Uncompetitive NMDA receptor antagonist · Off patent · ≈ $0.06 per 10 mg tablet · Evidence: thin, mixed, underpowered

What this document is

In 2015 I asked people who use mobility aids what would make their lives better. Some of them talked about pain, and about the bargain it forced on them every day: stoned past the point of finishing a sentence, or lucid and in agony.

That bargain is why I’m writing this. I wanted to know whether anything gets you out of it.

The candidate I keep coming back to is memantine, an Alzheimer’s drug. It plugs the NMDA receptor, which is the same ion channel that ketamine, dextromethorphan and PCP plug, except that it does it so gently that people take it every day for years and never get high. It is also prescribed at a far smaller dose than ketamine is when ketamine is used for pain: 20 mg a day by mouth, against 80 mg or more per infusion into a vein.⁴ It has been off patent for years, and a US pharmacy pays about six cents for a tablet.³

My guess is that memantine could stand in for opioids in the kinds of chronic pain that come from central sensitization (fibromyalgia, complex regional pain syndrome, migraine), and maybe in the pain that opioids themselves cause. Nobody knows yet whether that guess is right, and as far as I can tell the reason nobody knows is money, not science.

Memantine is not an established treatment for chronic pain. The only meta-analysis pooled eleven small trials and found no significant effect,¹ and the only Phase III trial a company ever paid for failed.² So this is not a story about a miracle drug that somebody buried. It is a story about a trial that would be cheap to run and that nobody can make money from running.

None of this is medical advice, and it is definitely not a protocol. It’s an argument with the references attached, written so that you can hand it to a doctor or a funder and ask whether it’s worth a look.

01 · Opioids are the wrong tool for most chronic pain

Opioids are very good at acute pain. For chronic pain, the best long trial we have says they do no better than a plan that starts with acetaminophen.

That trial is called SPACE. It took 240 veterans with chronic back pain, or hip or knee osteoarthritis, put half on opioids and half on non-opioid medication, and followed them for a year.⁵ How well people could function came out the same in both groups. Pain intensity was actually better without opioids (3.5 against 4.0 on the Brief Pain Inventory), and the opioid group had twice as many medication-related symptoms.

I bring this up because it changes what memantine has to beat. It isn’t up against a drug that works well for chronic pain, it’s up against one that did no better than the alternatives over a year and that people overdose on.

There is a mechanical reason opioids let people down here. Taken for months, they make the nervous system more sensitive to pain, not less. The dose that worked in March has stopped working by September (that’s tolerance), and the pain between doses ends up worse than the pain you started with (that’s opioid-induced hyperalgesia).

Both of those run through the NMDA receptor,⁶ which is the whole reason I think memantine deserves a test against opioids: it blocks the receptor that makes opioids stop working.

02 · Chronic pain is a glutamate problem

Chronic pain has a habit of carrying on after the injury has healed. What has happened is that the spinal cord and brain have turned up their own response to pain signals, and that turning-up depends on the NMDA receptor.

The NMDA receptor is a coincidence detector. While a neuron is at rest, the channel has a magnesium ion sitting in it like a cork. It only opens when glutamate is bound and the neuron is already depolarized, in other words when a signal lands on top of other signals. Then calcium floods in and the synapse gets stronger.⁹ This is how you form memories.

Your spinal cord does the same thing with pain. If pain fibers keep firing for long enough, the neurons in the dorsal horn stay depolarized long enough to pop the magnesium out. Calcium comes in, the synapse strengthens, and from then on the same input produces a bigger output. That is central sensitization: the gain on the pain system has been turned up, and it stays up after the original injury is gone.

I wrote about the neighbours of this idea in Biome of Overstimulation: Sensory Sensitivity as a Unifying Mechanism: excess free glutamate, sensory signals that never habituate, and the conditions that travel with them. Chronic pain is one of those conditions, and the hangover comparison from that post works here too.

An opioid turns down the signal coming into this system. It does nothing to the gain, and over months it actually turns the gain up. An NMDA antagonist goes for the gain.

03 · What memantine actually does

Memantine sits inside the open NMDA channel, in the spot where the magnesium sits, and blocks it. Plenty of drugs do that. Three things make this one different.

• It only blocks channels that are open. No glutamate, no block. Pharmacologists call this uncompetitive antagonism: the more pathologically active the receptor, the more of it gets blocked.⁹
• It leaves quickly. Its off-rate is fast enough that it doesn’t pile up in channels during brief, ordinary synaptic signaling.⁹
• It mostly blocks extrasynaptic receptors. At therapeutic concentrations it blocks extrasynaptic NMDA currents about twice as potently as synaptic ones.¹⁰

Put together, memantine damps down NMDA activity that is sustained and excessive, and mostly leaves ordinary signaling alone. A sensitized dorsal horn is sustained, excessive NMDA activity, which is the whole pharmacological case for trying it in chronic pain.

It also has a strange biography. Eli Lilly made it in 1963 hoping for a diabetes drug, and it didn’t lower blood sugar. Merz, in Germany, picked it up, noticed it did something in Parkinson’s disease, and ended up launching it for dementia in 1989, which happens to be the year somebody finally worked out that it acts on the NMDA receptor.¹² The EU approved it for Alzheimer’s in 2002 and the US in 2003.

So this drug has already been repurposed twice, and both times before anyone knew how it worked.

04 · Comparison with the dissociatives

PCP, ketamine and dextromethorphan all block the same channel as memantine, and every one of them has been tried for pain. Here they are side by side.

Memantine and ketamine are more alike than usually stated

You will often read that memantine is a “low-affinity” blocker and ketamine a “high-affinity” one. That turns out to be mostly wrong: their IC50, kinetics and voltage dependence at the NMDA receptor differ by less than a factor of two.¹⁷ And yet one of them is a dementia drug and the other is an anesthetic.

There are a few candidate explanations, and more than one could be true at once.

• Which receptors. Memantine favors extrasynaptic receptors,¹⁰ and it stabilizes a calcium-dependent desensitized state of the channel, which ketamine does not.¹⁸ It preferentially silences receptors that have been open too long.
• Magnesium. With physiological magnesium present, ketamine still blocks NMDA receptors at rest. Memantine does not. Ketamine’s block of resting receptors dephosphorylates eEF2 and releases a burst of BDNF synthesis; memantine does neither.¹⁹ This is the best current explanation for why ketamine is a rapid antidepressant and memantine is not.
• Exposure. Memantine is dosed to sit at 0.5–1 µM for weeks.¹¹ Ketamine is infused to a peak and is gone in hours. The receptor is the same but the exposure over time is very different. This is my own reasoning from the pharmacokinetics. I don’t have a citation for it.

Memantine is a dissociative at high doses

Take enough memantine and it behaves like the rest of the family. A French analysis of online self-reports found recreational users taking 156 mg on average, about eight times the maximum daily dose, and describing a dissociation that crept in over hours and then lasted, on average, 47 hours.¹⁶

A high that takes all afternoon to arrive and then stays for two days is not what most people are shopping for, which is probably why hardly anyone uses memantine recreationally. Compare OxyContin, which was sold with a slow-release coating you could defeat by crushing the tablet. Memantine is slow because of its half-life, so there is nothing to crush.

What the comparison says about pain

Of the four, ketamine has the best evidence for pain. But it means infusions and monitoring, and outside CRPS nobody has shown that the benefit lasts.⁴

Dextromethorphan and memantine have been compared head to head exactly once, in diabetic neuropathy. Pain fell 33% on dextromethorphan, 17% on memantine and 16% on the lorazepam control. None of those differences was significant in the main trial, but dextromethorphan, at a median 400 mg a day (several times what the cough-syrup label allows), showed a dose-response in the people who responded. Memantine could not be told apart from the control.¹⁴

The doses are nowhere near each other, and this matters more than anything else in this comparison. As a painkiller, ketamine is given at a minimum of 80 mg per infusion, straight into a vein over two hours or more, and the guidelines say that higher doses given for longer work better.⁴ For CRPS that means infusions every day for ten days. Memantine is prescribed at 20 mg a day, by mouth.

The two drugs block the channel about equally well.¹⁷ So memantine has been tried for pain at a small fraction of the dose at which ketamine works. When someone says memantine is a weaker painkiller than ketamine, they are comparing a small dose of one with a large dose of the other.

In the short trials we have, memantine is the weakest painkiller of the three, at a much smaller dose. Its advantage is that it’s the only one you could realistically take every day for years.

05 · The evidence

What we have clinically is a dozen or so small trials that don’t agree with each other. The table has the main ones, negative results included.

The pattern

When I look at that table I see a pattern. Memantine fails in peripheral neuropathic pain that has been there a long time: old shingles, old amputations, diabetic nerves. It works, in small trials, where the problem is central and either recent or still going on: fibromyalgia, CRPS, fresh amputations, surgery that hasn’t happened yet, migraine.

The fibromyalgia trial was built on spectroscopy showing raised glutamate in the insula, hippocampus and posterior cingulate of people with fibromyalgia.²⁵ And a 2024 network meta-analysis of 38 migraine trials put memantine among the more effective preventives, with the lowest dropout rate of any drug compared, although the interval on that last estimate includes no difference.²⁶

To be clear, this is me reading a table, not a finding. Nobody has designed a trial to test it.

Checking the pattern with numbers

So I tried to test it myself, as far as the published numbers allow. I pooled the trials that had usable numbers and split them by pain type. The split follows the 2025 Cochrane review of NMDA antagonists, which sorts these trials into peripheral neuropathic pain, CRPS and fibromyalgia;²⁷ I put CRPS and fibromyalgia together as the central group.

Negative means less pain on memantine. The difference between the two groups is −1.32 (95% CI −1.96 to −0.69, p = 0.003).

Lump all seven comparisons together and you get nothing significant, plus a lot of disagreement between trials (I² = 81%), which is exactly what the published meta-analysis found.¹ Split them by pain type and the disagreement disappears: I² is 0% in each group.

⚠ How much this is worth

Not much yet. The split came from looking at these same trials, so this is not an independent test.

The central group is two trials and 83 patients.²⁰²⁸ Without the fibromyalgia trial the difference is no longer significant (p = 0.07). With the stricter interval recommended for this few trials, the central estimate runs from −4.41 to +1.69.

Six trials had no numbers I could use. Three small ones reported no effect. The largest, 421 people with diabetic neuropathy, reported a benefit at 40 mg, and that goes against the pattern.³¹ The phase III that followed it failed.²

The two central trials also differ from the others in ways that have nothing to do with pain type. The fibromyalgia trial ran six months, and no peripheral trial in this analysis ran longer than nine weeks. The peripheral comparisons that carry the most weight used an active placebo, and the central trials did not. In the CRPS trial, everyone was also on morphine.

The number

In fibromyalgia the number needed to treat was 6.2.²⁰ For comparison, the pooled NNTs for the drugs we actually prescribe for neuropathic pain are 3.6 for tricyclics, 6.4 for SNRIs, 7.7 for pregabalin and 4.3 for strong opioids.³² Taken at face value, memantine in fibromyalgia is about as good as the SNRIs and better than pregabalin.

⚠ An honest caveat

That confidence interval runs from 3 to 47. It comes from one trial of 63 people in one Spanish city, and the comparison above is across different conditions. A number needed to treat of 47 is a drug that doesn’t work.

The pooled analysis is not significant.¹ The only large, well-funded trial ever run failed.² If a company were selling a patented drug on this evidence, I’d be writing a post about how they were overselling it.

I’m not saying memantine works. I’m saying we don’t know, it would be cheap to find out, and nobody has in more than twenty years.

On replacing opioids specifically

The claim in my own title has even less under it. There’s the mouse data in section 01, and one randomized trial in which memantine made morphine work in CRPS when morphine alone didn’t.²¹ There is also a single Medicare claims study suggesting memantine may be opioid-sparing in older adults with dementia and chronic pain, whose own authors say they can’t tell which came first.³³

That is everything I found. To me it’s enough to justify a trial, and nowhere near enough to justify a prescription.

06 · Why nobody ran the trial

Memantine did once have a commercial pain program. It ended in 2003, and what the company did afterwards is a tidy lesson in incentives.

Back in 2001, analysts expected Forest Laboratories to file memantine for neuropathic pain in 2003.³⁴ Pain was where the money was: Neurobiological Technologies, the small company that had done the early work, stood to collect a 13% royalty on pain sales against 1% on Alzheimer’s.³⁵ An earlier trial at 40 mg had even come out positive.²

Then, in May 2003, the confirmatory Phase III in diabetic neuropathy missed its primary endpoint.² Forest said it would run another Phase II and maybe file by 2006.³⁶ Five months later memantine was approved for Alzheimer’s, and no pain indication ever followed.

So: one failed trial, in one condition, in exactly the kind of pain where memantine looks worst, and that was more or less the end of industry interest. I don’t even think it was irrational. If you have a freshly approved blockbuster and a patent clock running, why would you pay for a second expensive attempt at figuring out which pain patients respond?

It made perfect business sense, and the question stayed unanswered.

What the company did before the patent expired

With generic memantine due in 2015, Forest (which Actavis bought in the middle of all this) announced in 2014 that it would pull the twice-daily tablet and leave only a newer once-daily capsule, Namenda XR. The idea was to move people with Alzheimer’s disease onto the new version before the cheap one existed, so that pharmacists would have nothing to substitute. The industry calls this a hard switch.

New York’s attorney general sued. In May 2015 the Second Circuit upheld an injunction, the first time an appeals court had ruled that a product hop can violate the Sherman Act.³⁷ A class action by direct purchasers, which also alleged a pay-for-delay deal with a generic manufacturer, settled for $750 million on the day the trial was supposed to open. The company admitted nothing.³⁸

So in the last years of memantine’s exclusivity, the pain program sat untouched while the company put its energy into moving dementia patients from two tablets to one capsule to keep the generics out.

I don’t think Forest was unusually wicked. Every step was what a profit-maximizing company would do. It’s just that none of those steps involved finding out whether the drug helps people in pain.

The problem of new uses

Once a drug goes generic, the incentive drops to zero. If one company pays for a trial in a new indication, every other manufacturer gets to sell the same tablets on the strength of it. Legal scholars call this the problem of new uses, and it applies to roughly 2,000 off-patent drugs.³⁹ Health economists have been saying for years that generic repurposing needs public money, because no private party is going to supply it.⁴⁰

Memantine’s price tells the same story. A US pharmacy buys a 10 mg tablet for 6.4 cents.³ The “average retail price” of 28 of them is $145, and a discount card gets that down to $4.68.⁴¹ So the same box costs $1.80, $4.68 or $145 depending on who’s paying and how, and not one of those numbers tells you what the drug is worth.

What happened to the related drugs

Now look at what happened to its cousins, ketamine and dextromethorphan.

Ketamine is generic. So its S-enantiomer was put in a nasal spray, wrapped in new patents, and launched as Spravato at $590 to $885 per session.¹³ ICER, the US cost-effectiveness body, priced a year of it at about $32,400, found too little evidence that it beat plain generic ketamine, and pointed out that it cost roughly ten times the intravenous generic.⁴²

Dextromethorphan is cough syrup. Add a little quinidine to slow its metabolism and you get Nuedexta, at around $1,235 a month.¹⁵ In 2019 its maker agreed to pay more than $108 million over kickbacks to doctors and misleading marketing in nursing homes, for use on dementia patients it was never approved for.⁴³

Both times, an old NMDA antagonist looked promising, and both times nobody tested the old generic drug. They built a version that could be patented.

In fairness, esketamine did get real trials out of this, and that is the best argument for patents there is: exclusivity pays for evidence. The flip side is that a drug with no exclusivity gets no evidence. Memantine is the example. Nobody owns it, so nobody paid for the trials, so a pain doctor wondering whether to try it has almost nothing to go on.

07 · CBD: the opposite case

Cannabidiol (CBD) is memantine’s mirror image: a non-opioid that is sold for pain, with an enormous market and almost no evidence behind it.

If incentives decide what gets tested and sold, which is what I argued in section 06, then there ought to be drugs that sell well without ever being tested. CBD is one. Anyone can sell it, nobody has to show that it works, and people in pain buy it.

What it does

CBD is not a dissociative and it doesn’t block the NMDA channel. What it does is hit a lot of targets at once: it facilitates 5-HT1A and glycine receptors and impairs GPR55 and several TRP channels, at concentrations anywhere from nanomolar to micromolar.⁴⁴ With that many targets you can propose a mechanism for almost any condition, and you’ll have a hard time testing any single one.

One of those targets does lead back to the NMDA receptor. In mice, CBD behaves like a σ1 receptor antagonist: it weakens σ1’s binding to the NR1 subunit of the NMDA receptor, enhances morphine antinociception and reduces NMDA-induced seizures.⁴⁴ So CBD may be turning down NMDA activity by a different route than memantine does. So far that has only been shown in mice.

What the trials say

A 2024 review rounded up every randomized trial that used pharmaceutical-grade CBD and measured pain. There were sixteen, across twelve pain conditions, with doses from 6 to 1,600 mg and durations from a single dose to twelve weeks. Fifteen found no benefit over placebo.⁴⁵ The authors’ conclusion was that CBD for pain is expensive, ineffective and possibly harmful.

Replacing opioids goes the same way. An updated systematic review of cannabinoids in general found that the animal and observational studies suggest an opioid-sparing effect and the better randomized trials don’t.⁵⁰

CBD does work for some things. It reduces seizures in two severe childhood epilepsies, and for those, proper trials were run.

What the market did

The untested version is sold to people in pain, and the tested version is sold to insurers.

Of 89 over-the-counter CBD topicals tested in 2022, only 24% contained what the label said, and THC turned up in 35% of 105 products, including some sold as THC-free.⁴⁶ An earlier study of extracts sold online got 31%.⁵¹ And when the FDA ran its own trial of a consumer-range dose in healthy adults, 5.6% had liver enzymes above three times the upper limit of normal within four weeks, against nobody on placebo.⁴⁷

The tested version is called Epidiolex. It’s the same molecule, purified, put through trials and patented, and it launched at about $32,500 a year.⁴⁸ It earned $1,059 million in 2025, and its owner has settled with generic challengers for entry in the very late 2030s.⁴⁹

The regulation went the way it went for opioids (section 08). For seven years after the 2018 Farm Bill there was no federal product standard at all. Then in November 2025 Congress redefined hemp inside a spending bill, with a cap of 0.4 mg total THC per container due to take effect on November 12, 2026. The law is aimed at intoxicating hemp THC, but it sweeps up most full-spectrum CBD products along the way. The industry says about 95% of hemp cannabinoid products will go, out of a market it values at $28 billion.⁵² First no standard, then a ban, and nothing resembling regulation in between.

Why this belongs in a post about memantine

The two drugs are the same problem seen from opposite sides.

Memantine has a mechanism and a handful of positive trials, and nobody markets it for pain. CBD is marketed for pain everywhere you look, and fifteen of its sixteen trials were negative. Either way, companies sold what it paid to sell.

And the customers are often the same people. If your opioid prescription has been cut and you’ve been told there is nothing else, of course you’ll try the product that promises natural pain relief. One of the review’s authors made the point that chronic pain makes people vulnerable to promises like these.⁴⁵

08 · Two policy failures

People with chronic pain in the US got screwed twice, in opposite directions, by the same institutions.

First, the overselling

Purdue Pharma has pleaded guilty to federal charges over OxyContin twice, once in 2007 and again in 2020.⁵³ McKinsey earned $93 million advising Purdue over roughly fifteen years,⁵⁴ which included a plan to “turbocharge” OxyContin sales while overdose deaths were climbing. In 2024 it paid $650 million to end the federal investigation, in a civil settlement that admits no liability.⁵³

In November 2025 a bankruptcy judge approved a $7.4 billion plan under which the Sackler family pays about $6.5 billion over fifteen years.⁵⁵ The individual victims share something over $850 million of that.⁵⁶

Then, the crackdown

When prescribing finally was restricted, it was patients who paid for it, far more than the companies. Doctors under regulatory pressure cut doses or stopped prescribing altogether.

A study of 113,618 patients on stable long-term opioids found that tapering was followed by significantly more overdoses and mental health crises.⁵⁷ That paper was corrected in 2022 for coding errors. The associations got smaller but held up, which is why I’m not quoting the original headline figure.

Prescriptions fell, and deaths kept rising for years anyway, because by then most of them were from illicit fentanyl. The CDC’s provisional count for 2025 is 69,973 overdose deaths, 44,564 of them involving opioids, down from 81,313 and 55,296 the year before.⁵⁸ That makes three years of decline, and it is still about 120 opioid deaths a day.

So if you have chronic pain in America, you have been told three things in a row. First: this drug is safe, take more. Then: this drug is dangerous, you can’t have it. And finally: there’s nothing else.

That last one is false. There are other options, some of them with decent evidence, and then there’s memantine, which has never been properly tested.

Germany as a comparison

I’m writing this from Germany, where it went differently. Germany has ranked second in the world for opioid consumption per person and, by every account I can find, has no opioid epidemic.⁵⁹ A systematic review of German prescribing found use going up and no sign of one.⁶⁰

The usual explanation is the way the system is built. Opioids were never a first-line treatment here, prescribers face extra screening and paperwork, and addiction treatment is covered.⁶¹ One German primary-care paper lists what was different in the US: higher doses per patient, a heavy cost burden on individuals, lighter regulation, and a health system oriented toward profit.⁶²

(Memantine itself, by the way, was on the market in Germany fourteen years before the FDA approved it.¹²)

I should admit that people use Germany to argue opposite things. The left reads it as proof that regulation and universal coverage work. Libertarians read it as proof that prescribing volume was never the cause, and that the real mistake was the American crackdown on prescribers. The German data fits both. The tapering study is evidence for the second reading, which is awkward for an argument like mine that blames the market, so you should know about both.

What both sides agree on is that the same drugs killed far fewer people in a different health system. So the crisis came out of how the drugs were sold and regulated, and not only out of the drugs.

09 · What a real trial would look like

The trial that would settle this is a completely ordinary one, and the money to pay for it already exists.

• Who. Fibromyalgia and CRPS first, because that’s where the signal is. A second arm in people tapering long-term opioids, because that’s where the need is.
• Dose. 20 mg and 40 mg against placebo. The only positive industry trial used 40.² Even 40 mg a day is far below the doses at which ketamine is given for pain (section 04).
• How long. Six months at least. Steady state alone takes weeks,¹¹ and the one positive long trial ran six months.²⁰
• Control. An active placebo. Sugar pills against a psychoactive drug are not blind.¹⁴
• Outcomes. Function and opioid dose, not only a pain score. SPACE used function as its main outcome.⁵
• Sponsor. Public money, since no company will pay.

About that money. The Purdue plan describes its payments as compensating victims and abating the opioid crisis, and it comes on top of a $26 billion settlement with the distributors and Johnson & Johnson in 2022.⁵⁵ Finding out whether a six-cent non-opioid works seems to me like a fair use of abatement money, and a memantine trial would cost a tiny fraction of either settlement.

There is even a possible sponsor. Under the bankruptcy plan, Purdue’s assets pass to a new public benefit company, Knoa Pharma, chartered to make overdose-reversal and addiction-treatment drugs.⁵⁵ A public benefit drug company is exactly the kind of organization that could test a drug nobody can own. Whether it will is a political decision.

10 · Taking this to your own doctor

If you live with chronic pain, here are the questions I would take to a doctor.

1. Is my pain plausibly centrally sensitized (fibromyalgia, CRPS, migraine, widespread pain out of proportion to any tissue damage) rather than an old peripheral nerve injury?
2. Have the drugs with real evidence been tried and failed: a tricyclic, an SNRI, a gabapentinoid?³²
3. What is my kidney function? Memantine leaves through the kidneys and the dose comes down in renal impairment.¹¹
4. Am I taking anything else that acts on NMDA receptors, such as amantadine, dextromethorphan or ketamine?¹¹
5. If I am on long-term opioids, could this be tried alongside a slow, supervised taper rather than instead of one?
6. This is off-label. Who pays, and who monitors?

⚠ Do not stop opioids abruptly

Tapering long-term opioids is itself associated with overdose and mental health crisis.⁵⁷ Nothing in this document is a reason to stop or cut a prescribed opioid by yourself.

⚠ Do not source it yourself

Memantine circulates online as a self-experimentation drug. In the French analysis, people self-medicating at an average of 23 mg a day for 15 weeks reported adverse effects in 77% of cases: dissociation, clouded consciousness, anxiety, insomnia.¹⁶ The half-life means a mistake takes days to wear off. One accidental overdose took close to 100 hours to resolve.⁶³

No doses are given here as a protocol

Deliberately. The numbers in this document are what trials used. Titration, kidney function and interactions are a prescriber’s job. This document exists to start a conversation, not to replace one.

I can’t tell you memantine works for pain, because it has never been properly tested. What I can tell you is that it costs six cents a tablet, that the last serious trial was twenty-three years ago, and that since then there has been an opioid epidemic and tens of billions of dollars in settlements. Somebody should run the trial.

Keyword summary

• central sensitization
• NMDA receptor, open-channel block
• opioid tolerance, opioid-induced hyperalgesia
• memantine, ketamine, dextromethorphan, PCP
• fibromyalgia, CRPS, migraine
• product hopping, hard switch
• the problem of new uses
• abatement
• cannabidiol (CBD), σ1 receptor

References

Domestic shorthair, neutered male, 9 years · 9 kg → 7 kg · Status: stable, at home

What this document is

Cake had megacolon that kept coming back. The standard fix is surgery, and by the time it was needed he was too unwell — and too heavy — to come through it safely. Enemas, a full manual clear-out, laxatives and a wet-food diet all failed to hold. At that point the realistic option given to me was putting him to sleep.

We found another option – my favorite vet suggested we try neostigmine, a drug that makes a weak colon push harder. He is still here – eating, grooming, using his litter tray, and happier than he was before this nightmarish adventure began. He just needs an injection every few weeks, and probably always will.

Neostigmine is not an established treatment for megacolon in cats. There is no trial behind it, no licensed indication, nothing in the standard textbooks. It is a folklore treatment — known to some vets, used quietly, never formally written up. That is exactly why it may not have been offered to you, and it is the first thing section 01 explains.

This is not a claim that it works for every cat, and it is not a protocol to follow at home. It is a written account of one cat’s treatment, detailed enough that you can take it to your own vet and ask whether it is worth considering.

01 · A folklore treatment, not an established one

When Cake’s vet suggested the neostigmine she used to treat baby horses, she was clear about what she was proposing: this is not a scientifically established treatment for megacolon in cats. It is not licensed for it. There is no clinical trial. It does not appear in the textbook chapters or the treatment guidelines on feline constipation.¹ A vet who says they have never heard of using it this way is not being careless — there is very little for them to have heard.

What does exist is elsewhere. Neostigmine is well established in horses for large colon impaction, where it is the most commonly chosen drug of its kind.²³ It is used in human medicine for a condition in which the colon dilates and stops moving.⁴ The principle — restore the push to a bowel that has stopped pushing — is old and well understood. It is the specific application to cats that nobody has written down.

Mathematicians have a name for a result that specialists know, pass around and rely on, but which nobody ever formally published: a folklore theorem. It is true, it is used, and you will not find it by looking it up. Neostigmine for feline megacolon sits in roughly that position. Some vets know of it. Some have used it and seen it work. But because it was never written up, knowledge of it spreads by word of mouth between practitioners rather than through the literature — which means whether it gets offered to your cat depends largely on who you happen to be standing in front of.

That is the gap this document is trying to close, and it cuts both ways. A folklore treatment is not a secret good one that the profession has ignored. It is a treatment nobody has tested properly, which is a real reason for caution and a legitimate reason for a vet to decline. Your vet may have grounds you are not aware of, specific to your cat. The point is only that the conversation should be possible.

⚠ An honest caveat

Sometimes euthanasia genuinely is the right decision. A cat in unrelieved pain, with a colon that has stopped responding, or with other serious illness alongside, may be at the end of what medicine can reasonably offer. This document is not an argument that nobody should ever make that choice.

It is an argument that you should get to make the choice knowing what the alternatives were.

02 · What megacolon actually is

The colon is the last stretch of bowel. It draws water out of waste and pushes what is left toward the exit in waves of muscular contraction.

In megacolon the colon stretches wide and loses much of its ability to push.¹ Waste sits there. The longer it sits, the more water is drawn out of it, and the harder it becomes. Harder waste is harder to move, so it sits longer still.

Almost everything that follows in this document is an attempt to break that loop at one point or another. It is worth holding onto, because it explains why single treatments keep failing: an approach that softens the waste but does nothing for the pushing, or improves the pushing while the waste is already concrete, addresses half a problem.

In most cats no cause is ever identified. This is called idiopathic megacolon.¹ When a cat can no longer empty the bowel at all, the word used is obstipation.

⚠ The single most important number in this document

In a 2022 study, cats whose signs had lasted less than six months responded to medical management in 66.7% of cases. Cats whose signs had lasted six months or more responded in 5.6% — the rest needed surgery.⁸

If your cat is early in this, that is the window. It is the strongest argument in the literature for pushing hard on diagnosis and treatment now rather than managing conservatively and seeing how it goes.

03 · The part that was missed: pain

Cake’s X-rays showed spondylosis — small bony spurs growing along the vertebrae of his lower spine. It is common in older cats and often causes no trouble at all, which is exactly why it is easy to dismiss.¹¹

But consider what a cat has to do to pass a stool: crouch, curl the lower back, hold that position, and strain. If that hurts, the cat puts it off. And putting it off is precisely what dries the waste out and hardens it. Pain and constipation feed each other.

This is not merely a plausible story. In a radiographic study of 1,365 cats, those with abnormal lumbosacral vertebrae were significantly more likely to have large bowel problems (p=0.0057, OR 1.73), and for acquired abnormalities specifically — the category spondylosis falls into — the odds ratio was 4.11 (p<0.0001).¹²

How we knew it was real

In my entire acquaintance with Cake, he had never washed the back half of his body. His front half was immaculate. This was put down to his size — he was 11 kg when I adopted him, and was around 9 kg by the time the symptoms of megacolon began. Even a 9 kg cat may simply not be able to bend that far.

He started grooming his back end again shortly after beginning pain relief every 6 weeks, before he had lost any meaningful weight. That timing is the whole argument. If bulk had been the obstacle, the behavior would have returned gradually as the weight came off. It did not. It returned when the pain was treated.

Worth checking on your own cat

A scruffy or matted coat over the lower back, dandruff you only see behind the ribs, or a dirty bottom on a cat who used to be fastidious. Grooming is one of four validated behavioral domains used to assess musculoskeletal pain in cats — alongside mobility, activity and temperament — and all four improve measurably after analgesia.¹³ It is easy for everyone in the room to attribute to age or weight. It is worth saying out loud at the appointment.

04 · What was tried, in order

Stage 1 — failed · A superficial enema

It cleared only the waste nearest the exit. Some stool came out, which looked like progress. The bulk of the blockage sat further forward, in the widened part of the colon, and fluid introduced from the rectum never reached it.

This is a common and instructive failure. If your cat has an enema and still seems wrong afterwards, or the belly still feels full, say so — partial output is easily mistaken for resolution.

Stage 2 — worked, did not hold · Full manual deobstipation under anesthetic

The vet emptied the entire colon by hand under general anesthetic. At the same time she confirmed that nothing was physically blocking the way — no narrowed pelvis, no growth, nothing swallowed.

That check turns out to be the single most important event in this whole account, for reasons that become clear in section 05.

Stage 3 — failed · Daily lactulose and exclusively wet food

Lactulose is a sweet liquid laxative that draws water into the bowel and keeps stool soft. Cake had 5 ml daily in his food, and ate wet food only. Dietary modification and hyperosmotic laxatives are the standard first-line components of multimodal management.¹

The blockage rebuilt anyway. This is the pivotal observation: his colon had been completely emptied, his stool was softened, his water intake was good, and waste still accumulated. The failure was therefore not one of hydration and not a one-off mechanical event. His colon was not pushing.

Stage 4 — unavailable · Subtotal colectomy

Where megacolon keeps recurring, the definitive treatment is an operation removing most of the colon. Cats generally do well afterwards — 72.2% favorable outcomes in the 2022 cohort,⁸ and a high rate of owner satisfaction across 18 hospitals in a 2021 series¹⁵ — and if it is genuinely available to your cat it is usually the better long-term answer.¹¹⁶

It was not available to Cake. By the time the question was live he was both markedly obese at 9 kg and unwell from the obstipation itself, and the combination made a long abdominal surgery an unacceptable risk. This is worth being precise about, because “not a surgical candidate” covers two very different situations: a cat who is too well to need it yet, and a cat who has become too unwell to survive it. Cake was the second.

There is a bitter logic in that which is worth naming. The disease that makes surgery necessary is the same disease that makes surgery dangerous, and the longer it runs the more true that gets.⁸ Waiting is not neutral.

On not having the surgery

For most of this document the non-surgical route reads as a consolation prize, and I want to correct that, because it isn’t only one.

Subtotal colectomy is major abdominal surgery, and the recovery is not nothing. Soft stools and sometimes frank diarrhea are expected for the first few months afterwards, stools become firmer with time but rarely return fully to normal, and some cats have occasional soft-stool leakage.¹⁷ Outcomes are good and most cats end up with an excellent quality of life¹⁵¹⁷ — I am not arguing against the operation. But a cat who never has it also never has the anesthetic, never has the laparotomy, never has the months of loose stool, and never carries the small risk of the anastomosis failing.

Cake has none of that. He has a dose every few weeks, a spoonful of syrup in his dinner, and an otherwise ordinary life. Measured against a cat who sailed through colectomy, that is second best. Measured against what was actually on the table for him — which was repeated anesthetics or euthanasia — it is a good outcome by any standard, and it is allowed to be one.

05 · Neostigmine, and why the timing is everything

Nerves in the bowel wall release a chemical messenger that tells the muscle to contract. Neostigmine slows the breakdown of that messenger, so the signal persists and the contraction is stronger.¹⁴ In plain terms: it helps a weak colon push. It is given by injection, because it is not absorbed if swallowed.¹⁸

⚠ The critical point

A drug that makes the bowel contract harder is dangerous if the bowel is genuinely blocked. Mechanical obstruction of the gastrointestinal tract is a listed contraindication to neostigmine.¹⁴ Contracting against a hard mass that will not move does not expel it. It raises pressure inside a colon that is already stretched and thin-walled. That is painful, and at worst the bowel can be damaged or perforated. The same objection is made explicitly in the equine literature: a prokinetic given in the presence of a physical obstruction pushes against an impassable point and increases pain.¹⁹

This is why the order of Cake’s treatment matters more than any individual element of it. The same drug is either useful or harmful depending entirely on the state of the colon when it is given.

Neostigmine was used only:

• after the colon had been completely emptied by hand;
• after mechanical obstruction had been excluded;
• and thereafter early, while the contents were still soft enough to move.

Why the human comparison only partly holds

Neostigmine is used in people for acute colonic pseudo-obstruction, which is defined as marked colonic dilatation in the absence of mechanical obstruction.⁴ But in that condition the distended colon is full of gas, which compresses and vents the moment motility returns. A megacolonic cat’s colon contains a solid mass whose deformability decreases by the day. Same drug, same organ, different mechanics — and the difference is the entire reason timing governs the outcome.

06 · Cake’s regime

Neostigmine, intermittently

• We started with a small test dose, under observation, to check he did not react badly to the drug itself.
• Then, proceeded to an induction course: twice daily for three to four days, started after the manual clear-out.
• Ongoing Maintenance: I feel his abdoment around twice a week to check if waste has built up. If it has built up, he gets a dose. If not, no dose. His current dosing rate is one dose every few weeks (recorded how much and when administered), two at the absolute maximum.
• As his weight fell from 9 kg to 7 kg, the dose was reduced — proportionally more than the weight loss alone would require, and it still works.

⚠ The stop rule

Now that he is in ongoing treatment: If two doses do not produce a bowel movement, he does not get a third. He goes in for imaging and manual clearance instead. So far this has not occured. However, failure to respond to dosing is the signal that the contents have already hardened past the point where more contraction helps — and at that point further dosing is pushing against something that will not move. This rule is not a formality. It is what keeps the treatment on the safe side of the contraindication.¹⁴¹⁹

No doses are given in this document

Deliberately. Neostigmine is supplied at several different concentrations, and a volume that is correct for one cat and one product can be a significant overdose with another. Dosing is calculated per kilogram by a vet who knows which vial is in their hand. This document exists to start a conversation, not to replace one.

Lactulose, daily

5 ml once daily, mixed into wet food. This is not an optional extra alongside the neostigmine — the two are co-essential, and this is the mechanistic heart of why the combination works where either alone did not:

• Lactulose keeps the stool soft enough to be moved.
• Neostigmine supplies the force to move it.

Soft stool with no propulsion simply sits there — that is what the months on lactulose alone demonstrated. Strong propulsion against a hardened mass only builds pressure — that is the danger in section 05. Neither element is sufficient. The aim is soft but formed stool; watery diarrhea means the dose needs revisiting with the vet rather than adjusting yourself.

Solensia, monthly

An injection for pain every 6 weeks. It is a felinized monoclonal antibody that binds nerve growth factor, a substance that sensitizes nerves to pain.²² It is licensed for osteoarthritis pain in cats;²³ using it for spinal pain is a slightly different application.

Its role here is not only comfort. If defecation stops hurting, the cat stops deferring it — which breaks the pain–avoidance loop from section 03 at the other end from the laxative and the prokinetic.

07 · Feeling the abdomen

Cake’s re-dosing is triggered by palpation — feeling the abdomen for waste building up, and treating before a hard mass has formed rather than after. That is what keeps the drug on the right side of the line described in section 05.

⚠ Read this before you try it

Ask your vet to show you on your own cat first, with their hands over yours. A description on a page is not a substitute for being taught, and the two structures you are distinguishing between feel more alike than this illustration suggests.

Never press hard, and never squeeze. A distended bladder can be ruptured by firm pressure. If your cat tenses, cries, or resists, stop.

The colon sits high and toward the cat’s left, running backwards through the rear half of the abdomen. The bladder sits low, central and further back. Both can feel firm. Telling them apart is the reason to be taught this in person rather than from a diagram.

How it is done

• Cat standing, or relaxed on your lap. Never when agitated.
• Fingers flat, using the pads rather than the tips. One hand each side, or one hand spanning the belly.
• Start behind the last rib, in the rear half of the abdomen, and work gently backwards until you are directly under the spine.
• Press slowly and lightly — enough to feel through the body wall, no more. The pressure is closer to resting your hand than to gripping.

What you are feeling for

• Normal: soft and yielding. Nothing firm that holds its shape.
• Filling up: a firm, sausage-shaped tube high on the cat’s left, running front to back. It may feel lumpy, like a row of hard beads through a sleeve.
• Not what you want: a hard, round, distinct ball sitting low and well back. That is more likely the bladder. Leave it alone.

⚠ A cat straining may not be constipated

A cat straining in the litter tray may be unable to urinate rather than defecate. From the outside the two look almost identical.

In a male cat, a urinary blockage is life-threatening within hours, not days. If you feel a tense, firm, round structure low and far back in a cat who is straining, do not press on it — phone the vet immediately and say you cannot tell which it is.

08 · Weight

Cake went from 9 kg to 7 kg. For an overweight cat this helps in several directions at once: less load on a painful spine, an easier crouch to pass stool, and better odds if surgery is ever reconsidered.

⚠ Do not do this quickly

Overweight cats must lose weight slowly and under veterinary supervision. Obesity and a period of reduced food intake are the classic setup for hepatic lipidosis, a serious and potentially fatal liver condition in cats.²⁵ No crash diets. If your cat stops eating, phone your vet. Weight loss that happens without you trying is also a reason to be seen, not reassured.

09 · Day to day, and what to watch

• Lactulose in wet food, daily. Aiming for soft but formed.
• Wet food only. Water always available.
• A written note of when he goes, what it looks like, and whether he strained. This is the single most useful thing to bring to an appointment.
• Regular abdominal palpation, to catch accumulation early rather than late.

The number that matters most

How often the injection is needed. If the gaps between doses stay roughly constant, the regime is holding. If they start shortening, the colon is likely deteriorating — and that is the moment for a fresh conversation about what comes next, including whether surgery has become possible again at a lower weight.

Writing that down honestly is uncomfortable, and it is the reason this document is a record rather than a recommendation. A treatment that is working is not the same as a cat that is cured. The duration data in section 02 make clear that this disease has a direction of travel.⁸

⚠ Phone the vet now if

• Straining in the tray with nothing coming out
• Nothing passed for more than a couple of days
• Vomiting, especially with a tight or swollen belly
• Not eating, or gone quiet and withdrawn
• Crying or yowling in the tray
• After a neostigmine dose: weakness, collapse, or difficulty breathing

10 · Taking this to your own vet

If your cat is where Cake was, these are the questions that opened up the useful parts of the conversation:

1. Has anything physical been ruled out — a narrowed pelvis, a stricture, a growth?
2. Could pain be part of why my cat avoids the tray? Has the spine, hips and pelvis been assessed, and would a trial of pain relief be reasonable?
3. If the colon has been fully emptied and it still rebuilds on laxatives, does that point to a propulsion problem rather than a hydration one?
4. Is my cat a surgical candidate — and if not, what specifically rules it out, and could that change?
5. If surgery is off the table, are there prokinetic options worth trying, including ones that are not standard?

References

My first play, Impaction, was performed in LA as part of Nothing to See Here, directed by my amazing friend Jacques Manjarrez. The last night was recorded and can be seen here.

Impaction

by Rin Ray

Cast

• Main Character (MC)
• Sick Creature — a human wrapped in blankets to look vaguely unidentifiable, but certainly alive
• Doctor — wearing a rhinoceros mask and a pink lab coat
• Creatures — a crowd in costumes

Props

• Lots of yellow and pink ribbon
• Two chairs
• Two phones
• A banana

Scene One

MC and Sick Creature on stage. A phone near the back of the stage.

MC: (Standing, staring out into space at the mostly lifeless body.)

Sick Creature: (Lying on their side, crying slightly, breathing labored.)

MC: (Backs up slowly and sinks down next to the phone without looking at it, still staring at Sick Creature. Tension slowly builds. Through shaky breaths, holding back tears, dials.) “Hi, does your clinic offer the procedure?”

Voice on the phone: (Disappointed) “Ah, I see. Okay, thank you.”

(MC crosses a name off a list. Approaches the creature and strokes its hair while calling once more. Increasingly agitated, MC throws the phone down.)

Sick Creature: (Flinches and curls in on itself, breathing faster and shallower, building toward panic.)

(MC picks the phone back up and calls the Doctor.)

MC: “Doctor, I can’t reach anyone who does the procedure. Do you know someone?”

(MC gets up to go outside. A flash of different styles and genres of music begins to play.)

(A parade emerges from behind the audience and weaves through the seats: minions, clowns, demons, nuns, a flamingo — all shitfaced drunk. Each holds a pink or yellow ribbon trailing behind them. The music keeps switching rapidly as they float by. Strings and ribbons flow through the space.)

(The characters pull to the side and play rock paper scissors in an uncanny, chittering clown language. The next batch comes out with ribbons in the same way, in front of MC, freestyling mischievously in their mayhem.)

(Suddenly MC is caught in a web of ribbons, dodging side to side, running their hands through them. They “trip” and grab one. The being holding that ribbon — the flamingo — stops, walks up to MC, and hugs them. MC holds on tight, then looks at the ribbon in their hand, then back at the flamingo. MC lets go of the ribbon, and the flamingo immediately turns around and rejoins the parade.)

Scene Two

(The parade mostly clears the stage as MC exits. A small parade keeps passing parallel to the stage, ribbons leading offstage. Upbeat, quickly shifting music continues.)

(Two chairs. The Doctor, in a pink coat, sits on one next to a phone. Hidden behind them: another phone and a banana.)

(MC opens a door, arriving.)

MC: “Help me.”

Doctor: “What? Okay.”

(MC and Doctor both start calling frantically while the parade continues, alternating the same line.)

MC & Doctor: “Hi, do you offer the procedure?”

(More characters join the parade in the background. MC and Doctor keep calling, now in different languages, building to a frantic pace.)

Doctor: “Hi, do you offer the procedure?”

MC: “Hallo, bieten Sie die Operation an?”

(More characters enter.)

Doctor: “Bonjour, proposez-vous l’OPÉRATION ?”

MC: “Hola, ¿ofrecen la OPERACIÓN?”

(The Doctor raises the banana to their other ear, as if on two phones at once.)

Doctor: “Salve, operationem offersne?”

Doctor: “They have always abandoned us. We must do it ourselves.”

(The parade grows louder and closer until it is walking in front of and around them, club music pounding. MC and Doctor are obscured.)

Scene Three

(While covered, the Doctor exits with their chair and phone. Sick Creature enters and lies on the floor, two ribbons already tied around its throat, each held by a costumed person.)

Sound: (The upbeat music continues, but slowed down, as if playing in another room.)

(The costumed people in front of MC part down the middle and move left and right behind them, revealing Sick Creature on the floor. MC sits blindfolded on a chair. Behind MC, the costumed people split the ribbons into two paths — yellow and pink — like a fork in the road.)

Doctor: (Walks out to stand behind MC.) “Here are the options.”

Doctor: “You can end its suffering — kill it.” (Points to the pink ribbon.) “Or you can try the procedure yourself, but act as its tormentor as you try.” (Points to the yellow ribbon.)

MC: “What will they lead to?”

Doctor: “You cannot know the future.”

(The two ribbons around Sick Creature’s neck feed into the two paths behind MC, so that MC sits inside a triangle of ribbon. The people holding the neck ribbons begin to pull.)

Sick Creature: (Fights the ribbons at its neck, clawing wildly and whimpering, in clear agony, growing more intense by the second.)

MC: (Breathing heavily, still blindfolded, gets up and crawls on hands and knees to stroke the creature’s face as it whimpers. The ribbon-holders move closer; the triangle shrinks and forces MC back into the chair.)

Sick Creature: (Screaming and seizing in pain, still clawing at the ribbons.)

(MC turns to the audience.)

MC: “Fuck! What do I do? Fuck!”

Offstage Narrator: “Those who want yellow, raise your hand. Those who want pink, raise your hand.”

(A TV in the background shows yellow on one side, pink on the other. The audience votes.)

In this post, I conjecture that the Frobenius eigenvalues of Artin-Schreier-Witt Curves (\( \mathbb{Z}/p^k \)-covers) are Gauss sums, generalizing an old and classic theorem of Davenport and Hasse (fuck nazis though). I explain and expand (i.e., make the paper type check and fill in details) a gorgeous reproof of this theorem using local systems by Robert Coleman.

This playful note is toward exploring this force, and outlining a conjectural approach for further exploiting it. Enjoy and click. Read about my Gauss Sums :P

Code newton (Newton polygons of Artin–Schreier–Witt curves)

• the euler characteristic \( \chi(SL_2(\mathbb{Z})) = \zeta(-1), \) and
• the tamagawa measure of \( \mu(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}) = \zeta(2)\).

I repeat, these apriori unrelated invariants of our beloved example case \( SL_2(\mathbb{Z}), \) are dual zeta values! Is it a general pattern? Let’s define and explain these results to spiritually prepare ourselves for the general picture!

Euler Characteristics of Arithmetic Groups – star of the show: \( \chi(SL_2(\mathbb{Z})) = \zeta(-1) \)

An arithmetic group is the integer points of an algebraic group (i.e. a group object in abelian varieties). For example, \( SL_2(\mathbb{Z}) \) is the arithmetic group associated to the algebraic group \( \mathbb{SL}_2 \). In general, \( \chi(G) \) for arithmetic groups is related to L-functions and zeta functions.

The euler characteristic is defined for a torsion free group \( G’ \) as follows: \(\chi(G') = \sum_i (-1)^i H^i(G', \mathbb{Z}).\) This is supposed to generalize the notion of euler characteristic for surfaces of polyhedra where \( \chi(P) = V - F + E \), the analogy is to consider \( \dim H^0(P, \mathbb{Z}) = V \), \( \dim H^1(P, \mathbb{Z}) = E \), and \( \dim H^2(P, \mathbb{Z}) = F\).

For example:

• if we consider \( F_n \) the free group of rank \( n \geq 0 \) then \( BF_n \) has one vertex and n \( 1 \)-cells, therefore, \( \chi(F_n) = 1-n \)
• free abelian group of rank \( n \geq 1 \) then \(n\)-torus \( (S^1)^n \) has euler characteristic \( \chi(S^1), …, \chi(S^1)=0.\)

The euler characteristic is defined for any group \( G \), in terms of \( G’ \) a torsion free group, \(\chi(G) = [G: G']^{-1}\chi(G')\)

Note: This doesn’t depend on choice of \( G \), consider a two different torsion free subgroups \( \Gamma’ \) and \( \Gamma’’\), and define \( \Gamma_0 := \Gamma’ \cap \Gamma’’ \) , then

It’s time to stare at the Gauss Bonnet Thm: Let \( Y \) be closed Riemannian manfiold, then there’s a canonical measure \( \mu \) in \( Y \) such that \(\chi(Y) = \mu(Y).\)

There’s a version for orbifolds, \( Y = BG\) and let \( X \) be its universal cover, there’s a canonical measure such that

Further, \( \chi(BG) = \chi(G). \)

Let’s look at \( SL_2(\mathbb{Z}) \). The free group \( F_2 \) is a torsion free group living inside of \( SL_2(\mathbb{Z}) \) via the ping pong lemma. It has index 12, not sure why tbh.

Thus, \(\chi(SL_2(\mathbb{Z})) = [F_2: SL_2(\mathbb{Z})]^{-1} \chi(F_2) = \frac{1}{12} (1-2) = -\frac{1}{12} = \zeta(-1).\)

The Tamagawa Number of Arithmetic Groups – star of the show: \( \mu_\infty(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{Q})) = \zeta(2) \)

Given an arithmetic group \( G \), it comes equipped with a canonical measure on its adelic points \( G(\mathbb{A}) \). All other measures are a constant times this measure. It is defined by considering the measure associated to a differential form.

Given an algebraic group \( G \), pick any \( \omega \) which is a top degree invariant differential form of \( G \) defined over \( \mathbb{Q}. \) Then, define the measure \( \mu \) place-wise (a la the adelic perspective of Tate) for each \( G (K_v) \) by \(\mu_v := |\omega|_v \) on \((\mathbb{Q}, +)\). We then renormalize such that:

• for finite places \(v \), \( \mu_v(\mathbb{Z}_v)=1 \),
• and at infinite places \(\mu_\infty([0,1]) = 1.\)

You might be like – we chose \( \omega \) there, seems noncanonical to me. Well it is noncanonical, if we consider it individually over each place, but all together, no matter which choice of \( \omega \) we take, we will get the same answer because for any number \( f \), the product of its valuation over global fields will always be one: \( \prod |f|_v = 1. \)

Non-archimedian case (family friendly) \( \prod_p \mu_p(SL_2(\mathbb{Z}_p)) = \zeta(2)^{-1} \)

In this section we wish to show that \(\prod_p \mu_p(SL_2(\mathbb{Z}\_p)) = \prod_{p} (1-p^{-2})^{-1} = \zeta(2)^{-1}.\)

We may compute the Tamagawa measure for \( SL_2(\mathbb{R}) \) using Iwasawa decomposition. We get that the Haar measure on \( SL_2(\mathbb{Q}_p) \) is \( |x^{-1}|_pdxdydz, \) where \( |x^{-1}|_p = p^{-v_p(x)} \). A nice reference on this is the notes of Paul Garrett.

Now consider the reduction map:

\(\mu\_p(SL_2(\mathbb{Z}\_p))= \# SL_2(\mathbb{F}\_p)\mu\_p(\text{ker } f)\) and \(\mu_p(\text{ker } f) = \int_{x,y,z} \|x^{-1}\|_p dxdydz = p^{-3}.\)

Lemma: \( \# GL_2(\mathbb{F}_p) = (p^2-1)(p^2-p), \),

Proof: This because when forming a matrix in \( GL_2 \), the first column \( r_1 \) of the matrix can be anything but the 0 vector (giving us \( p^2 - 1\) choices), and second column of the matrix can be anything other than a multiple of the first vector (it can’t be \( r_2 = cr_1 \) ).

Lemma: \( \# SL_2(\mathbb{F}_p) = 1-p^2, \)

Thus, we have proven: \( \mu(SL_2(\mathbb{Z}_p)) = \frac{1}{1-p^{-2}}, \) and we may conclude:

Let’s get Archimedian \( \mu_\infty(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{Q})) = \zeta(2) \) (to be continued…)

The Bernoulli numbers were defined by Faulhaber in terms of the following generating series.

But why? Where did this come from? Well, the mathematicians of the time were contemplating the following sorts of patterns:

The general formula requires \( B_k \) :)

To give you an idea of how hard this is, I don’t even know of a proof of the \( s = 2 \) case that that doesn’t involve guessing the formula then showing inductively (other than the Bernoulli version I’m about to show).

Today I’m giving a modern explanation of Faulhaber’s trick. This is based on notes of John Baez.

Consider \( \mathcal{E} \) to be the space of entire functions on \( \mathbb{C} \). We can consider \( \mathcal{E} \subset \mathbb{C}[[z]] \)

Next, consider the finite difference operator

Discrete Fundamental Theorem of Calculus: If \( \Delta F = f \), then \(\sum_{i = 0}^{n-1} f(i) = F(n) - F(0)\)

For example, here’s a big clue: If \( f(z) = z^s \), then if \( \exists F \) such that \( \Delta F = f \), then \(\sum^{n-1}_{i=0} i^s = F(n) - F(0).\)

Let’s set some groundwork before proceeding, we consider an operator (called the ‘‘annihilation operator” in conformal field theory):

Let us further define our friend the exponential

Lemma: \(e^{ta}f(z) = f(z+t)\)

Proof: I can’t help myself, I just love power series. We prove this by expanding both sides and observing that they are the same. Take \( f(z) := \sum_i c_iz^n) \), then

\(\begin{aligned} e^{ta}f(z) &= f(z) \\& + (t\frac{d}{dz})f(z) \\ & + (t\frac{d}{dz})^nf(z) \\ & + \cdots \end{aligned}\) i.e., \(\begin{aligned} e^{ta}f(z) &= c_0 + c_1z + c_2z^2 + \cdots + c_nz^n + \cdots \\& + c_1t + 2c_2tz + \cdots + tnc_nz^{n-1} + \cdots \\ & + n!\frac{c_nt^n}{n!} + (n+1)!\frac{c_{n+1}t^nz}{n!} + \cdots \\ & + \cdots \end{aligned}\)

Let’s expand the other guy, \(\begin{aligned} f(z+t) &= c_0 + c_1(z+t) + c_2(z+t)^2 + \cdots + c_n(z+t)^n + \cdots, \\ &= (c_0 + c_1t+ c_nt^n + \cdots) + (c_1t + 2c_2t + \cdots + nc_nt^{n-1} + \cdots)z + \cdots \end{aligned}.\)

It’s the same yay. That means we can write \(\Delta = e^{a} - 1.\)

So, given our big clue above, if we can find \( \Delta^{-1} \), we can solve our addition problem.

Let’s play…GUESS THAT INVERSE!!

If we consider \( e^x - 1 \) as a function, we want to find another function \( f(x)g(x) = 1 \), then:

GUESS #1: \( \frac{1}{e^x-1} \) beeping noise nope this is not entire, has a pole when \( x= 0 \).

GUESS #2: \( \frac{x}{e^x-1}) \) we can plug that pole, and get an entire function!! However, we have this extra \( x \) factor floating around (since we want to compose to 1, i.e., the identity operator).

If we were working non-commutatively, then, we’d want to look at either

NON-COMMUTATIVE GUESS #3 : \( x^{-1} \frac{x}{e^x - 1} \) or \( \frac{x}{e^x - 1} x^{-1} \)

Alright, we have some candidates. Let’s get back to operators to make this into real math. We define

Note, \( a a^{-1} f = f \), but the other order is not always true \( a^{-1} a f \neq f \) (because of the icky \( + c \) ).

That means our noncommutative guess \( \frac{x}{e^x - 1} x^{-1} \) is a good bet! In fact, we define the following operator:

and finally, we claim \(\Delta^{-1} = \frac{a}{e^a-1}a^{-1}.\)

Tying it all together to finish the job

We take: \( f(z) = z^s \), we need to calculate \( \Delta^{-1}f := \frac{a}{e^a-1}a^{-1} \).

Note in particular that \( \Delta^{-1}f(0) = 0 \). Using our discrete fundamental theorem of calculus, we see the following:

Since \( \Delta(\Delta^{-1}f) = f\), let \( f = z^s \), then

Tada! We did it!

There is also the following fabulous quick derivation trick of the Bernoulli numbers, which we discussed several years ago in Umbral Calculus Derivation of the Bernoulli Numbers.

A quick way to remember the sum formula is to set \( B^i \) equal to the Bernoulli number \( B_i \). This can be earnestly done if one uses umbral methods.

Graph Laplacians in More Generality

The definition of the graph Laplacian for a graph \( G \) is the following. Let \( v \in V \) be the vertices in the graph, and let \( N(v) \) be the set of vertices neighboring the vertex \( v \).

The directed graph Laplacian rather looks at \( N^+(v) \), the set of vertices the vertex \( v \) maps to with edge length 1:

We can define the graph Laplacian to act on entire functions on \( \mathbb{C} \):

Notice that if we consider the lattice of \( \mathbb{Z} \) in \( \mathbb{R} \) to be directed (going toward positive numbers), we must make a slight type change.

This is quite interesting because the left inverse of a graph Laplacian is the Green’s function associated to the uniform random walk on the graph where you hop to each of your neighbors of distance one away with probability one. The Bernoulli numbers are derived from the right inverse of \( \Delta = -\nabla_{\mathbb{Z}, 0} \). That’s a big difference in description of the inverses on both sides!

Let’s use this language to boogie. Take the lattice \( \Lambda := \mathbb{Z} + \tau\mathbb{Z} \subset \mathbb{C} \) , with the directions pointing toward the positive imaginary and positive real directions. The neighbors of the vertex

If we allow ourselves full license to fuck around, let’s set \( \tau = i \) and play GUESS THAT INVERSE. We consider the function \(1-\frac{1}{2}(e^x + e^{\tau x})\) And look at it’s inverse: \(\begin{aligned} \frac{1}{1-\frac{1}{2}(e^x + e^{\tau x})} &= \frac{2}{e^x + e^{\tau x}-2} \\ &= \frac{2}{\sum_n \frac{(1+\tau^n)x^n}{n!}-2}\end{aligned}\)

This function has a pole of order 3 at 0, and for the simplest case \( \tau = i \), the remaining poles are of order one at roots of \( i \). We can plug those poles, no problem:

I’m not sure where to go from here, I’d love for this to in some way compare to the Kronecker-Eisenstein series (which tantalizingly involves the character \( \phi(z) = e^{\frac{z-\overline{z}}{A}} \)). Here, \( A = \frac{\text{im } \tau}{\pi} \) is the area of the fundamental domain of the lattice \( \Lambda \) divided by \( \pi \). The Kronecker-Eisenstein series is defined as follows, (typically the role of the letter \( b \) is played by \( a \), but we are already using \( a \) to stand for annihilation operator).

where the \( * \) means that the summation excludes \( \lambda = -z \) if \( z \in \Lambda \).

A good reference on these is Elliptic Functions according to Eisenstein and Kronecker : An Update. See in particular the elliptic analogs of Bernoulli numbers discussed on page 8.

This is a nostalgic topic for me which feels like home to step back to. I previously contemplated the analogs of such shifts in 2015 while I was still a robotics inventor sitting in the back of lectures at Berkeley On Detiling Polynomials: A Generalization of the Euler MacLaurin Formula.

Let \(X\) be a curve over a complete local ring, and let us decompose it into two affine pieces \(X = A \cup B.\)

Given a sheaf \(\mathcal{F}\) over \(X\), we have the following Mayer-Vietoris sequence.

In particular, if take the decomposition of \(X\) into the point at infinity and its complement \(X = U \cup {\infty}\), we get the following sequence, let \(T\) be a divisor at \(\infty\).

A global section \(H^0(X, \Omega^1_X)\) can be specified by either a basis in terms of \(\mathcal{F}(U)\), in terms of \(\mathcal{F}^\wedge_\infty.\) This is because we are considering \(H^0(X, \mathcal{F})\) as a subset of \(\mathcal{F}(U) \times \mathcal{F}^\wedge_\infty\).

Lyrics:

I’ll cut you into manageable pieces
I hope you’re not too hard to glue back together
There’s so many ways to form an affine cover
Yet I keep slicing open your stomach
And I pull out your heart
I don’t know why
that’s where I like to start

It’s an easy thing to fall in love with a hand
When its lifeless and the fingers are missing
I can trace the truth
As it sits motionless
beneath my glass, slowly twitching

And when I stitch it back together
It jumps and motions toward the sky
There’s a hanging sequence of all of your pieces
And your other hand is waving hi

I’ll cut you into manageable pieces
I hope you’re not too hard to glue back together
There’s so many ways to form an affine cover
Yet I keep slicing open your stomach
And I pull out your heart
I don’t know why
that’s where I like to start

Those intersections between your skin
Reveal raw flesh that I’m kissing
There’s so many ways to stitch you back
I GIVE YOU MORE EYES MORE EYES MORE EYES MORE EYES
YOUR DICK IS MISSING, i needed it to make more eyes

I love to cut you apart to find the ways you combine
How many times can I do this before you fall apart forever?

If I was braver then, I would have added this drawing to the paper which I wrote at an Arizona Winter School. I still think that paper is good resource for learning quickly about the Fargue Fontaine curve. A Global Crystalline Period Map

All of this is on my mind because I have had the great privledge and joy of returning to this topic after years away.

While we are sharing old things, in 2018 or 2019, Artem, Kolya and I made an outline of an approach to rational Chromatic Vanishing \(H^*(J_h, W(k)) \times \mathbb{Q} \simeq H^*(J_h, W(k)[[u_1, ..., u_{h-1}]][u^{\pm}]) \otimes \mathbb{Q}\) using the two tower isomorphism which reduces the problem to directly calculating the \( GL_h(\mathbb{Q}_p) \) action on constant functions on the Drinfeld projective space Chromatic Vanishing Approach.

At the time, the tools to compute the p-adic cohomology of the Drinfeld projective plane were not available to us so we ulitmately were scooped. I was initially sad about this because I felt then I could not prioritize thinking about these awesome and gorgeous concepts. However, recently I have begun with many others a spin-off project, which I am delighted and healed by. :)

From 5 to 9 August 2024, the Young Topologists Meeting came to the University of Münster. YTM is a conference run by and for early-career topologists: PhD students and postdocs give most of the talks, and two lecture series anchor the week. This year they were Ishan Levy (Copenhagen) on Telescopic computations in algebraic K-theory and Arunima Ray (MPI Bonn) on From knots and links to 4-manifolds.

I was on the organizing committee, together with Konrad Bals, Etienne Batelier, Edith Hübner, Marin Janssen, Alexander Kutzim, Lukas Stöveken, Maximilian Tönies and Thomas Tony. I also drew the logo, and Natalia Pacheco-Tallaj did the graphic design around it.

The logo

The letters Y, T and M are drawn as flames, hot pink with cyan shadows, and the T in the middle is split by a lightning bolt. They sit inside a yellow seal with a scalloped edge, somewhere between a sun and a sticker. Everything has a heavy black outline, so the logo reads both large on the banner and small in the corner of each day of the schedule.

The schedule

Natalia and I made the schedule together. Each day is its own lime-green card with the logo in the corner, speakers in cyan, and titles in black; the coffee breaks, lunches and the poster session (“with finger food!”) get their own bands. Yellow smiley stickers and pink and cyan bursts are scattered across the page, and a last panel says where things are: talks in M1 and M2, coffee breaks and the poster session in the Cluster Lounge, and a silent room in the SRZ. Here is the full schedule, page by page:

The palette is the same everywhere: hot pink, cyan, lemon yellow and lime on teal and lilac.

YTM 2024 was supported by the Cluster of Excellence Mathematics Münster, the K-theory Foundation and the Foundation Compositio Mathematica. Banner: © Natalia Pacheco-Tallaj (graphic design) & Rin Ray (logo). Schedule: © Natalia Pacheco-Tallaj & Rin Ray. Photos of the week, by Arye Deutsch, are on the conference page.