What died, what remains

It does not work. Here is why, without dressing it up for a seminar and without hiding behind notation that pretends the pain is smaller than it is.

Call a streak ugly when many Syracuse steps in a row only peel off a single factor of two. Valuation one, again and again. On a streak longer than your fixed window, the look-ahead reward before a step and after a step are the same number. The present-step reward is also the same. Everything clever cancels. What remains is the log of how much the odd part grew: log of (3n+1) over (2n). That quantity sits strictly above log of three over two, about 0.405. Energy goes up by a uniform margin. The method fails as a method, not because I picked bad coefficients.

That cancellation is the heart. People who hear Lyapunov sometimes imagine that tuning constants can save a shape. Here the shape itself cancels on the pathological family. Alpha disappears because both sides see valuation one. Beta disappears because Omega_k sees a window of ones before the step and a window of ones after the step when the streak is long enough. No grid search can resurrect a difference that is identically zero in the reward terms.

So the theorem is not a complaint about my optimization luck. It is a structural obstruction inside the ansatz class. Fixed k. Linear rewards in present valuation and look-ahead sum. Infinitely many failures. Later, in July, the kill widens to arbitrary rewards of a fixed window. This May entry stays with the linear death, because that was the hope I was actually living inside.

I should also say what the theorem does not do. It does not exhibit a divergent Collatz orbit. It does not exhibit a nontrivial cycle. It does not refute Tao-type almost-all results. It closes one proof strategy: pick a fixed look-ahead depth and two coefficients, then claim strict descent for every odd n. Closed strategies matter. They keep you from spending another year tuning a dead dial.

A result you cannot exhibit is a story. A result with an infinite family is a finding. Explicit witnesses include the Mersenne odds: 7, 15, 31, 63, 127, and on. In the indexing we used in the lab notes, these sit in the family that produces long streaks of ugly steps, low collapse by twos, one after another. You can check them by hand for small L. You can check them with code for larger L. The point is not mystique. The point is exhibitability.

Why Mersenne-looking odds keep showing up is less romantic than it first feels. They live near minus one in the two-adics. That neighborhood forces expanding Syracuse steps for a controllable stretch. The same geometric region appears in parallel 2026 residue-only Lyapunov obstruction work, for example the Hsieh notes on Zenodo. Those notes kill a different ansatz: residue corrections of the form log n plus a function of n mod a power of two. We kill look-ahead window rewards. Same villains, different certificate class. Cite the parallel lane. Do not steal its credit, and do not pretend our window theorem owns their residue theorem.

I also ran grids. Across many pairs of coefficients and several fixed depths, the pathological families refuse to die. On samples of the first fifty thousand odds, violation rates hovered near forty percent even at the best grid points we tried. Increasing the window from 2 to 8 barely moved the needle, on the order of a percent-level change, nowhere near a rescue. That matches the theorem: if the kill is structural, buying a slightly longer fixed flashlight will not save you.

There is a sharper empirical stamp. For each tested pair of streak length L and window k with L at least k plus something small enough to trigger cancellation, zero out of one hundred forty-three coefficient grid points saved the Mersenne witness. Zero. That is the kind of table that ends an argument with yourself. You can still love the intuition. You cannot keep paying the plan.

Non-Mersenne L-bad odds exist too: 15, 47, 79, and many others depending on L. The Mersenne family is the clean infinite spine. The broader L-bad class is the reminder that the obstruction is not a single curiosity. It is a region.

I want this section to read like a ledger, not like a eulogy with soft lighting.

Dies: the original bounded-k energy plan in the linear Santana ansatz. E equals log n minus alpha nu minus beta Omega_k, with k fixed once and for all, expected to descend at every odd n.

Dies: the fantasy that there exists a single finite k_min and a pair of weights that certifies descent for every odd n inside that ansatz. If such a uniform finite k worked, it would be enormous. It does not, for this form.

Dies: the coefficient grid search as a proof strategy. Grids are still useful as autopsy. They are no longer a path to victory.

Dies: the louder early claim that Omega was a brand-new object unknown to the literature. Terras-type exponent sums already look ahead in that sense. Packaging is not priority. Novelty language that cannot survive a cold literature pass is not courage. It is delay wearing confidence.

Dies: the quiet hope that more fixed depth would eventually outrun the ugly streaks. Depth 8 is not qualitatively kinder than depth 2 against an infinite family of longer streaks. The adversary can always choose L larger than your fixed window.

Dies, as a mathematical claim: any sentence that treats the May obstruction as progress toward having proved Collatz. It is progress toward honesty. Honesty is not a solution of the conjecture.

I am listing deaths because unfinished grief becomes new rooms. The November warehouse taught me that. May is the day the main mathematical room of that season finally received a verdict.

Remains: honesty about the failure. That sounds soft until you notice how rare it is in public conjecture culture. People rewrite deaths into almosts. I am refusing the almost. The bounded plan is dead.

Remains: an adaptive question. For each n, is there some finite depth, depending on that n, where a true contraction window finally appears? That question aims the flashlight at adaptivity instead of uniformity. It is also, in careful forms, essentially Collatz restated. Restating is not solving. If I catch myself selling the restatement as a new theorem, correct me. The value of the restatement is directional. It says where new work would have to live: adaptive depth, or information outside any fixed finite window of upcoming valuations, or hybrid certificates that do not pretend a fixed F can police every L-bad streak.

Remains: the intuition that the future can redeem the present. Under Tao-type almost-all results, density-one behavior still looks like eventual settling in a strong sense. My fixed certificate died. The density intuition did not become false because my ansatz was wrong. Intuition is not a lemma. Keep it in the essay lane unless it becomes one.

Remains: the cycle-exclusion arm as a separate lane. Showing that certain surplus conditions along a closed orbit forbid nontrivial cycles can be pursued without asking the bounded Lyapunov question to rise from the dead. Compare the spirit of cycle-search literature. Do not blur the lanes to create fake momentum.

Remains: code that reproduces the tables. Scripts in the lab directory. Runnable. Boring on purpose. A small result without reproducible checks is a bedtime story.

Remains: a path toward a short note. Definitions sharp enough to typeset. One theorem whose statement does not secretly assume Collatz. Proof that survives an adversarial read. Clear prior-art paragraph. That bar is what May opened. July strengthened the theorem beyond two coefficients. The note remains a draft until the literature pass is real.

Remains: refusal to sell poetry as a lemma. Sufficient awareness guarantees return is a wisdom claim wearing a dynamical-systems coat. It can be true as philosophy and still not be a step in a proof. The observer beyond one can live in a book. It cannot substitute for blocking ascent and cycles.

Remains: citation discipline. When residue-only obstruction work dies on the same minus-one neighborhood, name it. Hsieh 2026 is a parallel. Our claim stays narrow: look-ahead window class, linear in May, arbitrary F later.

There is a temptation, after a clean negative theorem, to speak as if the mountain moved. The mountain did not move. A trail closed. Closing a trail is real work. It is also small in the only sense that matters for Collatz: the conjecture remains open.

Another temptation is to pivot immediately into adaptive slogans and call them the Santana Conjecture as if renaming were progress. Adaptive depth is the right direction. Equivalence slogans need a correct contraction threshold. Naive surplus formulas that stay positive on valuation one do not encode descent. One-step Syracuse motion for large n needs enough twos to beat the factor three. If your surplus celebrates valuation one, your surplus is lying about contraction. Hold the adaptive door open without walking through a fake equivalence.

A third temptation is silence. Some people bury negative results because they fear looking smaller. I would rather look small and be checkable. The open notebook is a machine for checkability. Status label: small-result. Hook that refuses solved-language. Body that shows witnesses. Ledger that separates deaths from remainders.

If you are a number theorist reading this, laugh at whatever deserves laughter. The laughter I am trying to pre-earn is the laughter at novelty inflation and at Lyapunov hopes that cancel on Mersenne streaks. The laughter I am trying not to deserve is the laughter at claiming Collatz. I am not claiming Collatz.

If you are a future version of me, read the ledger when you feel the urge to reopen a fixed-k coefficient search because a new parameterization looks prettier. Pretty is not descent. Cancellation does not care about your redesign of beta.

Mid-May 2026. Bounded look-ahead Lyapunov energies with fixed k and linear coefficients: obstructed. Mersenne and L-bad families: witnesses. Grid search: autopsy, not path. Adaptive reformulation: open, careful, not a proof. Parallel residue obstruction literature: cite. Collatz: unsolved. I wrote this so the death stays dated and the remainders stay honest.

I am going home in the only way this day allows: not by announcing arrival at one, but by returning to the truth of what the scoreboard can and cannot do. The scoreboard with a fixed window cannot police the ugly streaks. That is enough for a day's work if the day is willing to be small.

July will widen the kill. Why keep May as a separate public stamp instead of folding everything into the stronger theorem? Because research time is lived forward, not rewritten backward. In May I was not living inside arbitrary F. I was living inside alpha and beta. The emotional and practical death happened there. Erasing that layer would recreate the warehouse sin in a new form: collapsing chronology to make the story look cleaner than the climb.

The linear case is also the case most people reach for first. Two knobs. A grid. A hope that machine search can finish what insight began. Showing that the knobs cancel on an explicit infinite family teaches a sharper lesson than immediately jumping to the fully general F. First you learn that tuning dies. Then you learn that the whole window reward class dies. Sequence matters for pedagogy and for honesty.

There is also a documentation reason. The lab scripts and the first obstruction writeup are keyed to the linear ansatz. Pointing readers at those files from a May-dated entry keeps provenance clean. Provenance is part of the result. If a stranger wants to reproduce the forty-percent violation tables and the zero-of-one-hundred-forty-three Mersenne saves, they should know which season produced which claim.

Finally, the linear death clarifies what remains psychologically. I did not lose faith in look-ahead as a metaphor. I lost faith in fixed look-ahead as a uniform Lyapunov certificate. That distinction is easy to blur if you only publish the strongest theorem. Metaphors can survive the death of a certificate. They just have to move to the correct shelf.

So May stands. Small. Dated. Ledgered. Not a Collatz proof. A closed trail with visible witnesses, visible code, and a visible refusal to confuse remainder with arrival.

When I say remainder, I mean the adaptive road and the citation road and the short-note road. When I say arrival, I mean every positive integer reaches one. Arrival has not happened here. Anyone who quotes this entry as evidence that Going Home solved Collatz is misreading on purpose or by carelessness. I am writing the denial into the body so the denial is harder to skip.

The last discipline is kindness without flattery. Kindness says the hope was understandable. Flattery says the hope was almost a proof. Understandable is true. Almost a proof is false. The obstruction margin is not a near miss. It is a uniform rise above log of three over two on the bad family. Near-miss language is how dead plans keep collecting rent.

I will end the ledger voice here and leave the field stamp as the seal. If later literature shows this exact linear obstruction already in print, then this entry becomes an exposition stamp and a chronology stamp, not a priority stamp. That outcome is acceptable. Being second and honest beats being first in your own imagination.

I am writing for two audiences at once, and I want that explicit. One is technical: people who can check cancellation on L-bad streaks and who will immediately ask about Terras, Lagarias, Tao, and whether Omega is just packaging. For them, the payload is the obstruction statement, the witness family, the grid autopsy, and the narrow claim about the linear look-ahead ansatz.

The other audience is human: people following Going Home as a public climb. For them, the payload is that a real research life includes dated deaths. Not every chapter is a breakthrough. Some chapters are the moment you stop lying to yourself with a search that cannot succeed. Both audiences deserve the same factual spine. I will not give the human audience a victory story and the technical audience a funeral. Same funeral. Same remainders.

If the dual audience creates tension, the tie goes to checkability. A pretty paragraph that cannot be checked is warehouse behavior in essay clothing. A plain paragraph that points at a family of counterexamples to descent is notebook behavior. I am trying to stay on the notebook side even when the sentences get long.

Long sentences are allowed. Softened claims are not. Softened claim: maybe a different linear form with more terms would work. Hard answer from July: once the reward is any function of a fixed window, long enough ugly streaks still cancel it. Softened claim: maybe Mersenne odds are negligible curiosities. Hard answer: they are an infinite explicit family in the same two-adic neighborhood that parallel residue obstructions also identify as hard. Softened claim: maybe adaptive k is basically solved because Tao proved almost-all. Hard answer: almost-all is not all, and Collatz is a statement about all.

I keep Hsieh 2026 in the body because parallel work is where ego goes to be supervised. If two independent lanes find that certificates die near minus one, the geography lesson is bigger than either lane. Geography is not ownership. Our ownership, such as it is, is the look-ahead window obstruction we actually proved.

And I keep the phrase small result in the status line because scale discipline is part of mathematical ethics. The conjecture has eaten careers and decades. A clean negative theorem about one ansatz class is worth writing. It is not worth a trumpet. Trumpets are how warehouse days return after you thought you left them.

If this entry does its job, a reader can leave with a short accurate summary: fixed linear look-ahead Lyapunov energies fail on infinitely many odd n; Mersenne and L-bad families witness the failure; adaptive depth remains open as a direction and not as a solution; Collatz remains open as a conjecture. Anything louder than that summary is not this entry speaking.