vacuum_has_zero_strain
plain-language theorem explainer
The vacuum ledger carries no strain by construction, so the J-cost of its derived strain is identically zero. Ledger-based gravity models cite this to set the zero baseline before computing curvature from non-vacuum densities. The proof is a direct reflexivity that follows from the definitions of vacuumLedger and strainFromLedger.
Claim. Let $V$ be the vacuum ledger and $S$ the strain map from ledgers to their associated $J$-cost. Then $J(S(V)) = 0$.
background
In the RRF module, gravity is defined as the geometric manifestation of ledger constraint density, with mass identified as transaction density per volume and curvature as the resulting constraint pressure. The $J$-cost function, introduced via the structure in PhiForcingDerived.of and calibrated on the multiplicative monoid of positive reals in LedgerFactorization.of, quantifies strain or defect. The vacuum ledger is the zero-density base case, distinct from massiveLedger and constructed so that strainFromLedger returns a value whose $J$ is zero.
proof idea
The proof is a one-line term-mode reflexivity that unfolds the definitions of vacuumLedger and strainFromLedger to obtain the equality directly.
why it matters
This result supplies the zero-strain baseline required for the scalar curvature construction that follows in the same module, supporting the claim that gravity is ledger curvature rather than an independent force. It aligns with the Recognition Science mapping of curvature to constraint pressure and the phi-ladder mass formulas that begin from the vacuum case. No downstream uses are recorded yet, leaving its role as a foundational lemma for later curvature derivations.
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