ledger_self_grounding
plain-language theorem explainer
The theorem shows the ledger is self-grounded because every J-cost on a positive real is non-negative. Workers on the simulation hypothesis in Recognition Science cite it to confirm that the ledger requires no external validator. The proof is a one-line wrapper that applies the non-negativity lemma for J-cost after introducing the positive argument.
Claim. For every positive real number $x$, the J-cost satisfies $J(x) = (x-1)^2/(2x) > 0$ unless $x=1$, so the ledger is self-grounded.
background
The J-cost function is the recognition cost of a positive configuration, obtained from the multiplicative recognizer or observer forcing; its non-negativity follows from rewriting as a square over a positive denominator. RSExists holds exactly when this cost is zero, which occurs only at the fixed point $x=1$. The module IC-004 treats the ledger as identical to physical reality, so the simulation hypothesis collapses because any external substrate would itself be a ledger entry.
proof idea
The proof is a one-line wrapper that applies the lemma Jcost_nonneg after introducing the positive real $x$.
why it matters
This supplies the non-negativity step required by the downstream certificate ic004_certificate, which records that any universe reproducing RS dynamics must operate on reals and that the simulation/reality distinction has no content. It closes the self-grounding clause of IC-004, confirming the ledger is the unique physical substrate. The result rests on the upstream non-negativity of J-cost and aligns with the Recognition Composition Law by keeping costs non-negative under composition.
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