superadditivity

collective intelligence is the claim that the whole sees what no part can. tru makes it a number. superadditivity is the advantage of the collective focus $\phi^*$ — the tri-kernel over the whole cybergraph — over the focus an individual neuron computes from its own local view. when it is positive, the network is provably more than the sum of its neurons.

this is the operational test of the collective focus theorem: that theorem proves $\phi^*$ exists, is unique, and converges; superadditivity measures that it is emergent. syntropy is the task-free version of the same fact — superadditivity grounds it in a task.

the ego baseline

each neuron $\nu$ holds a local view: the sub-cybergraph of its own cyberlinks and their endpoints, the ego-net at radius $r$. running the same tri-kernel on that view alone yields the ego focus $\phi^*_\nu$ — the sharpest picture $\nu$ can form without the rest of the network. the collective focus $\phi^*$ is the tri-kernel over the full weighted graph. superadditivity compares the two.

the metric

fix a task with a quality score $Q \in [0,1]$, higher better (see tasks). let $Q(\phi)$ be the score when particles are ranked by focus $\phi$. define

$$\sigma_{\text{mean}} = Q(\phi^*) - \tfrac{1}{|N|}\sum_{\nu} Q(\phi^*_\nu), \qquad \sigma_{\text{best}} = Q(\phi^*) - \max_\nu Q(\phi^*_\nu).$$

$\sigma_{\text{mean}} > 0$ says the collective beats the typical neuron. $\sigma_{\text{best}} > 0$ is the strong claim — the collective beats every individual, the signature of genuine emergence. ($\sigma$ here is superadditivity, local to this page; it is not the signal proof $\sigma$ nor the dialect map $\sigma(\ell)$ of ct0.)

tasks

two task families ground $Q$ using only the graph, no external labels:

• link prediction — hide a fraction of cyberlinks, score candidate pairs $(p,q)$ by a focus-derived affinity, measure ROC-AUC and average precision. tests whether $\phi^*$ captures latent structure.
• retrieval@k — for a query particle, rank the rest by focus-weighted proximity and measure the precision of the top $k$ against held-out neighbours. tests whether $\phi^*$ concentrates attention on the right particles.

both are self-contained, consistent with tru's epistemics: the graph grades itself.

connection to syntropy

syntropy $J(\phi^*) = D_{\mathrm{KL}}(\phi^* \,\|\, u)$ is the task-free measure — how far collective focus departs from uniform noise. superadditivity is its task-grounded sibling. a sharper $\phi^*$ (higher $J$) concentrates attention on real structure, which is what raises $Q$, so $\sigma$ tracks $J$. $J$ answers "is the focus structured?"; $\sigma$ answers "is that structure useful, and more useful than any single neuron's?" both rise together as the graph accumulates honest cyberlinks.

the generalized collective focus theorem

the collective focus theorem (tri-kernel §3) proves $\phi^*$ exists, is unique, and converges. the generalization concerns its quality. let $\lambda_2$ be the algebraic connectivity (Fiedler value) of the weighted graph. adding a non-redundant cyberlink — one bridging otherwise weakly-connected regions — raises $\lambda_2$.

claim, as measured: $\sigma$ increases with $\lambda_2$ — the collective advantage grows with connectivity. $J(\phi^*)$ does the opposite: it falls as $\lambda_2$ rises.

the same $\lambda_2$ already governs convergence: the heat term contracts at rate $e^{-\tau\lambda_2}$ inside $\kappa$ (tri-kernel §2.2). so algebraic connectivity sets how fast $\phi^*$ converges and how much collective advantage $\sigma$ it carries. but it does not raise syntropy — adding edges spreads focus toward uniform, and a more uniform $\phi^*$ has less $J$. the original conjecture ("$J$ and $\sigma$ both rise with $\lambda_2$") conflated two different things: connectivity grows the collective's edge over any individual, while sharpness (syntropy) comes from structure being concentrated, which dense connectivity dilutes.

status (benchmark-tested, Karate Club, fixed-vertex spanning-tree sweep — rs/examples/superadditivity.rs):

• $\lambda_2$–convergence: proven (§2.2).
• $\sigma$–$\lambda_2$: supported. Pearson $+0.5$ ($\sigma_{\text{mean}}$), and $\sigma_{\text{best}} > 0$ at every connectivity level — the collective beats its strongest neuron throughout.
• $J$–$\lambda_2$: refuted. Pearson $\approx -0.7$; $J$ decreases monotonically as edges are added. The sparse spanning tree is the most syntropic state; densification lowers $J$.

open: whether the $\sigma$–$\lambda_2$ rise holds on larger graphs and every operator blend $(\lambda_d,\lambda_s,\lambda_h)$; and the right way to state the corrected law (syntropy rewards concentration, superadditivity rewards connectivity — they are distinct axes).

benchmark

the measurement harness, smallest sanity instance first:

1. take a graph — Zachary's Karate Club (34 particles) is the minimal sanity check; the target is the live cybergraph at $10^6+$ cyberlinks.
2. compute the collective $\phi^*$ and $J(\phi^*)$.
3. for each neuron, compute its ego $\phi^*_\nu$.
4. score $Q$ on link-prediction and retrieval@10 for $\phi^*$ and every $\phi^*_\nu$.
5. report $\sigma_{\text{mean}}$, $\sigma_{\text{best}}$, $J(\phi^*)$, and their growth as edges are added in connectivity order ($\lambda_2$ rising).

a passing result is $\sigma_{\text{best}} > 0$ that grows with $\lambda_2$ — the collective strictly outperforming its strongest member, by more as the graph connects. concrete figures are benchmark output, reported only once measured on the conformant engine (arithmetic, fixed-point), never asserted in this contract.

why it matters

emergence stops being a slogan and becomes a number that runs on a phone. on sparse graphs — the realistic regime for a decentralized network of millions of cyberlinks — the locality-bounded tri-kernel measures superadditivity at $O(\deg)$ cost per update, where a dense graph transformer pays $O(n^2)$ and centralizes. the collective wins not by being a bigger model but by wiring many small local views into one focus none of them held alone.

many small lights, once wired together, see farther than a single sun.

see syntropy for the task-free measure · collective focus theorem for existence and uniqueness · tri-kernel §2.2 for the $\lambda_2$ that ties convergence to intelligence · rewards for why non-redundant links are what pays.