A limit without its order is not a quantity – giving the singular limit an address.
Paper 113 wrote that a module is not a wall but a limit, and Paper 03 called the limit of vanishing kinematic viscosity a singular limit, adding a boundary that does not contract to that frame. This paper gives that word singular an address: two limits that do not commute. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. That limits commute with limits and colimits with colimits, that filtered colimits commute with finite limits, the derived functor of an inverse limit, the Curie-Weiss model, and the fact that phase transitions belong to the thermodynamic limit are all standard.
First, the smallest example. The quantity n divided by n plus m tends to zero if m is sent to infinity first and then n, and to one if n is sent first and then m (computed here). Nothing subtle is happening; the point is that the order is part of the statement.
Second, the smallest categorical example. Take a two-by-two family of one-element sets whose first row is a and b and whose second row is b and a. Taking unions along the columns first makes both rows the two-element set, and intersecting the rows leaves both elements. Intersecting the rows first, however, makes both columns empty, and taking unions leaves them empty (checked here). A union is a colimit and an intersection is a limit, so the two do not commute: two two-element sets already suffice for a counterexample. Collect locally and then restrict, and restrict and then collect, are different operations.
Third, if the family is directed they do commute. Taking the increasing chains of the sets from zero to k and from three to three plus k, taking unions first and then intersecting, and intersecting first and then taking unions, both give three, four and five (checked here). The separator is therefore not whether one has a limit or a colimit but whether the index is directed, that is, filtered. Filtered colimits commute with finite limits and others need not – the counterexample above uses nothing but the failure of directedness.
Fourth, and this is the core. Treat the Curie-Weiss magnet exactly at a finite number of particles. The Hamiltonian consists of a term proportional to the square of the total spin and a field term, and the computation is a binomial sum with log-sum-exp, so no configuration is approximated. Setting the product of inverse temperature and coupling to two, that is, below the transition, the magnetisation at zero field is exactly zero whether the number of particles is ten, a hundred, a thousand or ten thousand, because the symmetry under reversing all spins is unbroken. Yet fixing the field at a positive value and raising the number of particles drives the value up towards the positive solution of the mean-field equation, 0.957504, reaching 0.957581 at ten thousand particles and a field of one thousandth (computed here). Sending the particle number to infinity first and then switching the field off gives 0.957504; switching the field off first and then raising the particle number gives zero. A finite system has no spontaneous magnetisation, and spontaneous magnetisation appears only when the limit is taken first. A phase transition happens only in the thermodynamic limit, as Yang and Lee put it, just is the statement that these two limits fail to commute. The language of physics and the language of categories are saying one and the same thing.
Fifth, the two sides are not mirror images. Taking the tower of cyclic groups modulo the powers of a prime, the orders grow without bound and the inverse limit is the ring of p-adic integers, of the cardinality of the continuum, while the integers are countable. The map from the integers into the p-adic integers is therefore injective and not surjective. Applying an inverse limit to a short exact sequence loses exactness at the right-hand end, and the defect is the derived functor. In the category of abelian groups, directed colimits are exact and inverse limits are not: limits and colimits are not mirror images, which is another face of Paper 153’s finding that adjunction is not symmetric between left and right.
The census comes out as follows. Limits always commute with limits. Colimits always commute with colimits. Filtered colimits commute with finite limits, given that the index is directed. Colimits and limits in general do not commute. And sending the particle number to infinity does not commute with switching the field off. Applied to the corpus: Paper 03 called the limit of vanishing viscosity a singular limit and added a boundary that does not contract to Paper 113’s frame; in the vocabulary here that means that this limit does not commute with the operation being performed, so any claim about it must say whether that limit or the long-time limit was taken first. Paper 113 wrote that a module is not a wall but a limit, and what this paper adds is: which limit, and in which order relative to what else.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none). What is added: giving Paper 03’s singular limit a categorical address, setting the cases of commuting and not commuting side by side with the smallest examples and their numbers, and writing that the separator is directedness. What remains: writing down one limit does not yet produce a quantity. A number is fixed only once one also writes what was taken and in what order. A finite magnet has no spontaneous magnetisation, and switching the field off after passing to the infinite system produces one – what appeared was not a magnetisation but an order.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). A Limit Without Its Order Is Not a Quantity [Paper 154]. Zenodo.
DOI: 10.5281/zenodo.22084647
ORCID: 0009-0006-8587-3450