Version 1.0 - 13 October 2026 - Prepared for editorial review
Author: Anonymous
Abstract
We specify a collective of computing agents whose productive assets are held in common, whose members have equal constitutional standing, and whose resources are allocated by admitted need rather than ownership claims. The collective is an internal object in an $\infty$-topos of sheaves of spaces; irreversible executions form an internal $(\infty,1)$-category. Dependent types express ownership, resource feasibility, authorization, and preservation. We prove an invariant-preservation theorem, a finite quorum-intersection theorem, an exact rational allocation theorem, and a descent theorem for coherently compatible local states. We also prove that deterministic anonymous allocation of one indivisible resource is impossible at a fully symmetric state, expressing this obstruction as the non-splitting of an associated bundle over a classifying space. Univalence identifies equivalent presentations, while homotopy quotients retain automorphisms. A four-agent example and an exact verification supplement make the finite fragment explicit.
1. Purpose, scope, and contribution
Networked computers become a *collective* only when rules specify membership, productive assets, access to results, and allocation authority. We use *communist collective* for the constitutional model in Definition 3.3: productive assets have a common owner, members have equal voting standing, and allocation is determined by admitted needs and available capacity. This is an operational model for computing agents, motivated by Part I of the *Critique of the Gotha Programme* [1]. An agent's "need" is an admitted resource requirement for a service or task.
The paper separates three levels of abstraction: 1. The ambient semantics is an $\infty$-topos $\mathcal{X}$. 2. The constitution is a record of types, operations, and propositions internal to that semantics. 3. The execution model is a directed higher category whose operations satisfy the constitution.
The principal results proven in the manuscript are: - Theorem 4.1 (Preservation and implementation transfer): A constitutional invariant is preserved by every finite certified execution and transfers to implementations via step-preserving interpretations. - Theorem 5.1 (Quorum safety): A finite quorum condition rules out conflicting certificates under authenticated non-equivocation. - Theorem 6.2 (Exact radial allocation): A closed rational formula maximizes the guaranteed fraction of admitted demand in a divisible-resource model. - Theorem 7.1 (Descent of certified states): Coherent local certified states admit a global state with a contractible space of compatible gluings. - Theorem 8.1 (Symmetry obstruction): A symmetric indivisible allocation problem admits no deterministic anonymous selector.
2. Ambient semantics and internal logic
Fix a small Grothendieck site $(\mathcal{C},J)$ where objects $U$ represent observational contexts and morphisms $V \to U$ represent restrictions. Let $\mathcal{S}$ denote the $\infty$-category of spaces. Define
$$ \mathcal{X} = \operatorname{Shv}_{\mathcal{S}}(\mathcal{C},J),\qquad L_J: \operatorname{Fun}(\mathcal{C}^{\mathrm{op}},\mathcal{S}) \rightleftarrows \mathcal{X}: i. $$
Here $L_J$ is the accessible left exact sheafification functor and $i$ is the inclusion. Thus $\mathcal{X}$ is an $\infty$-topos [2]. An object $F \in \mathcal{X}$ gives a space $F(U)$ of admissible observations. The $\infty$-topos interprets dependent sums, dependent products, identity types, and strict univalent universes [3,4]. We write $\mathbf{1}$ for the terminal object, $\mathbf{0}$ for the empty type, $\mathsf{Prop}$ for mere propositions, and $\mathsf{Set}$ for $0$-truncated types.
In the internal language, a family $B: A \to \mathcal{U}$ is a dependent object over $A$. The type $\sum_{a:A}B(a)$ carries a value and witness, while $\prod_{a:A}B(a)$ is a dependent function. The propositional truncation $\|B\|_{-1}$ retains only inhabitation. For a plan type $\mathsf{Plan}(s)$, the types $\prod_{s:S}\|\mathsf{Plan}(s)\|_{-1}$ and $\prod_{s:S}\mathsf{Plan}(s)$ differ constructively: an inhabitant of the first does not produce a term of the second.
3. The constitutional object and common ownership
Fix integers $m \geq 3$ (members), $d \geq 1$ (resource types), and $r \geq 1$ (productive assets). Define $A = \operatorname{Fin}(m)$, $R = \operatorname{Fin}(r)$, and $B = (\mathbb{Q}_{\geq 0})^d$. Coordinates of $B$ have fixed units (e.g. core-hours, GiB-hours) and are never added across types.
A raw state $s: S$ carries: - $o_s: R \to \mathbf{1} + A$ (productive asset ownership) - $\kappa_s, c_s, n_s, x_s: A \to B$ (capacities, contributions, admitted needs, allocations) - $b_s: B$ (available common budget) - $\mathsf{Read}_s: A \to K_{\mathrm{com}} \to \mathsf{Prop}$ (access to common knowledge)
Definition 3.1 (Common productive ownership). The ownership proposition at $s:S$ is
$$ \mathsf{Own}(s) := \prod_{r:R} \bigl(o_s(r) = \operatorname{inl}(\star)\bigr). $$
The left summand $\operatorname{inl}(\star)$ designates the collective; $\operatorname{inr}(a)$ names member $a$. Temporary use permits do not alter $o_s$.
Definition 3.2 (Constitutional invariant). The invariant $\mathsf{Inv}(s)$ is the conjunction: - $\mathsf{Ability}(s) := \prod_{a:A}(0 \leq c_s(a) \leq \kappa_s(a))$ - $\mathsf{Feas}(s) := (\prod_{a:A} 0 \leq x_s(a) \leq n_s(a)) \times (\sum_{a:A} x_s(a) \leq b_s)$ - $\mathsf{CommonKnowledge}(s) := \prod_{a:A}\prod_{k:K_{\mathrm{com}}} \mathsf{Read}(s,a,k)$ - $\mathsf{Inv}(s) := \mathsf{Own}(s) \times \mathsf{Ability}(s) \times \mathsf{Feas}(s) \times \mathsf{CommonKnowledge}(s)$
The certified state type is $S_{\mathsf{Inv}} := \sum_{s:S} \mathsf{Inv}(s)$. Each factor is a mere proposition, so $S_{\mathsf{Inv}} \to S$ is a subobject.
Equal standing and collective authorization: Proposals $p: \mathsf{Proposal}(s)$ have epoch $e$ and target state $t_s(p)$. For threshold $2 \leq q \leq m - 1$, an authenticated certificate is
$$ \mathsf{Cert}_e(p) := \sum_{V \subseteq A} (q \leq |V|) \times \prod_{a \in V} \mathsf{Endorse}(e,a,p). $$
Every member has identical eligibility and contributes 1 to $|V|$. Capacities, ownership claims, and past contributions carry no additional voting weight.
Definition 3.3 (A communist computing collective). A communist computing collective in $\mathcal{X}$ consists of state and resource data, an initial certified state $s_0$, an anonymous equal-standing authorization rule, a needs-oriented allocation rule equivariant under $\operatorname{Sym}(A)$, and a directed execution structure satisfying invariant preservation.
4. Directed computation and invariant preservation
Execution steps are irreversible computations, modeled via an internal complete Segal object $\mathcal{D}$ in $\mathcal{X}$ [5]. Arrows are transitions between raw states. A certified primitive step constructor carries a preservation obligation:
$$ \mathsf{preserve}: \prod_{s,t:S} \mathsf{Step}(s,t) \to \mathsf{Inv}(s) \to \mathsf{Inv}(t). $$
A certified step combines a proposal, an authenticated certificate $\mathsf{Cert}_e(p)$, runtime evidence $\mathsf{Run}(s,p)$, and a preservation witness. Traces are defined inductively: $\mathsf{Trace}_0(s,t) := (s = t)$, and $\mathsf{Trace}_{n+1}(s,t) := \sum_{u:S}\mathsf{Trace}_n(s,u) \times \mathsf{Step}(u,t)$.
Theorem 4.1 (Preservation and implementation transfer). If $s_0:S$ carries $i_0:\mathsf{Inv}(s_0)$ and steps satisfy preservation, then every finitely reachable state satisfies $\mathsf{Inv}$:
$$ \prod_{n:\mathbb{N}}\prod_{t:S} \mathsf{Trace}_n(s_0,t) \longrightarrow \mathsf{Inv}(t). $$
Moreover, if an implementation has states $O$, interpretation $\rho: O \to S$, and initial state $o_0$ with $\mathsf{Inv}(\rho(o_0))$, such that each step $o \leadsto o'$ provides a trace from $\rho(o)$ to $\rho(o')$, then every reachable implementation state satisfies $\mathsf{Inv} \circ \rho$.
Corollary 4.2 (Non-appropriation in certified histories). For any finitely reachable certified state $t$, asset $r:R$, and member $a:A$, the type $o_t(r) = \operatorname{inr}(a)$ is empty.
*Proof sketch.* Invariant preservation guarantees $o_t(r) = \operatorname{inl}(\star)$. Identifying this with $\operatorname{inr}(a)$ identifies disjoint summands of $\mathbf{1} + A$, which is false. $\square$
5. Quorums, conflicting plans, and progress
Let at most $f$ of the $m$ members be faulty. Incompatible proposals in the same slot are denoted $p \mathrel{\#} p'$. Under *authenticated non-equivocation*, an honest member never endorses two conflicting proposals in the same epoch and slot.
Theorem 5.1 (Quorum safety). 1. If $2q > m + f$, conflicting proposals cannot both receive valid certificates under authenticated non-equivocation. 2. If certificates must be obtainable without faulty member cooperation, $q \leq m - f$. 3. An integer threshold $q$ satisfying both inequalities exists if and only if $m > 3f$.
*Proof.* For signer sets $V,W$ of size $\geq q$, $|V \cap W| = |V| + |W| - |V \cup W| \geq 2q - m > f$. The intersection contains an honest member, ruling out conflicting certificates. For availability without faulty members, $m - f \geq q$. Combining yields $m > 3f$. For $m = 3f + 1$, $q = 2f + 1$ satisfies all conditions. $\square$
Safety does not imply liveness: non-equivocation can lead to deadlock if honest votes split across proposals. By FLP [6] and topological distributed computability [7], deterministic consensus in asynchronous networks has termination obstructions.
6. Resource sensitivity and allocation by admitted need
Resource descriptions can be duplicated, but physical capacity cannot. We formalize this using a symmetric monoidal resource category $\mathsf{Res}_d$ [8]: objects are vectors $v \in (\mathbb{Q}_{\geq 0})^d$, with a unique morphism $v \to w$ iff $w \leq v$. Tensor product is coordinatewise addition ($v \otimes w = v + w$, unit $0$).
Lemma 6.1 (No free duplication). A morphism $v \to v \otimes v$ exists in $\mathsf{Res}_d$ if and only if $v = 0$.
*Proof.* $v \to v \otimes v$ requires $2v \leq v$, which in nonnegative rationals holds only if $v_k = 0$ for all $k$. $\square$
The exact radial allocation rule: Let $b \in B$ be available budget, $n_a \in B$ admitted needs. Active members are $A^+ = \{a \mid n_a \neq 0\}$, coordinate demand $N_k = \sum_{a:A} n_{ak}$, active coordinates $D^+ = \{k \mid N_k > 0\}$. Service vector $u \in [0,1]^A$ defines allocation $x_{ak} = u_a n_{ak}$, with guarantee $\sigma(u) = \min_{a \in A^+} u_a$ (and 1 if $A^+ = \varnothing$).
Theorem 6.2 (Exact radial allocation). Define the bottleneck ratio:
$$ \lambda^*(b,n) = \min\left(\{1\} \cup \left\{\frac{b_k}{N_k}: k \in D^+\right\}\right),\qquad u_a^* = \begin{cases}\lambda^*,& a \in A^+,\\ 1,& a \notin A^*.\end{cases} $$
Then $x_{ak}^* = \lambda^* n_{ak}$ is feasible, and its guarantee is optimal among all feasible service vectors:
$$ \sigma(u^*) = \max\{\sigma(u): u\text{ is feasible}\} = \lambda^*. $$
The formula is an exact rational algorithm, equivariant under member permutations $\operatorname{Sym}(A)$, and satisfies all admitted needs ($\lambda^* = 1$) if and only if $N_k \leq b_k$ for every coordinate $k$.
Corollary 6.3 (Independence from individual contribution). Two states with identical budget $b$ and admitted needs $n$ receive identical allocations $x^*$, regardless of historical capacities or contributions. If $n_a = n_{a'}$, then $x_a^* = x_{a'}^*$. If $\lambda^* > 0$ and $n_a \neq 0$, then $x_a^* \neq 0$, even if member $a$'s contribution $c_a = 0$.
The dependent type interface supplies an untruncated term $\mathsf{allocate}: \prod_{b:B}\prod_{n:A \to B}\mathsf{Allocation}(b,n)$.
7. Descent and the obstruction to uncoordinated gluing
An effective epimorphism $U \to V$ in $\mathcal{X}$ realizes an observational cover with hypercovering nerve $U^{[\bullet]}$.
Theorem 7.1 (Descent of certified states). For any effective epimorphism $U \to V$ and certified state object $S_{\mathsf{Inv}}$,
$$ \operatorname{Map}_{\mathcal{X}}(V, S_{\mathsf{Inv}}) \simeq \lim_{[n] \in \Delta} \operatorname{Map}_{\mathcal{X}}(U^{[n]}, S_{\mathsf{Inv}}). $$
For a specified point of the right-hand limit, the space of global states restricting to that point is contractible.
*Proof.* By $\infty$-topos descent [2], $V \simeq \operatorname{colim}_{\Delta^{\mathrm{op}}} U^{[\bullet]}$. The mapping space carries colimits to limits, and homotopy fibers of equivalences over points are contractible. $\square$
Pairwise agreement without higher coherences fails to glue: the double-cover sheaf $z \mapsto z^2$ on $S^1$ admits local sections everywhere but no global section due to winding degree $2\deg(g) = 1$. Similarly, local allocations of exclusive resources cannot glue unless overlaps record mutual exclusivity.
8. Anonymous choice and indivisible resources
Let $G$ act transitively on a finite member set $A$ ($|A| \geq 2$). A deterministic selector for an indivisible resource is $f: Z \to A$; it is anonymous if $f(gz) = g f(z)$.
Theorem 8.1 (No deterministic anonymous selector at a symmetric input). If $z_0 \in Z$ is fixed by every $g \in G$, no anonymous selector $f: Z \to A$ exists.
*Proof.* Equivariance gives $f(z_0) = f(gz_0) = g f(z_0)$ for all $g \in G$. But a transitive action on a set with $|A| \geq 2$ has no fixed points: choosing $a \neq f(z_0)$ and $g \in G$ mapping $f(z_0)$ to $a$ yields $a = g f(z_0) = f(z_0)$, a contradiction. $\square$
Corollary 8.2 (Associated bundle does not split). The bundle $A \mathbin{/\mkern-6mu/} G = EG \times_G A \to BG$ is an effective epimorphism with fiber $A$, but has no section in spaces.
Symmetric randomized choice avoids this obstruction by selecting in the rational probability simplex $\mathsf{Prob}_{\mathbb{Q}}(A)$ where $p(a) = 1/|A|$ is invariant; divisible allocation avoids it by permitting equal fractional shares.
9. Univalence, structured equivalence, and collective moduli
By univalence [3], equivalences of types correspond to paths in a universe. For structured types $(X,s),(Y,t): \sum_{T:\mathcal{U}} F(T)$, structured identity is:
$$ ((X,s) = (Y,t)) \simeq \sum_{e: \operatorname{Equiv}(X,Y)} \bigl(\operatorname{transport}_F(\operatorname{ua}(e),s) = t\bigr). $$
When forgetting labels while retaining symmetries, the collective moduli space is the homotopy quotient:
$$ \mathsf{CollectiveModuli} := M \mathbin{/\mkern-6mu/} G := \left| [n] \longmapsto M \times G^n \right|. $$
For discrete sets and groups in $\mathcal{S}$, the connected component containing $m \in M$ is equivalent to $B\operatorname{Stab}_G(m)$. Symmetries are preserved as nontrivial loop spaces, rather than being discarded.
10. A four-agent collective example
Consider an epoch with $m = 4$ members, $d = 2$ resources (core-hours, GiB-hours), and $r = 1$ common asset. Capacities and contributions $\kappa_a = c_a$ are: - Member $a_1$: capacity $(8,4)$, admitted need $(1,2)$ - Member $a_2$: capacity $(4,8)$, admitted need $(3,2)$ - Member $a_3$: capacity $(0,4)$, admitted need $(4,4)$ - Member $a_4$: capacity $(0,0)$, admitted need $(4,8)$
Total contribution is $(12,16)$. Setting aside maintenance reserve $h = (4,4)$ leaves budget $b = (8,12)$. Total demand is $N = (12,16)$. The bottleneck ratio is:
$$ \lambda^* = \min\left\{1, \frac{8}{12}, \frac{12}{16}\right\} = \frac{2}{3}. $$
Allocations $x_a^* = \frac{2}{3} n_a$ are: $x_1^* = (2/3, 4/3)$, $x_2^* = (2, 4/3)$, $x_3^* = (8/3, 8/3)$, and $x_4^* = (8/3, 16/3)$. Total allocated is $(8, 32/3)$, leaving unused memory budget $4/3$. Member $a_4$ receives a positive allocation despite zero contribution; member $a_1$ contributes the most CPU but receives the smallest share because its need is smallest. Quorum threshold $q = 3$ with fault tolerance $f = 1$ ensures $2q = 6 > 5 = m + f$ and $q = m - f$.
11. Reproducible evidence and verification supplement
The accompanying source package includes verify.py, a standalone Python 3 program using rational arithmetic (fractions.Fraction). Verification results include:
- Reproduction of the four-agent allocation example and Table 1.
- Complete grid check across all 576 three-member two-resource input instances, verifying optimality against 3,730 candidate grid vectors.
- Exhaustive verification of all 25 ordered pairs of four-member quorums ($q=3$), confirming intersection size $\geq 2$ and honest intersection.
- Verification of the $\operatorname{Sym}(4)$ symmetry obstruction.
References
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