In this paper we explore a simple but uncomfortable question: could the universe fail to “fit together” if its fundamental constants had the wrong values? We usually treat numbers like the strength of electromagnetism and the cosmological constant as independent inputs. They are measured and then inserted into our equations. But there is another possibility. Maybe those constants are not freely choosable at all. Maybe they are jointly constrained by a deeper geometric requirement — a consistency rule that says only certain combinations of numbers produce a universe that closes without contradiction.
The argument begins at the smallest scale. I model space as emerging from triangular transport loops in a simplicial structure — essentially, space built from tiny relational pieces that must glue together coherently. If you demand that these triangles remain stable under small perturbations and that they glue consistently without hidden extra structure, something striking happens: the minimal symmetry capable of carrying the required phase information is a single, compact, one-parameter phase symmetry. In more familiar language, you are forced into a structure mathematically equivalent to the symmetry that underlies electromagnetism. In this framework, a small “phase twist” per fundamental cell becomes the natural unit of coherence.
Then we zoom out to the largest possible scale. In a universe that asymptotically approaches a dark-energy-dominated state, there is a natural closed surface: the de Sitter event horizon. Its size is determined entirely by the cosmological constant. If you imagine tiling that surface in Planck-sized patches, you obtain an enormous but definite number of cells. A deep result from geometry tells us that the total phase twist accumulated over any closed surface must be an integer — a winding number. That leads to a simple but powerful constraint: the number of Planck-scale cells multiplied by the phase twist per cell must equal a whole number. In other words, the size of the universe and the strength of electromagnetism cannot vary independently if the geometry is to close properly.
There is a subtle point here. The electromagnetic coupling changes slightly with energy scale, while a topological winding number cannot. The resolution is that the closure condition must be anchored to a fixed physical reference scale. Once that is done, the familiar running of the coupling is not a problem — it simply reshuffles how we express the same invariant quantity.
The most dramatic part of the story comes when this geometric closure condition is combined with two independent discrete mechanisms developed elsewhere in the VERSF program. One provides a discrete expression for the electromagnetic coupling as a function of an integer constraint count. The other links the cosmological constant to a coherence scale that depends exponentially on that same integer. When the pieces are put together, the resulting winding number becomes extraordinarily sensitive to that discrete count. Changing the integer by just one unit shifts the prediction by fifty to a hundred orders of magnitude. Only one value — seven — places the winding number in the observed cosmological regime.
The claim is not that the constants are magically derived from nothing. It is subtler than that. The claim is that if space emerges from a minimal coherence structure with a single phase degree of freedom, and if the universe asymptotically approaches de Sitter geometry, then geometric admissibility links the strength of electromagnetism, the size of the universe, and the underlying discrete constraint count. In this picture, those numbers are not independent accidents. They are mutually restricted by the requirement that the universe closes consistently at both the smallest and the largest scales.
This new paper builds directly on A Geometric Closure Condition Linking Fundamental Constants, but it strengthens and clarifies the structure in several important ways. The earlier work established the core idea: that the universe must satisfy a single geometric admissibility condition linking the fine-structure constant and the cosmological constant through a global winding number. That paper derived the closure equation and showed how it numerically aligns with the observed cosmological hierarchy. The new paper goes deeper into the structural foundations. It shows why a minimal coherence model forces a U(1) phase symmetry rather than assuming it, clarifies how Chern–Weil integrality enters formally, tightens the renormalization-group consistency argument, and — most significantly — demonstrates that when closure is combined with independent discrete mechanisms from the broader VERSF programme, the constraint count is not arbitrary. Only K = 7 produces a winding number in the observed regime, while nearby integers fail catastrophically. In short, the earlier paper established the existence of a closure condition; the new one shows how that condition emerges from deeper coherence principles and how it rigidly selects the discrete structure underlying the constants.