rs_energy_unity
plain-language theorem explainer
The Recognition Science energy of the all-ones (unity) configuration vanishes for every positive system size N. Thermodynamic and variational arguments that need a zero-energy ground state cite this identity. The proof is a one-line wrapper: energy is defined as log-charge, and the unity configuration already has log-charge zero.
Claim. For every positive integer $N$, if $c$ is the unity configuration on $N$ entries (every entry equal to $1$), then the RS energy $E(c) := \sum_i \log(x_i)$ satisfies $E(c) = 0$.
background
Module F-011 builds temperature and the canonical ensemble from the ledger's J-cost and finite observer resolution. Entropy is total defect $S(c) = \sum_i J(x_i)$; energy is the conserved log-charge $E(c) = \sum_i \log(x_i)$. Both come from earlier foundation modules (InitialCondition, VariationalDynamics).
The unity configuration is the zero-defect state: every entry equals $1$. Upstream, unity_log_charge_zero already records that this state has vanishing log-charge (and vanishing total defect). RS energy is defined as that same log-charge, so the thermodynamic ground-state energy is forced to zero once the identification is made.
In the module narrative, $E$ plays the role of internal energy because it is conserved under the variational dynamics, extensive, and selects equilibrium through the variational principle.
proof idea
One-line term proof. Unfold the definition of RS energy as log-charge of the configuration, then apply the upstream lemma that the unity configuration has log-charge zero (proved by unfolding entries to $1$, using $\log 1 = 0$, and summing zeros). No new algebra is introduced here.
why it matters
Thermodynamics needs a definite zero of energy before temperature $T = \partial E/\partial S$ and equilibrium entropy-as-a-function-of-energy can be stated. This identity pins $E=0$ at the unique zero-defect (unity) state, matching the module claim that $S=0$ at unity and $S>0$ off unity.
It sits beside the sibling non-negativity and zero-iff-unity facts for entropy, and feeds the equilibrium-entry and equilibrium-entropy development in the same file. In the broader RS chain it is the energy half of the ground-state package already forced for defect/cost: the consensus (all-ones) configuration is the zero-cost, zero-charge reference against which heat-bath coarse-graining and temperature are measured. No downstream consumers are recorded yet; the immediate use is local to the thermodynamic ladder in this module.
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