Appendix · The Proofs

The Proven Results

The three theorems the book leans on, stated and proved

Most of the book is plain reasoning that carries itself. A few claims are marked Proven — genuine theorems the argument rests real weight on. This appendix is where they are proved, so the badge points at a derivation rather than an assertion. Each result below is linked from the chapter that uses it.

One multiplication distributes over every addition

The addition dial sweeps a whole family of additions with a single knob $p$ :

S_{p}(x, y) = (x^p + y^p)^(1/p)

The claim is that ordinary multiplication distributes over every setting of the dial at once — not merely over ordinary $+$ . The proof is a single line: pull the common factor through the reshaping and back.

a*S_{p}(x, y) = (a^p*(x^p + y^p))^(1/p) = ((a*x)^p + (a*y)^p)^(1/p) = S_{p}(a*x, a*y)

So for every $p$ , the triple $(ℝ₊, S_p, \times)$ is a field, isomorphic to $(ℝ₊, +, \times)$ through the reshaping $x \mapsto xᵖ$ . The resistor rule ( $p = -1$ ) is not a special case that happens to cooperate; it is one rung of an infinite family, and the one multiplication cannot tell the rungs apart.¹

What this backs This is the result carried by the Proven badge in The Addition Dial — "multiplication passes cleanly through all of them."

The sign of the knob chooses where "nothing" sits

Every addition needs an identity — the value you can combine with anything and change nothing. For $S_p$ , the identity $e$ is the value that vanishes under the reshaping, because $S_p(x, e) = (xᵖ + eᵖ)^(1/p)$ equals $x$ exactly when $eᵖ = 0$ . Solving $eᵖ = 0$ puts the identity at one of the two poles, and the sign of $p$ decides which:

p > 0 ⟹ additive identity = 0 p < 0 ⟹ additive identity = ω

Turn the knob from positive to negative and the additive zero jumps from $0$ to $ω$ . These are not two arbitrary values: they are the reciprocal pair the whole program is built on, welded by $1/0 = ω$ . The addition's choice of zero is governed by the same $0 \leftrightarrow ω$ involution as the multiplication — the sign of the addition-parameter is that reciprocal, read off the additive side.

What this backs This is the result carried by the Proven badge in The Addition Dial — "the sign of the knob decides where nothing sits."

You cannot dial a new field into being

Re-chart the numbers so the addition becomes ordinary $+$ , and ask which re-charted multiplications distribute over it. Fix one input $a$ and let $m_a(x)$ be "multiply by $a$ ." Distributivity says $m_a(x + y) = m_a(x) + m_a(y)$ — the map is additive. A additive map on the line is linear (barring the non-measurable monsters that need the axiom of choice to build), so $m_a(x) = c(a)\cdotx$ , and chasing the constant back forces the multiplication to be ordinary $\times$ .²

Rigidity. If the addition is any reversible re-charting of

+

, the only re-charted multiplication that distributes over it is ordinary multiplication. Every field you reach this way is

ℝ

in disguise — there was never a new field to find.

There is exactly one way out, and it is not in the interior of the dial. The tropical addition $min$ (or $max$ ) is idempotent: $x ⊞ x = x$ . No reversible re-charting of $+$ can be idempotent — a cancellative operation with $x ⊞ x = x$ forces $x$ to be the identity for every $x$ , which is impossible. So tropical is not a rung; it is the limit at the edge, the point where the addition stops being invertible. There the rigidity theorem no longer applies and a genuinely new object appears — but with no inverses it is a semiring, not a field.³ Keep reversibility and you are trapped at $ℝ$ ; give it up and you escape only as far as a semiring.

What this backs These two results carry the Proven badges in Where the Choices Show — the rigidity theorem and the tropical edge.

The $S_p$ family is the classical power mean (Kolmogorov–Nagumo mean); its homogeneity is elementary. What is claimed here is the reading — one multiplication indifferent to the whole family — not the formula. ↩
The step "additive ⇒ linear" is the Cauchy functional equation: any solution that is measurable, or bounded on a set of positive measure, or monotone on an interval, is $x \mapsto cx$ . See J. Aczél, Lectures on Functional Equations and Their Applications (Academic Press, 1966), ch. 2. ↩
For the semiring structure at the tropical limit see D. Maclagan & B. Sturmfels, Introduction to Tropical Geometry (AMS, 2015), ch. 1. ↩

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