a quantity that remains constant through every transformation. the constraint that shapes where convergence can go
without conservation, a system can collapse to zero, explode to infinity, or drift without limit. conservation forces the dynamics onto a bounded surface where the banach fixed-point theorem can find equilibrium
in physics
three conservation laws hold across all known physics:
energy — the total energy of an isolated system never changes. it transforms between kinetic, potential, thermal, electromagnetic — but the sum is constant. discovered empirically, later understood as a consequence of time-translation symmetry (Noether's theorem, 1918)
momentum — total momentum is conserved in the absence of external forces. consequence of space-translation symmetry
charge — electric charge is neither created nor destroyed. consequence of gauge symmetry
every conservation law corresponds to a symmetry of the system (Noether's theorem). conservation is not a rule imposed from outside — it is structure that the dynamics cannot violate
in cyber
the cybergraph has three conservation laws enforced at every state transition:
focus conservation
$$\sum_i \text{focus}(i) = 1 \quad \text{always}$$
focus can flow between neurons, be consumed by computation, and regenerate proportionally to stake. it cannot be created from nothing, destroyed, or exceed 1 in total
this single constraint does the work that other systems split across gas models, fee markets, and priority auctions. it forces the tri-kernel onto the probability simplex $\Delta^{|P|-1}$, where convergence produces a unique Boltzmann distribution as equilibrium
enforced in nox by stark circuit constraints — an invalid conservation proof means an invalid state transition, rejected by every verifier
balance conservation
$$\sum_i \text{balance}(i) = B_{\text{total}} \quad \text{for non-minting transactions}$$
tokens move between neurons but the total supply is fixed outside minting events. enforced by polynomial commitment structure
energy conservation (privacy layer)
$$\sum(\text{record values}) = \text{initial} + \text{minted} - \text{burned}$$
enforced by ZK circuit constraints. the network verifies conservation without seeing individual values — private ownership with public aggregates
why conservation shapes convergence
conservation is not a side constraint. it is the reason convergence produces something meaningful
without $\sum \phi_i = 1$: the tri-kernel could push all focus to zero (everything becomes irrelevant) or to infinity (everything becomes infinitely important). both are meaningless. conservation eliminates these degenerate outcomes and forces the system to make choices — emphasizing one particle necessarily defocuses others
this is why focus works as both attention and fuel simultaneously. a conserved quantity that represents attention is automatically scarce. scarcity forces prioritization. prioritization creates structure. structure is syntropy
in thermodynamics: energy conservation forces the system to find the Boltzmann distribution — the unique distribution that maximizes entropy subject to fixed total energy. in cyber: focus conservation forces the system to find $\pi^*$ — the unique distribution that minimizes free energy subject to fixed total focus. same mathematics, same principle
conservation and costly signals
conservation is what makes cyberlinks meaningful. because focus is conserved, spending it on a link is a real sacrifice — the neuron cannot spend the same focus elsewhere. this is the costly signal property
without conservation, signaling is free. free signals carry no information (cheap talk). conservation transforms every cyberlink into an economic commitment — a statement backed by finite resources. this is the bridge between physics and game theory: conservation laws create the scarcity that makes incentives work
conservation and proof by simulation
the cybics postulate: every truth accessible to intelligence is a fixed point of convergent simulation under conservation laws
the last three words are load-bearing. convergence without conservation is unconstrained optimization — it can find any fixed point, including trivial ones. conservation constrains the space of admissible states, ensuring the fixed point is physically meaningful
in the formal definition: a simulation-proof of property $P$ requires a dynamical system $(Ω, T, C)$ where $C(T(ω)) = C(ω)$ for all $ω$. the conservation law $C$ is part of the proof. remove it and the proof loses its anchor
the symmetry beneath
Noether's theorem: every continuous symmetry of a system implies a conserved quantity
in the cybergraph, focus conservation corresponds to a symmetry: the tri-kernel is invariant under relabeling of time steps. it does not matter when a cyberlink is created — the same graph structure produces the same $\pi^*$. this time-invariance is the symmetry; focus conservation is the consequence
see convergence for why conservation shapes the destination. see focus for the conserved quantity. see costly signal for the economic consequence. see cybics for the philosophical role