Chapter 15
Conversion, Enzymes, and the Virtual Token
Chapter 14 left a question open and then made it unavoidable. Once phlogiston is decomposed into a spectrum, a computation holds balances in several kinds at once, and the natural question is whether those kinds have a common measure — whether, having so much storage and so much bandwidth and so much authority, one can say how much the computation has.
The answer is that a common measure exists exactly when the ways of converting one kind into another contain no free lunch, and that when it exists it is essentially unique. This chapter states that, because the ecology chapters need it: a scientist eats, and eating converts what was inside another computation into something this one can spend. Part Part IV states it again, at length, and reads the same theorem as a statement about energy. Here it is only a statement about exchange rates.
15.1 Conversion systems
Work over the signature monoid \((\Sigma, \ast, ())\) of Chapter 14, whose elements index the layers of a token stack.
A conversion system over \(\Sigma\) is a triple \(\hypK = (\Tok, E, r)\) where \(\Tok \subseteq \Sigma\) is a finite set of token types; \(E \subseteq \Tok \times \Tok\) is a symmetric set of enzymes, presented with both orientations; and \(r : E \to \Rpos\) is a rate satisfying \[r(a \to b) \cdot r(b \to a) = 1 .\] An enzyme \(a \to b\) is read operationally: one unit of the \(a\)-token is consumed and \(r(a \to b)\) units of the \(b\)-token released. We write \(\Gamma_{\hypK} = (\Tok, E)\) for the underlying graph, which we may take connected without loss of generality.
The paradigm enzyme is a persistent exchange process — a contract sitting on a channel that swaps one kind of token for another and back again. That such a thing is expressible as an ordinary term, rather than as a privileged operation of the runtime, is the whole point of stacks being terms. An enzyme is a computation, and it can be eaten like any other.
\(\hypK\) is monoidal when the rates respect composition of signatures: whenever \(a \ast a'\) and \(b \ast b'\) are tokens and the component enzymes exist, \[r(a \ast a' \to b \ast b') = r(a \to b) \cdot r(a' \to b') .\]
Monoidal compatibility says that the worth of a compound authority is built from the worths of its components in the same way that \(\ast\) builds the authority.
15.2 No arbitrage is the virtual token
Pass to logarithms. The log-rate is the edge \(1\)-cochain \(\ell(a \to b) := \log r(a \to b)\), antisymmetric by reversibility.
A cycle is a closed edge path \(a_0 \to a_1 \to \cdots \to a_k = a_0\); its yield is the product of the rates along it. The system exhibits arbitrage if some cycle has yield \(\neq 1\), and is arbitrage-free if every cycle has yield \(1\).
A valuation, or virtual token, for \(\hypK\) is a function \(\nu : \Tok \to \Rpos\) with \[r(a \to b) = \frac{\nu(a)}{\nu(b)} \qquad \text{for every enzyme } a \to b .\] Writing \(p = \log \nu\), the condition is \(\ell = -\,\delta p\): the log-rate is minus the graph gradient of a potential.
For a connected conversion system \(\hypK\) the following are equivalent:
\(\hypK\) is arbitrage-free;
\(\ell\) is exact, \(\ell = -\,\delta p\) for some \(p : \Tok \to \Real\);
a valuation \(\nu = \exp p\) exists.
The potential, and hence the valuation, is unique up to an additive (respectively multiplicative) constant — a global choice of unit. If \(\hypK\) is monoidal, \(\nu\) may be taken to be a monoid homomorphism, and is then unique up to a character of \(\Sigma\).
(i)\(\Rightarrow\)(ii): fix a basepoint \(a_\star\) with \(p(a_\star) = 0\) and define \(p(a)\) by summing \(-\ell\) along any path from \(a_\star\) to \(a\). Path-independence is exactly the vanishing of \(\ell\) around cycles, so \(p\) is well defined, and \(\ell = -\delta p\) by construction. (ii)\(\Rightarrow\)(iii) is \(\nu = \exp p\). (iii)\(\Rightarrow\)(i) telescopes: a cycle has yield \(\prod_i \nu(a_i)/\nu(a_{i+1}) = 1\). Uniqueness up to a constant is connectedness of \(\Gamma_{\hypK}\). For the monoidal refinement, additivity of \(p\) on generators propagates along \(\ast\)-compatible enzymes, leaving as residual freedom a homomorphism \(\Tok \to \Real\).
Theorem 15.1 is the combinatorial core of the fundamental theorem of asset pricing, and what matters here is which way the implication is read. The virtual token does not have to be built once arbitrage is removed. Its existence is the removal. A single measure of what a computation has is not an additional fact about the world; it is the shadow cast by a conversion structure with no cycles that pay.
The freedom in \(p\) up to an additive constant is the freedom to choose where zero sits. Naming an existing token type as the reference — pricing in one of the flavors, as one prices in dollars — fixes the gauge at a vertex. The virtual token is the refusal to privilege any vertex.
15.3 When arbitrage is present
When \(\hypK\) is not arbitrage-free, \(\ell\) is not exact, and the graph Hodge decomposition splits it into an exact part and a cyclic part: \[\ell \;=\; \underbrace{-\,\delta p}_{\text{the valuation that does exist}} \;+\; \underbrace{\ell^{\mathrm{cyc}}}_{\text{the arbitrage}} .\] The first summand is the best common measure the system admits; the second is a cohomology class, and it measures precisely how much the system fails to have one. It is not noise. It is a quantity, it is computable from the rate data, and a computation that can find a cycle with yield greater than one can profit from it without doing any work at all.
Part Part IV identifies the exact part with energy and the cyclic part with the failure of energy to be a state function, and Chapter 31 draws the consequences: a first law, a second law, and a bound on erasure. Chapter 24 does something different with the same decomposition — it observes that when several communities of learners each maintain their own token and must convert between them, the cycles of the conversion graph carry redundancy, and redundancy is an error-correcting code. The reader who wants the physics should go to the former and the reader who wants the sociology to the latter. Both are reading the same short theorem.