The Tostines Paradox: A Fixed-Point Resolution of Apparent Circular Causation

The Tostines slogan — *"Vende mais porque é fresquinho, ou é fresquinho porque vende mais?"* ("Does it sell more because it's fresher, or is it fresher because it sells more?") — is routinely cited in logic and philosophy pedagogy as a textbook case of vicious circularity. This paper argues that the classification is mistaken. The slogan is not a logical circle but a compressed description of a stable feedback system with two coupled rates: a restocking rate \(r\) and a staling rate \(\Gamma\). Read as a description of a physical process rather than as a propositional argument, the apparent interchangeability of cause and effect dissolves into a unique fixed point, \(P^* = r/(r+\Gamma)\), the equilibrium freshness fraction of the shelf population. The "two arrows" of the slogan are the two rates of a two-state system; the "circle" is the loop \(r \to P \to r\); and the fixed point is the system's operating point. The paper makes three contributions beyond the basic resolution. First, it gives a **formal criterion** distinguishing vicious from benign circularity: a circular structure \(F(x)=0\) is vicious if it has no solution, or only unstable solutions, and benign if it has at least one asymptotically stable solution. The Tostines slogan is benign by this criterion in both its linear and its feedback reading, and the pedagogical classification is shown to be a category error — a benign circle misread as a vicious one. Second, it **closes the feedback loop explicitly**: when restocking responds to freshness, \(r = r_0 + \alpha P\), the linear relaxation becomes a Riccati equation. It is proved that the physical system has exactly one fixed point in \([0,1]\) for all \(\alpha \ge 0\), that this fixed point is globally asymptotically stable with eigenvalue \(-\sqrt{(\alpha - r_0 - \Gamma)^2 + 4\alpha r_0}\), and that **no bifurcation occurs** — the second root of the quadratic lies outside the state space. The self-regulation is structural: the feedback enters through the restocking term, whose marginal benefit is suppressed at high freshness. Two physically motivated variants that *do* bifurcate (staling feedback and threshold demand) are identified and explicitly placed outside the slogan's minimal reading. Third, it locates the resolution within a broader class of **stochastic substrate models**: the two-rate system is the \(n=2\) case of the absorber race studied in Part III of the SFIT series, its stationary distribution is the \(n=2\) Luce form, and its feedback loop is a solvable toy model of the rate back-reaction that Part III flags as an open problem. The paper is explicitly classical and makes no quantum claim; speculative extensions to superposition, entanglement, or Hawking radiation are excluded. The resolution is dynamical, not logical: the circle is real, stable, and non-vicious. The closing argument is that the slogan should be reclassified in logic pedagogy as a textbook example of apparent circularity resolving into a stable equilibrium — a distinction worth making explicit, since conflating the two trains students to see circles where there are loops, and loops where there are grounds. Includes a bifurcation diagram (Figure 1) of the Riccati fixed points and a new formal definition of vicious versus benign circularity.