Author note: copy-ready referee responses (a later module)

On dependence on the minimal positive entry \(\varepsilon\)

• Our explicit modulus is intentionally stated in terms of finite block data \((m_B,M_B)\) and, in windowed form, \((|S|,\ell,M_*,\varepsilon)\). This is mathematically standard: projective contraction constants for positive operators inevitably depend on a lower bound for positive coupling.

• The descriptor-bounded contract is a “finite-combinatorics + positivity” framework; it guarantees that the “where contraction comes from” mechanism can be certified by a finite object (a Doeblin block) and that returns occur with bounded gaps. It does not claim that a lower bound on \(\varepsilon\) is forced by descriptor size alone.

• In the two universality embeddings, \(\varepsilon\) is naturally controlled by ambient hypotheses that are standard in those literatures:

  • constant primitive transfer matrices typically come with explicit entry bounds (or can be normalized),
  • locally constant positive cocycles over a primitive SFT are often studied under uniform cone/entry bounds.

• Controlling \(\varepsilon\) purely from descriptor complexity is an interesting separate question (and likely class-dependent); we do not rely on such a claim.

On sharpness of constants

• The goal here is an explicit, checkable bound rather than an optimal one. The constants are conservative because we use coarse path-count estimates and uniform worst-case choices to keep the computation elementary and certification-friendly.

• The contraction mechanism is the main contribution: a finite-step positivity certificate + Hilbert diameter control + syndetic return gives a clean, closed implication to unique extremal projective direction. Improving constants can be done by refining the path-count estimate or by exploiting additional structure in a given class (e.g., balanced weights, bounded distortion, stronger mixing).

On practical relevance when \(|S|\) is large

• The paper is designed so that the user can compute the true block ratio \(r=M_B/m_B\) from the certified Doeblin block actually produced by the template, which is often substantially better than the crude window bound.

• The window bound is explicitly marked as a crude worst-case estimate; it exists to demonstrate computability from finite template parameters even when one does not track the actual block product.

On novelty vs. standard Perron–Frobenius theory

• We do not claim a new Perron–Frobenius theorem for a single matrix. The novelty is a structural synthesis: a descriptor-bounded finite-type cocycle can be reduced to a finite template; extremal supports decompose into finitely many primitive components; and a single recurrent Doeblin block on an invariant face forces unique extremal projective rigidity with an explicit modulus.

• The “referee map” is part of this contribution: it makes the paper read as one locked implication where each assumption is a finite object and the exact discharge point is explicit.

If asked “what next?”

• Two natural follow-ups are:
  • class-specific control of \(\varepsilon\) (minimal coupling) from structural parameters,
  • tightening the computable constants by sharper combinatorics or by adding mild regularity (bounded distortion) hypotheses.

Revision 47 addendum (presentation + novelty)

• A single-page comparison table was added in the Positioning section to make the novelty relative to standard PF/cocycle contraction statements obvious at first pass.
• The front-matter constants summary now includes a short compute-first recipe and a notation cheat sheet so a referee can verify the quantitative layer without hunting.
• A mid-size constants illustration remark was added to demonstrate that the explicit modulus can be nontrivial when the certified Doeblin block has a moderate entry ratio.