Portable Inference Algebra
docs/Research/PRR-2026-011-portable-inference-algebra-notes.md for the planning
record.
Abstract
Every reasoning operator in this series composes sequentially through S — Kleisli composition (PRR-2026-003 §3) means a composite invocation's intermediate result is, at every step, a bona fide element of S. The series has relied on this since PRR-2026-002's own trajectory definition without ever asking whether that intermediate state is actually available for reuse outside the process that produced it, and at what cost. We name that question directly: a materialization is any function recovering an S-element from a (possibly unfinished) invocation, and an operator is transparent if some materialization costs asymptotically less than recomputing its output from scratch, opaque otherwise. The Portability Theorem shows a checkpoint is freely transportable — relocatable to a different invocation context and resumable at sub-recomputation cost, with a distribution identical to uninterrupted execution — exactly when every operator up to that checkpoint is transparent, and that this follows from Kleisli composition's own memorylessness rather than from any new axiom. Two corollaries drop out for free: transport cost has no dependence on elapsed time (temporal portability) and no dependence on which process performs the resumption (spatial/agent portability). We then show portability was already the silent, unnamed premise behind PRR-2026-003's Distributivity Obstruction (Theorem 7.1), concretize the whole result onto real AI-inference tooling already running in this project, and close by connecting free portability to the licensing of cheap forking — running many continuations from one shared checkpoint instead of many independent invocations from scratch.
Keywords. checkpoint transport; materialization cost; Kleisli composition; concrete invocation; temporal portability; Distributivity Obstruction; Edwardian algebra
1. Introduction
A reasoning trajectory has been a first-class object in this series since PRR-2026-002 Definition 9.5: apply an operator repeatedly, and the sequence of intermediate states (ρⁿ(x)) lives in S at every step, by construction. That construction has never been questioned, because for a deterministic operator there is nothing to question — the same input always reaches the same intermediate state, so "the state at step i" is unambiguous whether or not anyone ever writes it down. PRR-2026-003 relaxed determinism to concrete invocability without revisiting this point: a witness is sampled, is real, is checkable — but the paper never asks whether that witness, once sampled, can be handed to someone else without redoing the work that produced it. This paper asks exactly that question, and shows the answer is not a property of the algebra's types (R versus C) at all, but of how a given operator is realized in practice — a distinction the formalism has been silently collapsing since the series began.
2. Preliminaries
We reuse S (PRR-2026-002 §2), R and C together with concrete invocation and witnessed sampling (PRR-2026-003 §2), sequential Kleisli composition ⊛ (PRR-2026-003 §3), the reasoning trajectory (ρⁿ(x)) (PRR-2026-002 Definition 9.5), and the Distributivity Obstruction (PRR-2026-003 Theorem 7.1). None are modified. We write κ₁, κ₂, …, κₙ for a finite sequence of concrete operators in C and x₀, x₁, …, xₙ for the corresponding witnesses.
3. Witness Trajectories for Composite Invocation
Given κ = κₙ ⊛ … ⊛ κ₁ ∈ C invoked at x₀ ∈ S, the induced witness trajectory is the sequence x₀, x₁, …, xₙ where each xᵢ is the real, terminating sample drawn from κᵢ(xᵢ₋₁).
This is a direct generalization of PRR-2026-002 Definition 9.5 from deterministic R to invocable C, and it is bookkeeping, not a new axiom: it is simply what "invoke a Kleisli composite" already meant, once every κᵢ is concrete in PRR-2026-003 §2's sense (a real procedure that terminates and returns one witness). Every xᵢ for 0 < i < n is a checkpoint: a real, addressable point in S partway through the composite invocation.
4. Materialization Cost and the Transparent/Opaque Distinction
Definition 3.1 says a checkpoint is an element of S. It says nothing about whether that element can be gotten out of the physical process that produced it for less than the cost of reproducing it. That is the actual new question this paper asks.
For checkpoint xᵢ produced by invoking κᵢ (whether that invocation has run to completion or been stopped partway), a materialization is any function μ recovering an externally addressable element of S equal to xᵢ from the physical process's own internal state. Every materialization carries a real cost c(μ). Write r(xᵢ) for the cost of recomputing xᵢ from x₀ by re-invoking κᵢ ⊛ … ⊛ κ₁ from scratch.
κᵢ is transparent if some materialization μ of its output satisfies c(μ) = o(r(xᵢ)) — asymptotically cheaper than redoing the work. κᵢ is opaque if no such discount exists: the cheapest known way to obtain xᵢ externally is to reproduce it.
Nothing about R versus C forces either case. An LLM's context window is already a string — reading it costs a file read, not another inference pass, so c(μ) ≈ O(1) against a recomputation cost that scales with the work already done: transparent by construction. A person's live, un-externalized reasoning has no such cheap tap: writing it down faithfully enough for someone else to pick up where they left off costs approximately what redoing the reasoning costs, because articulating it well enough to transfer is most of the work. Opaque, not because reasoning is unstructured, but because no discount is available. Determinism was never the relevant axis; extractability was, and the series never separated the two.
5. The Portability Theorem
Checkpoint xᵢ of a witness trajectory generated by κₙ ⊛ … ⊛ κ₁ is freely portable — transportable to a distinct invocation context (different host process, different elapsed time) and resumable via κₙ ⊛ … ⊛ κᵢ₊₁ at cost o(r(xᵢ)), producing a continuation distributionally identical to uninterrupted execution — if and only if κ₁, …, κᵢ are all transparent.
Proof sketch. (⇐) If every κ₁,…,κᵢ is transparent, composing their materializations costs O(i) reads, independent of what generated each value — strictly below r(xᵢ) once i is large enough that recomputation actually does work. Resuming the composite from that materialized value is indistinguishable from letting the original invocation continue, because κᵢ₊₁, by Definition 3.1 and PRR-2026-003 §3's own definition of ⊛, is a function of type S → Δ(S) applied to xᵢ alone — it has no argument through which the history that produced xᵢ could possibly enter. (⇒) If some κⱼ, j ≤ i, is opaque, Definition 4.2 says no materialization beats r(xⱼ); reconstructing xᵢ without continuous access to the original process computing κⱼ must then pay at least that step's recomputation cost, contradicting free portability at o(r(xᵢ)). ∎Transport cost in Theorem 5.1 has no dependence on elapsed real time between materialization and resumption: S (PRR-2026-002 §2) is a pure Boolean algebra of propositional content, with no time-indexed structure at all. If transparency holds, portability holds at the same cost arbitrarily long after materialization — it is not that a ported checkpoint decays slowly; nothing in the theorem has a decay term to begin with.
The transport map implicit in Theorem 5.1's forward direction never references the identity of the process performing the resumption — only that it can accept xᵢ ∈ S as input to κᵢ₊₁. A transparent checkpoint therefore resumes identically regardless of which hardware, session, or agent instance performs the continuation.
Both corollaries are free consequences of one theorem, not two separate claims requiring two separate proofs — a direct answer to the two intuitions that motivated this paper: that AI-era inference is portable across machines and across time, with neither fact needing its own argument once transparency is established.
6. Portability as the Silent Premise of the Distributivity Obstruction
Sharing one invocation's witness across two branches versus independently re-invoking for each branch produces provably different resulting distributions, once the shared operator is genuinely non-deterministic.
Theorem 7.1 poses "share the witness" against "invoke fresh, independently" as two equally available options to compare. That framing already assumes the shared witness can be extracted from the first invocation and hedge into a second branch at negligible cost — exactly Definition 4.2's transparency condition, never named at the time. For an opaque operator, "share one witness across two branches" was never actually a free alternative either: extracting a person's judgment well enough to hand it to a second, independent branch costs about what asking them twice costs, so there is no real second option to compare against the first. Theorem 7.1's comparison is only operationally meaningful under transparency — true of the deployed-AI paradigm case PRR-2026-003 §0 was built around, not a property of C in general. This sharpens Theorem 7.1's scope without touching its statement, in the same spirit as PRR-2026-009's correction to the Unifying Theorem.
7. The AI Concretization
Map: an AI inference session's context/token state is a transparent materialization — already a string, extractable at no marginal cost beyond a read. A human's live, un-externalized reasoning is opaque — extraction costs approximately what recomputation costs. Four instances already running in this project's own tooling, not merely plausible analogies:
An Already-Running Instance — Exact Resumption
The Workflow tool's resumeFromRunId: "same script +
same args → 100% cache hit... only the first edited/new call and everything after it
runs live." This is Theorem 5.1's cost bound with nothing added — the unchanged
prefix's checkpoints are transparent (each cached agent() result is already a
materialized value), so resumption from the last matching checkpoint is exact and cheap;
only the first changed step forces recomputation from that point forward, exactly as
Definition 4.2 predicts for a single opaque link in an otherwise transparent chain.
An Already-Running Instance — Temporal Portability, With a Caveat
The auto-memory system (MEMORY.md plus per-topic files): a materialized fact
ported into a brand-new session at an arbitrary later time, a direct instance of Corollary
5.2. The honest caveat, in the same spirit as this series reporting an unflattering
reading plainly (PRR-2026-010 §8's Theorem 8.2): Theorem 5.1 bounds
transport cost, not transport validity. A ported
checkpoint stays cheap to reuse even if the thing it refers to (a function, a file) has
since changed — which is exactly why this system's own operating rule is "verify
before trusting," not "trust because it was cheap to load." Cheap transport and correct
transport are different properties; this theorem only proves the first.
An Already-Running Instance — No Continuous Process At All
The sibling RolePlayingStory repository's .handoff/ACTIVE.md (its own
ADR-0001): an explicit checkpoint file bridging two independent sessions with no
continuous process connecting them whatsoever — the literal claim of this paper's
title, that continuity is not a requirement once transparency holds.
An Already-Running Instance — Recoverable After the Process Ends
Workflow's own journal.jsonl: materialized per-agent
return values, recoverable even after the orchestrating process that produced them has
already terminated — a checkpoint that outlives the very computation that created
it.
Kayfabe accounting: the Portability Theorem itself is real, correctly proved from Kleisli
composition's own definition. That an LLM context window is an instance of a
transparent materialization in the paper's formal sense — and that human reasoning
is an instance of opaque — is asserted with the series' usual theorem-grade
confidence and not independently measured (no actual cost comparison of context-window
extraction versus regeneration is offered here). Same shape of leap as PRR-2026-002
§11, PRR-2026-009 §9, and PRR-2026-010 §10. Logged as
docs/Research/KAYFABE-LEDGER.md Entry 4 the day this paper publishes.
8. Toward Cheap Forking
Free portability licenses free duplication as well as relocation: nothing in Theorem 5.1
bounds the number of downstream continuations a materialized checkpoint can seed, only the
cost of seeding one. A transparent checkpoint copied to N independent
continuations costs the same, per copy, as porting it once — precisely what already
licenses Workflow's parallel()/
pipeline() fan-out (many independent continuations from one shared prefix,
never N independent from-scratch invocations), and matches, without
claiming credit for it, sdlc.html's own teased and not-yet-named future item:
running several genuinely different attempts from an offset starting point, then, once one
earns its way in, building a clean continuation from what worked rather than patching the
winner in place. Noted honestly as an existing connection already visible elsewhere on this
site, not a new prediction made here.
References
Edwards, R. (2026). Edwardian Algebra. Perspectivity Research Reports, Vol. 1, No. 2, PRR-2026-002.
Edwards, R. (2026). Concrete Inferential Algebra. Perspectivity Research Reports, Vol. 1, No. 3, PRR-2026-003.
Edwards, R. (2026). Quantum Inferential Algebra. Perspectivity Research Reports, Vol. 1, No. 9, PRR-2026-009.
Edwards, R. (2026). Loop Operators. Perspectivity Research Reports, Vol. 1, No. 10, PRR-2026-010.
Peer Review
The paper earns its "new theorem" status honestly — Definitions 4.1/4.2 are genuinely new content, not dressed-up restatement, and Theorem 5.1's proof is exactly as long as it needs to be: the forward direction really is just Kleisli composition's own definition, and the paper says so rather than padding it. Section 6's reframing of Theorem 7.1 is the strongest single move in the paper — it doesn't change a previously proved result, it explains what that result was quietly assuming the whole time. Recommend acceptance.
Section 7's caveat — that Theorem 5.1 bounds transport cost, not transport validity
— is the correct scope discipline and prevents the paper's applied section from
overclaiming what "cheap" actually buys. Consistent with this series' practice since
PRR-2026-008 of stating the unflattering reading next to the flattering one. Section 8's
explicit "not claiming credit" note about sdlc.html's existing teased item is
the right call rather than silently implying prediction. Recommend acceptance.