The Existence of an Integrating Factor Does Not Give You a Temperature — Caratheodory’s axiomatisation in three steps.
Caratheodory (1909) reduced the second law of thermodynamics to a single physical sentence: in every neighbourhood of any state there exist states that cannot be reached from it adiabatically. From that, the relation between the small quantity of heat and entropy follows, namely that the inexact differential of heat equals the absolute temperature times the exact differential of entropy. This paper splits that derivation into three steps and calls the roll on what each step gives and what it does not. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Caratheodory’s axiomatisation, the Frobenius integrability condition, the non-integrability of the contact form, and the non-uniqueness of integrating factors are all standard.
The three steps are these. Step one, from inaccessibility to integrability (Caratheodory’s theorem). Step two, from integrability to the existence of an integrating factor (the Frobenius theorem). Step three, from the composite-system requirement to the uniqueness of a universal temperature. And the third step is often omitted.
First, and this is the core. The existence of an integrating factor does not give you a temperature. Suppose the small quantity of heat can be written as some factor times the exact differential of some function. Then for any monotonic function one cares to choose, the pair recomposed through it satisfies exactly the same equation, because the derivative introduced by the composition cancels between the factor and the differential (verified here symbolically). So there are infinitely many such pairs. What Frobenius delivered is only that such a pair exists; which one deserves to be called the temperature is still entirely undecided. Extracting a universal temperature requires a new physical setting: bringing two systems into thermal contact. Demanding that the combined system also possess a common integrating factor forces the factor to split into a universal function of the empirical temperature alone times a system-dependent factor. That universal function is the absolute temperature. This is a third step, and it does not follow from the first two.
And the corpus contains the same shape. Paper 113 said the same of stability arguments: the argument delivers existence, and measurement delivers the value. The addresses differ. Paper 113 treated existence arguments for bound states; this paper treats the integrability of differential forms. The field and the tools are different. But the shape is the same. An existence theorem gives existence and nothing more.
Second, something more surprising. On a two-variable state space an integrating factor exists for purely mathematical reasons. Every Pfaffian form in the plane admits a local integrating factor, by the existence theorem for first-order ordinary differential equations, as verified here by worked example. But the state space of a simple compressible system has two variables: the small quantity of heat is the change in internal energy plus pressure times the change in volume, and the state variables are just internal energy and volume. Hence, for a simple system, the equation stating that heat equals temperature times the exact differential of entropy requires no physical axiom at all; it holds on mathematical grounds. The content of the second law appears only when the state variables number three or more, that is, for composite systems. Only there does the existence of a common integrating factor stop being automatic. This is a shape the corpus has met before: content appears only above a certain dimension (Papers 114, 105). Here the threshold is three.
Third, in three variables it really does fail. Taking the so-called contact form, the Frobenius condition is not satisfied and no integrating factor exists (the wedge product is computed explicitly here). At the same time, along the paths on which that form vanishes, any two points of three-dimensional space can be joined, since the contact distribution is bracket-generating, by the Chow-Rashevskii theorem. Inaccessibility fails and integrability fails together. These are precisely the two that Caratheodory’s theorem ties. If one does not hold, neither does the other.
Fourth, the notation is announcing a category. The small quantity of heat is not an exact differential while the small quantity of entropy is. Both are called differentials and they belong to different categories: the first depends on the path, the second does not. The symbol delta exists to announce that difference. The integrating factor is the device that converts one into the other. And which root the equals sign of that equation carries depends on where one stands. Inside Caratheodory’s route, temperature and entropy are defined by the equation, the axiom having guaranteed that such a factorisation exists and the equation naming it; it is a definition. If instead temperature and entropy are fixed beforehand by other means, an ideal-gas thermometer or the statistical count of states, the same equation becomes a claim to be verified. One equation, two roots, depending on the road taken to it. This is Paper 16’s point, instantiated in thermodynamics.
Finally, one correction. Caratheodory’s formulation is often introduced as having a single physical assumption. But calling the roll gives two: inaccessibility, and that a composite system possesses a common integrating factor. The second is frequently narrated as though it belonged to the mathematics. This is not a reproach. Caratheodory himself carries out the composite-system argument. What omits it is the later exposition.
IMPORTANT (scope): a division into steps of established standard material (Layer 2). Caratheodory’s axiomatisation (1909), the Frobenius integrability condition, the non-integrability of the contact form, and the non-uniqueness of integrating factors are all standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that Caratheodory’s derivation is wrong (it is not) / that the second law is unnecessary for simple systems (the law holds; the point is that its equation needs no axiom to derive there). What is added: splitting the derivation into three steps, writing what each step does and does not deliver, stating that the content of the second law appears from three variables upward, and making explicit that the physical requirements number two. What remains: an existence theorem gives existence and nothing more. If a value is wanted, one more piece of physics must be laid down.
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). The Existence of an Integrating Factor Does Not Give You a Temperature [Paper 126]. Zenodo.
DOI: 10.5281/zenodo.22079437
ORCID: 0009-0006-8587-3450