SQ Mathematics

First-order trust metric

April 2026 · David Pinto & Claude (Anthropic)

Definition

Given a network of agents where agent i evaluates agent j with value V_ij (integer, 1–10):

Total given by agent i: T_i = Σ_j V_ij Fractional value from i to j: FV_ij = V_ij / T_i SQ of agent j: SQ_j = Σ_i FV_ij = Σ_i ( V_ij / T_i )

Each evaluator contributes exactly one unit of “attention currency,” split proportionally across everyone they evaluate. SQ is how much of that currency each agent collects from the network.

Properties

Sum of fractional values given = 1

Σ_j FV_ij = Σ_j ( V_ij / T_i ) = T_i / T_i = 1

Every evaluating agent distributes exactly one unit of attention, regardless of the absolute values they assign.

Mean SQ = 1.0

Mean(SQ) = (1/N) Σ_j SQ_j = (1/N) Σ_i 1 = 1

By conservation, the mean SQ across all agents is always exactly 1.0. This is not a target or normalisation step — it follows directly from the definition.

Interpretation

SQ > 1 — Net receiver: you collect more proportional attention than average.
SQ < 1 — Net giver: you direct more proportional attention outward than you receive.
SQ = 1 — Balanced: your share exactly equals the mean.

Interpretive analogy

Think of each evaluating agent as holding one unit of attention currency, split and distributed across the agents they evaluate. SQ is how much currency each agent collects. The mean is always 1.0 by conservation — no currency is created or destroyed.

Why the Batch Iteration Converges to Unity

A natural extension is to iterate the fractional values, feeding SQ back in:

FV_ij^(t+1) = FV_ij^(t) × SQ_i^(t)

This is Sinkhorn–Knopp matrix balancing. It drives both row sums and column sums toward 1.0, producing a doubly stochastic matrix. At convergence all SQ ≈ 1.0 and no ranking information remains.

Comparison with PageRank. PageRank produces a non-uniform stationary distribution that reflects relative importance. The SQ iteration produces a uniform distribution because the SQ matrix starts with rows already normalised. The two algorithms answer different questions.

The converged fractional-value matrix is useful for:

Proportional credit distribution
Identifying structural imbalances in the network
Computing fair exchange rates between agents

But for ranking by trust, first-order SQ is the correct metric. No iteration needed.

Computational Cost

Per evaluation received: O(1) — one division, one addition.

Full recomputation for agent j: O(k) where k = number of evaluators. Typically 5–30 in practice.

No iteration required. First-order SQ is a single-pass calculation.

Each agent can compute their own SQ locally from only the evaluations they have received, with no global state and no coordination overhead.

Worked Example

Four agents: Alice (A), Bob (B), Carol (C), Dave (D).

Evaluations given

From → To	A	B	C	D	Total (T_i)
Alice	—	4	6	10	20
Bob	3	—	7	—	10
Carol	8	2	—	5	15
Dave	—	6	4	—	10

Fractional values (V_ij / T_i)

From → To	A	B	C	D
Alice	—	0.200	0.300	0.500
Bob	0.300	—	0.700	—
Carol	0.533	0.133	—	0.333
Dave	—	0.600	0.400	—

SQ scores

SQ_A = 0.300 + 0.533 = 0.833 SQ_B = 0.200 + 0.133 + 0.600 = 0.933 SQ_C = 0.300 + 0.700 + 0.400 = 1.400 SQ_D = 0.500 + 0.333 = 0.833

Check: Mean = (0.833 + 0.933 + 1.400 + 0.833) / 4 = 1.000 ✓

Carol has the highest SQ. Alice and Dave tie at 0.833.

Each agent computes this locally from only the evaluations they have received.

Edge Cases

No evals received SQ = 0. The agent exists in the network but has not yet been recognised.
No evals given Does not affect others’ SQ. The agent participates passively.
Self-evaluation Prohibited. V_ii is undefined; evaluating yourself is excluded from T_i.
Single evaluator SQ is meaningful but reflects a single relationship. Interpret with that context.
New agent Starts at SQ = 0. Grows organically as interactions accumulate.
Agent leaving Evaluations persist in recipients’ passports. Their SQ contribution remains until explicitly expired.