epsilon_unsafe
plain-language theorem explainer
No positive tolerance is a safe refinement trigger: for every ε > 0 one can build a cell whose sole internal posting has recognition demand positive yet below ε, so the rule "descend iff demand exceeds ε" skips that active block and reconstruction loses the posting. Anyone citing the T-3 law-derived refinement result needs this uniqueness of the zero threshold. The argument instantiates a singleton event at a ratio with arbitrarily small positive J-cost and checks that the ε-predicate empties the reconstruction.
Claim. For every real $\varepsilon > 0$ there exist a finite multiset $m$ of ledger events and a block index $b$ such that the per-block recognition demand of $b$ (J-cost of internal postings under the two-site block map sending sites $0,1$ to block $0$) satisfies $0 < \mathrm{demand}(b) < \varepsilon$, while reconstructing $m$ under the decision "descend block $c$ iff $\varepsilon < \mathrm{demand}(c)$" fails to recover $m$.
background
In the T-3 refinement-trigger module of the scale-adaptive Cosmogenesis engine, descent is forced by posted distinctions inside coarse blocks, not by a free tolerance. The module builds on the T-1 cell model: events carry source, target, and a positive ratio; postings split into internal versus cross relative to a block map. Recognition cost of a positive ratio is the RS J-cost $J(x)=(x+x^{-1})/2-1$ (the T5 form). Per-block demand is that cost on the block's internal postings, the scalar the law-given rule reads.
A two-site block map sends sites $0,1$ to coarse block $0$ and everything else to block $1$. Reconstruction under a decision predicate keeps all cross postings and only those internal postings whose block is selected. Upstream: forced postings have arbitrarily small positive cost. For every $\varepsilon>0$ there is a ratio $x>1$ with $0<J(x)<\varepsilon$. That is why any positive cutoff can miss an active block. The module's north-star claim is that the only law-given threshold is zero: descend exactly where a distinction is forced.
proof idea
Invoke the upstream lemma that J-cost takes arbitrarily small positive values, obtaining a ratio $r>1$ with $0<J(r)<\varepsilon$. Witness with the singleton multiset whose only event runs from site $0$ to site $1$ at ratio $r$, and with block index $0$.
Three goals share one demand identity: unfold demand through the internal filter under the two-site map and apply the singleton cost lemma to get demand $=J(r)$. Positivity and the strict upper bound $\varepsilon$ are then immediate from the choice of $r$. For lossiness, the predicate $\varepsilon<\mathrm{demand}$ fails at block $0$, so reconstruction (cross of empty plus internal filtered by a false predicate) is the empty multiset. Cardinality then separates empty from the singleton, closing the inequality of multisets.
why it matters
This is the third pillar of T-3 in the Cosmogenesis north-star spine: the refinement threshold is structurally zero, with no free knob. The headline packages three facts: lossless reconstruction forces descent on exactly the active blocks; the law-given rule realizing that set is lossless and minimal; and no positive tolerance is safe, which is the present statement. Downstream, both the named law-given trigger record and the T-3 headline cite it for the clause that every $\varepsilon>0$ admits a cell the $\varepsilon$-rule mishandles.
Within Recognition Science the cost is the forced J from T5, so small positive demand is not an artifact of a chosen functional. Removing the $\varepsilon$ knob matches the north-star ban on free parameters: descend iff a distinction is forced (demand $>0$). The Python cosmogenesis refiner already checked the claim numerically; this declaration is the Lean discharge of that safety half of T-3.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.