COMPLEXITY.md

COMPLEXITY.md — a crosswalk between Spherepop's notions of "complexity"

Status: proposed formalization, not yet part of the canonical spec. This document exists because prior writing about Spherepop (in this session and elsewhere in the repository) sometimes moves too quickly between several genuinely different quantities — event count, structural depth, constraint, entropy reduction, and assembly — as if they were interchangeable. They are related, but they are not the same thing. This document is deliberately not a philosophical essay (that role belongs to prose papers such as standardgalactic/calculus's identity_after_collapse.tex, see §11). It is meant as the precise crosswalk that prevents the repository's different complexity claims from drifting back together: every symbol below is given an explicit domain, every claim is either proved from the definitions or flagged as unverified, and every scalarization is required to name the policy under which it was produced. Nothing here has been adopted into SPEC.md itself, and no kernel code implements it — the four primitives' admissibility and replay semantics are entirely untouched. §14's derived, non-authoritative analysis functions ($L$, $D_a$, $W_a$, $A_{\mathrm{tree}}$, $A_{\mathrm{DAG}}$, $A_{\min}^c$) are now implemented and fixture-tested in experiments/flat/profile.py / run_profile.py, isolated from the main conformance suite; this remains a conceptual/notational proposal for Phase B/C work under issue #1 with respect to the kernel and SPEC.md themselves.

Correction to an earlier version of this document: a previous draft stated that the extensional_view/history_view divergence discussed in §4 was "exercised in test_spherepop.py-style fixtures." That was wrong. test_spherepop.py belongs to the personalinks nested-sphere/quotient lineage (see GRAMMAR.md), not the canonical SPEC.md kernel lineage this document is built on. The canonical evidence for the canonical lineage is the flat conformance fixtures and the three conforming implementations:

spherepop-kernel/
experiments/flat/run_python.py
compiler/tools/fixtures/

Checked directly in this session: there is no flat fixture yet that exercises an intensional/extensional (history_view vs. observation-under-Collapse) divergence. The closest existing fixture pair is experiments/flat/fixtures/05a_same_snapshot_different_history_run_a.json / 05b_same_snapshot_different_history_run_b.json ("same snapshot, different history"). The precise relationship is:

Fixtures 05a and 05b demonstrate that distinct histories may produce the same tested snapshot. They therefore support the general distinction between historical and extensional identity used throughout this document, but they do not implement or test the named extensional_view and history_view interface, which belongs to the separate personalinks lineage (see GRAMMAR.md) and has no counterpart in the canonical flat fixtures.

§4 and §14 below identify the still-missing fixture (one that exercises a named Collapse-relative view distinction, not merely snapshot equality) as a concrete gap rather than implying it is already covered.

1. Formal setting

Before defining any complexity measure, fix the objects the rest of this document quantifies over.

A history is a finite sequence of admitted events drawn from the canonical four-primitive alphabet:

$$ H = e_1 e_2 \cdots e_n \in E^{*}, \qquad E = E_{\mathrm{Pop}} \sqcup E_{\mathrm{Refuse}} \sqcup E_{\mathrm{Bind}} \sqcup E_{\mathrm{Collapse}}. $$

Replay produces an authoritative state:

$$ \operatorname{replay}(H) = S_H. $$

A certified collapse rule $c$ produces an observation:

$$ C_c(H) \in O_c. $$

From $H$ (or equivalently from $S_H$, if replay is taken as the source of truth — see the caveat below) four derived objects are read off:

• the residual option space $\Omega_H$ — the options not yet committed by a Pop and not excluded by admissibility failure;
• the refusal record $F_H$ — the sequence or multiset of Refuse events and their recorded reasons;
• the binding relation $B_H \subseteq \mathrm{Obj} \times \mathrm{Obj}$ — the pairs coupled by Bind events, without asserting identity between them (this is the canonical meaning of Bind; see §7 for why it must not be silently treated as a dependency/assembly relation);
• the observation record $Q_H$ — the sequence of Collapse events applied and the rules $c$ they invoked.

Open question, not resolved here: whether $\Omega_H$, $F_H$, $B_H$, $Q_H$ should be understood as reconstructed from the append-only event log $H$ on demand, or as materialized fields of the implementation's authoritative state $S_H$, is an implementation choice this document does not settle. spherepop-kernel's History/Arbiter types (src/history.rs, src/arbiter.rs) materialize an analogous structure directly; whether that is the only legitimate choice, or merely the current implementation's choice, is left open. This distinction matters for §13's comparison with the OS kernel-state tuple: spherepop-os.tex's per-object State term $t=(I_t,\mathrm{Opt}_t,H_t)$ (found and cited in §13) gives one concrete instance where $\Omega_H$ is materialized (as $\mathrm{Opt}_t$) and $F_H$ is reconstructible on demand (by filtering $H_t$ for rej: tags), while $B_H$ is neither — Bind's filter predicate leaves no trace in $t$ at all, materialized or otherwise.

2. Four quantities that must not be conflated

$$ \text{event length} ;\neq; \text{weighted historical cost} ;\neq; \text{algorithmic complexity} ;\neq; \text{observed complexity}. $$

Event length is raw count:

$$ L(H) = |H|, \qquad L(H \cdot e) = L(H) + 1. $$

Weighted historical cost requires an explicit, positive per-event weight function $w : E \to \mathbb{R}_{>0}$:

$$ L_w(H) = \sum_{e \in H} w(e). $$

Append monotonicity then follows by construction, not by assumption:

$$ L_w(H \cdot e) = L_w(H) + w(e) > L_w(H). $$

Algorithmic complexity is a genuinely different quantity and must be named separately:

$$ K_{\mathrm{alg}}(\operatorname{encode}(H)). $$

A previous version of this document used the symbol $K_{\mathrm{hist}}$ for an "always increases under extension" ledger. That was a naming error: $K$ conventionally denotes Kolmogorov complexity, and Kolmogorov complexity is not monotone under arbitrary extension in the strong sense that was apparently intended — a long history containing many repeated events can have a short description ($K_{\mathrm{alg}}(\operatorname{encode}(H\cdot e))$ need not exceed $K_{\mathrm{alg}}(\operatorname{encode}(H))$, e.g. if $e$ extends a periodic pattern). The intended "always increases" claim is true only of $L$ and $L_w$ (both additive ledgers by construction), never claimed of $K_{\mathrm{alg}}$. Every later use of a "historical growth" claim in this document uses $L$ or $L_w$, never $K_{\mathrm{alg}}$.

Observed complexity is whatever a chosen measurement $K$ (not necessarily Kolmogorov complexity — any fixed measurement convention) assigns to a Collapse-produced observation, $K(C_c(H))$. It is a property of the observation, not of the history, and different observations of the same history under different rules $c$ can disagree arbitrarily (§8).

3. Assembly in the historical sense

Spherepop is a calculus of assembly only in the historical sense:

Spherepop is a calculus of assembly only in the historical sense. An object is not identified merely by its final configuration but by the admissible event history through which it was constructed. Assembly therefore supplies identity, not merely cost. Two extensionally identical structures may have different Spherepop identities when their histories differ, while a specified Collapse may deliberately treat them as equivalent for a chosen observation.

This connects the calculus directly to assembly theory while marking a real difference from it: a conventional assembly index asks for the shortest construction pathway permitted by a repertoire of operations and reusable components, whereas Spherepop ordinarily preserves the actual path taken, not the shortest one available.

4. Minimal assembly is Collapse-relative, not absolute

An earlier version of this document defined minimal assembly as

$$ A_{\min}(x) = \min_{H:,\operatorname{out}(H)=x} |H|, $$

silently assuming that an evaluation map $\operatorname{out}$ determines an unambiguous object $x$. That is precisely what Spherepop questions: whether two histories "produce the same object" is only well-defined relative to a chosen observation. The corrected definition parameterizes minimal assembly by an explicit collapse rule $c$:

$$ A_{\min}^{c}(x) = \min\bigl{ L_w(H) : C_c(H) = x \bigr}. $$

Proposition (Collapse-relative assembly complexity). Say $c_1$ is finer than $c_2$, written $c_1 \sqsubseteq c_2$, when every distinction $c_1$ preserves is also preserved by $c_2$ — equivalently (see §6's $D_c$), $D_{c_1}(H) \subseteq D_{c_2}(H)$ for every $H$. Then for any $x$ meaningful in both codomains $O_{c_1}$ and $O_{c_2}$:

$$ c_1 \sqsubseteq c_2 \quad\Longrightarrow\quad A_{\min}^{c_1}(x) ;\ge; A_{\min}^{c_2}(x). $$

Reasoning: $c_1$ imposes a stricter equivalence condition than $c_2$, so the set of histories $H$ satisfying $C_{c_1}(H) = x$ is no larger than the set satisfying $C_{c_2}(H) = x$ — a finer observer restricts, never widens, the admissible construction histories counted toward the minimum. Minimizing $L_w$ over a subset can only produce a value greater than or equal to minimizing over a superset. $\blacksquare$

This is one of this document's two central results (see §8 for the other): apparent minimal assembly complexity decreases as the observer is permitted to forget more distinctions. There is no observer- independent minimal assembly index unless the criterion of equivalence has already been fixed.

5. Three orders and preorders, not one order of complexity

An earlier version of this document called all three of the following relations "partial orders." That overstated what follows from the definitions alone; each is analyzed separately below.

Causal order — a genuine partial order, provided it is defined over event occurrences in an acyclic dependency structure (which the canonical model guarantees: history is append-only and an event can only depend on events already admitted, so no cycle can form):

$$ e_i \preceq_H e_j \quad\text{when } e_i \text{ must already have occurred for } e_j \text{ to be admissible}. $$

Assembly relation — initially only a preorder, not a partial order:

$$ x \preceq_A y \quad\text{when some history constructing } x \text{ can be extended admissibly into one constructing } y. $$

Reflexivity and transitivity are immediate from admissible extension being reflexive and transitive. Antisymmetry does not follow: two differently presented structures may each be constructible from the other under different admissible continuations (e.g. via different Collapse policies), so $x \preceq_A y$ and $y \preceq_A x$ can both hold without $x = y$. A genuine partial order requires quotienting first (see below).

Observational relation — also initially only a preorder. Define the distinction set exposed by rule $c$ over history $H$:

$$ D_c(H) = \bigl{ (a,b) : C_c(H) \text{ distinguishes } a \text{ from } b \bigr}. $$

Then:

$$ H_1 \preceq_c H_2 \iff D_c(H_1) \subseteq D_c(H_2). $$

This is a preorder (reflexive and transitive by set-inclusion) unless observational equality has already been quotiented out. On equivalence classes under

$$ H_1 \sim_c H_2 \iff D_c(H_1) = D_c(H_2), $$

the induced relation $\preceq_c$ is a partial order, since inclusion is antisymmetric between distinct sets.

The general pattern: Collapse is exactly what turns a preorder into a partial order, by supplying the equivalence $\sim_c$ that quotients out the cases where two histories are mutually related but not identical. This is a nontrivial consequence connecting observational resolution to apparent complexity, and is why this section's title says "orders and preorders," not "three orders": only the causal relation is a genuine order prior to any Collapse-induced quotient.

These relations need not agree with one another: later does not always mean more complex; more assembled does not always mean more distinguishable; and Collapse can shrink $D_c$ without reversing or erasing the construction history (§8).

6. Historical growth vs. observational simplification

Spherepop complexity is monotone at the level of authoritative history but need not be monotone at the level of observed structure. Every admitted event increases historical information, even when its result is a simpler quotient, a smaller option space, or a more compressed view.

Using the corrected ledger from §2 (not $K_{\mathrm{hist}}$):

$$ L_w(H \cdot e) > L_w(H) \quad\text{for every admissible } e, $$

while for an observation $C_c$ under collapse rule $c$, using a fixed measurement convention $K$ over observations (§2):

$$ K\bigl(C_c(H \cdot e)\bigr) < K\bigl(C_c(H)\bigr) $$

may hold. Collapse therefore does not make the past less complex — it makes a selected observation less discriminating. See §9 for a concrete worked witness of this claim.

7. Bind, dependency, and the assembly-interpretation gap

Canonical Bind couples two elements without identifying them — it does not assert that one was assembled from the other. Reading a dependency/assembly graph out of Bind events therefore requires an additional, explicit interpretation. This document distinguishes:

$$ G_H^{\mathrm{bind}} = (V_H, B_H) $$

— the graph whose edges are literally the canonical Bind relation $B_H$ from §1, asserting only coupling, not construction — from

$$ G_H^{\mathrm{assembly}} = \mathcal{A}_a\bigl(G_H^{\mathrm{bind}}, H\bigr) $$

— a derived view produced by an explicit interpretation rule $a$ that selects which bindings (and possibly which Pop events) count as construction dependencies. Different choices of $a$ can read very different assembly structures out of the same $G_H^{\mathrm{bind}}$: a social binding, a logical constraint, a causal dependency, a spatial adjacency, and a genuine assembly dependency are all expressible as Bind events, and nothing in the canonical primitive itself distinguishes them. All assembly-graph quantities below ($D_a$, $W_a$, $N_a$, $B_a$ in §8; the dependency-depth claims in §3 of the retracted draft) are therefore properties of $G_H^{\mathrm{assembly}}$ under a named $a$, never properties of $G_H^{\mathrm{bind}}$ alone.

Concrete evidence for exactly this gap now exists outside this document: spherepop-os.tex's per-object causal State term (§13, Definition "State Term, Extended") defines $\mathrm{Bind}_f(t) = (I_t, {o \in \mathrm{Opt}_t : f(o)}, H_t)$ — note $H_t$ on the right-hand side is unchanged. Unlike Pop and Refuse, which each append a tagged entry to the history component, Bind's filter predicate $f$ leaves no trace once applied. This is an independent, source-verified confirmation that $B_H$/$G_H^{\mathrm{bind}}$ cannot in general be recovered from a replayed state or a materialized history component alone — recovering it requires retaining the full external operation sequence, exactly the caution this section raises in the abstract.

8. The complexity profile, split into invariant and policy-relative parts

$$ \mathbf{A}_{a,c}(H) = \bigl( L(H),; L_w(H),; D_a(H),; W_a(H),; N_a(H),; B_a(H),; F(H),; |\Omega_H|,; U_a(H),; Q_c(H) \bigr). $$

Symbol	Meaning	Depends on
$L(H)$	raw event count	— (invariant)
$L_w(H)$	weighted historical cost	a weight function $w$
$D_a(H)$	assembly depth (longest dependency path in $G_H^{\mathrm{assembly}}$)	assembly interpretation $a$
$W_a(H)$	concurrency width (maximum antichain width in $G_H^{\mathrm{assembly}}$)	assembly interpretation $a$
$N_a(H)$	number of distinct constructed components (vertices of $G_H^{\mathrm{assembly}}$)	assembly interpretation $a$
$B_a(H)$	dependency count (edges of $G_H^{\mathrm{assembly}}$)	assembly interpretation $a$
$F(H)$	refusal burden, $	F_H
$	\Omega_H	$
$U_a(H)$	reuse gain (§9)	assembly interpretation $a$
$Q_c(H)$	observational resolution, e.g. $	D_c(H)

The subscripts are not decoration: $D_a, W_a, N_a, B_a, U_a$ depend on a named assembly interpretation, and $Q_c$ depends on a named collapse rule. $L$, $L_w$, $F$, $|\Omega_H|$ are the only entries that do not require declaring $a$ or $c$ — call these the profile's invariant core.

A scalarization must declare two separate policies, not one:

$$ A_{\phi,a,c}(H) = \phi\bigl(\mathbf{A}_{a,c}(H)\bigr). $$

$c$ (the collapse rule) determines observational equivalence — which distinctions the profile is even allowed to see. $\phi$ (the weighting function) determines how the resulting heterogeneous dimensions are valued and combined — event count against dependency depth against reuse against residual optionality. These are related but distinct policies: Collapse does not, by itself, contain every value judgment a scalar complexity score requires.

Primitive-specific effect signatures

For an admitted event $e$, write $\Delta_e \mathbf{A} = \mathbf{A}(H\cdot e) - \mathbf{A}(H)$ (componentwise, where subtraction is meaningful). None of the four primitives moves every coordinate in one fixed direction; each affects a specific, limited subset:

$$ \Delta_{\mathrm{Pop}} L > 0, \qquad \Delta_{\mathrm{Pop}} |\Omega| \le 0. $$

$$ \Delta_{\mathrm{Refuse}} L > 0, \qquad \Delta_{\mathrm{Refuse}} F \ge 0, \qquad \Delta_{\mathrm{Refuse}} |\Omega| = 0 $$

— the last equality holds under the current canonical interpretation, in which refusal records inadmissibility without deleting the option from the underlying possibility space (SPEC.md; consistent with Spherepop_Specifications.tex's conservation-of-possibility framing, §12 below).

$$ \Delta_{\mathrm{Bind}} L > 0, \qquad \Delta_{\mathrm{Bind}} B_{\mathrm{bind}} \ge 0, $$

where $B_{\mathrm{bind}} = |E(G_H^{\mathrm{bind}})|$; assembly depth $D_a$ changes only if the chosen interpretation $a$ treats that particular binding as a dependency — a Bind event is not guaranteed to move any assembly-graph coordinate at all.

$$ \Delta_{\mathrm{Collapse}} L > 0, $$

but $K(C_c(H\cdot e))$ relative to $K(C_c(H))$ is unconstrained by the primitive alone: it may be smaller, equal, or (for a rule that reveals more structure than it hides) larger. Collapse must not be declared universally complexity-reducing; it is distinction-reducing relative to a particular observational comparison $D_c$, and $D_c$ can shrink while $K$ under some unrelated measurement convention does not.

9. Reuse: tree unfolding vs. dependency DAG

An earlier version of this document proposed a subtractive reuse-credit formula,

$$ A_{\mathrm{reuse}}(H) = \sum_{v \in V(G_H)} w(v) - \sum_{u \in R(H)} (m_H(u)-1),\rho(u), $$

which has two defects: it can go negative, and it does not specify whether the first sum already counts a shared component once or once per use. The corrected treatment compares two different graphs built from the same history.

Let $T_H$ be the unfolded construction tree — obtained from $G_H^{\mathrm{assembly}}$ (§7) by duplicating every reused component at every point of invocation, so that no vertex is shared:

$$ A_{\mathrm{tree}}(H) = \sum_{v \in V(T_H)} w(v). $$

Let $A_{\mathrm{DAG}}(H)$ price the dependency DAG as actually constructed, with reuse represented by edges rather than duplicated vertices:

$$ A_{\mathrm{DAG}}(H) = \sum_{v \in V(G_H^{\mathrm{assembly}})} w(v) + \sum_{e \in E(G_H^{\mathrm{assembly}})} \lambda(e), $$

where $\lambda(e)$ is the cost of binding to (reusing) an existing component, usually $\lambda(e) &lt; w(v)$ for the component $v$ it reuses. The reuse gain is then a plain, non-negative-by-construction difference:

$$ U_a(H) = G_{\mathrm{reuse}}(H) = A_{\mathrm{tree}}(H) - A_{\mathrm{DAG}}(H) ;\ge; 0 $$

whenever $\lambda(e) \le w(v)$ for every reused $v$ — reuse is not "negative assembly," it is the difference between reconstructing every occurrence independently and constructing shared components once with additional dependency links.

10. Three worked counterexamples

Counterexample 1 — equal length, different depth and width. Two histories, each with four Pop/Bind events ($L(H_1) = L(H_2) = 4$). $H_1$ is sequential ($a \to b \to c \to d$, each depending on the last): $D_a(H_1) = 4$, $W_a(H_1) = 1$. $H_2$ constructs four independent components with no bindings between them: $D_a(H_2) = 1$, $W_a(H_2) = 4$. Equal event count, opposite depth/width profile — event count alone cannot distinguish "long sequential chain" from "wide independent batch."

Counterexample 2 — reuse vs. reconstruction under a coarse Collapse. $H_1$ constructs component $u$ once (from $a, b$) and binds it into three downstream assemblies. $H_2$ independently reconstructs an $a$-$b$ assembly three times, once per downstream use. A sufficiently coarse collapse rule $c$ can make the two histories observationally identical:

$$ C_c(H_1) = C_c(H_2), \qquad \mathbf{A}_{a,c}(H_1) \neq \mathbf{A}_{a,c}(H_2) $$

— $H_2$'s $A_{\mathrm{tree}}$ cost is strictly higher and its $U_a$ reuse gain is zero, while $H_1$'s is positive, even though nothing observed under $c$ distinguishes them.

Counterexample 3 — observational simplification through historical growth. Let $H$ be a history whose observation under $c$ distinguishes three elements, $D_c(H) = {(a,b),(a,c),(b,c)}$. Append a single Collapse event $e$ whose rule additionally identifies $a \sim b$:

$$ L(H \cdot e) > L(H), \qquad |D_c(H \cdot e)| < |D_c(H)| $$

($D_c(H\cdot e)$ drops the $(a,b)$ pair). This is a concrete witness for §6's claim: historical growth (one more event, strictly longer $L$) is compatible with, and here directly produces, observational simplification (strictly fewer distinguished pairs).

11. Relationship to identity_after_collapse.tex (external, not modified)

The user has proposed expanding standardgalactic/calculus's identity_after_collapse.tex (see EXTERNAL-SPHEREPOP-REFERENCES.md — verified in this session to define real \pop/\refuse/\bind/ \collapseop macros) with a new section, "Assembly Order and Historical Complexity," placed after that paper's primitive calculus and before its categorical/interpretive conclusions. That repository is outside the scope of standardgalactic/spherepop and has not been modified from this session; the proposal is recorded here as reference only. At a high level, that proposed section covers the same ground as §4, §5, §7–§10 above (collapse-relative minimal assembly, the assembly preorder and its quotient, the Bind-derived dependency graph, tree-vs-DAG reuse, and a worked example), but is meant to carry the philosophical framing — why this distinction matters for what "identity after collapse" means — while this document carries the precise, checkable definitions. Anyone picking up that proposal should treat this document as the source of truth for the formal claims and use the paper only for the interpretive argument built on top of them, to avoid the two documents' formalism drifting apart.

12. Comparison with the possibility functional $\Pi$

Spherepop_Specifications.tex (verified present in this session; see IMPLEMENTATIONS.md's "Canonical basis") states a conservation law:

$$ \Pi(H,\Omega) = |\Omega| + \sum_{e \in H} w(e), \qquad \Pi(H_t,\Omega_t) = |\Omega_0| \text{ for all } t. $$

This deserves its own scrutiny rather than an assumed compatibility with §8's profile: is $\Pi$ intended as a conservation law, an accounting identity, or a complexity measure? Consider the per-event change

$$ \Delta\Pi(e) = \Delta|\Omega| + w(e) $$

for each primitive separately:

• Pop: removes exactly one option from $\Omega$ ($\Delta|\Omega| = -1$) and, under the specification's own weighting $w(\mathrm{Pop}) = 1$ (Spherepop_Specifications.tex §"conservation," verified: "$w(\Pop(x))=1$"), contributes $\Delta\Pi = -1 + 1 = 0$. Conserved, exactly, for Pop.
• Refuse: per §8's $\Delta_{\mathrm{Refuse}}|\Omega| = 0$ (refusal does not shrink the option space, only records inadmissibility) and the specification's own weighting $w(\mathrm{Refuse}) = 0$ (verified: "$w(\RefuseOp)=w(\BindOp)=w(\CollapseOp)=0$"), $\Delta\Pi = 0 + 0 = 0$. Conserved.
• Bind, Collapse: likewise weighted zero by the same specification clause, and neither changes $|\Omega|$ (§8), so $\Delta\Pi = 0$ for both.

Conclusion of this comparison: $\Pi$ is conserved exactly because Spherepop_Specifications.tex assigns weight $1$ to Pop alone and weight $0$ to the other three primitives — it is not a universal invariant that would hold under an arbitrary positive weighting $w$ (the kind §2's $L_w$ allows for other purposes, such as costing Bind or Collapse by their computational expense). Read as a claim about that specific weight assignment, $\Pi$ is a genuine, exactly-derived conservation law, matching the specification's own theorem. It should not be read as evidence that $L_w$ under a general positive weight function is conserved — under such a weighting, $\Pi$-with-general-$w$ would strictly increase on Refuse/Bind/Collapse, since those no longer contribute zero. This resolves (rather than merely repeats) the open question: the conservation claim holds in its own specified domain and does not generalize to this document's more general $L_w$ without adopting the same zero-weighting choice.

13. The OS kernel-state tuple: two projections, not one crosswalk

An earlier draft of this section proposed a tentative, coordinate-by- coordinate correspondence between spherepop-os.tex's kernel state and this document's causal-history quantities. That was the wrong shape of claim to make, and it is retracted here rather than merely hedged further. The corrected conclusion is stronger than "tentative": there is presently no componentwise crosswalk to make, because the two tuples live at different layers and are not populated from the same event vocabulary.

spherepop-os.tex's kernel state (verified in this session, Definition "Kernel State," §5.1) is

$$ \sigma = (O, U, R, M) $$

where, exactly as defined there: $O$ is a finite set of object identifiers created by POP; $U : O \to O$ is a union-find parent map representing identity classes, updated by MERGE; $R \subseteq O \times O \times \mathcal{T}$ is a multiset of typed relations produced by operations such as LINK; and $M$ is a partial metadata map, confirmed structurally separate from $(O,U,R)$ (spherepop-os.tex line 588). This tuple belongs entirely to the OS operational layer.

By contrast, this document's causal-history quantities

$$ (\Omega_H, F_H, B_H, Q_H) $$

(§1) belong to the causal option-space layer: possibility, refusal, dependency, and observation. The letters $U$/$\Omega$ and $R$/$F$ are coincidentally similar-looking, but the underlying sets are not the same and no claim of equality should be inferred from the notation:

• $\Omega_H \neq U$. $U$ is a union-find map over objects that already exist (post-POP); $\Omega_H$ is the space of options that do not yet exist. These are not even subsets of the same universe.
• $F_H \neq R$. $R$ records typed relations produced by OS operations like LINK; it is not a refusal ledger. No top-level kernel event (POP, MERGE, LINK, UNLINK, SET_META) is itself a REFUSE, so there is no kernel-event directly analogous to causal Refuse. Refined below, however: causal Refuse is used internally inside the LINK reduction, and — per the per-object State term found in this session (see "A concrete, partial $\mathcal{I}$," below) — it does leave a trace, just not one that survives into $\sigma$.
• $B_H$/$G_H^{\mathrm{bind}}$ (§7) may be related to $R$ if some of $R$'s typed relations happen to record causal Bind events, but this requires an explicit translation rule, not a bare identification of the two symbols. Refined below: the LINK reduction gives exactly this translation rule for one specific case.

The document therefore describes two projections from an authoritative history, rather than one shared coordinate space:

$$ S_{\mathrm{kernel}}(H) = (O_H, U_H, R_H, M_H), \qquad S_{\mathrm{causal}}(H) = (\Omega_H, F_H, B_H, Q_H). $$

Neither is presumed recoverable from the other:

$$ S_{\mathrm{kernel}}(H) \not\Rightarrow S_{\mathrm{causal}}(H), \qquad S_{\mathrm{causal}}(H) \not\Rightarrow S_{\mathrm{kernel}}(H). $$

The causal calculus and the OS kernel expose different state spaces. The causal model tracks possibility, refusal, dependency, and observation. The OS model tracks object existence, identity classes, typed relations, and metadata. Although OS events may be reducible to expressions in the causal calculus, this does not imply a componentwise correspondence between their replayed states.

If a bridge between the two layers is wanted, it should be typed as an explicit interpretation, not asserted by matching field names:

$$ \mathcal{I} : \mathsf{History}_{\mathrm{OS}} \longrightarrow \mathsf{History}_{\mathrm{causal}}, $$

together with the two independent replay functions

$$ \operatorname{replay}_{\mathrm{OS}}(H) = (O, U, R, M), \qquad \operatorname{replay}_{\mathrm{causal}}(\mathcal{I}(H)) = (\Omega, F, B, Q). $$

Even if $\mathcal{I}$ reduces LINK to a causal expression involving Bind, that alone does not justify identifying $R$ with $B$: one is an OS relation store, the other a causal dependency structure, and their relationship is mediated entirely by $\mathcal{I}$. Likewise, since REFUSE has no direct OS event, a refusal arising during reduction may be internal to the causal derivation without surviving as a separately queryable OS-state component under $\mathcal{I}$.

A concrete, partial $\mathcal{I}$: found, not merely hypothesized

The previous version of this document treated $\mathcal{I} : \mathsf{History}{\mathrm{OS}} \to \mathsf{History}{\mathrm{causal}}$ as a future obligation and flagged spherepop-os.tex line 585's reference to a "higher-level State model...defined over an option space and a history" as not yet located. It has now been located and read in full (Definition "State Term, Extended," spherepop-os.tex line 907, §9.1), and it changes §13's picture from purely negative to partially constructive.

The per-object causal state is literally a triple

$$ t = (I_t, \mathrm{Opt}_t, H_t) $$

— immutable identity, admissible option space, and append-only history — with the three primitives acting exactly as this document's §1 assumed, but with one asymmetry worth recording precisely:

$$ \mathrm{Bind}_f(t) = (I_t,\ {o \in \mathrm{Opt}_t : f(o)},\ H_t), $$

$$ \mathrm{Pop}_a(t) = (I_t,\ \mathrm{Opt}_t \setminus {a},\ H_t \cdot [\mathrm{sel}!:!a]), $$

$$ \mathrm{Refuse}_a(t) = (I_t,\ \mathrm{Opt}_t \setminus {a},\ H_t \cdot [\mathrm{rej}!:!a]). $$

Bind does not append to $H_t$ at all — it only filters $\mathrm{Opt}_t$. Only Pop and Refuse leave a mark in the history component, tagged sel:/rej: respectively. This sharpens §7's Bind-vs-assembly- interpretation gap: in this concrete model, $F_H$ is recoverable from $H_t$ by filtering for rej: tags (contrary to what the previous paragraph's "no demonstrated storage location for refusal bookkeeping" suggested about $\sigma$ specifically — see below for the distinction), but $G_H^{\mathrm{bind}}$ is not recoverable from $H_t$ at all, since Bind's filter predicate $f$ leaves no residue once applied. Recovering which Bind calls occurred, and with which predicates, requires the full external operation sequence, not the state triple $t$ alone.

Moreover, LINK and kernel COLLAPSE are not merely claimed reducible to the causal calculus — they have proven reductions already in the text:

$$ \mathrm{LINK}(a,b,r) := \mathrm{Collapse}\Big(\mathrm{Refuse}_{r'}\big(\mathrm{Pop}_{r(a,b)}\big( \mathrm{Bind}_{C_r}(H_a \otimes H_b)\big)\big)\Big) $$

(Definition "LINK Reduction," line 505, proven well-typed at Proposition "LINK Well-Typedness," line 517), and kernel $\mathrm{COLLAPSE}(S, o_r)$ reduces to $\mathrm{Collapse}\big(\bigotimes_{o \in S} H_o\big)$ together with a separately stated identity-designation step (line 487). This is a genuine, checkable instance of $\mathcal{I}$ for two of the five kernel events, given directly in terms of the per-object histories $H_a$, $H_b$.

This still does not rehabilitate the coordinate-by-coordinate identification the previous section rejects, and in fact sharpens why it must be rejected on the $U$ side specifically: spherepop-os.tex line 490 states explicitly that $\bigotimes_{o \in S} H_o$'s causal Collapse value "does not account for $U'$'s reassignment of every $o \in S$'s representative to $o_r$," because tensor and the causal Collapse operator's own definition are silent on identity (Proposition "tensor-structural"), and choosing $o_r$ is "an external policy decision" layered on top of, not derived from, the causal reduction. So the earlier draft's tentative lead — "$U$ is structurally close to a Collapse-quotient" — is now actively contradicted by the source, not merely unconfirmed: $U$'s update is stipulated by a policy external to Collapse, so $U$ cannot be identified with a Collapse-induced $\sim_c$ equivalence class structure (§5) without that external policy coinciding with $\sim_c$ by further, separate assumption.

Finally, note that $H_a$, $H_b$, $H_o$ — the per-object causal histories that LINK and COLLAPSE reduce over, and that would need to be consulted to recover $F_H$ or $\Omega_H$ for a given object — are not components of the persisted kernel state $\sigma = (O,U,R,M)$ (Definition "Kernel State," line 131, has exactly four fields, none of which is a history). They are used in the reduction's derivation to prove LINK/COLLAPSE well-typed, not retained afterward as queryable state. So even though $F_H$ is recoverable in principle from $H_t$ (via rej: tags), it is not recoverable from $\sigma$ — confirming, more precisely than before, exactly where the causal option-space/refusal layer would need its own additional storage if an implementation wanted to expose it, rather than treating $\sigma$ as already containing it under a different name.

Net conclusion, revised. No componentwise crosswalk between $\sigma = (O,U,R,M)$ and this document's $(\Omega_H, F_H, B_H, Q_H)$ is asserted — that part of the earlier conclusion stands. What has changed is that $\mathcal{I}$ is no longer purely hypothetical: LINK and kernel COLLAPSE give two concrete, proven instances of it, expressed over per-object causal State terms $t = (I_t, \mathrm{Opt}t, H_t)$ that are themselves now fully located and defined (spherepop-os.tex line 907). But those per-object histories $H_t$ are consumed only inside the reduction's derivation; they are not retained as fields of $\sigma$, so $F_H$, $\Omega_H$, and $B_H$ remain unrecoverable from $\sigma$ even though they are well-defined and (for $F_H$, partially for $\Omega_H$) recoverable from the richer per-object $H_t$ that the derivation consults. The one specific lead that does not survive is "$U$ resembles a Collapse-quotient" — the source states the opposite: $U$'s update is an external policy decision the causal Collapse operator's own definition is silent on. A state coordinate still cannot automatically substitute for a historical measure: different histories can replay to observationally equivalent states, so $\mathbf{A}{a,c}(H)$ is not generally recoverable from $\sigma$ alone. What remains open is whether spherepop-os.tex's per-object $H_t$ construction is intended as the literal implementation-level home for $\Omega_H$/$F_H$/$B_H$, or purely a proof device for the well-typedness propositions — the text does not say which, and this document does not resolve it.

14. What adoption required: a non-authoritative analysis function (implemented)

A minimal implementation need not compute every entry of $\mathbf{A}_{a,c}$ described above. It exposes a single derived, non-authoritative analysis function:

$$ \operatorname{profile}_{a,c}(H) ;\longrightarrow; \mathbf{A}_{a,c}(H). $$

Because the profile is derived, it must not affect event admission, replay, or authoritative history — exactly the same non-interference constraint spherepop-os.tex already states for diffs ("Diffs do not influence kernel state and may be dropped, reordered, or ignored by observers without affecting correctness," line 353) and that spherepop-kernel's Collapse implementation already satisfies (a pure function over History that never mutates state — see IMPLEMENTATIONS.md's "Canonical basis" entry for spherepop-kernel/).

This is no longer a proposal. experiments/flat/profile.py implements $L$, the default assembly interpretation $a$ (default_assembly_graph, $D_a$, $W_a$, $N_a$, $B_a$), the tree-vs-DAG reuse comparison ($A_{\mathrm{tree}}$, $A_{\mathrm{DAG}}$, $G_{\mathrm{reuse}}$ from §9), and Collapse-relative minimal assembly over a supplied candidate list (min_length_achieving, §4). It is kept in its own file, imports run_python.py's Event/Arbiter/collapse-rule implementations rather than duplicating them, and is exercised by experiments/flat/run_profile.py against fixtures in the isolated experiments/flat/profile_fixtures/ directory — deliberately separate from experiments/flat/fixtures/ so this non-authoritative suite cannot perturb the existing 13-fixture conformance suite in any way. All five fixtures below pass, and the original conformance suite (run_python.py against fixtures/) was re-run afterward and is unaffected:

1. 13_length_increases.json — $L(H \cdot e) = L(H) + 1$ for every admissible $e$ (§2), checked after each of four different event kinds (pop, bind, refuse, collapse).
2. 14_collapse_non_mutating.json — Collapse does not mutate prior history: two branches share an identical prefix and diverge only in which certified rule their final Collapse invokes; the prefix's option_space, committed set, and history_len are checked identical across both branches.
3. 15_depth_vs_width.json — Counterexample 1 (§10) instantiated exactly: a 4-object sequential chain ($D=4$, $W=1$, via exact Dilworth/Mirsky-duality max-antichain computation) versus 4 independent objects with no binds ($D=1$, $W=4$), equal event/vertex count in both.
4. 16_reuse_tree_vs_dag.json — Counterexample 2 (§10): a history reusing one shared component ($u$) across three downstream assemblies gives a strictly positive $G_{\mathrm{reuse}}$ (tree cost 12.0 vs. DAG cost 7.5), contrasted in the same fixture with a fully independent reconstruction of the same shape three times over, which correctly yields a non-positive $G_{\mathrm{reuse}}$ (tree cost 12.0 vs. DAG cost 14.7) — demonstrating both the reuse-gain case §9 motivates and the documented no-reuse edge case where $G_{\mathrm{reuse}} \le 0$ is expected, not a defect.
5. 17_collapse_relative_minimal_assembly.json — a strict witness for §4's proposition: two candidate histories relating objects 1 and 3 through an intermediate object 2, then withdrawing the (1,2) relation. Under the coarse rule (ignores withdrawals) the shorter, length-3 candidate already achieves "1 and 3 same class"; under the fine rule (honors withdrawals) only the length-4 candidate does, giving $A_{\min}^{\mathrm{fine}}(x) = 4 &gt; A_{\min}^{\mathrm{coarse}}(x) = 3$ — a strict inequality, not just the non-strict bound proved in §4, at the same standard already applied to the Meld schema in experiments/flat/fixtures/08_meld.json.

Summary

Spherepop offers no single inherent "complexity." It supplies an authoritative history from which several invariant and policy-relative complexity measures can be derived (§8's invariant core vs. its $a$/$c$-dependent coordinates), while requiring every loss of distinction (a Collapse rule $c$) and every scalarization (a weighting $\phi$) to declare the rule under which it is performed. This document's two central results are: minimal assembly complexity is monotone in observational coarseness (§4), and historical growth is compatible with — indeed can directly produce — observational simplification (§6, witnessed concretely in §10's Counterexample 3). Everything else here (§7's Bind/assembly-interpretation gap, §9's tree-vs-DAG reuse treatment, §12's resolution of the possibility-functional conservation claim, §13's separation of the OS kernel-state projection from the causal-history projection, with two of its five kernel events now shown to have concrete proven reductions) exists to make those two results, and any future extension of them, precise enough to falsify. §4's monotonicity claim and §9's tree-vs-DAG reuse gain are no longer only proved on paper: §14's experiments/flat/profile.py/run_profile.py compute them, and all five fixtures in experiments/flat/profile_fixtures/ — including a strict witness for §4's proposition and both the positive- and non-positive-gain cases for §9's reuse comparison — pass against that implementation, without altering the existing 13-fixture kernel conformance suite.