Chapter 31
Conservation, and What It Costs to Have It
The previous chapter gave the dynamics a charge per step. This chapter asks what is conserved, and the answer is more interesting — and more demanding — than the earlier presentation of this material allowed. Three things have to be separated that the earlier text ran together: a bookkeeping identity that holds by construction, a conservation law that holds only under a condition, and a generator of time evolution that we do not yet have. Physical energy is all three at once, and the coincidence is substantive. Here they come apart, and watching them come apart is the point.
31.1 The bookkeeping identity, and why it is not enough
We begin with the claim the earlier presentation made, stated honestly. Extend the reversible envelope of Chapter 28 to account-aware terms, so that a trace entry is a triple \((r, \phi, \vect{c})\) recording the rule, its witness, and the amount debited, and extended terms are triples \(\langle P, \trace, \vect{A}\rangle\). The backward rule undoes a step by popping the entry and returning the debit: \[\langle P', (r,\phi,\vect{c}) \cdot \trace, \vect{A} \rangle \;\rewrite_{r^-}\; \langle P, \trace, \vect{A} + \vect{c} \rangle.\]
In the resource-aware reversible envelope, any closed path — one beginning and ending at the same extended term — has zero net change in the vectorial account.
Each forward step debits \(\vect{c}\); the corresponding backward step credits exactly \(\vect{c}\); on a closed path each forward step is matched by its own inverse, so the debits and credits cancel.
Proposition 31.1 was previously called the conservation of energy for GSLTs. It is not. The backward rule was defined to credit what the forward rule debited, so the proposition says only that the account is a function of the extended state — which was true by construction of the extended state. Nothing has been learned about the dynamics. A real conservation law must be capable of failing, and must name the condition under which it does not. The rest of this chapter supplies both, following the companion notes [63, 31].
31.2 Enzymes, conversion, and the virtual token
Resources come in kinds — the components of \(\vect{A}\) — and the kinds are convertible. A process that turns one kind into another at some rate is an enzyme.
What follows in this section was already proved, in Chapter 15, as a fact about exchange rates: a set of enzymes admits a single price list exactly when no cycle of conversions pays for itself. It is restated here because the reading changes, and the change of reading is the content.
There the question was a bookkeeping one — can a computation holding balances in several kinds say how much it has? — and the answer was a condition on a graph. Here the same condition is what forces an energy into existence. That is a strong claim and it is worth pausing on why it is not a pun. Energy in physics is characterized by two properties: it is conserved under the dynamics, and it is a single number even though the world contains many kinds of stuff. The second of those is exactly what a valuation provides, and the theorem says the second is not an extra assumption but a consequence of the first once “conserved” is spelled out as “no closed loop of conversions is profitable”. A world in which some cycle of exchanges pays is a world in which there is no energy function — not because energy is hidden there, but because the quantity does not exist to be found.
The intuition worth carrying forward is therefore this. Energy is not a substance the conversion structure happens to move around. It is the shadow the conversion structure casts when it has no free lunches in it, and the size of the free lunches is precisely the extent to which the shadow fails to be well defined. The Hodge split of the next section makes that quantitative.
Fix a finite set \(\Tok\) of token flavors. A conversion system \(\mathcal{K}\) is a directed graph \(\Gamma_{\mathcal K}\) on \(\Tok\) together with a rate \(r(a \to b) \in \Rpos\) on each edge: one \(a\)-token converts to \(r(a\to b)\) many \(b\)-tokens. The log-rate is the \(1\)-cochain \(\ell(a \to b) = \log r(a\to b)\). A cycle admits arbitrage if the composite rate around it exceeds \(1\).
The following are equivalent: (i) \(\mathcal{K}\) is arbitrage-free; (ii) \(\ell\) is exact, i.e. \(\ell = \delta\nu\) for a potential \(\nu : \Tok \to \Real\); (iii) conversion is path-independent. The potential \(\nu\), unique up to an additive constant, is the virtual token: a single scalar value assigned to each flavor.
This is the discrete, computational analogue of the fundamental theorem of asset pricing, and it is the first honest thing that can be called energy in this framework. Energy is not a component of \(\vect{A}\). It is a valuation on the components, and it exists exactly when the conversion dynamics has a symmetry — path-independence. A system whose enzymes can be run around a cycle at a profit has no energy function at all.
The valuation is also a compression. A conversion system carries up to \(O(n^2)\) rates for \(n\) flavors; Theorem 31.1 collapses these to \(O(n)\) numbers, one price per flavor, modulo a global gauge constant. Conservation and compression are the same fact seen from two sides, and we will meet this again as the memory price of history in Section 31.7.
31.3 The Hodge split: energy and arbitrage
When arbitrage is present, \(\ell\) fails to be exact, and the failure is measured exactly.
Orient the edges of \(\Gamma_{\mathcal K}\) and equip the cochain groups with the standard inner products, with coboundary \(\delta\) and adjoint \(\delta^{\ast}\). Then \[C^1 \;=\; \underbrace{\operatorname{im}\delta}_{\text{exact}} \;\oplus\; \underbrace{\ker\delta^{\ast}}_{\text{cyclic}},\] orthogonally. Writing \(\ell = \ell_{\mathrm{en}} + \ell_{\mathrm{arb}}\) along the splitting, \(\mathcal{K}\) is arbitrage-free iff \(\ell_{\mathrm{arb}} = 0\), and the space of arbitrage components has dimension \(\bt(\Gamma_{\mathcal K}) = |E| - |\Tok| + 1\).
A tree of enzymes admits no arbitrage; each independent cycle contributes exactly one degree of arbitrage freedom. Orthogonal projection onto the exact part is the operation of shaking the arbitrage out, and it is idempotent: one squeeze suffices.
Arbitrage-removal is an idempotent reflection; the cost monad [62, \S9] is emphatically not idempotent, since its multiplication merges two resource accounts rather than collapsing a redundant copy, so metering a metered calculus yields a finer account. Metering accumulates; arbitrage-removal saturates. The two constructions live on the same material with opposite idempotence, and it is worth noticing that the framework makes both natural.
31.4 The first law
Extend \(\nu\) additively from flavors to token stacks. In an arbitrage-free conversion system the total value \(\nu\) of a token supply is invariant under every enzyme operation: an enzyme \(a \to b\) replaces one \(a\)-cell of value \(\nu(a)\) by \(r(a\to b)\) many \(b\)-cells of total value \(r(a\to b)\,\nu(b) = \nu(a)\).
This is a conservation law in the proper sense: it has a hypothesis (arbitrage-freedom), it can fail, and when it holds it says something the construction did not put there by hand. It upgrades the conservation of authority of [61, Prop. 4.7] — invariance of the consumed signature multiset across every partition of a join — from a multiset invariant to a scalar one.
31.5 The second law
Real conversion is lossy, and the platform this framework was built for takes that as a design principle rather than a defect: every resource-consuming operation carries an explicit charge [61]. Model a lossy enzyme by a rate strictly below the value-preserving one, \(\rho(a\to b) = r(a\to b)\,e^{-\theta(a\to b)}\) with dissipation \(\theta \geq 0\).
With lossy enzymes the log-rate cochain \(\hat\ell = \ell - \theta\) satisfies, around every oriented cycle, \[\oint \hat\ell \;=\; -\oint \theta \;\leq\; 0,\] with equality iff the cycle is frictionless. Hence no exact potential exists in general; instead the total value \(\nu\), computed against any fixed reference potential, is non-increasing along enzyme operations and strictly decreasing whenever a lossy enzyme fires. \(\nu\) is a Lyapunov function for the token economy.
This is the second law arriving on schedule, and it changes what the conserved quantity of the previous section is called. The frictionless virtual token is a conserved energy; the friction-bearing one is a monotone free energy — exergy, the part of the value still able to do computational work, dissipated by \(\oint\theta\) as an entropy analogue. A cost-accounted calculus is frictionful by design, so it lives in the dissipative regime, and its honest scalar is free energy rather than energy.
The consequence is not merely thermodynamic bookkeeping. A process whose free energy is exhausted deadlocks by starvation [65]: the Lyapunov descent of Proposition 31.4 is a certificate that value degrades monotonically toward an absorbing set. Mortality is not an extra assumption about computation in this framework. It is what the second law says about a computation that has to pay for its own steps.
31.6 The ledger law: potential and kinetic
Beneath the cohomology sits an elementary invariant, and it is the cleanest thing in the chapter. Recall from Chapter 30 that the temporal monoid — the token stack — reifies the clock: its consumed prefix is the run length.
Run a cost-accounted term under the history functor from an initial stack of depth \(\sigma_0\). Let \(\sigma(t)\) be the stack depth remaining and \(\kappa(t)\) the number of events recorded in the history after \(t\) steps. Each forced step pops one cell and appends one event, so \[\sigma(t) + \kappa(t) \;=\; \sigma_0 \qquad \text{for all } t.\]
Read \(\sigma\) as potential energy — fuel not yet expended — and \(\kappa\) as kinetic — work done and logged. Their sum is the conserved total, and reversibility is precisely the invertibility of the transfer: undo a step, and one unit returns from spent to remaining. This is the frictionless case of the previous sections read on the nose, and it is also the honest version of what Proposition 31.1 was groping toward, with one crucial difference: it names the resource whose motion between two ledgers constitutes time.
The token stack runs one way — it only drains. The history runs the other — it only accumulates. The ledger law balances them. Neither alone is a time coordinate; the pair is.
31.7 Conservation bounds erasure from below
Now the result that makes conservation cost something. Transport the valuation onto the reduction graph: let the energy form \(\omega\) assign to each step the change in \(\nu\)-value of the configuration’s token content, inclusive of the dissipation of any lossy enzyme invoked. By Proposition 31.2, \(\omega = \delta\phi \oplus \omega_{\mathrm{cyc}}\), and the holonomy of a loop is \(\hol(\gamma) = \oint_\gamma \omega = \oint_\gamma \omega_{\mathrm{cyc}}\), the exact part contributing nothing around closed paths.
A history may be folded — coarsened by an erasure that identifies distinct pasts. The erasures form a lattice indexed by subgroups of the fundamental group of the reduction graph [31]: to erase is to close loops.
Let an erasure close the loop-subgroup \(\Lambda\). Energy descends to a single-valued state function on the folded quotient iff \(\hol(\gamma) = 0\) for every \(\gamma \in \Lambda\). Consequently:
if \(\omega\) is exact — arbitrage-free and frictionless — then every erasure qualifies, down to and including total erasure: a conservative system tolerates arbitrary forgetting;
if \(\omega\) carries a nonzero holonomy class, conservation forbids folding the holonomy-carrying loops. The history recording them must be retained.
The naive expectation is inverted. One might have supposed that coarser erasure buys conservation, by throwing away the detail that spoils it. The opposite holds: conservation is a lower bound on how much history must be kept, and the bound is set by the holonomy class. Zero holonomy puts the bound on the floor; nonzero holonomy forces retention up to the cover that trivializes it — the Riemann-surface move, unwinding a multivalued potential by climbing onto the sheet where it becomes single-valued, remembering the winding rather than collapsing it.
Among the erasures on which energy is conserved there is a coarsest: fold the tree wherever the potential is single-valued and leave it unfolded exactly where the holonomy obstructs. This is the natural physical state space of the computation — the reduction graph quotiented as far as energy conservation permits, and no further.
31.8 Landauer: the two thresholds coincide
Erasure is what makes the ledger law’s transfer one-way. Forget a recorded event and \(\kappa\) falls, but \(\sigma\) cannot rise — one cannot un-spend fuel — so the invariant breaks irreversibly.
Discarding a history event that is not recoverable from the current state destroys information and, by Landauer’s principle [49], dissipates energy as heat; discarding an event that is recoverable is logically reversible and free. The recoverable events are exactly those determined by the current configuration — the exact part \(\delta\phi\) — and the irrecoverable ones are the holonomy-carrying part \(\omega_{\mathrm{cyc}}\). Hence the history one may erase for free is exactly the history whose erasure preserves energy: erase-freely and conserve-energy-freely are the same permission, indexed by one and the same cohomology class.
Remark 31.7 supplies the mechanism under Remark 31.5: dissipation is the Landauer heat of erasing the irrecoverable, holonomy-carrying history, and the free energy that decreases monotonically is what remains recoverable.
31.9 A charge, not a substance — and a missing generator
We can now say what the conserved quantity is. It is not a substance, not a stuff held or transferred or pointed at. It is the potential forced into existence by a symmetry of the conversion dynamics (Theorem 31.1), and its ontological status is that of a Noether charge [104]: the conserved scalar associated with an invariance, in the deflationary reading under which energy is the conserved quantity associated with a symmetry rather than a substance the world contains.
Honesty requires the counter-reading. Physical energy earns a thicker ontology because it also generates the dynamics: the same object that is conserved is the Hamiltonian that pushes the state forward in time, and the coincidence is substantive. What has been established here discharges only the bookkeeping role. The generative role is open, and it is open twice.
In an arbitrage-free, frictionless cost-accounted GSLT, the total virtual-token value \(\nu\) is the conserved quantity conjugate to translation along the temporal monoid: the path-independence symmetry of Theorem 31.1 is the internal symmetry whose Noether charge is \(\nu\), and its coupling to the stack makes \(\nu\) the generator of a step of reduction. Under friction the correspondence degrades to the inequality of Proposition 31.4: \(\nu\) generates a descent rather than a symmetry.
In the free history, reduction is time-reversal symmetric: each forward step has a unique inverse, so the augmented dynamics admits an involution exchanging forward reduction with backward replay. This involution is the symmetry whose Noether charge is the conserved energy of Theorem 31.2(i); the two-sided time of the ledger law is the parameter it translates; and the conserved energy is its generator. The erasure grades then measure how much of the symmetry, and hence how much of the conservation, has been spent.
Both are conjectures, and this chapter does not upgrade them. What the framework offers that physics does not is the ability to hold the two hats apart — the charge proven, the generator open — and watch whether they want to merge. That is the sharp form of the ontological question: is energy a conserved charge that happens to generate time, or two coincident things?
31.10 Located authority, and how space and time get entangled
One defect remains, inherited from Definition 30.6: a single account says what may be spent but not who may spend it.
A token stack written into the ambient process soup is consumable by whatever needs it. But parallel composition is associative–commutative: the soup is an unordered bag, so any process in it is adjacent to every stack in it. Taking adjacency as the right to spend is ambient authority — the antithesis of an object-capability discipline, and a latent double-spend.
The repair is to locate every stack, and in the rho calculus a location is nothing more exotic than a name. To locate a stack is to hold it on a channel: the stack becomes a message on a channel \(c\), and the right to spend from it is exactly the right to receive on \(c\). Because channels minted fresh are unforgeable, possession of \(c\) is a genuine capability — a process never handed \(c\) cannot name it, cannot receive on it, and so cannot touch the stack, even though the bag being associative–commutative it sits in the same soup. The name does the work that position cannot: it says who may draw, while the signature at the head of the stack says which token is spent. Authority is possession of a name, never adjacency.
This is a correctness fix, and it would be worth making for that reason alone. But it also carries the most suggestive physics in this part of the book, and we state it as such.
The framework carries two combination operators of different character. The interaction constructor is spatial: it records what is in contact with what. The stack is temporal: it records what is spent next, and next. Without an authority discipline the two are separable coordinates — nearness and consumption order may be varied independently. But a located stack ties a temporal resource, what may be spent next, to a spatial surface, who is near. A draw then requires both spatial proximity and temporal availability, and the two structures can no longer be varied independently. Authority entangles them. What was a product of a space and a time becomes a single coupled structure gating interaction — a spacetime [62, \S13].
This is the first point in the book at which the claim made in the Pledge — that rods and clocks must be inside the model rather than outside it — is discharged by a mechanism rather than asserted. The clock is the stack; the rod is the nearness relation; and what welds them together is the discipline that says who is allowed to spend. Whether the resulting nearness evolution obeys anything resembling a field equation is left open, and named as such in Chapter 36.