Definition 1.1 (Graph). A graph is

where g is represented state, r is operative transformation structure, and Γ is the admissibility grammar governing graph-originated transformations.

Nothing in that is typed, and the omission is the definition’s content rather than its brevity. What the word graph denotes here is the floor — difference, and what differing consists in — while the typed object the word ordinarily calls to mind is derived and arrives with anything that runs (Remark 3.11 of the long paper (What plurality forces, and what it does not)). A reader supplies the typing regardless, which is why this is said here and again where it matters rather than left to the definition to carry alone.

Definition 1.2 (Grounding). A grounding is a particular assignment of rule content and initial conditions to the substrate — one (g,r,Γ) with its laws and its starting configuration fixed — together with the domain it is taken to model. The substrate is what every grounding shares; a grounding is what any one of them adds. Claims made about the substrate are accordingly claims about all groundings, and are correspondingly weak; claims made within a grounding carry that grounding’s contingent rule content and are where this account’s falsifiable commitments mostly sit.

Definition 1.3 (Admissible continuation). A continuation δ belongs to P(G_t) iff it is expressible in the current graph vocabulary, satisfies the relevant antecedent conditions, and is admissible under the operative grammar for the transformation class in question.

Definition 1.4 (Collapse). Collapse is the frame-relative passage from a pre-realisation possibility structure to realised continuation:

It is not identified with deliberative choice.

Hypothesis 1.5 (Law of Graphic Equalisation). For realised graph G_t, present graph structure determines admissible continuation structure P(G_t). Interaction and collapse realise successor G_t+1, and that realised successor becomes the antecedent graph for the next local transition. Recursive dynamics are generated by repeated equalisation between present represented structure, admissible transformation, containing constraints, and realised continuation.

Definition 1.6 (Residual). The residual is

where ⊖ is the grounding’s difference relation preserving relevant discrepancy.

Proposition 1.7 (Residual is what a containing grammar does not admit). Because the graph is a subgraph of its containing frame W, a realised graph-originated deformation is already a change in W — it is not submitted to W for entry, and there is no step at which W grants or withholds it. The delta is not offered; it is. What the change grammar of W — write it Γ_Wc, the change face of W’s grammar (Proposition 3.3) — admits coheres with the rest of W and merges into its structure; what Γ_Wc does not admit is not thereby undone, but remains in place, present and unintegrated. Admissibility sorts a change that already exists; it does not gate one seeking to occur. The residual is exactly this non-admitted remainder:

Corollary 1.8 (Growth is grammar extension). A residual is absorbed either by a containing frame extending Γ_Wc to admit it, or by paying to copy it out (Section 7 of the long paper (The frame interface and its economics)); otherwise it persists in place and is re-encountered each time the merge runs. The first route is the substrate form of minting a new distinction: the residual does not fit because it is inadmissible, so admitting it requires extending the grammar, and extending the grammar is the act by which a previously inadmissible deformation becomes recognisable. Recognition and growth are therefore one mechanism joined at one place — the residual is where the shape a graph decodes and the grammar that admits it meet.

Remark 1.9 (Only g is required; the triple is what a running world costs). The triple is not the minimum. Strip r and Γ and something remains: structure, with relations among its parts. Nothing happens in it — no transformation, so no collapse, no residual, and nothing it individuates — but it has not thereby become nothing, and it has not stopped being individuable by something else. What r and Γ buy is occurrence, and everything this account is about — change, collapse, observation, cost, experience — requires them. Γ is definitional — an everything-admitting grammar is still the grammar P is written over (Definition 1.3) — and what Remark 8.2 calls an optimisation is the restrictiveness of its two faces, not its existence. The triple is the price of a world in which anything happens rather than the price of there being anything, and Remark 8.2’s “only that rules run is compulsory” should be read in that narrower sense.

A distinction is doing the work there. Being a frame is being individuated (Definition 3.1), and individuation is performed by the container, not by the thing: a boundary belongs to the frame that draws it. So a structure with no rules of its own is still a frame — something else instantiated its boundary — and what it lacks is not framehood but the capacity to individuate anything itself. Stripping r does not leave “no frame”, and reading it so would invalidate the mind companion’s result, which reasons from a rock’s being a frame to the universe’s being one (its Proposition 2.1).

Proposition 2.1 (Antecedent-state principle). The conditions under which a transition is admissible are functions of its antecedent state. A consequence that exists only because a transition has occurred cannot retroactively serve as an antecedent cause of that same transition.

Proposition 2.2 (Graph-law determinism). For a fully specified antecedent configuration—including the graph state, operative transformation structure, admissibility grammar, and every causally operative containing relation relevant to the collapse—the realised continuation is determined:

The Law (Hypothesis 1.5) contains no primitive random actualisation operator.

Hypothesis 2.3 (Minimal graph sufficiency). No parent pointer, world model, affordance catalogue, Narrative, Prospect, Value, or self-model is primitive; such structures may exist in g when the graph actually represents them. Any additional structure required only by a particular capability class is represented content or grounding, not a primitive of graphhood, unless a graph lacking it cannot participate in the basic realised transition relation.

Hypothesis 2.4 (Discrete sufficiency). Every quantity required internally by the present substrate can be represented by discrete distinguishable states at the operative frame and resolution.

Axiom -1 (Relativity of grounding). There is no operational view from nowhere. Every distinction, measurement, identity relation, admissibility judgement, and represented quantity is made in some frame (Definition 3.1). Frames may represent one another, but no quantity used by the theory requires a frame-independent evaluator.

Definition 3.1 (Frame). A frame is an instantiated boundary relation — the edge B_W:G itself, and not the subgraph G it demarcates. The subgraph is the relatum; the frame is the relation. To individuate is to instantiate such an edge, which is why individuation is per frame rather than a fact about the graph.

Corollary 3.2 (Frame and dimension are one edge type in two roles). A dimension is the same construction read from the other end. The boundary B_W:G is a dimension of W — one of the properties along which W separates what it contains — and each dimension W instantiates is a frame to the values it separates. So frame and dimension name two roles one edge type plays relative to another edge, not two kinds of thing, and the asymmetry between them is positional in the same way that the asymmetry between a parent and a child is.

Proposition 3.3 (One interface, two gates). A situated graph meets its containing frame through a single relation, and what an observation from that frame delivers is not distinct in kind from what the containing frame imposes: to a subgraph, being observed is being deformed by its container. That one relation is gated twice, by the same kind of admissibility grammar on its two faces — the aperture grammar Γ_a on the input and the change grammar Γ_c on the output:

Γ_a limits, by grain, which of the container’s deformation enters the rules; Γ_c determines which of the rules’ realised deltas cohere into the container and which remain as residual (Proposition 1.7). Both are faces of the frame’s one grammar Γ, and instantiate for a specific container W as Γ_Wa and Γ_Wc; it is Γ_Wc, the change face, that the residual proposition uses. Execution runs between the two.

Definition 3.4 (Aperture). A frame’s aperture is its input gate Γ_a together with what that gate fixes: which dimensions it can separate along at all, how finely it separates within each (its resolution, per dimension, for which this paper also uses grain), and how far along each it reaches. The three are properties of one edge — existence, grain, and extent — and are named separately because they vary independently, not because they are different kinds of limit. What falls outside an aperture on any of the three is absent to that frame rather than held-but-unresolved.

Definition 3.5 (Change grammar). A frame’s change grammar Γ_c is its output gate: it determines which of the realised deltas its rules produce cohere into the containing frame and which do not. What it declines has entered the frame and been operated on, so it is residual (Definition 1.6) rather than absent, and it persists in place rather than being discarded. Γ_a and Γ_c are the two faces of one grammar Γ on one boundary edge, and the asymmetry between them is which direction they gate, not what kind of thing they are.

Definition 3.6 (States of a dimension). For a frame F, a dimension is in exactly one of four states:

NULL: the dimension is not instantiated; there is nothing to have a value. Undetermined: the dimension is instantiated and no value has been realised — collapse has not occurred. Undisclosed: a value has been realised but is not held by this frame. Resolved: a realised value, held.

Definition 3.7 (Capability and accessibility). A frame’s capability is what its gates admit — what it can represent and what it can do. Its accessibility is what it can reach from where it stands, given what it holds to spend. A disclosure moves a gate and changes capability; whether the reachable point can then be crossed to depends on the Narrativium measured for the frame. Narrativium, which defines that quantity, types it as a capacity rather than as a cost (its Remark 1.1), and keeps measurement and expenditure distinct; the phrasing here follows it.

Definition 3.8 (Information hazard). A disclosure changes admissibility when it moves a frame’s aperture gate, its change grammar gate, or both (Proposition 3.3), and what it changes when it moves one is that frame’s capability (Definition 3.7). An information hazard is a capability change whose negative sign is admissible in the reference frame, above the point at which the impact forces a write. The class is disclosures that change admissibility; the sign is applied afterwards by a frame with preferences, and the magnitude is a ratio to what the frame can absorb rather than an absolute.

Proposition 3.9 (The threshold is a ratio, and sits where the impact forces a write). A minimum magnitude is required, or every capability change would be a hazard. The threshold cannot be an absolute, since an absolute would be a magnitude fixed independently of any frame, which Axiom -1 forbids; it is a ratio — impact against the scale of the frame bearing it. And the substrate supplies where the ratio sits rather than leaving it to stipulation: the threshold is the point at which the impact forces a write. An impact a frame can absorb coheres with what it holds, forces no write, and costs it nothing; an impact it cannot absorb leaves residual and compels change. That is the same line every other quantity here is measured against, and it is a ratio by construction, because absorption capacity is the frame’s own.

Theorem 4.1 (No frame-free identity). The substrate alone does not determine whether two represented instances are the same graph independently of an individuating frame.

Definition 4.2 (SELF identity). SELF identity is the continuity of the graph as a whole — its rules together with the state they act on — across realised transitions. A graph is the same self while its present remains congruous with its own history: if its rules changed abruptly, the structure carried forward would no longer cohere with what produced it, and that incongruity is the discontinuity. Represented self-description — content inside g by which a graph identifies itself — is one optional part of this and not the whole of it; a graph may have SELF identity without representing itself at all.

Definition 4.3 (SUPERGRAPH identity). For graph G individuated in frame W,

is the identity relation under which W tracks that situated instance.

Proposition 5.1 (Cost characteristically reduces to the write). In the substrate as characterised here, the boundary relation itself is free, and the standing structure persisting is free to the frame; the operation that costs is a delta that writes over that structure, and r is what writes. Whether persistence is free simpliciter is set by the rules of the superframes, which may sustain a frame without charge or require a write to keep it standing, and the substrate does not decide between them. Free here means information-preserving rather than absent: evaluating a rule is itself a write, and is unpaid because it destroys nothing (Remark 7.17 of the long paper (Where the check goes, and why the free case is the empty one)). To that extent

Proposition 5.2 (Self-change is direct and gated; a graph is a subset of its container). A graph changes itself and its subgraphs directly. Self-change is real, but it is rule-governed, with its own cost and limits (an instance of the non-uniform writeability of Remark 5.3); a substrate’s plasticity is exactly the rules for how its rules may change. What a graph does not do is change its container directly: the container applies deformations to it — being observed is being so deformed — and gates its changes through its own change grammar (Proposition 1.7). But a situated graph is a subset of its container, not a thing set against it, so a self-change is already a change in the container, in the region that is the graph, propagating outward through whatever couples it to the rest.

Remark 5.3 (Writeability status is contingent, not uniform). The write-economics stated here — reads free, cost carried by writes, and no direct alteration of the container (Proposition 5.2) — should be held at the strength the evidence supports, and the two halves of that need separating. The structural claim — that a graph affects its container as a subset of it and not by some other route — is a strict restriction and is committed to as one (Section 19 of the long paper (Falsifiability commitments)). What is a pattern rather than a universal is the distribution below: which writes are free, which are costly, and how much is direct. What is freely writeable, what carries a cost, and what a graph can write directly rather than only across a boundary, need not be uniform; different structures may have different writeability profiles. The strict forms above characterise a common and efficient configuration (Remark 8.2) rather than a necessity of graphhood, and asserting a single strict write-economics everywhere would be the same over-specification Remark 8.2 warns against, a fixed value put where the account has only a setting. These are accordingly carried as falsifiability commitments, open to counterexample, and a structure with a different writeability profile refines the characterisation rather than breaking the substrate.

Definition 5.4 (Well-formedness). A graph is well-formed to the degree that it pays for nothing that does no work for it. Four costs bear on that and they behave differently, so the notion is graded along each rather than binary.

Writes are paid when made (Proposition 5.1), which prices doing. Retention is nearly free per unit, which is why an unused distinction is not immediately pruned. Search is paid on every operation over what is held and scales with how much is held, so inert structure is cheap to keep and expensive to keep in quantity.

The fourth is the worst and is not a variant of the third. A write may land on structure that cannot affect anything — a modification to a branch nothing reaches. It is paid in full: decided, made, checked, risked. It returns nothing. And unlike a search, which is expensive and then yields, this failure is silent: the graph has spent, has no effect to show for it, and no signal that the effect is missing. What it acquires instead is a false model of itself, in which something was changed that was not — residual it cannot see, produced by its own inert contents, and the share of writes exposed to it grows with how much is held.

Proposition 5.5 (Evaluation is a write, and free because information-preserving). A read is an operation that returns a datum; the returned datum has to be somewhere to be used, and in a substrate that is graph all the way down there is no return channel outside the structure, so the value materialises as structure and the write is the return. Evaluation is a write in that plain sense: a match outcome, a position in an ordering, and the fact of having tried are returned values, and returned values are structure. It is nonetheless unpaid, because it destroys nothing — free here means information-preserving rather than absent. What it costs does not follow from that: a rule’s cost is set by whatever supplies its primitives and is inherited along provision, so reversibility is a structural fact about a map while cost is a fact about what supplies a primitive, and the two must not be run together.

Proposition 5.6 (Primitives leak, and complete provenance is still unrecoverable). A primitive discloses the structure beneath it: leaking is ordinary rather than exceptional (Remark 14.2 of the long paper (Exposure is excluded; leakage is produced)). What does not follow is that provenance can be recovered entire. Every act of recovery is itself performed by primitives, so a graph inferring what lies beneath one operator does so using others whose own provenance is not thereby in view, and there is no position from which the whole supply is inspected at once. The result is the standing shape: partial always, complete never — leaks give a graph real and sometimes decisive knowledge of what supplies it, and never the closure of that question.

Proposition 6.1 (Decoherence is one operation; the frames are what differ). Decoherence is one operation. It does not come in kinds and it does not come in degrees, and the two cases above fail identically. What differs between them is the frame — arithmetic in one, kinematics in the other — and the frames are what make the cases different, not the failure. “Everything in the superframe” is a constant expression whose value is frame-relative, which is Axiom -1 applied to incoherence rather than to observation.

Proposition 6.2 (A handler relocates a failure; it does not close it). A handler is installed policy that determines what a frame coheres a refused write to, and it can be set to cohere it to anything, including success. Swallowing does not undo the inadmissibility: the containing grammar still refused and the write still did not happen, so what is removed is the frame’s record of the refusal rather than the refusal. Detection is available and never free — a frame holding a record of the attempt separately from the handler’s report can compare the two — but it is unavailable from the handler’s own report, which is self-consistent by construction. The audit is therefore installed structure, and being itself a policy it is abusable in the same way one level in, so what its purchase buys is relocation rather than closure, terminating only in a containing frame.

Proposition 6.3 (Dependents are held by the referring frames, not the referent). A thing does not carry a record of what depends on it: the dependencies live in the frames holding the references and not in the referent, so from the position of the structure being removed they are not visible at all, and knowing what would break requires the containing frames. Removal therefore yields three states rather than two. Absent — nothing ever cohered, nothing is held and nothing owed, and it is free. Vacated with the references left standing — a hole, unpaid, with consequences determined by whatever the dangling relations do next. Vacated with the absence represented — a tombstone, a cascade rule, a retained count of dependents — which resolves cleanly and is installed machinery paid for under Proposition 5.1.

Proposition 6.4 (Termination propagates along supply, not containment). Being contained by something and being run by it are two relations, and they need not coincide. Termination propagates along the supply relation and not the containment relation: downward it need not reach at all, a container being able to stop while what it held runs on, and upward the supplier’s stop is total, at whatever containment distance and with no handler available, since the operations a handler would run are the ones withdrawn. This splits the dependents of a stopped structure into three kinds rather than two. A dependent that merely refers to it is left dangling and keeps running, wrongly; a dependent wholly run by it simply stops; and a dependent only partly supplied by it keeps running with an incomplete operation set, and its behaviour is undefined.

Proposition 6.5 (A boundary is known only by crossing it). A frame that never fails discloses nothing about what it admits. Admissibility is invisible from inside its own range, since everything works, and the boundary is legible only where it is crossed — so the way to learn a rule is to force the failure and read the residual. The test and the change are accordingly the same operation: knowledge of a boundary is bought by crossing it, and there is no cheaper instrument. It follows that every handler installed reduces what failure can teach, a frame that coheres its failures to tidy values reporting success and disclosing nothing, so that robustness and legibility are traded against one another rather than being two names for quality.

Axiom 1 (No substrate time dimension). Time is not a first-class dimension of the substrate. What the substrate has is change: realised collapses. It has no ratchet over and above them — no mechanism that manufactures succession — because the three properties usually bundled under that word are readings over changes or properties of them, not a further thing the substrate runs. The ordinal event order of Section 11 of the long paper (Causal order and local time) is carried by the changes’ own enabling-relations; direction is the asymmetry of enablement itself — if one change makes another possible, the second does not thereby make the first possible, which holds whether or not a change preserves what it followed; and persistence is only that further changes occur. A time dimension exists only where a frame instantiates one to represent that order.

Proposition 7.1 (Time is inferred, not observed). No collapse presents a temporal coordinate. Succession is a relation between realised states, and a relation between states is not a datum of any one of them. A frame therefore acquires a time dimension only by inferring it from the relations among what it holds. Consequently

Proposition 7.2 (The unit of inferred time is a cycle, and requires closure). A frame does not count its individual changes — those are below its aperture — but its own cycles, each one traversal of a closed rule-loop. The unit of inferred time is therefore a cycle, not a change, and a cycle has no fixed length: each does a variable amount of change, so inferred time is non-uniform in underlying change (a nonlinear grounding, which Hypothesis 2.4 already permits) and a fuller cycle reads as a longer moment (subjective dilation). A cycle exists only for a graph with closure — a graph without it runs to fixed point once and stops, one pass and no repeated tick — so having time requires closure as well as inference (Proposition 7.1): a one-shot computation, lacking a loop to iterate, has no time, and a frame’s experienced “now” is one cycle of its own loop.

Definition 7.3 (The present, structurally). Within a frame, the present is the collapse boundary at which every instantiated undetermined dimension relevant to that collapse has been determined:

U_F counts undetermined dimensions only. NULL dimensions are excluded — they are not variables. Undisclosed dimensions are also excluded: they are already determined, and a frame’s failure to hold them is an epistemic deficit rather than an unresolved collapse. A frame with outstanding undisclosed structure is still at its own present.

Proposition 7.4 (No implicit historical state). A previous success, failure, attempt, or residual has no later causal force merely because it occurred. It affects later dynamics only insofar as its consequences are present in later graph or boundary state.

Corollary 7.5 (History principle, strengthened).

The past is surviving consequence instantiated now, not a stack of still-existing historical frames.

Definition 8.1 (Primitives). A graph’s primitives are the operations it invokes without performing: the operator set its rules bottom out in. Every level has them, and having them is what being a level consists of — a graph with no primitives would have to perform everything it invokes, which is the regress this substrate does not contain. That is the whole of the definition, and in particular it says nothing about what a graph can learn of where they come from.

Remark 8.2 (What is required and what is optimisation). Only that rules run is compulsory for anything to occur — structure alone is what is required for there to be anything at all (Remark 1.9). Applying a delta is itself a rule, not a primitive operation beneath the rules, so the specific dynamics — inertia, the merge behaviour, closure — are contingent rule content, and a different r or Γ is a different physics rather than a violated law (this is why the initial and boundary conditions of a grounding are supplied separately from its laws; Section 11 of the long paper (Causal order and local time)). The two gates of Proposition 3.3, and even the retained standing structure itself, are optimisations on bare rule-running: the aperture grammar Γ_a bounds the input, the change grammar Γ_c sorts the realised output into what coheres and what is residual, and the retained structure buys free continuation. A substrate whose grammar admits everything still runs — effectively ungated, without momentum, wastefully. It is not a substrate lacking Γ, since Definition 1.3 defines the possibility structure through the grammar and a collapse needs something to select from. The two gates are accordingly what an efficient graph accretes under a budget, not constituents of graphhood. The retained structure specifically is what distinguishes a self-sustaining loop from a purely reactive one, since coasting is the cached structure.

Proposition 8.3 (Primitives are supplied, not intrinsic). A graph is data — nodes, relations, and rules — and the operations its rules bottom out in are supplied to it rather than contained in it. The same graph may therefore be run against different supplies of the same operator set, and nothing supplying an operator need be the substrate the graph describes or is “in.” Where the work is done is a fact about that supply and not about the graph.

Proposition 8.4 (A well-formed runner does not expose itself, and no level is self-transparent). The asynchronous, locally synchronous structure a frame observes (Remark 11.3 of the long paper (Globally asynchronous, locally synchronous)) is the graph’s internal view and does not fix its runner’s actual mode. Where a frame has no boundary to the substrate that runs it, its runner lies beyond the frame’s epistemic boundary (Section 1 of the long paper (Scope and epistemic status)); a hidden runner may execute the graph serially while the graph appears asynchronous to itself (Remark 11.3 of the long paper (Globally asynchronous, locally synchronous)). And no level is self-transparent: reading is an operation whose product lands one level up (Remark 7.17 of the long paper (Where the check goes, and why the free case is the empty one)), so a level cannot hold the reading of itself while being the level that is running — a frame may be given sight of what runs it and cannot be given sight of its own running.

Proposition 8.5 (A primitive is another graph’s running, exposed). Nothing supplying a primitive is a substance apart from graph data. An operator is the observable running of some other graph, which has primitives of its own, supplied in the same way. A graph’s “primitives” are therefore the observable running of other graphs, exposed as an operator set, with no primitive runner at the base. The regress bottoms not at a floor but at the epistemic frontier of Section 1 of the long paper (Scope and epistemic status): below the lowest measured level is the unclaimed pre-relational substrate.

Proposition 8.6 (The single prohibition). The account forbids one thing: a frame cannot get outside itself. Concretely, and confined to what is actually derivable here, it takes three forms. A frame cannot be transparent to itself at the level it is running at (Proposition 8.4). It cannot measure its own throughput in units independent of it, since the measurement is funded by the throughput — which leaves a number obtainable and its absolute scale not. And it cannot hold a view from nowhere, which is Axiom -1.

Hypothesis 9.1 (Near-uniqueness of the cheap cut). For a given task under a given binding constraint, the cheapest arrangement that performs it is nearly unique. This is an empirical posit and not a result of this account: nothing here derives it, and what is available without it is weaker — a binding constraint narrows the solution space, and the more binding it is the narrower it gets.

Proposition 9.2 (Convergence is observed, not enforced). Two frames doing the same work converge on similar cuts as each becomes better formed, because the cheap cut for a given task is nearly unique. This is a tendency of cost and not the discovery of a privileged boundary, and the two are distinguishable in principle: a privileged boundary would enforce convergence, and enforcement would extend to the hard cases; cost-shaped cuts converge where the work is shared and come apart where it is not. The step from shared work to similar cuts assumes Hypothesis 9.1.

Proposition 9.3 (Convergence under compression). A structure operating under a constraint, and under pressure to spend nothing on what does no work, approaches the arrangements that constraint forces — asymptotically, and only over the region the constraint touches. Two such structures therefore come to resemble each other without either having access to the other, because both are downstream of the same forcing rather than converging on a target.

Hypothesis 9.4 (Located degradation). Where a classification’s purpose can be specified independently and in advance of inspecting its failures, the failures concentrate near the boundary of that purpose more than a uniform distribution over the domain would produce. The claim is comparative and is not that failures occur nowhere else.