Same Entropy, Different Borders: A Certified Study of the Period-Tripling Quadratic Cascade

We reconstruct the entropy of three real quadratic maps from the border: first the open system, then the scalar. The maps are f_c(x) = x² + c at the certified centres of the period-3, period-9 and period-27 components of the period-tripling cascade (N1, N2, N3). For each map we construct, without using entropy, a capture region generated by a certified boundary point, a finite forward-invariant set of endpoints, a Markov partition of the surviving set and its transition graph. Entropy is introduced only afterwards, as the exponential growth rate of admissible continuations. With validated interval arithmetic we certify that the three graphs have one, two and three recurrent classes, of periods (1), (1,3) and (1,3,9), that a class of period d has spectral radius φ^(1/d), and therefore that the symbolic entropy equals log φ in all three cases. That the three maps share this value is classical (the entropy formula for tuning). What the paper adds is the certified, border-first reconstruction of three different recurrent profiles behind the one value. Under explicitly cited results (Singer, Mañé, the lap formula) the same value is the topological entropy of the map. One level deeper, we certify the restrictive renormalisation interval of N2 and N3 and report three pre-registered tests: the top class is conserved; the predicted orientation flip and the predicted "one added class" rule are falsified, the latter by an additional recurrent component of entropy zero made visible by the refinement. We define the boundary-entropy profile Π(G, U), the object that the entropic reading compresses, and state which of its properties are classical, which are certified here and which are open. The profile is not a new entropy and not a complete invariant. A single pre-registered certified run of the metric reading shows that the expansion criterion holds for N1 but does not close within the declared depth for N2 and N3: an operative limit of the certifier, not a property of the dynamics. The paper can be read and checked without knowledge of Sub-Limit Dynamics (SDL); the SDL reading is given in a separate section. Files. The PDF is the paper. C18_support_package_v1.0.zip contains: the executable package (certifier scripts and certified JSON outputs, Python ≥ 3.13 with python-flint, mpmath, numpy); the audit package (proofs, pre-registrations, result reports, frozen milestones); the first floating-point run, marked as history and not a certificate; the record of an independent re-execution of the three certifiers, in which the structure certificate is regenerated byte for byte and the other two certificates differ only in the field recording the Python version; the LaTeX source. MANIFEST_SHA256.txt lists the SHA-256 of every file in the archive; README.md gives the commands. Status of the results. Certified: Theorem A and the structural, renormalisation and metric-run certificates. Proved under cited hypotheses: Theorem B (topological entropy), conditional on the statements of the lap formula and of the symbolic-dynamics and variational-principle references, whose original texts are listed as residuals in Appendix D. Open: the metric quantities for N2 and N3 and the existence and identification of the conditionally invariant and bilateral-survival measures. AI assistance. AI tools were used for drafting, for code support and for technical review. Scientific responsibility, verification of the results and publication decisions remain with the author.