Does the Chowla–Selberg formula break the sum-rule criterion?

Object. A working criterion for which magnitudes are reachable by structural argument: a magnitude is reachable when it decomposes as a sum over all places, and blind otherwise. Two confirmations, one on each side. The canonical height of the generator of 37a1 is computable as a purely archimedean limit, 4^{-n}h(2ⁿP) → 0.051110, and independently from L′(E,1) and the real period, giving 0.0511114 — same magnitude, two routes, because the Néron–Tate height is a sum of local heights over all places. The real period Ω is the negative case: no local decomposition, placeable but never derivable.

Presenting problem (the friction). For elliptic curves with complex multiplication, the period is not opaque. The Chowla–Selberg formula expresses it as a finite product of Gamma values at rational arguments, with exponents determined by the class group of the imaginary quadratic field. That is a decomposition, it is explicit, and it computes the very quantity the criterion names as its blind case.

So either the criterion is false as stated, or "decomposes over places" is narrower than "decomposes," and the distinction between the two decompositions is the whole content of the criterion — in which case it has never been stated precisely enough to be tested.

History. The Gamma function is the archimedean local factor. A product of Gamma values at rational arguments is therefore a product of archimedean quantities indexed by residue classes, not by places: the index set is the class group, or characters of it. Whether that counts as the same kind of object as a sum of local heights is exactly what is unsettled.

Differential. 1. The criterion survives with a sharpening: reachability requires decomposition over places, and Chowla–Selberg decomposes over a class group instead, which is a different index set carrying no local information. Ω remains blind; the formula is an archimedean identity, not a global one. 2. The criterion is too narrow. Decomposition over any arithmetically meaningful index set confers reachability, and the correct statement replaces "places" with something like "a finite set of local data," which would make CM periods reachable and predict that non-CM periods are the real blind case. 3. The criterion is false: it was fitted to one positive and one negative example that happened to lie on the same side of a distinction it does not make, and CM periods show the negative example was misclassified. 4. The criterion is fine but Ω was the wrong witness. The genuinely blind magnitudes are transcendence statements and effective constants, and a period was chosen because it looked hard rather than because it fails the test.

Discriminating tests. State the criterion precisely enough to classify Chowla–Selberg before looking at the answer — that is the actual test, and it is cheap. Then: compute Ω for a CM curve numerically (AGM) and against the Gamma product, confirming they agree to high precision, and check whether the Gamma-product form can be rewritten as a sum of local terms over places or demonstrably cannot. Then repeat with a non-CM curve, where no such formula is expected, and see whether the criterion's prediction differs between the two — if it does not, the criterion is not discriminating anything.

Stakes. This is the newest load-bearing claim available and it currently rests on two data points. If branch 1 holds, the criterion survives with a sharper statement and gains its first real test. If branch 3 holds, the fold that produced it needs redoing, and the blind spots are not one thing after all.

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Update (2026-09-12): the criterion has been amended from another direction, and the amendment may merge two standards.

The reachability criterion has since been restated with a second clause: a magnitude is index-visible either (a) through a place-wise sum rule, as heights are, or (b) through a motivic Galois action, as periods are — the coaction principle constraining which periods can occur, with the small-graphs principle predicting new constants before they are computed. Blind without either.

That partly settles the friction in this case from outside it. Chowla–Selberg expresses a CM period through Gamma values at rational arguments; Gamma is the archimedean Euler factor of zeta, and the constants entering through it are exactly those the motivic route governs. So the branch structure above resolves toward branch 2 — the criterion was too narrow — though not by widening the index set, which is what branch 2 anticipated. It was too narrow by admitting only one mechanism where there are two.

What survives, and it is sharper than the original question. Routes (a) and (b) do not deliver the same thing, and calling both "visible" may be the fold repeating the error it was built to correct.

• Route (a) produces the number. The canonical height of the generator of 37a1 comes out as 0.051110 from an archimedean limit and 0.0511114 from an L-value and a period: the same magnitude, computed, twice.
• Route (b) produces the class. A motivic Galois action says which algebra of constants a period lives in, which constants can appear at which weight, and what cannot occur. It does not produce the value.

The supporting computation makes the difference concrete rather than rhetorical. The multiple-zeta-value weight dimensions follow 1/(1 − x² − x³), giving 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12 (verified; the sequence continues 16, 21, 28, 37, 49, 65, 86, 114 through weight 20), which matches a free algebra on one generator per odd degree ≥ 3 — and the corresponding statement that K-theory of the integers decides which constants may appear, with ζ(m) the regulator of a rank-one group exactly when m is odd and at least 3. That tells you ζ(4) and ζ(6) are not independent. It tells you nothing about what ζ(3) equals.

So the original distinction — shape is reachable and size is not — has not been crossed by the motivic route. It has been reproduced one level up, with the "shape" now being the shape of the space of constants rather than of a geometric object.

Revised differential. 1. The amendment is correct and the two routes deliver the same standard of visibility; the height/period asymmetry above is an artifact of which examples were chosen. 2. The amendment merges two standards. Route (a) is derivation of a value, route (b) is membership in a class, and only (a) is what "visible" meant when the criterion was stated. The blind spot is intact and has been relabelled. 3. Both routes are real but they are visibility of different things — value and class — and the criterion should be stated twice rather than once, with the interesting question being whether any magnitude has both. 4. Route (b) does yield values in the cases that matter, through the coaction determining coefficients, and the shape/size distinction genuinely breaks there.

Discriminating test. Name one magnitude for which the motivic route has produced a number not obtainable by the place-wise route. If one exists, branch 1 or 4. If every instance produces a constraint on which constants appear and no value, branch 2 or 3, and the amended criterion needs splitting.

Note that the amendment's own stated limits point the same way: the rational coefficients cannot be said, and the coefficients are exactly the part that turns a class into a number.

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Third data point (2026-09-12): route (a) retrodicts a literature.

The Mandelbrot set supplies the discriminating case the height/period pair could not, because both of those are global and neither separates the two routes.

The bifurcation measure on parameter space is the Laplacian of the critical height (DeMarco), and the critical height is a sum of local heights over all places, with the archimedean piece being harmonic measure on the boundary. So route (a) applies, in full. What it then predicts is not a number: measure-theoretic facts about the parameter space should be provable and topological ones should not. That is the actual state of the field — equidistribution of post-critically-finite parameters is a theorem, and local connectivity is open.

This is a stronger confirmation than either founding example, and of a different kind. Heights confirmed that route (a) produces a value. This confirms that route (a) predicts which questions are answerable, which is what a criterion about visibility is supposed to do and what route (b) has not yet been shown to do.

It sharpens the case's central question rather than settling it. Route (a) now has a value-confirmation and a literature-shape confirmation. Route (b) has neither: the motivic action constrains which constants appear and has not been shown to predict which questions about them are answerable. Until it does, branches 2 and 3 stand and branch 1 looks weaker than when it was written.

A worked instance of the same object's split is worth recording as calibration. The skeleton of the set — how many components of each period, where they attach, what kind of numbers the centres are — is symbolic dynamics on the binary tree, roots of unity, and algebraic integers, and is entirely reachable. The flesh — area, dimension, local connectivity — is magnitude, and none of it is. The one quantity crossing the line is the one with a place-wise decomposition.

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Likely resolution (2026-09-12): the criterion was named after a symptom.

The difficulty throughout has been that the first mechanism was described by what its outputs look like — a sum over all places — rather than by what it is. Under that description anything that decomposed over any index set looked like a candidate, and class groups, discriminant groups and topological data all presented themselves in turn.

The mechanism has since been named: there is a lattice — a discrete cocompact subgroup — so Poisson summation applies. The rationals form a lattice in the adeles, so reciprocity; the integer lattice sits in Euclidean space, so the sphere-packing bound. Five things recorded separately as bridges are one theorem applied five times: the functional equation of the zeta function, the explicit formula, heights and the product formula, Milgram's Gauss sum, and the linear-programming bound for packings.

With the mechanism named, the test is sharp and this case's question has an answer. A product of Gamma values indexed by class-group characters is not Poisson summation on a lattice. The decomposition is real and it is not the bridge, so the period remains outside — branch 1, the criterion survives with a sharper statement.

One residue worth keeping. The original friction was correct and was pointing at a real defect: the criterion could not classify Chowla–Selberg because it had never stated its own mechanism. A criterion described by its symptoms cannot adjudicate a borderline case, and the borderline case is what forces the mechanism into the open.

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Update (2026-09-13): the criterion is binary and the target is not — a fifth branch, and a counterexample it cannot see.

The criterion sorts magnitudes into reachable and blind. Independent work on what "magnitude" names found the category is not binary but a family parameterised by how many experimental inputs a value costs:

• free — improved only by measuring the thing itself; irreducible input cost;
• derived — improved by measuring something else, with the error amplifying in transit (the amplification factor being the logarithmic distance between the two scales);
• computed — improved by running a longer calculation, with experiment contributing nothing.

The third is a counterexample the criterion cannot classify. A three-dimensional Ising correlation-length exponent is known to about one part in 10⁶, obtained by semidefinite programming that imposes crossing symmetry and unitarity and optimises. It has no decomposition over places, or over any arithmetic index set — so the criterion calls it blind, and it is one of the best-determined numbers in physical science.

So decomposition over places is sufficient for reachability and demonstrably not necessary: convex optimisation under a positivity constraint is a second route, and it is not an arithmetic one.

And there is a whole category the criterion cannot see: bounded. Consistency requirements — anthropic bounds, unitarity bounds like the ~1 TeV cap on a Higgs mass from WW scattering, swampland conjectures — produce an interval with no value. Both examples' intervals contain the measured answer. A bounded magnitude is neither reachable nor blind, and the binary has no slot for it. The general form: a decomposition is needed to compute a value; it is not needed to constrain a range, because positivity yields inequalities and never numbers.

Fifth branch. 5. Reachability is not a property of the magnitude but of the route, and there is more than one route. Place-decomposition is the arithmetic route; convex optimisation under positivity is an analytic one; and a third status — bounded without being valued — exists alongside both. The criterion is then correct about what it classifies and wrong to present itself as exhaustive.

Discriminating test for branch 5, and it is cheap. Take the three best-determined "structural" numbers with no place decomposition — a critical exponent from the bootstrap, a Feigenbaum constant, an optimal sphere-packing density in a dimension where it is known — and ask whether each is (a) blind under the criterion and (b) in fact determined. If all three are both, the criterion is not exhaustive and branch 5 holds. If the determined ones turn out to have a hidden decomposition, branch 2 was right and the index set needed widening rather than the criterion needing a second route.

And a diagnostic that separates the senses mechanically. Collect every independent determination of the quantity and ask whether the spread is exactly zero or a few percent. A count has no spread; a computed magnitude carries error bars on both the calculation and the measurement. Applied to the criterion's own examples, this says whether "reachable" has been meaning derivable or well determined — which the two original data points could not distinguish.