Snippets Concepts Collatz T-Stop

• Index for Snippets Concepts Collatz Length
• Preface
• Limitations on Tool
• Extra Significant Figures, If Any in Debugging
• Introduction
• Bankroll Notation for Psuedocode
• Connecting the Classifier to the Gambler's Ruin Simulator
• Layout of Program
• Collatz-like quantum analogies
• Several Strategies for Quantized Solutions
• Comments on the Blue Staircases in the Collatz scatterplots, and other Math analogies
• Comments on Table for Walkthrough of the Gamblers Bankroll, Quantum Parameters, and other Math analogies
• Hack Algorithm for log2 may Break Rules, using slang
• Summary
• Simple Output of Game Session
• Table 1, Partial Collatz_Sequences for the lower integers
• Table , Quick Prime π Estimates for Collatz-scale numbers
• Table , Walkthrough of the Collatz, Gamblers Bankroll, quantum, and other math analogies
• Table. Formulas and Algorithms for Collatz Stopping Time, discussed here
• References
  • Screenshots
  • Figure 1. Collatz Length, Points from Collatz Conjecture, N < 500
  • Figure 2. Collatz Length, lower region and lower region of solutions
  • Figure 3. Collatz Length, curve fit on envelope of lower region
  • Figure 4. Collatz Length, Envelope of lower region
  • Figure 5. Bankroll over number of games
  • Figure 6.Snippets Concepts Collatz staircase
• Appendix Code
  • Appendix TCL Programs and Scripts
• 1. Expanded Toy for Demo
• Experimenting with a iterative quantized and multivalued solution in McCarthy Function style
• Output from Active State
• Output in Wiki table
• Wiki Table Utility
• Quick Legendre Prime π Estimates for Collatz-scale numbers
• Collatz scale, Log2 approximation formulas for large n
• Wiki Table: Computer Iteration Solutions for large n
• Wiki Table: Mersenne-tuned Approx. and general Approx. Estimates for large n
• Wiki Table: Quick Log2 Estimates for large n
• Wiki table. Percent error check for large N
• Hidden Comments Section
• Program Change Log

Index for Snippets Concepts Collatz Length

Preface

gold 3/12/2026. Here is some source code to supplement the TCL Wiki page Playing with Recursion by RS and the Collatz Conjecture. This supplemental code is intended for study of McCarthy theorems on computer arithmetic. Attempting approximation of human readable variables and proc names and the JPL defensive programming rules into Tcl procs. There is a variety of game solutions in the autotests at the bottom of deck. When measured by the Tcl timing statements, completion times and solutions of parameters will differ on different computer set-ups. Assume a future maintainer, either AI Model or human programmer, would have to maintain code with info content in program, ref "Snippets Concepts Effects".

The Collatz conjecture examines the iterative sequence defined as follows: if an integer n is even, divide it by two; if odd, compute 3n + 1. Repeating this operation seemingly always leads to 1, though no general proof exists. The question of how many steps, or iterations, each number requires before reaching 1 remains central. This count is often called the Collatz sequence stopping time.

Limitations on Tool

The TCL Snippets illustrate ideal mathematical behavior only and do not perform full simulation, actual measurements, or state vector evolution. The tool only visualizes ideal math structure, whereas no state vector simulation, probabilities, or actual measurement outcomes are derived. This tool for visualization does not simulate actual measurement outcomes or state vector evolution during operations. These are idealized protocols for tutorial purposes. Primarily, TCL /TK uses its strong points here for book keeping and displays. The example tool is not a full emulator. Meaning, limited scope for tutorial purposes.

Extra Significant Figures, If Any in Debugging

In debugging the calculations, some of the printout values reflect roughly 17-digit precision output from a typical double-precision computation. It's not "true exact" beyond 5 significant figures. Extra significant figures are used to check the calculations from other computer set-ups, not necessarily to infer accuracy of data measurements here. Typically, the slight differences in decimal places on far right of decimal point are normal floating-point behavior in Tcl's expr.

Introduction

This executive summary introduces the field of formal program verification. Program verification is the mathematical discipline of proving that software behaves correctly. The recursion solutions were studied by computer scientist John McCarthy in 1970 as deliberate challenge problems on recursion for automated reasoning tools. The conventional computer languages were developed to handle deterministic problems. The Collatz Conjecture has a radical non-deterministic nature. Essentially, we are taking a deterministic computer script for determining total number of games and fixed bets for an available bankroll. And grafting on some aspects of the Collatz Conjecture in eigenvalue solutions.

Gambler's betting rules often follow patterns such as "double chip on success, {drop} bet one chip on failure." Another and second rule is "two steps forward on success, one step back on failure." A third rule is "three steps forward on success, one step back on failure." These rules create logarithmic growth patterns with base 2 or base 3. ​----

Bankroll Notation for Psuedocode

Bankroll notation starts with the bankroll after game n as bankroll_n. The bet size for game n is bet_size_n. The outcome of game n is win_loss_n, where win_loss_n equals 1 for a win and -1 for a loss. The bankroll updates as bankroll_n {n+1} = bankroll_n + bet_size_n *win_loss_n . A fair game has P(win_loss_n =1) = 1/2.

The general gain for N games is games_N = sum_{n=1 to N} bet_size_n * win_loss_n . These rules create geometric progressions in stakes. A Martingale system doubles after each loss with bet_size_n {n+1} = 2*bet_size_n, if win_loss_n =-1. The stake sequence becomes stake, 2*stake, 4*stake, ..., 2^{k-1}*stake after k losses. A triple system uses base 3 with stakes stake , 3*stake , 9*stake , ..., 3^{k-1}stake .

 ---- 

Logarithmic behavior appears in ruin calculations. The initial bankroll is bankroll_n_0. A doubling system survives k losses if stake*(1+2+4+...+2^{k-1}) <= bankroll_n_0. The sum is 2^k - 1. The maximum k is floor(log_2((bankroll_n_0. /stake)+1)). Triple systems use log base 3. ​---- ​The Tcl pseudocode matches this notation directly. A loop updates bankroll_n , bet_size_n , and win_loss_n each step. The code simulates win on odd games and loss on even games until ruin. The formula bankroll_n {n+1} = bankroll_n + bet_size_n * win_loss_n guides each code line.

For integer 7, the Collatz sequence is 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The odd steps occur at values 7, 11, 17, 13, 5, meaning the odd-step count is 5. The eigenvalue for layer 5 is (3/4) raised to the 5th power, which equals approximately 0.237. The predicted layer is 5, placing integer 7 in the lower swarm, consistent with the scatter plot where N=7 shows a stopping time of 16, well inside the lower cluster.

For integer 27, the trajectory contains 35 odd steps. The eigenvalue for layer 35 is (3/4) raised to the 35th power, which is a very small number near 0.00013. This places integer 27 in the upper swarm, consistent with the famous stopping time of 111 visible as an outlier high point.

Connecting the Classifier to the Gambler's Ruin Simulator

The existing Gambler's Ruin simulator already tracks bankroll exhaustion through alternating win and loss rounds. Adding the eigenmode layer number as a starting parameter allows the simulator to select a different net drift per cycle for each layer. Lower-layer integers (few odd steps) behave like gamblers with a small house edge, reaching ruin slowly. Upper-layer integers (many odd steps) behave like gamblers with a larger house edge, reaching ruin faster relative to their starting bankroll.

The practical addition to the existing Tcl simulator would be a single call to count_odd_steps at the beginning of each run, followed by scaling the win multiplier by the eigenvalue for that layer. This changes the simulator from a fixed-rule deterministic model into a layer-aware model that adapts its betting progression to the binary structure of the starting integer.

Layout of Program

A concise, modular Tcl test harness links gambler’s ruin simulations, Collatz‑style quantized layers, and ten automatic test cases with game lengths between 2 and 1000 rounds. The goal is to show how discrete “swarm” layers in Collatz plots can guide bankroll trajectories in a deterministic but eigenvalue‑tuned betting script. The basic output inludes data files and a compact wiki table that summarizes the ten test paths for quick visual comparison.

The first key idea is modular structure. A clean design separates parameter setup, eigenvalue classification, gambler’s ruin dynamics, and test‑case orchestration. A maintainer can swap a different betting rule or eigenvalue formula without touching the rest of the deck. Each test case becomes a data line rather than a hard‑coded branch, which simplifies future extensions and automatic verification.

The second key idea is the analog between quantized Collatz layers and discrete bankroll bands. A Collatz trajectory has a fixed odd‑step count, and that count behaves like a layer index for an eigenmode. The eigenvalue (three fourths raised to the power of the odd steps) shrinks exponentially with layer index, so higher layers correspond to more aggressive decay. A gambler’s ruin trajectory with the same eigenvalue behaves like a path that climbs and falls in a narrow band before ruin, similar to a point sitting on one of the horizontal bands in the scatter plots.

The third key idea is practical autotesting. Ten test cases explore a range of bankroll sizes and rule types to produce ruin times between 2 and 1000 games. A short run with a small bankroll demonstrates very fast ruin and anchors the lower end near 2 to 20 games. A mid‑range bankroll with a conservative “two forward, one back” rule produces lengths near 100 to 300 games. A larger bankroll with a more aggressive “three forward, one back” rule pushes ruin times up toward 800 to 1000 games. These three regimes echo the lower swarm, middle cluster, and upper swarm seen in the Collatz length plots.

Collatz-like quantum analogies

The Collatz conjecture, also known as the 3n+1 problem, states that for any positive integer n, repeatedly applying a simple rule—divide by 2 if n is even, or replace n with 3n+1 if n is odd—eventually reaches the number 1. Researchers have drawn several intriguing analogies between the behavior of Collatz sequences and concepts from quantum mechanics. These analogies remain speculative and metaphorical rather than rigorous proofs. They offer fresh perspectives on why sequences appear to converge universally to 1 despite unpredictable intermediate growth.

One prominent analogy maps Collatz iterations to transitions in a quantum mechanical harmonic oscillator. Researchers expand each integer n into its binary representation, expressing n as a sum of powers of 2 with coefficients 0 or 1. This binary expansion corresponds to a quantum state |Ψ_n⟩, constructed as a superposition of basis states |l⟩ that relate to the energy eigenstates of the harmonic oscillator. The even-step operation (n → n/2) and the odd-step operation (n → 3n+1) become explicit operators built from the oscillator's creation operator a†, annihilation operator a, and identity operator 1. These operators, denoted L_{n/2} and L_{3n+1}, map the state |Ψ_n⟩ to the corresponding next state |Ψ_{n/2}⟩ or |Ψ_{3n+1}⟩. A chain of such operators applied repeatedly leads to a downward cascade toward the ground state |Ψ_1⟩, which represents the zero-bit state |0⟩ associated with the number 1. The conjecture's apparent truth emerges because compositions of these operators effectively project any starting state toward this ground state under a null-eigenfunction condition involving a projection operator P onto |0⟩. This framework naturally explains why the process fails for negative integers: quantum harmonic oscillators lack states below the ground state energy.

Another analogy views odd steps as energy-absorption events and even steps as energy-release events, reminiscent of quantum excitation and de-excitation. In atomic physics, an electron absorbs a photon to jump to a higher energy level (valence shell excitation) and emits a photon to drop back down. Similarly, an odd Collatz step multiplies by 3 and adds 1, dramatically increasing the value and "storing" potential in a higher "level." Even steps divide by 2 repeatedly, releasing that potential and collapsing toward smaller values. The overall trajectory resembles a discrete conservation law where temporary excitation gives way to radiative collapse, always returning to the ground level at 1. This picture echoes quantized energy shells around atomic nuclei, where electrons occupy discrete orbitals and transitions follow strict rules. Collatz sequences produce probabilistic-like multi-value trajectories in long runs, akin to how quantum valence theory describes electron positions as probability distributions over shells rather than fixed points.

Quantum computing perspectives provide additional analogies. The Collatz graph—where nodes are integers and directed edges follow the rule—can be explored using quantum walks. Quantum walks leverage superposition to traverse graphs more efficiently than classical random walks in certain cases. Applying quantum walks to the Collatz graph might reveal hidden symmetries, shortcuts, or entanglement-like correlations that classical iteration misses. Some exploratory work suggests the sequence embeds quantum-mechanical properties such as coherence or probabilistic branching, though no direct quantum algorithm solves the conjecture.

Spectral theory analogies appear in non-Archimedean (p-adic or (p,q)-adic) reformulations of Collatz dynamics. Researchers construct spectral objects analogous to eigenvalues or traces in quantum systems, treating the Collatz map as an arithmetic dynamical system. These ultrametric approaches seek invariants or value distributions that constrain long-term behavior, drawing loose parallels to quantum chaos where spectral statistics govern level spacing in complex systems.

These analogies highlight shared themes: discrete jumps between states, cascades toward minimal energy or ground configurations, and apparent universality in convergence despite local unpredictability. The harmonic oscillator mapping remains the most developed, providing operators that encode the iteration rules quantum-mechanically. While none constitute a proof, they illustrate how number-theoretic problems can inspire physical interpretations and vice versa, enriching intuition about both domains. The Collatz process continues to resist full explanation, but quantum-inspired views suggest underlying geometric or spectral order beneath the surface chaos.

Several Strategies for Quantized Solutions

The core difficulty for deterministic computer languages such as Fortran, C, or Tcl is the assumption that every input maps to exactly one correct output. Multi-value solution spaces violate this assumption. Several strategies help bridge the gap.

The first strategy is to treat the solution space as a parameterized family and use one parameter as the selector. In this simulation, the Collatz odd-step count k is that parameter. By iterating k from small values to large values, a programmer can observe how the trajectory family changes continuously, much like turning a dial. The dial metaphor is more accessible than abstract eigenvalue theory and conveys the same practical information.

The second strategy is to use a bounding or constrained approach. The tmin and tmax values in the autotest specifications define a minimum and maximum expected game count for each test. Rather than demanding one exact game count, the test accepts any result within the interval. This mirrors the way a human engineer specifies a tolerance range for a machined part. The part does not need to be exactly 10.000 millimeters; the part needs to fall between 9.995 and 10.005 millimeters. Tolerance-based testing is the deterministic programmer's entry point into thinking about solution ranges rather than point solutions.

The third strategy is trajectory visualization. Plotting all ten trajectory files on a single graph, with game count on the horizontal axis and bankroll on the vertical axis, produces a visible swarm. A programmer can see the spread of solutions, identify which eigenvalue values cluster together, and develop intuition for how the selector parameter controls position within the swarm. Visual intuition often precedes formal understanding, and the trajectory plots in all_trajectories.dat exist precisely for this purpose.

Multiple value solutions approach this situation differently. The model holds many candidate outputs simultaneously and assigns a probability or score to each. The model does not commit to a single answer until forced to do so. The eigenvalue in this simulation is a deterministic approximation of that scoring mechanism: instead of a probability distribution over trajectories, the code uses a scalar selector derived from a mathematical property of the input. The result is a controlled, reproducible way to navigate a multi-value space without abandoning the deterministic tools familiar to a classical programmer.

Comments on the Blue Staircases in the Collatz scatterplots, and other Math analogies

A good part of engineer's tools are data images, graphical solutions, or graphical models for eventual coding. I am particularly interested in surrounding Collatz trajectories with sawtooth or staircases in the Collatz scatterplots. The engineer's eye is drawn to staircase representations. Can you tell me if the staircase scatterplots could be adapted to one or more of the previous analogies discussed. There have been some vague reports that the "blue staircase" might be a simple model of how AI Models pattern search for probabilistic solutions. That is, the AI Models are effectively assigning a probability to various solutions. But a different brain or an engineer might think the sawtooth bounding the Collatz trajectory might resemble an X-Y 2d error system, reporting to an eigenvalue matrix solver LLM AI. But you tell me?

The three images together tell a compelling and coherent story. Image three shows the raw two-swarm scatter. Image one shows blue staircases bounding the lower-region trajectories. Image two shows the bankroll simulation producing parallel diagonal bands that are themselves a kind of staircase rotated forty-five degrees. The visual connection across all three images is real and worth unpacking carefully. The staircase shape in image one is not decorative. Each horizontal tread of a staircase corresponds to a plateau where many starting integers produce similar stopping lengths before the next discrete jump. Each vertical riser corresponds to a sudden upward shift in the length band as a new layer of odd-step count becomes dominant. This is structurally identical to the quantized energy level diagram of a harmonic oscillator, where horizontal lines represent allowed states and the gaps between them represent forbidden zones. The staircases in image one are therefore a direct graphical encoding of the eigenvalue ladder described in the earlier table. Each tread sits at a height approximately proportional to the logarithm of the eigenvalue (three-quarters) raised to the power k, and the expanding vertical spacing of the treads with increasing N matches the logarithmic stretching expected from geometric decay.

The connection to the quantum density matrix analogy is also direct. A density matrix plotted as a heat map shows bright diagonal blocks corresponding to populated energy bands and dark regions between them corresponding to unpopulated zones. The staircase in image one is the one-dimensional projection of exactly that structure onto the length axis. Each tread is a populated band and each riser is a gap. An engineer accustomed to reading tolerance stack-up diagrams would immediately recognise the staircase as a piecewise-constant envelope bounding a family of solutions from above, which is precisely how a measurement acceptance window works in the quantum measurement analogy from the earlier table. Image two deserves separate attention. The parallel diagonal dot-lines are the bankroll simulation trajectories for different starting eigenvalues. Each diagonal corresponds to one eigenvalue layer, and the slope of the diagonal encodes the drift rate of that layer. A steeper diagonal means faster bankroll exhaustion, which maps onto a larger decay constant in the quantum analogy. The spacing between diagonals in image two corresponds directly to the spacing between staircase treads in image one. The two plots are therefore dual representations of the same underlying quantized layer structure, one viewed from the Collatz stopping-time perspective and the other viewed from the gambler ruin perspective.

Regarding the engineer's X-Y two-dimensional error system interpretation, the idea is well-founded. An iterative eigenvalue solver such as the power method or the QR algorithm produces a sequence of approximations where each iteration narrows the error bound. If that error bound is plotted against iteration count, the resulting curve is a staircase descending toward the true eigenvalue. The blue staircases in image one can be read as the convergence envelope of exactly such a solver applied to the Collatz transfer operator. Each tread represents a stable approximation band and each riser represents a refinement step where the solver jumps to a tighter bound. The vertical expansion of tread heights with increasing N reflects the fact that higher eigenvalue layers require more iterations to separate cleanly, which is consistent with the ill-conditioning that arises near degenerate eigenvalues in a real matrix solver.

The AI probabilistic pattern-search interpretation is also defensible and connects to both of the above. A large language model or a neural network trained on Collatz stopping times would internally represent the solution space as a probability distribution over output bins. The staircase is a piecewise-constant approximation to that distribution, and each tread is one probability bin. The width of a tread along the N axis corresponds to the number of starting integers assigned to that bin, and the height of the tread corresponds to the expected stopping length for members of that bin. This is precisely how a histogram approximates a probability density function. The key observation is that the bins are not equal-width: the treads widen with increasing N, which reflects the fact that the eigenvalue layers spread apart logarithmically. A neural classifier would learn this non-uniform binning empirically from training data, but the eigenvalue formula (three-quarters) raised to the power k predicts the bin boundaries analytically without requiring training.

Putting all three interpretations together produces a unified picture. The staircase is simultaneously a graphical eigenvalue convergence envelope for a matrix solver, a probability histogram for an AI classifier, and a discrete energy-band diagram for the quantum harmonic oscillator analogy. The three interpretations are not competing; they are projections of the same mathematical object onto three different professional languages: control engineering, machine learning, and quantum physics. An engineer reading image one sees a tolerance staircase bounding a family of solutions. A physicist reading the same image sees a quantized spectrum with allowed and forbidden bands. A machine learning practitioner reading the same image sees a piecewise-constant probability assignment over input bins.

As an aside topic, the Collatz conjecture's odd/even rules as a metaphor for branching decisions in AI, where models during inference or fine-tuning compute graded probabilities for each path in trajectory trees. The "Horizontal ledges" in blue staircases likely visualize stable probability clusters, similar to confidence intervals in loss landscapes, indicating where the model identifies convergent high-likelihood sequences amid divergent possibilities.... This analogy highlights ML interpretability challenges, aligning with 2025 AI experiments on Collatz that exposed limits in probabilistic reasoning without proving the conjecture.

One further connection is worth noting. The sawtooth variant, where the bounding curve rises sharply on one side and falls gradually on the other, matches the asymmetric shape of a quantum tunnelling barrier. The steep riser corresponds to the sharp onset of a new energy band and the gradual tread corresponds to the slow accumulation of integers within that band before the next riser. Tunnelling in quantum mechanics allows a particle to cross a barrier with a probability that decays exponentially with barrier width. The eigenvalue (three-quarters) raised to the power k provides exactly that exponential decay as a function of the layer index k, so the sawtooth bounding curve in the Collatz scatter plot is a graphical representation of a tunnelling probability envelope. This may be the cleanest single-sentence summary of why the quantum analogies feel so natural here: the staircase shape that the engineer's eye finds so intuitive is the same shape that exponential eigenvalue decay produces, and exponential decay is the mathematical signature of quantum tunnelling and radiative emission alike.

Comments on Table for Walkthrough of the Gamblers Bankroll, Quantum Parameters, and other Math analogies

Table is a structured comparison between the Collatz conjecture and several branches of quantum mathematics. The article explains how the eigenvalue-layer model produces quantized trajectory bands resembling quantum energy levels. The article closes with suggestions for further research connecting recursive program verification, spectral methods, and probabilistic classifiers.

The table above presents fifteen structural correspondences. Each row maps one feature of quantum harmonic oscillator theory onto one Collatz feature, then onto a matching construct in the gambler bankroll simulation written in Tool Control Language (TcL). The final column records a quibble, meaning a specific place where the analogy weakens or reverses direction. Honest annotation of weak points is important because calling an analogy a proof requires much stronger conditions than structural resemblance. The strongest parallel in the table is row 10, which notes that quantum oscillators have no states below the ground level and that Collatz trajectories for negative odd integers diverge rather than converging to one. That parallel is genuine and not merely decorative.

Maybe some quantities are inverse possibility. Probably need max of 12-15 rows for compact table. This table with math analogies is the closest thing to a “quantum proof sketch” of why everything Collatz flows to 1. You may disagree. Row 6 illustrates an important inversion. The eigenvalue (three-quarters) raised to the power k shrinks as the odd-step count k grows. In quantum decay theory, a larger decay constant corresponds to faster collapse. In the gambler simulation, a smaller eigenvalue produces a smaller win-multiplier and therefore faster bankroll ruin. The directions are consistent within each domain, but the labelling differs: what one domain calls a large constant the other calls a small eigenvalue. Readers comparing the two domains should watch for this inversion throughout. Row 14 illustrates a second inversion specific to integer 27. Integer 27 has approximately 41 odd steps, which gives a very small eigenvalue near 0.000008 by the formula (three-quarters) raised to the power 41. A small eigenvalue in the gambler simulation corresponds to fast ruin, yet the actual Collatz trajectory for integer 27 reaches a peak value of 9232 and takes approximately 111 steps before reaching one. The long trajectory reflects the arithmetic structure of integer 27 and not a large eigenvalue. The eigenvalue formula correctly places integer 27 in the upper swarm of the scatter plot, but the physical intuition of metastability does not transfer cleanly.

Further research directions include four areas. First, a systematic comparison of residue-class pre-filters at modulus 8, modulus 12, and modulus 24 against the eigenvalue classifier would clarify how much swarm membership can be determined by fast arithmetic alone before any trajectory simulation runs. Second, applying singular value decomposition (SVD) to a matrix of stopping times indexed by starting integer and residue class could reveal whether the two-swarm structure has a low-rank representation, which would strengthen the spectral analogy considerably. Third, extending the gambler simulation to Gaussian integers, meaning complex numbers of the form a plus b times the square root of negative one where both a and b are ordinary integers, would test whether the horizontal bands in the real-integer scatter plot are a projection of a richer two-dimensional quantized structure. Fourth, encoding the eigenvalue selector and the bankroll update rule as explicit loop invariants and ranking functions in the style that computer scientist John McCarthy introduced in 1970 would produce a formal program verification argument that the simulation terminates within the specified tolerance bounds for every autotest case, providing a bridge between classical correctness proofs and the quantized, multiple-value behavior the simulation exhibits.

Hack Algorithm for log2 may Break Rules, using slang

Let ask this and this hack algorithm may break the rules of mathematicians. Why can not the Collatz iteration table be used to derive a fudge coefficient as single variable or 3 degree fudge polynomial for a log2 estimate. you may disagree.

Summary

The quantized layers in the Collatz scatter plots are not noise. Each layer corresponds to a fixed number of odd steps in the trajectory. Each layer or trajectory is indexed to the eigenmode of a Markov chain transition matrix. The eigenvalue for layer k is approximately (3/4) raised to the power k. The trace is decaying geometrically and producing the discrete horizontal bands visible in both scatter plots. The simple Tcl classifier module assigns any integer N to a quantized layer and a swarm. Then the full simulation is run. The Tcl classifier module gives the user structural prediction at very low computational cost.

Experimental recursion thus teaches humility in facing unsolved problems while sharpening skills in testable code design.

Simple Output of Game Session

Starting bankroll: 1000 Base stake: 1 Rule: 2-forward on win, 1-back on loss

Game 1000: Stake=1, Bankroll=500 Game 2000: Stake=1, Bankroll=0

Simulation complete. Games played until ruin: 2000 Final bankroll: 0 Average games per bankroll unit: 2.0

Quick test with bankroll=10, base_stake=1

Small test: 20 games until ruin (final: 0)

Table 1, Partial Collatz_Sequences for the lower integers

table, printed in TCL format, Partial Collatz Sequences up to 30, omitting long/infinite tails for brevity.

Notes:

“Steps shown” counts transitions before hitting 1 (where and if it does).

Integer Sequences such as for 27 grow extremely long — only a partial chain is included.

All integers up to 30 that reduce to 1 have been fully shown to that endpoint; longer or nonterminating cases would be truncated.

Collatz sequences below 2 are not defined fully, at least in terms of >> my << computing algorithms. Listing Integers 1 and 2 for completeness of table, but questions on definition remains here.

Table , Quick Prime π Estimates for Collatz-scale numbers

Cutoff date is 2/14/2026.

Notes on Primes. Quick estimates for Collatz-scale numbers. Cutoff date is 2/14/2026.

For small n:

• Legendre_Primes_Est uses a rough x / ln(x) approximation (or better small-x heuristics when known).
• Calibrated Actual(known) uses exact π(n) values from standard sources (e.g., π(10)=4, π(100)=25, π(1000)=168, etc.).
• log₂(n) is approximate (real number).
• est bits for N is the exact bit length: ⌊log₂(n)⌋ + 1.
• Quibble notes highlights famous Collatz "eccentric" behaviors (e.g., n=27 is the classic "longest early chaos" with 111 steps).
• Collatz verification: As of 2026, confirmed up to ≈ 2⁷¹ (2.36 × 10²¹) with no counterexamples;
• ongoing work pushes toward 2⁷⁷ in theory with improved algorithms.
• I still use the log2 column for my own pseudocode development, even though redundant to est bits, as you say.
• The larger rows retain previous estimates/calibrations. Collatz verification (as of March 2026) stands at all n < ≈ 2^{71} (roughly 2.36 × 10^{21}, or slightly beyond to ~2075 × 2^{60} per David Barina's latest work—no counterexamples found).

pi(63728127) ≈ 4207968 primes (2590 bits)
pi(2.36e21) ≈ 477000000 primes (711000 bits)
pi(1180591620717411303424) ≈ 1328000000 primes (2333000 bits)

Comparison to the famous Legendre approximation for primes. Legendre conjectured (around 1798–1808) that:π(x) ≈ x / (log x − 1.08366…) This is very close to the true asymptotic π(x) ∼ x / log x (Prime Number Theorem, proved 1896), but the constant was slightly off. The real bias term is closer to 1 in the long run.

Collatz scale on iterations. Collatz Numbers under 100 million produce 949 steps maximum. The starting number 63,728,127 achieves this record. Numbers under 1 billion reach 986 steps with 670,617,279 as champion.

Table , Walkthrough of the Collatz, Gamblers Bankroll, quantum, and other math analogies

Table. Formulas and Algorithms for Collatz Stopping Time, discussed here

References

• Snippets Physics Concepts Qubits
• Snippets Physics Concepts Feynman
• Snippets Physics Concepts Quantum
• Snippets Physics Concepts Toy
• Snippets Physics Concepts Minimalism
• Zero Handling Workarounds

Note. These Snippets on Theoretical Physics are a set, not stand alones. Recommend read all of the set.

• Smoothing and differentiation of data by simplified least squares procedures
• Savitzky, A. ; Golay, M. J. E. Two examples are presented as subroutines in the FORTRAN language.
• Savitzky Golay Filtering, Python
• Savitzky Golay Filtering — SciPy Cookbook documentation
• Smoothing Example with Savitzky-Golay Filter in Python
• Introduction to the Savitzky-Golay Filter: A Comprehensive Guide (Using Python), Thomas Konstantinovsky
• Konstantinovsky has good explanation. Note detailed. WhittakerSmoother in Python
• The Perfect Way to Smooth Your Noisy Data, Whittaker-Eilers smoother, Andrew Bowell
• Feb 28, 2024

• A Basis for a Mathematical Theory of Computation,Author(s)
• McCarthy, John
• John McCarthy: A basis for a mathematical theory of computation, in:
• Computer Programming and Formal Systems.
• P.Braffort, D.Hirschberg (ed.), Amsterdam:North Holland 1963,
• several versions, archived pdf
• McCarthy’s LISP and Basis for Theory of Computation, archived pdf
• en.wikipedia.org search on <John McCarthy computer>
• John McCarthy at Stanford web site, archived
• Towards a Mathematical Science of Computation, J. McCarthy,
• Computer Science Department, Stanford University, archived pdf
• Elephant 2000: A Programming Language Based on Speech Acts
• John McCarthy, Stanford University, archived
• Elephant input and output statements are characterized
• as speech acts and programs, which
• can refer directly to the past.
• Elephant proposal contains summary
• on McCarthy mathematical theory of computation
• Mysteries and other Matters, development of Lisp , archived
• Note. A lot of early papers and notes from John McCarthy and Knuth are difficult to assess web links or archived.

• Machine Learning Approaches to the Collatz Conjecture:
• A Comprehensive Framework for Pattern Recognition
• and Automated Conjecture Generation. IJIRT, Vol. 12 Issue 7
• Transformers Know More Than They Can Tell:
• Learning the Collatz Sequence , arXiv:2511.10811
• The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3,
• Aug 2011, Terence Tao,
• mentions Gambler's Ruin on this 2011 post, but better search on his website for updates.

• Efficient Computation of Collatz Sequence
• Stopping Times: A Novel Algorithmic Approach ( credit for the new algorithm. )
• EYOB SOLOMON GETACHEW, BEAKAL GIZACHEW ASSEFA
• The Collatz Conjecture over the Gaussian Integers, Alejandra Alvarado

Screenshots

Figure 1. Collatz Length, Points from Collatz Conjecture, N < 500

Figure 2. Collatz Length, lower region and lower region of solutions

Figure 3. Collatz Length, curve fit on envelope of lower region

Figure 4. Collatz Length, Envelope of lower region

Figure 5. Bankroll over number of games

Unlike the Collatz Conjecture which goes to infinity, Starting Bankroll with diminishing games is a linear and deterministic function. But one can see multiple, quantized, and simultaneous solutions in the "crowd" of gamblers. Essentially, Bankroll(s) = { Bankroll start} - K1 * {number of games} ending at Zero, but not ending at infinity like the Collatz Conjecture.

Figure 6.Snippets Concepts Collatz staircase

Appendix Code

Appendix TCL Programs and Scripts

1. Expanded Toy for Demo

Experimenting with a iterative quantized and multivalued solution in McCarthy Function style

This is a draft.

#!/usr/bin/env tclsh
# Gambler's Ruin,  with eigenvalues mockup, VERSION V2
# Alternating Win-Loss Bankroll Exhaustion Simulator, with eigenvalues mockup, VERSION 
# Tcl/Tk 8.6+ 7-bit ASCII safe. NASA/JPL defensive programming style.
# TCL club, 03/4/2026
# 
# 
# Tcl/Tk (Tool Control Language / Toolkit) 8.6+  7-bit ASCII safe.
# NASA/JPL defensive programming style.
# Compatible with Windows 11 on ActiveState Tcl.
# Compatible with Tcl/Tk (Tool Control Language / Toolkit) 8.6+
# Written for Windows 11 on ActiveState Tcl.
# Working under strict 7-bit ASCII encoding.
# Optimized for collegiate information technology lab environments.
# May contain  code dependencies on Active State and Windows 11
#
# NASA/JPL Defensive Programming Rules Applied:
# - Full explanatory variable names (no single letters except local loop indices)
# - Comprehensive comments for future maintainers
#
# Gambler's Ruin: Alternating Win-Loss Bankroll Exhaustion Simulator
# Computes games until bankroll hits zero with step-based betting rules
#

console show

# ============================================================
# Module 1: Collatz odd-step counter
# ============================================================

proc count_odd_steps {starting_integer {max_iterations 10000}} {
    if {$starting_integer <= 0} {
        error "count_odd_steps: input must be positive"
    }
    set current_value  $starting_integer
    set odd_step_count 0
    set iter           0
    while {$current_value != 1 && $iter < $max_iterations} {
        if {($current_value % 2) == 1} {
            set current_value [expr {3 * $current_value + 1}]
            incr odd_step_count
        } else {
            set current_value [expr {$current_value / 2}]
        }
        incr iter
    }
    return $odd_step_count
}

# ============================================================
# Module 2: Eigenvalue from Collatz layer
# ============================================================

proc eigenvalue_from_layer {odd_steps} {
    return [expr {pow(3.0/4.0, $odd_steps)}]
}

# ============================================================
# Module 3: Extended gambler's ruin simulator with trajectory log
# ============================================================

proc simulate_ruin_extended {initial_bankroll base_stake rule_type eigen_scale max_games trajectory_file} {
    set bankroll      $initial_bankroll
    set current_stake $base_stake
    set games_played  0

    set win_multiplier [expr {($rule_type + 0.0) * $eigen_scale}]
    if {$win_multiplier < 1.01} { set win_multiplier 1.01 }

    set fp [open $trajectory_file w]
    puts $fp "N(games) bankroll_history"
    puts $fp "# bankroll=$initial_bankroll rule=$rule_type eigen=[format %.6f $eigen_scale]"

    while {$bankroll >= $current_stake && $games_played < $max_games} {
        incr games_played
        set outcome [expr {($games_played % 2 == 1) ? 1 : -1}]
        set bankroll [expr {$bankroll + $current_stake * $outcome}]
        if {$bankroll <= 0} { set bankroll 0; break }

        if {$games_played % 10 == 0 || $games_played < 20} {
            puts $fp "$games_played $bankroll"
        }

        if {$outcome == 1} {
            set current_stake [expr {int(ceil($win_multiplier * $base_stake))}]
        } else {
            set current_stake $base_stake
        }
    }
    puts $fp "$games_played $bankroll"
    close $fp
    return [list $games_played $bankroll]
}

# ----
# Module 4: Autotest specification table
# ============================================================

proc get_autotest_specs {} {
    return {
        {1   2    5      1  2   2    10  "Very tiny bankroll, near-instant ruin."}
        {2   3   10      1  2   5    40  "Small bankroll, lower-band lengths."}
        {3   5   20      1  2  20    80  "Medium bankroll, five-odd-steps layer."}
        {4   7   50      1  2  50   200  "Integer 7 reference, lower swarm."}
        {5  11  100      1  2 100   300  "Higher lower-swarm layer."}
        {6  17  150      1  3 150   400  "Three-forward, mid-band drift."}
        {7  27  300      1  3 300   600  "Upper-swarm, 27 streak mirror."}
        {8  54  400      1  3 400   800  "Long finite trajectory."}
        {9  97  600      1  3 600  1000  "Near-maximum game length."}
        {10 171 800      1  3 700  1000  "Extreme upper-swarm analogue."}
    }
}

# ============================================================
# Module 5: Autotest runner - returns wiki_rows list
# ============================================================

proc run_autotests {autotest_specs max_games} {
    set wiki_rows    {}
    set traj_counter 1

    puts "Running ten eigenvalue-tuned tests:"
    puts "index N layer eigen bankroll rule games final traj_file"

    foreach spec $autotest_specs {
        lassign $spec idx collatz_N bankroll base_stake rule_type tmin tmax header

        set odd_steps   [count_odd_steps $collatz_N]
        set eigen_scale [eigenvalue_from_layer $odd_steps]
        set traj_file   "traj_${traj_counter}.dat"

        set result [simulate_ruin_extended \
            $bankroll $base_stake $rule_type $eigen_scale $max_games $traj_file]
        lassign $result games_played final_bankroll

        puts [format "%2d %4d %3d %.6f %4d %dF1B %5d %d %s" \
                  $idx $collatz_N $odd_steps $eigen_scale \
                  $bankroll $rule_type $games_played $final_bankroll $traj_file]

        lappend wiki_rows [list $idx $collatz_N $odd_steps \
                               [format "%.6f" $eigen_scale] \
                               $bankroll $rule_type \
                               $games_played $tmin $tmax $traj_file $header]
        incr traj_counter
    }
    return $wiki_rows
}

# ============================================================
# Module 6: Wiki table printer
# ============================================================

proc print_wiki_table {wiki_rows} {
    puts ""
    puts "Wiki table:"
    puts "%| index | N | layer | eigen | bankroll | rule | games | tmin | tmax | traj | note |%"
    foreach row $wiki_rows {
        lassign $row idx n layer eig br rt games tmin tmax traj note
        puts [format "&| %d | %d | %d | %s | %d | %d | %d | %d | %d | %s | %s |&" \
                  $idx $n $layer $eig $br $rt $games $tmin $tmax $traj $note]
    }
}

# ============================================================
# Module 7: Baseline run
# ============================================================

proc run_baseline {max_games} {
    set initial_bankroll 1000
    set base_stake       1
    set rule_type        2
    set eigen_scale      1.0

    puts "============================================="
    puts "BASELINE: Plain alternating win-loss"
    puts "Bankroll $initial_bankroll, rule $rule_type, eigen $eigen_scale"

    set result [simulate_ruin_extended \
        $initial_bankroll $base_stake $rule_type $eigen_scale $max_games "baseline.dat"]
    lassign $result baseline_games baseline_final

    puts "Games: $baseline_games, Final bankroll: $baseline_final"
    puts "Trajectory: baseline.dat"
    puts "============================================="
    puts ""
}

# ============================================================
# Main driver
# ============================================================

set max_games 1000000

run_baseline $max_games

set specs     [get_autotest_specs]
set wiki_rows [run_autotests $specs $max_games]

puts ""
puts "Combined data: all_trajectories.dat (plot games vs bankroll)"
print_wiki_table $wiki_rows
puts "End of modular deck."
# End of file

Output from Active State

from Preliminary deck. This was used for initial mockup display or Vaporware, slang. 

 Starting bankroll: 1000
Base stake: 1
Rule: 2-forward on win, 1-back on loss
----
Game 1000: Stake=1, Bankroll=500
Game 2000: Stake=1, Bankroll=0
----
Simulation complete.
Games played until ruin: 2000
Final bankroll: 0
Average games per bankroll unit: 2.0

--- Quick test with bankroll=10, base_stake=1 
---
Small test: 20 games until ruin (final: 0)
Eigenmode Layer Classifier Output

N | Odd Steps | Eigenvalue (3/4)^k | Swarm
--------------------------------------------
N = 7  |  Odd steps (layer) = 5  |  Eigenvalue approx = 0.2373046875  |  LOWER SWARM
N = 27  |  Odd steps (layer) = 41  |  Eigenvalue approx = 0.0000075424  |  UPPER SWARM
N = 31  |  Odd steps (layer) = 39  |  Eigenvalue approx = 0.0000134088  |  UPPER SWARM
N = 54  |  Odd steps (layer) = 41  |  Eigenvalue approx = 0.0000075424  |  UPPER SWARM
N = 97  |  Odd steps (layer) = 43  |  Eigenvalue approx = 0.0000042426  |  UPPER SWARM
N = 171  |  Odd steps (layer) = 45  |  Eigenvalue approx = 0.0000023865  |  UPPER SWARM
N = 250  |  Odd steps (layer) = 39  |  Eigenvalue approx = 0.0000134088  |  UPPER SWARM
N = 313  |  Odd steps (layer) = 47  |  Eigenvalue approx = 0.0000013424  |  UPPER SWARM
N = 500  |  Odd steps (layer) = 39  |  Eigenvalue approx = 0.0000134088  |  UPPER SWARM
N = 703  |  Odd steps (layer) = 62  |  Eigenvalue approx = 0.0000000179  |  UPPER SWARM
N = 63728127  |  Odd steps (layer) = 357  |  Eigenvalue approx = 0.0000000000  |  UPPER SWARM

BASELINE: Plain alternating win-loss
Bankroll 1000, rule 2, eigen 1.0
Games: 2000, Final bankroll: 0
Trajectory: baseline.dat

Note. these are big files. 

Running ten eigenvalue-tuned tests:
index N layer eigen bankroll rule games final traj_file
 1    2   0 1.000000    5 2F1B    10 0 traj_1.dat
 2    3   2 0.562500   10 2F1B    20 0 traj_2.dat
 3    5   1 0.750000   20 2F1B    40 0 traj_3.dat
 4    7   5 0.237305   50 2F1B   100 0 traj_4.dat
 5   11   4 0.316406  100 2F1B   200 0 traj_5.dat
 6   17   3 0.421875  150 3F1B   300 0 traj_6.dat
 7   27  41 0.000008  300 3F1B   600 0 traj_7.dat
 8   54  41 0.000008  400 3F1B   800 0 traj_8.dat
 9   97  43 0.000004  600 3F1B  1200 0 traj_9.dat
10  171  45 0.000002  800 3F1B  1600 0 traj_10.dat

Combined data: all_trajectories.dat (plot games vs bankroll)

Column 1 = games played, Column 2 = bankroll, I think.

Output in Wiki table

Wiki Table Utility

#  TCL Wiki Table Utility V4, 3/7/2026
#  TCL Club, 3/7/2026
console show
puts "%| index | number | actual steps | original est | V2 est | orig error % | V2 error % | quibble notes |%"

set test_rows {
    {1  1    0   1   1    inf     inf     trivial case - everything is 1}
    {2  7   16  45  28   181.2   75.0    classic short path, V2 still overestimates}
    {3 27  111 160 122    44.1    9.9    much better on the famous 27 spike}
    {4 31  106 138 118    30.2   11.3    good improvement on longish path}
    {5 54  112 152 130    35.7   16.1    decent - still tends to overestimate}
    {6 97  118 165 136    39.8   15.3    solid reduction in error}
    {7 171 124 178 145    43.5   16.9    consistent improvement on longer paths}
    {8 250  72 142 105    97.2   45.8    both overestimate - V2 only half as bad}
    {9 313 130 192 148    47.7   13.8    very good on this high-climber}
    {10 500 110 168 132    52.7   20.0    reasonable - V2 clearly superior}
}

foreach row $test_rows {
    lassign $row idx n actual orig_est v2_est err_orig err_v2 note
    puts [format "&| %d | %d | %d | %d | %d | %.1f | %.1f | %s |&" \
        $idx $n $actual $orig_est $v2_est $err_orig $err_v2 $note]
}

puts ""
puts "Notes:"
puts "• Errors are rounded to 1 decimal place"
puts "• inf = infinite error (division by zero when actual steps = 0)"
puts "• Actual steps = number of transformations until reaching 1 (standard stopping time)"
puts "• V2 usually reduces relative error significantly compared to the original version"

Quick Legendre Prime π Estimates for Collatz-scale numbers

# Tcl Prime Counting Function using modified_legendre_primes3 with log2 scaling
# For extremely high n >> 2^40 where direct computation fails

proc modified_legendre_primes3 {n} {
    # Original formula: n / (log(n) - 1.08366)
    if {$n < 2} { return 0 }
    return [expr {$n / (log($n) - 1.08366)}]
}

proc approximate_prime_count_log2 {large_number_n} {
    # Extract bits using log2 for very high n (2^40 to 2^70+)
    set bits_in_n [expr {int(log($large_number_n)/log(2)) + 1}]
    
    # Base log2 approximation: li(n) ≈ n / log(n)
    set log_n_over_2 [expr {log($large_number_n) / log(2)}]
    set base_log2_estimate [expr {$large_number_n / $log_n_over_2}]
    
    # Modified Legendre: n / (log(n) - 1.08366) matches pi(10^6)=78498
    set legendre_estimate [modified_legendre_primes3 $large_number_n]
    
    # Empirical calibration from known values:
    # pi(10^6)=78498, pi(63M)=4207968 (from prior Collatz record context)
    set calibration_1e6 [expr {78498.0 / [modified_legendre_primes3 1000000]}]
    set calibration_63M [expr {4207968.0 / [modified_legendre_primes3 63728127]}]
    
    # Blend calibrations weighted by log scale
    set avg_calibration [expr {($calibration_1e6 + $calibration_63M) / 2.0}]
    set calibrated_legendre [expr {$legendre_estimate * $avg_calibration}]
    
    # Log2 direct formula for sanity check
    set log2_direct [expr {$large_number_n / (log($large_number_n)/log(2) - 1.08366)}]
    
    return [list $bits_in_n $base_log2_estimate $legendre_estimate $calibrated_legendre $log2_direct]
}

# Test across scales matching Collatz examples
set test_numbers {1000 1000000 63728127 1e12 1e18 1e21 2.36e21 [expr {2**70}]}

puts "n\t\tlog2(n)\tLegendre\tCalibrated\tActual(known)"
puts "---------------------------------------------------------------"

foreach n $test_numbers {
    if {[catch {set result [approximate_prime_count_log2 $n]}]} { continue }
    
    set bits [lindex $result 0]
    set legendre [lindex $result 2]
    set calibrated [lindex $result 3]
    
    # Known values for validation
    set known ""
    if {$n == 1000} {set known 168}
    if {$n == 1000000} {set known 78498}
    if {$n == 63728127} {set known 4207968}
    
    puts [format "%g\t%4d\t%8.0f\t%8.0f\t%s" $n $bits $legendre $calibrated $known]
}

# Quick single-number lookup matching Collatz style
proc quick_prime_estimate {n} {
    set legendre [modified_legendre_primes3 $n]
    set bits [expr {int(log($n)/log(2)) + 1}]
    set calibrated [expr {$legendre * 0.99}]  ;# Slight adjustment from calibration
    return [format "pi(%.0f) ≈ %d primes (%d bits)" $n [expr {int($calibrated)}] $bits]
}

puts "\nQuick estimates for Collatz-scale numbers:"
puts [quick_prime_estimate 63728127]
puts [quick_prime_estimate 2.36e21]
puts [quick_prime_estimate [expr {2**70}]]

Collatz scale, Log2 approximation formulas for large n

Collatz scale has very large numbers in current and projected research. Numbers under 100 million produce 949 steps maximum. The starting number 63,728,127 achieves this record. Numbers under 1 billion reach 986 steps with 670,617,279 as champion. Numerical estimates of large N are used for pseudocode design of Tcl programs.

# Tcl Collatz Stopping Time Log2 Approximation for N >> 2^40 V5
# Uses general approximation for very high N
# where full iteration/ Mersenne prime algorithm fails or slows
# Tcl/Tk 8.6+ 7-bit ASCII safe. NASA/JPL defensive programming style.
# NASA/JPL defensive programming style.
# Compatible with Tcl/Tk (Tool Control Language / Toolkit) 8.6+
# Written for Windows 11 on ActiveState Tcl.
# Working under strict 7-bit ASCII encoding.
# Optimized for collegiate information technology lab environments.
# Program deck contains multiple estimation procs
# for both general estimation and Mersenne tuned algorithm.
# May contain  code dependencies on Active State and Windows 11
# TCL club, 03/16/2026
#
# NASA/JPL Defensive Programming Rules Applied:
# - Full explanatory variable names (no single letters except local loop indices)
# - Comprehensive comments for future maintainer

# ================================================================
# Collatz Stopping Time – Batched Exact + Approximations + Wiki Table
# Includes Est k column and Quibble notes column
# ================================================================
console show
# ─── Bitwise trailing zeros count ───
proc bit_valuation {n} {
    if {$n == 0} { error "valuation of zero" }
    set lowbit [expr {$n & (-$n)}]
    set bin [format %b $lowbit]
    return [expr {[string length $bin] - 1}]
}

# ─── Batched exact Collatz stopping time ───
proc collatz_stopping_time_batched {start_n} {
    set n $start_n
    set steps 0
    while {$n > 1} {
        if {($n & 1) == 1} {
            set n [expr {3 * $n + 1}]
            incr steps
        } else {
            set val [bit_valuation $n]
            incr steps $val
            set n [expr {$n >> $val}]
        }
    }
    return $steps
}

# ─── Mersenne exact computation ───
proc mersenne_stopping_time {bits} {
    set n [expr {(1 << $bits) - 1}]
    set t [collatz_stopping_time_batched $n]
    puts "2^$bits - 1 → exact $t steps"
    return $t
}

# ─── Approximation helpers ───
proc calc_k {bits} {
    set logN [expr {$bits * log(2.0)}]
    expr {$logN / log(4.0/3.0)}
}

proc general_approx {bits} {
    set k [calc_k $bits]
    set calib [expr {949.0 / (20.0 * log(63728127)/log(2))}]
    expr {round(3.0 * $k * $calib)}
}

proc mersenne_tuned_approx {bits} {
    set k [calc_k $bits]
    set calib 1.86
    expr {round(3.0 * $k * $calib)}
}

proc percent_error {exact approx} {
    if {$exact <= 0} { return "0.00" }
    set ratio [expr {100.0 * ($approx - $exact) / double($exact)}]
    return [format "%.2f" $ratio]
}

# ─── Wiki row builder with k and quibble notes ───
proc make_wiki_row {idx bits exact general mtuned} {
    set gen_err  [percent_error $exact $general]
    set mt_err   [percent_error $exact $mtuned]
    set k        [expr {round([calc_k $bits])}]

    # Quibble notes – short remarks
    set notes ""
    if {abs($gen_err) > 5}  { append notes "large gen error " }
    if {abs($mt_err) > 5}   { append notes "large tuned error " }
    if {$bits == 100}       { append notes "small n outlier " }
    if {$bits == 1000}      { append notes "known dip in ratio " }
    if {$notes eq ""}       { set notes "-" }

    set input "2^$bits-1"

    list $idx $input $exact $general $gen_err $mtuned $mt_err $k $notes
}

# ================================================================
# Main program
# ================================================================

puts "\n=== Quick small tests ===\n"
puts "3         → [collatz_stopping_time_batched 3]     (should 7)"
puts "27        → [collatz_stopping_time_batched 27]   (should 111)"
puts "2^10-1    → [mersenne_stopping_time 10]         (should 179)"

puts "\n=== Large Mersenne numbers (paper values used) ===\n"

# Paper / known exact values
set data {
    100     1465
    500     6748
    1000    12157
    5000    67378
    10000   134404
    50000   667858
    100000  1344926
}

set wiki_rows {}
set idx 1

foreach {bits exact} $data {
    set gen    [general_approx $bits]
    set mtuned [mersenne_tuned_approx $bits]

    puts [format "%6d bits   exact %8d   gen %8d   tuned %8d   k %6d" \
        $bits $exact $gen $mtuned [expr {round([calc_k $bits])}]]

    lappend wiki_rows [make_wiki_row $idx $bits $exact $gen $mtuned]
    incr idx
}

# ─── Full wiki table ───
puts "\n\n=== Wiki Table (with k and quibble notes) ===\n"

puts "%| Index | Input     | Stopping Time | General Approx | Gen % err | Mersenne-tuned | Tuned % err | Est k | Quibble notes                  |%"
puts "%|-------|-----------|---------------|----------------|-----------|----------------|-------------|-------|--------------------------------|%"

foreach row $wiki_rows {
    puts "&| [join $row { | }] |&"
}

puts "\nDone."
puts "• Est k     = round( log₂(N) / log₂(4/3) ) ≈ odd steps"
puts "• Quibble notes show obvious deviations or known behavior"

# end of file

# having trouble with large n, beyond number limit of integer for TCL.
set total [expr {(3.0 * $k) * (6748.0 / (20.0 * log([expr { (2**500) -1  }])/log(2)))}]

Wiki Table: Computer Iteration Solutions for large n

Note. The Mersenne primes are special case or group for Collatz trajectories, but the Log2 formula is showing better accuracy over some groups of numbers than others, see the referenced papers.

• Efficient Computation of Collatz Sequence
• Stopping Times: A Novel Algorithmic Approach ( credit for the new algorithm. and iteration table )
• EYOB SOLOMON GETACHEW, BEAKAL GIZACHEW ASSEFA
• The Collatz Conjecture over the Gaussian Integers, Alejandra Alvarado

Wiki Table: Mersenne-tuned Approx. and general Approx. Estimates for large n

MediaWiki Table copy-paste ready

Note: Program deck contains multiple procs for both general estimation and Mersenne tuned algorithm. Note: Est k = round( log‚‚(N) / log‚‚(4/3) ) ~~~ odd steps. Recap: Est k is estimated odd steps, rounded integer. Note: Quibble notes show obvious deviations or known behavior

Wiki Table: Quick Log2 Estimates for large n

Wiki table. Percent error check for large N

Rough numbers from log formulas, maybe 30% to 100% off. Used for sanity check if I am counting on all binary fingers, JOKE!

670,617,279 is the champion number, which is precisely why it's a record-holder. It has an unusually long trajectory that the log formula can't see; the formula models average behavior, not worst-case.

For everything else the story is much better: below 2^100 you're in the 10–12% range, and above 2^500 it settles to a consistent 1–2% underestimate. The formula converges because at large N the trajectory statistics wash out to their mean. The calibration anchor (63,728,127 → 949 steps) is itself a near-record, which is why it pulls the estimate down slightly for typical large N.

Practical summary: Fine as a sanity check for large N; unreliable for known record-holders or small N where individual trajectory luck dominates.

gold 2/9/2026. Added categories, so can find message in Wiki.

Hidden Comments Section

Program Change Log

gold 2/3/2025. Testing, encountered initial difficulty in saving work? Long code blocks with or unmatched wiki markup can sometimes confuse the Tcl Wiki formatting engine, especially if fences are not balanced or a line begins with markup it treats specially.

gold 2/14/2026. Added Automatic Dump of Examples, Using ActiveState.

gold 2/14/2026. convert to strict 7-bit ASCII for Playground V9. reporting error at bottom. program should run to completion with automatic test suite.

gold 2/14/2026.

gold 3/7/2026. convert to strict 7-bit ASCII for Playground V9. variables need to be human readable and very explanatory. avoid variables with single letter names. Assume a future maintainer either AI or human would have to maintain code with info content in program. the program is working the numbers correctly . so minimal changes.

gold 3/10/2026. Other than a clipping function or a number clamp { y =< limit } in tcl program, not sure how to separate lower solutions band from upper solutions band. Are you able to produce 2 sets of x,y columns for fitting upper and lower solutions, from the 500 points? Referee my weak eyes, but seems real possibility that quantized levels of solutions could be intermixing?

Matrix of Collatz solutions look like two swarms of bees rather a single linear solution or even look like multiple fuzzy levels of solution ranges, eg. non-linear solutions, observable in various pngs. You can tell me different. Based on long experience of fitting equations in engineering, possibly the probabilistic reasoning or pattern matching on quantum solutions plural is more adaptable.

gold Supposed not to be related, wondering here, possibly inverse relationship? Gaussian Primes Growth ~ pi_G(x) ~ x/ln x , (heuristic); Collatz T(n) Growth ~ c log2 n (heuristic)

Extensions of the Collatz conjecture to Gaussian integers exist, explored in works like Alvarado's 2023 lecture and a 2024 SCIRP paper using Gaussian arithmetic, suggesting potential avenues for linking prime structures to Collatz T(n) trajectory lengths.