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  (1/5 Changing the Geometry of Arithmetic) What is a number? We usually learn to see numbers as points on a line. It is an extraordinarily powerful representation: order, distance, addition and subtraction become natural. But the number line is a representation of numbers, not necessarily the only geometry through which we can understand them. Suppose instead that we look at a number through its irreducible multiplicative structure. Each irreducible base becomes an independent direction. A n...

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  (2/5 Changing the Geometry of Arithmetic) If irreducible bases define the directions of a multiplicative geometry, those directions need a scale. There is a natural one: λₚ = log p For n = ∏ₚ p^νₚ(n), the weighted coordinates recover the logarithmic magnitude: log n = Σₚ νₚ(n) log p Multiplicative composition has become additive both in the coordinates and in their natural scale. Now something familiar appears. Weight each state by n⁻ˢ. Independence of the prime directions allows the g...

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  (3/5 Changing the Geometry of Arithmetic) A geometry becomes more interesting when we stop asking only where its states are and start asking what can move between them. In multiplicative coordinates, multiplication already acts as translation: ν(mn) = ν(m) + ν(n) So instead of treating arithmetic states as static points, we can ask whether multiplicative structure can be understood as transport through irreducible directions. This introduces a distinction that does not appear in the number-...

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  (4/5 Changing the Geometry of Arithmetic) The geometry we have built is multiplicative. But many of the hardest questions about primes are additive. Consider two constraints: p + q = N and q − p = h. Neither is native to multiplicative coordinates. So rather than trying to turn addition into multiplication, keep both structures and let them interact. An additive constraint selects a family — a fibre — of states inside the multiplicative geometry. Now the question changes. We are no lo...

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  (5/5 Changing the Geometry of Arithmetic) Suppose an arithmetic configuration is absent. Traditionally, that may look like a failure of construction: we searched for something and did not find it. But in a transport picture, persistent absence is much more demanding. If the geometry keeps providing admissible transport while the arithmetic never realizes the corresponding configuration, the discrepancy cannot simply disappear. It must be carried somewhere. So permanent absence requires a p...

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  (1/5 Testing the Geometry) From Euler to symmetry Euler's product was our first checkpoint. Independent prime directions, equipped with logarithmic weights, naturally reconstruct ζ(s) = ∏ₚ (1 − p⁻ˢ)⁻¹. But reproducing Euler is not a serious test of the geometry. That structure is already built into unique factorization. The harder question begins when zeta develops a symmetry that is not visible in the Euler product alone. After completion, its functional equation relates s to 1−s...

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  (2/5 Testing the Geometry) From logarithmic weight to Hamiltonian The logarithmic weights log p entered naturally because they turn multiplicative composition into additive magnitude. But once we have states, transport and a logarithmic scale, we can ask whether that scale can become more than a weight. Can it be promoted to an operator acting on the multiplicative space? This is where a Hamiltonian becomes natural. Not as a physical analogy imposed on arithmetic, but as an operator associ...

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  (3/5 Testing the Geometry) Goldbach: width as a resource Now impose an additive constraint on the multiplicative geometry: p + q = 2N. For fixed N, this selects a fibre of candidate configurations. What matters is that the fibre has width. As N grows, the additive condition leaves an expanding family of configurations through which multiplicative transport can be tested. Suppose the required prime configurations were permanently absent beyond some scale. Then every available channel in a...

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  (4/5 Testing the Geometry) Twin primes: remove the width The Goldbach fibre had a hidden resource: width. For p + q = 2N, the available additive range grows with the scale. Now replace it with q − p = 2. The situation changes abruptly. The separation remains fixed. There is no expanding additive region through which transport can spread. Any mechanism that fundamentally relied on increasing width should therefore fail here. And that failure is informative. A common geometry does not...

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  (5/5 Testing the Geometry) Mersenne: remove abundance We have been progressively removing resources. First, an expanding additive fibre. Then, a fixed gap with no growing width. Now consider M_q = 2^q − 1. Here the arithmetic is more rigid still. A prime divisor of a Mersenne number cannot have arbitrary form. Its possible location is constrained by the exponent q and by congruence conditions. The admissible arithmetic states are sparse. There may be no wide fibre across which to distr...