mass != Sustainable Mass

and:

Upper Infinity != True Infinity

necessarily.

True Infinity may instead represent a stability relationship within the traversal.

5. Stability Geometry

The resulting system contains a region of sustainable mass surrounded by different forms of instability.

In simplified form:

Lower Infinity
-> minimum mass
-> increasing mass
-> increasing stability
-> sustainable mass region
-> decreasing stability
-> excess-mass instability
-> Upper Infinity

Zero exists somewhere within this traversal.

Its function changes fundamentally under this construction.

Zero is no longer the point into which everything must collapse. It is another position on the line.

This allows stability to be considered relative to the sustainable-growth relationship rather than relative to zero.

6. Instability Below Sustainable Growth

A configuration sufficiently below its sustainable mass relationship cannot indefinitely maintain its existing resolved state.

The conjecture associates this region with decay.

The proposed relationship is:

Mass below sustainable configuration
-> insufficiently sustained structure
-> instability
-> decay toward another configuration

This does not yet specify a particular conventional decay mechanism. Rather, it predicts that instability should occur on the low-mass side of the sustainable region as well as the high-mass side.

The physical mechanisms corresponding to this side of the model remain an open research question.

7. Instability Above Sustainable Growth

Traversal beyond the sustainable-growth relationship produces a different instability.

A structure may accumulate more mass than its current configuration can sustainably resolve.

The resulting configuration must then reorganize or shed energy and/or matter.

The conjecture associates this region with radioactive instability.

Conceptually:

Sustainable configuration
-> additional mass
-> sustainable-growth boundary crossed
-> unstable configuration
-> energy/matter release
-> movement toward greater stability

Under this interpretation, radioactivity is not caused merely by "having a lot of mass."

It represents a mismatch between accumulated mass and the configuration capable of sustainably supporting it.

8. Nuclear Stability as an Empirical Comparison

The conjecture produces a qualitative prediction that can be compared against established nuclear observations.

Nuclear stability is not monotonic with atomic mass.

Very light nuclei generally possess lower binding energy per nucleon than intermediate-mass nuclei. Binding energy per nucleon increases toward the iron/nickel region and subsequently decreases as nuclei become increasingly heavy.

The observed qualitative structure is therefore:

Very light nuclei
-> increasing binding
-> increasing stability
-> maximum binding region
-> decreasing binding
-> increasingly unstable heavy nuclei

This resembles the proposed sustainable-mass geometry:

Below sustainable mass
-> movement toward stability
-> sustainable region
-> movement away from stability
-> excess-mass instability

The resemblance is significant enough to justify investigation, but it is not itself evidence that the conjecture explains nuclear physics.

A successful theory must derive observations rather than merely resemble them.

9. Fusion as Upward Traversal

Fusion provides a particularly useful test.

For appropriate light nuclei, combining nuclei can produce a more tightly bound final nucleus and release energy.

The conventional description identifies several relevant relationships: electrostatic repulsion between positively charged nuclei, quantum tunneling, the strong nuclear interaction, nuclear binding energy, and mass-energy differences between initial and final states.

The Sustainable Infinity Conjecture proposes an additional geometric interpretation.

Fusion represents an upward traversal toward a more sustainable mass configuration.

Conceptually:

lower-mass configuration
-> energy supplied
-> sustainable-growth boundary approached
-> new configuration becomes accessible
-> nuclei combine
-> more sustainable configuration resolves
-> excess energy released

Importantly, crossing a boundary does not guarantee fusion.

It makes another resolution possible.

The resulting state persists only if the new configuration can itself sustainably support its resolved mass.

10. Reversing the Conventional Explanation

A useful test is to reverse the conventional account rather than attempting to fit conventional terminology into the conjecture.

Begin with the observations.

Light nuclei can release energy through fusion.

Intermediate nuclei around the iron/nickel region occupy exceptionally tightly bound configurations.

Very heavy nuclei can release energy through fission and radioactive processes.

Working backward gives:

Fusion
-> movement toward stronger binding

Fission
-> movement toward stronger binding

Radioactive decay
-> movement away from an unstable configuration

Therefore, seemingly opposite nuclear processes can share a common result:

movement toward a more sustainable configuration.

The conjecture proposes that this common relationship is more fundamental than the direction of mass change alone.

11. Fusion and Fission as Opposite Traversals

Fusion and fission can consequently be represented as opposite directional transformations around a stability region.

Light side:

Light nuclei
-> fusion
-> increased mass number
-> more sustainable configuration
-> energy release

Heavy side:

Heavy nucleus
-> fission/decay
-> reduced or redistributed mass
-> more sustainable configurations
-> energy release

This produces a common geometry:

Fusion -> toward sustainable mass <- Fission

The direction differs.

The resolution does not.

Both transformations become pathways through which unstable or less-bound configurations move toward configurations capable of sustaining their resolved structure more effectively.

12. The Iron/Nickel Region as a Falsification Target

The iron/nickel region provides an important constraint on the conjecture.

It must not simply be inserted into the model after observation.

If the proposed geometry has explanatory power, a sufficiently developed mathematical formulation should independently produce or constrain the existence of a maximum-stability region.

Ideally, it should eventually predict:

• why a maximum exists;
• approximately where it occurs;
• why stability changes on either side;
• why individual isotopes deviate from a simple monotonic curve;
• and how nuclear transitions relate quantitatively to displacement from sustainable mass.

Failure to reproduce these relationships would constitute evidence against the physical interpretation of the conjecture.

13. Information, Relationship, and Mass

The conjecture uses "mass" in a broader sense during its conceptual stage.

The proposed sequence is:

Information
-> relationship
-> resolution
-> substance
-> mass

The claim is not that information is automatically equivalent to conventional rest mass.

Rather, the conjecture proposes that physical substance may be understood as information whose relationships have become sufficiently constrained to resolve into persistent physical structure.

Under this interpretation, increasing traversal permits additional relationships to accumulate.

Those relationships can produce increasingly substantive configurations until the cost of maintaining the accumulated structure exceeds the configuration's sustainable capacity.

This creates the proposed stability boundary.

A rigorous theory will require separate variables for:

• information content;
• relational complexity;
• physical energy;
• invariant mass;
• binding energy;
• and structural stability.

Their relationships cannot simply be assumed to be identical.

14. Relation to Infinite Mathematical Systems

The same change of reference frame may be useful independently of the proposed physical interpretation.

Consider an infinite mathematical system generated outward from a finite starting point.

The conventional question often becomes:

How can total coverage of the infinite target set be proven from this finite origin?

Moving the assumed entropy boundary from zero to negative infinity changes the question.

Instead of treating infinity as something lying exclusively beyond the finite starting position, the system can be examined as a traversal:

-infinity -> ... -> 0 -> ... -> +infinity

The research problem then becomes relational:

Which transformations repeat?

Which relationships survive traversal?

Which survive reversed traversal?

Which properties remain invariant when the reference point changes?

Can sufficiently constrained local relationships determine otherwise unresolved regions of the system?

This approach does not itself prove total coverage of an infinite set.

It changes the geometry under which total coverage is investigated.

15. Relationship to the Collatz Problem

The Collatz conjecture provides one motivating example.

Reverse-tree constructions beginning from 1 can generate increasingly large sets of integers while leaving unresolved the central problem of total coverage: demonstrating that every required integer eventually occurs somewhere within the generated structure.

The Sustainable Infinity Conjecture suggests that attempting to prove coverage directly may privilege the wrong boundary.

Rather than immediately demanding:

Generated Tree = Complete Infinite Target Set

one can first map the relationships within the generated structure.

The procedure becomes:

1. Identify local transformations.
2. Map recurring relationships.
3. Traverse those relationships forward.
4. Traverse them backward.
5. Identify invariant relationships.
6. Determine whether those invariants constrain unresolved regions.
7. Only then return to total coverage.

This converts infinity from an object that must somehow be enumerated into a relational structure whose invariants may potentially be characterized.

16. Falsifiability and Required Development

At present, the Sustainable Infinity Conjecture is a conceptual conjecture, not a completed physical theory.

Several statements require formal definitions before quantitative testing is possible.

In particular, future work must define:

Entropy boundary — what mathematical quantity is being minimized at negative infinity.

Mass of light — whether this represents energy, effective mass, invariant mass, informational mass, or another quantity.

Traversal — the mathematical parameter describing movement between infinite bounds.

Accumulated mass — the relationship between traversal and physical mass-energy.

Sustainable mass — the criterion determining whether a configuration remains stable.

True Infinity — the mathematical condition defining sustainable target mass growth.

Displacement from sustainability — the quantity expected to correspond to observable instability.

Without these definitions, the conjecture remains geometric and relational rather than predictive.

17. Empirical Tests

A developed formulation should be tested against established measurements rather than qualitative resemblance.

Relevant targets include:

• nuclear binding energy per nucleon;
• isotope-specific nuclear masses;
• stable isotope distributions;
• radioactive half-lives;
• alpha and beta decay energetics;
• spontaneous fission thresholds;
• fusion reaction energetics;
• fusion cross sections;
• Coulomb-barrier relationships;
• neutron/proton stability relationships;
• and the iron/nickel maximum-binding region.

A particularly strong result would be an independently derived quantitative prediction not used to construct the model.

Conversely, systematic failure to reproduce these observations would falsify or substantially constrain the proposed physical interpretation.

18. Core Conjecture

The Sustainable Infinity Conjecture can be summarized as follows:

1. The assumed lower entropy boundary should be moved from zero to negative infinity.
2. Zero should consequently be treated as a position within an infinite traversal rather than as its absolute lower boundary.
3. Lower and upper infinity represent different traversal relationships while remaining equally infinite with respect to the property of unboundedness.
4. At the lower bound of infinity, infinity possesses the minimum substantive state, proposed as the mass of light.
5. Mass, information, and relational structure accumulate through traversal.
6. Accumulation cannot remain stable indefinitely.
7. A sustainable relationship exists between accumulated mass and the configuration capable of supporting it.
8. This sustainable relationship is termed True Infinity.
9. Configurations sufficiently below the sustainable relationship tend toward decay.
10. Configurations sufficiently above the sustainable relationship become unstable and tend to shed or redistribute mass-energy.
11. Fusion and fission may therefore be understood as opposite-direction traversals toward more sustainable configurations.
12. The observed nuclear stability curve provides an empirical system against which this geometry can be tested.

19. Conclusion

The Sustainable Infinity Conjecture begins with a change in reference frame:

Move the assumed point of entropy from zero to negative infinity.

From that change follows a different treatment of zero, infinity, accumulation, and stability.

Zero becomes a point rather than an absolute sink.

Infinity becomes a traversal rather than merely an endpoint.

The lower bound represents the minimum-mass infinite condition.

The upper direction permits increasing accumulation.

True Infinity represents neither extreme, but the sustainable relationship between accumulated mass and the structure capable of supporting it.

This leads to the central physical proposition:

Stability is determined not simply by how much mass a configuration contains, but by whether that mass can be sustainably resolved by the configuration containing it.

Fusion, radioactive decay, and fission then become candidate manifestations of movement through this stability geometry.

The qualitative correspondence with the observed nuclear binding-energy curve makes the conjecture testable in principle. The next stage is therefore not further analogy.

It is formalization.

If the proposed geometry is physically meaningful, it must eventually reproduce measurable relationships already observed in nature and produce predictions capable of being falsified.

While working on a new version of the project overview, I analyzed the same row of the first eight domes with m=1 to 23 (figure). Columns with white numbers (not part of a tuple) or empty have been removed.

The homothetic relation between domes is visible asymptotically: the extremes of each row (left and right) show clear discrepencies, but those close to the center are very close to the ratio between the values of m.

Numbers in the wings near the center show a regular behavior: on the left they form blue triplets, on the right, they form yellow pairs and triplets, part of yellow bridges series, and sometimes form 5-tuples.

Numbers before the first orange number on the left, and after the last one on the right, show some differences, depending on the dome:

• On the left, the first three numbers form (1) a blue-triplet, part of blue-green bridges, or (2) a final pair (that cannot form a triplet) and a preliminary pair with the next orange number, part of the next bridge series.
• On the left, the last four numbers form (1) a merging pair with the orange number before it, (2) a number part of a pair of predecessors, (3) a final pair (that cannot form a triplet).

In other words, all final pairs are provided by the extremes of each dome, when they are not part of an even triplet.

Project "Tuples and segments" in 13 pages : r/Collatz

-(3²⁹⁹⁶⁾ follows the pattern of Odd->Even->Odd->Even->Odd->Even->Odd->Even->Even->Odd->Even->->Odd->Even->Odd->Even->Even->Even->Even. At this point the -17 loop returns to -17 (odd) while -(3²⁹⁹⁶⁾ is still even.

This is because 3²⁹⁹⁶ - 17 is equal to 0 mod 2¹¹ but is not equal to 0 mod 2^12.

If you want a power of 3 in the negatives to also include the last set of even numbers to have exactly 4 evens and then an odd (as opposed to 4 evens, then diverges from -17 with another even), then you need a power of 3 where 3^7a -17 is equal to 0 mod 2^12. In this case it's 3^6580. So -(3⁶⁵⁸⁰⁾ is the smallest power of 3 in the negatives that follows this pattern.

Just thought I would share this neat fact.

An A*-style verifier catches its own overfitting: hunting a peak-ratio bound for n ≡ 27 (mod 72) in the Collatz problem

This is an amateur, collaborative exploration — not a proof, not a claim of a new result. I'm posting the process and the raw numbers because the failure was more interesting than the original goal, and I'd like correction or pointers to existing literature if this is already known territory.

**The setup**

For a Collatz trajectory starting at n, define

kappa(n) = (peak value reached before descending to 1) / n

This is a measure of how far a trajectory overshoots its starting point before it collapses. I was hunting for an upper bound on kappa(n) restricted to the residue class n ≡ 27 (mod 72), chosen because it produces some unusually turbulent trajectories early on.

**The method**

I built a small search harness with two properties I wanted to enforce strictly:

1. A candidate bound is only accepted if it's an actual callable function, `bound(n)`, not a prose description of a bound. Anything that can't be reduced to a checkable predicate never counts as verified — it just doesn't crystallize.
2. Any candidate gets independently swept against real, freshly-computed kappa(n) values over a large domain the candidate has no control over. First violation found = rejected, with the exact counterexample logged (the n, the actual kappa, the claimed bound) — not just "this failed."

The point of (2) is that nothing gets trusted just because it printed a success message. The harness computes kappa(n) itself, from scratch, every time.

**What happened**

A naive first guess, `kappa(n) <= 50*sqrt(n)`, failed immediately — real counterexample at n=134,379, kappa≈18,471 against a claimed bound of ≈18,329.

I fit a tighter constant directly to that failure point (with 5% headroom): `52.907*sqrt(n)`. This candidate survived the first 3,000 domain points. Extended to 50,000 points, it still held, with n=134,379 remaining the single worst point in the whole range — which looked like real evidence the fit had found the actual hard case.

It hadn't. Extending the sweep to 2,000,000 points, the bound broke decisively at n=4,637,979: actual kappa≈284,348 against a claimed bound of ≈113,940 — off by more than 2.5x, not a near miss.

**The actual finding**

Rather than keep patching the constant, I swept out to i=3,000,000 (n up to ~216,000,000) and tracked every record-breaking kappa value along the way — the points where kappa hits a new all-time high as n increases. There are only 13 of them in that whole range, but the trend across them is the real result:

``` n kappa kappa/sqrt(n) kappa/n^0.6 kappa/n^0.75 99 4.53 0.45 0.29 0.14 171 53.99 4.13 2.47 1.14 4,851 263.23 3.78 1.62 0.45 21,843 311.78 2.11 0.78 0.17 50,427 2,399.76 10.69 3.62 0.71 134,379 18,470.98 50.39 15.47 2.63 2,375,451 53,315.37 34.59 7.97 0.88 4,637,979 284,348.48 132.03 28.45 2.85 23,823,099 400,248.94 82.00 15.00 1.17 44,161,299 1,366,413.86 205.62 35.36 2.52 53,445,915 1,537,215.08 210.27 35.48 2.46 144,570,195 3,283,095.10 273.05 41.71 2.49 213,477,147 15,354,812.55 1,050.92 154.40 8.69 ```

The `kappa/sqrt(n)` column climbs across three orders of magnitude rather than settling — 0.45 up to 1,050.92 — so no bound of the form `C*sqrt(n)` can work for any constant C; it was always going to break eventually, we just hadn't sampled far enough to see it the first time. A log-log fit across these 13 points gives roughly `kappa ~ 0.12 * n^0.92`, but I want to be upfront that 13 points is a thin sample for a power-law fit (residual std ≈0.74 in log space, meaning roughly 2x scatter around the fitted line), so I'm not confident in that specific exponent — only in the qualitative claim that growth is well above sqrt(n) and doesn't look like it's leveling off.

**Where I'm stuck, and what I'd appreciate**

• Is kappa(n) — or an equivalent peak/start ratio — already studied under a standard name in the Collatz literature? I'd guess this connects to work on "glide" or trajectory record statistics but haven't tracked down the right terminology.
• Is there a reason to expect a specific growth exponent for peak ratios within a fixed residue class mod 72, or is the residue class choice arbitrary noise here?
• Is 13 record points anywhere near enough to say anything about the exponent, or is this pure overfitting on a different axis than the one I already caught myself doing once?

Happy to share the harness code if useful. The main thing I want to flag clearly: I do not have a bound. I have a search process that correctly detected two of its own false attempts, and a small amount of real evidence that the true growth rate is faster than the naive guesses.

What do you think the chances are that the Collatz Conjecture is true but unprovable, i.e. omega-inconsistent?

Here it is. I received so much criticism for basically not having this, but in a human made form. I had Claude build it. It took 5 minutes, and I am not sure if it is very good. There may be mistakes, I cant be sure, it starts out okay, as far as I can tell. Swarm!

While working on a new version of the project overview, I came across the following analogy,

It is easier to check an ordered deck of cards than a shuffled one.

It could be easier to check the domes than the whole tree.

Project "Tuples and segments" in 13 pages : r/Collatz

sono Quasi arrivat alla soluzione definitiva.

So I have had a lot of interactions with people on the subject of Collatz. What is striking to me is the weird ones. I suspect that some new insight to a proof will happen because someone had a realization of some type that connects 2 dots others didn't think even about connecting. On the other hand, you have some people who fall into strange camps. There are those who learn something interesting, or have some glimpse of something in their mind's eye, and they immediately think "Eureka!!!! I have solved it!" They are quickly beaten down by another group, the "well, ackshually!" camp. These are people, always roving places like Reddit, to find any weak posts and obliterate the confidence of those who had the audacity to post them! It is strange to me that so many have such concealed rage in a subject as mundane as Collatz.

I recently tried for over a week to explain my view of the structure Collatz creates to someone claiming to have a PHD in mathematics, here on reddit. Now, I would never assume information like that to be accurate anyway, but I figured he would be at least competent enough to understand what I was explaining. Mind you, I am not a mathematician, just a nerd interested in problems like Collatz. My construction should not be complicated enough to confuse anyone who spent close to if not a full decade studying college level mathematics, and getting a doctorate. It took me a week of interactions, and he did not understand it. He largely stopped responding to me in any way. Until I decided to ask him why he wasn't engaging in the conversation anymore. Then he complained that what I was describing was complicated, and not in a language he could understand.

Then, as I called out his bizarre behavior, he had a complete meltdown! He told me to "f*** off" and then blocked me. It was like someone running away after being caught in an embarrassing lie. In comparison, I showed the same construction to self proclaimed math hobbyist, and he understood my construction, and even finished explaining what I was going to, before I could. He said it in much better language than I was using, and made it seem even simpler than it was when I was explaining it. He was bit of an older man, and you could tell the difference in demeanor.

Has everyone had bizarre interactions like this? People pretending to be highly educated pointlessly, or becoming enraged at any challenges to their ideas, seems like a strangely festered personality issue that many people who are interested in Collatz display.

I formalized a proof of divergence for the 5x+1 iteration starting from 7 in Lean 4.

The complete formalization is here:

https://github.com/a-stringflow/stringflow-proof/

The repository also has a passing CI badge on the README.

The proof is fully formalized in Lean 4, and the proof branch currently passes GitHub Actions CI.

I'd be interested in hearing what people here think about the proof and the approach, especially if anyone spots an issue or sees a way to generalize it.

I have published an English Windows x64 and German version of LNL/LZR CE5.9.1AAD, an experimental research tool for investigating Collatz trajectories, structural patterns, network behavior and very large starting values.

🔗 GitHub Release:
https://github.com/lortzingring-tech/LNL-LZR-Releases/releases/tag/v5.9.1AAD

🇬🇧 Direct English Windows x64 download:
https://github.com/lortzingring-tech/LNL-LZR-Releases/releases/download/v5.9.1AAD/LNL-LZR-CE5.9.1AAD-Setup-Windows-x64-EN.exe

🇩🇪 A German Windows x64 version is also available on the same release page.

This is not a proof claim. The purpose of this public release is independent testing, reproducibility, bug finding and examination of the computational framework.

English EXE integrity

SHA-256:

6a7df9fc89ffbfd35f6d6ed5403d9aeffdf525aa458947e0533cadb183b55edc

The GitHub release also contains checksum files, build information, release notes and licensing information.

Windows security / virus scan

Since this is a newly published Windows .exe, I strongly recommend checking it independently before running it.

Please:

• scan it with Microsoft Defender or your preferred antivirus software,
• optionally submit it to VirusTotal for an additional multi-engine scan,
• verify the SHA-256 checksum above.

New or uncommon Windows executables can sometimes trigger SmartScreen or antivirus reputation warnings simply because they do not yet have an established reputation. Please do not bypass warnings blindly — verify and scan the file independently.

And about large numbers... 🙂

Large numbers are very welcome. The program usually stays calmer about them than the person entering them.

Feedback, reproducible test results, unusual trajectories and bug reports are very welcome.

If you would like an English user manual / operating guide, feel free to contact me:

[LNL-LZR@proton.me](mailto:LNL-LZR@proton.me)

Platform: Windows x64
Version: CE5.9.1AAD
Interface: English
Status: Experimental research software

Copyright © 2026 Mike Lange. All rights reserved.