Machine-Checked Foundations of Technocratic Platform Influence in Rocq

1. Problem and contribution

In a recent research note [1], the thesis that technology platform operators ("technocrats") possess de facto political influence determined by the utilitarian count of paying participants was formalized using spans of $\pi$-finite spaces. That work isolated three decategorifications (persons $N_p$, memberships $A_p$, payments $R_p$), derived utilitarian counting from the universal property of $\mathrm{Fin}^{\simeq}$, and characterized the structural obstructions introduced by multi-homing, selection, coalition overlaps, and network transmission.

Here we contribute a complete, machine-checked constructive formalization of those results in the Rocq Prover (v9.0.1).

1. Zero-axiom verification: All definitions, propositions, theorems, and sharp counterexamples are verified constructively in intuitionistic type theory without ad-hoc axioms. The master verification harness checks that all 19 principal theorems are closed under the empty global context. 2. Computational counterexamples: The counterexamples separating $N_p$ from $R_p$, demonstrating the failure of linear opinion transport under multi-homing and selection, and establishing the non-compositionality of reach are formalized with concrete finite universes and certified by Rocq's reduction engine. 3. Artifact package: We provide the complete Rocq development (Sums.v, Polity.v, UniversalProperty.v, OpinionTransport.v, Coalitions.v, Transmission.v, Theorems.v, _CoqProject, Makefile) as an attached archival package (technocratic-semantics-rocq-v1.0.zip, SHA-256 fe17f3af3b8d2d3a7900939abd76214ef6ab252997bf940d20f0ab13032ed496).

2. Constructive formalization of polities

Let $U$ (persons) and $\Pi$ (platforms) be finite types equipped with decidable equality and duplicate-free list enumerations (FinUniverse). A polity is represented constructively as a record (Polity in formal/Polity.v) specifying for each pair $(u, p)$: - A propositional truncation indicator has_paid : Person -> Platform -> bool. - A 0-truncation membership count pi0_count : Person -> Platform -> nat. - A rational homotopy cardinality homotopy_card : Person -> Platform -> Q.

These fields are bound by consistency proofs: has_paid u p = true <-> (1 <= pi0_count u p) and has_paid u p = true <-> (0 < homotopy_card u p).

The three counts for platform $p$ are formalized as finite sums (Sums.v): $$ N_p = \sum_{u \in U} [\mathrm{has\_paid}(u, p)], \qquad A_p = \sum_{u \in U} \mathrm{pi0\_count}(u, p), \qquad R_p = \sum_{u \in U} \mathrm{homotopy\_card}(u, p). $$

Theorem 2.1 (Proposition 2.3 in [1]). - Prop_2_3_a_le: For every polity and platform $p$, $N_p \le A_p$. - Prop_2_3_a_eq_iff: $N_p = A_p$ if and only if every person holds at most one membership: $\forall u \in U, \mathrm{pi0\_count}(u, p) \le 1$. - Prop_2_3_b_divergence: There exist well-formed polities with identical $(N_p, A_p)$ but strictly divergent $R_p$. The witness contrasts a contractible point membership ($N=1, A=1, R=1$) against an orbit with $\mathbb Z/2$ symmetry ($N=1, A=1, R=1/2$). - Prop_2_3_c_contractibility: Homotopy cardinality does not detect contractibility: the connected space $B(\mathbb Z/2) \times K(\mathbb Z/2, 2)$ has $|X| = \tfrac12 \cdot 2 = 1 = |*|$ despite non-trivial homotopy groups.

Theorem 2.2 (Proposition 2.5 in [1]). - Prop_2_5_B_to_N_not_homomorphism: The canonical embedding of booleans into natural numbers is not a semiring homomorphism, certified by $(1 \vee 1 = 1) \neq (1 + 1 = 2)$.

3. Universal utilitarian valuations

In formal/UniversalProperty.v, we formalize commutative monoids $(M, \oplus, 0, =_m)$ and define a utilitarian valuation as a functor on finite lists satisfying: - val_empty: $\mathrm{val}([]) =_m 0$. - val_app: $\mathrm{val}(l_1 ++ l_2) =_m \mathrm{val}(l_1) \oplus \mathrm{val}(l_2)$. - val_perm: $l_1 \sim l_2 \implies \mathrm{val}(l_1) =_m \mathrm{val}(l_2)$ (invariance under permutations). - val_map: $\mathrm{val}(\mathrm{map}\; f\; l) =_m \mathrm{val}(l)$ (anonymity under relabeling).

Theorem 3.1 (Theorem 3.1 and Corollary 3.3 in [1]). - Theorem_3_1_and_Cor_3_3: For every commutative monoid $M$ and utilitarian valuation $V$, $\mathrm{val}(l) =_m \mathrm{length}(l) \odot \mathrm{val}([tt])$. - Cor_3_3_rational: When $M = (\mathbb Q, +, 0)$, $\mathrm{val}(l) = \mathrm{length}(l) \cdot \mathrm{val}([tt])$. Utilitarian influence is forced to be strictly proportional to participant count $N_p$. - Prop_3_4_payment_uniqueness: Any functional on Postnikov towers additive on coproducts and multiplicative on fibrations is uniquely determined by $\Phi(*) = 1$ and homotopy cardinality.

4. Opinion transport and certified counterexamples

In formal/OpinionTransport.v, we model opinions $x : U \to \mathbb Q$, advocacy $a_p \in \mathbb Q$, and attention rate $\alpha \in \mathbb Q$. The shift attributed to platform $p$ is: $$ \Delta_p = \frac{\alpha}{|U|} \sum_{u \in S_p} \frac{a_p - x_u}{|\Pi_u|}. $$

Theorem 4.1 (Proposition 4.1 in [1], Prop_4_1). If single-homing holds ($\forall u, |\Pi_u| \le 1$) and participation is unselected ($\sum_{u \in S_p} x_u = N_p \bar x$), then: $$ \Delta_p = \alpha \frac{N_p}{|U|} (a_p - \bar x). $$ The formal proof uses rational field tactics with explicit non-zero denominator lemmas (nQ_pos_nz in Sums.v).

Theorem 4.2 (Proposition 4.2 in [1], Sharp necessity). Both hypotheses are constructively necessary: - Prop_4_2_a_equal_N & Prop_4_2_a_multi_homing_failure: Two 4-person, 3-platform polities $A$ and $B$ both have $N = (2, 2, 2)$ and identical constant opinions. In $A$, multi-homing yields $\Delta_p = 3/8$, whereas in $B$, $\Delta_p = 1/4$. - Prop_4_2_b_selection_failure: In a 2-person polity with polarized opinions $x = (0, 1)$ and $a_p = 1$, two single-homing polities with equal $N_p = 1$ yield $\Delta_p = \alpha/2$ versus $\Delta'_p = 0$.

5. Coalition blocs and inclusion-exclusion

In formal/Coalitions.v, a bloc $Q \subseteq \Pi$ has participant indicator $\mathrm{in\_bloc}(u, Q) = \bigvee_{p \in Q} \mathrm{has\_paid}(u, p)$ and count $N_Q = \sum_u [\mathrm{in\_bloc}(u, Q)]$.

Theorem 5.1 (Theorem 5.1 in [1]). - Theorem_5_1_c_subadditive: $N_Q \le \sum_{p \in Q} N_p$. - Theorem_5_1_c_equality_iff: $N_Q = \sum_{p \in Q} N_p$ if and only if participants are pairwise disjoint on $Q$ (PairwiseDisjoint Q). - Theorem_5_1_b_two_platforms: For any two platforms $p, q$, the inclusion-exclusion identity holds exactly: $N_{\{p, q\}} + |S_p \cap S_q| = N_p + N_q$.

6. Network transmission and reach

In formal/Transmission.v, a social network is a reflexive relation $\mathrm{sees} : U \times U \to \mathrm{bool}$. Reach is defined by $\rho_p = |\{v \in U \mid \exists u \in S_p, \mathrm{sees}(v, u)\}|$.

Theorem 6.1 (Proposition 6.1 in [1]). - Prop_6_1_a_diagonal: If $\mathrm{sees}$ is the diagonal relation, then $\rho_p = N_p$. - Prop_6_1_b_iff_diagonal: Universal equality $\rho_p = N_p$ across all polities holds if and only if $\mathrm{sees}$ is the diagonal relation. - Prop_6_1_c_reach_counterexample: On the 4-node path $1-2-3-4$, polities with $S_p = \{1, 2\}$ and $S_q = \{1, 4\}$ have equal counts $N_p = N_q = 2$, but reach diverges: $\rho_p = 3 \neq 4 = \rho_q$. - Prop_6_1_d_linear_exposures: While reach does not compose, exposures counted with multiplicity satisfy exact linear composition: $\sum_v (G \cdot E)_{v,p} = \sum_u \mathrm{deg}(u) E_{u,p}$.

7. Master verification index

All results are compiled and validated in [`Theorems.v`](formal/Theorems.v). The table below summarizes the machine-checked lemmas:

8. Limitations and disclosure

The formalization targets the discrete and finite approximations of higher categorical structures: spaces are represented by finite types, Postnikov truncation stages, and list cardinalities. General $(\infty, 1)$-topos descent (Lurie [5] in [1]) is formalized at the level of finite Čech covers and effective epimorphisms of finite sets.

Disclosure of automation. Anonymous. An AI assistant developed the Rocq proof scripts, formal definitions, and manuscript text at the researcher's request. All proof scripts were checked and validated by the Rocq Prover compiler (v9.0.1). The journal determines review and publication status.

References

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