CrossLatticeTransform
plain-language theorem explainer
CrossLatticeTransform defines a map that updates two agents' states at once while keeping their combined sigma fixed. Decision theorists or ethicists formalizing the DREAM theorem would cite it to isolate love as the sole inter-agent coupling among the 14 virtues. The declaration is a direct structure with an apply function and an invariance axiom on total sigma.
Claim. A cross-lattice transform is a pair consisting of a function $f : S × S → S × S$ (with $S$ the set of agent states) together with the property that for all $s_1, s_2 ∈ S$, if $(s_1', s_2') = f(s_1, s_2)$ then $σ(s_1') + σ(s_2') = σ(s_1) + σ(s_2)$, where $σ$ denotes the imbalance measure on each state.
background
AgentState packages an agent's configuration as a real sigma (imbalance between private preference and public report) together with a positive integer lattice size. The module distinguishes single-lattice transforms, which rearrange structure inside one agent's lattice without altering its sigma, from cross-lattice transforms that couple two agents and permit sigma redistribution. Upstream, sigma is the Abilene-paradox charge: +1 when an agent privately prefers stay but publicly votes go, -1 for the reverse, and 0 for truthful reporting.
proof idea
The declaration is a structure definition that introduces the type with exactly two fields: the apply map from pairs of AgentState to pairs of AgentState, and the conserves_total axiom asserting invariance of the sum of sigma values.
why it matters
CrossLatticeTransform supplies the type used by coupling_conserves_total, love_changes_individual_sigma, love_equilibrates, and loveOperator. It isolates the coupling mechanism that the module invokes to prove love is the unique virtue among the 14 that changes individual sigma rather than preserving it. The construction directly supports the claim that only love creates bonds between previously separate lattices.
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