Symmetric ribbon numbers: two settled cases (12n706, 12n504) and the computed set R^s_5

Two symmetric ribbon numbers determined by explicit witnesses, and the set R^s_5 computed. NO THEOREM OF OURS IS CLAIMED AND NONE OF THE MATHEMATICS IS OURS: the notion of a symmetric ribbon number, the lemmas used, the bounds and the tables belong to Akash, Corrado, Placke, Sanketh, Starns, Timothy and Zupan, 'Symmetric ribbon numbers of low-complexity knots', arXiv:2606.02390v1 (1 June 2026; still v1 when checked on 22 September 2026). What is deposited is a set of computed objects together with the checks they survived and an explicit statement of what they do and do not establish. RESULTS. Table 3 of that paper lists four prime non-alternating symmetric ribbon knots with 12 crossings whose symmetric ribbon number is undetermined: 12n23, 12n49, 12n504, 12n706. Two are settled here by exhibiting an object. r_s(12n706) = 4: the lower bound is the paper's own (Table 3 gives r = 4, Lemma 2.2 gives r_s >= r; independently Corollary 5.1 excludes r_s = 2 and 3 for det = 49), the upper bound is a labeled knotoid diagram with four crossings via Lemma 2.3. r_s(12n504) = 5: lower bound again the paper's (Table 3, Theorem 1.5; independently Corollary 5.1, since det = 121 > 49 excludes r_s = 4), upper bound a labeled knotoid diagram with five crossings. Each witness is given in full - Gauss word of the arc diagram, twist labels, DT code of the resulting symmetric union, crossing number after simplification. HOW IDENTITY IS ESTABLISHED: not by matching polynomials. For every witness deposited, SnapPy's isometry_signature(verified=True) agrees bit-for-bit with that of the named knot - an interval-arithmetic certificate - and verify_hyperbolicity() returns True on both sides; for 12n706 the verified volume is 13.4173743919? on both sides. A witness without that certificate is rejected by the build script and does not enter the deposit. The search discriminates: for 12n706, of 720 label combinations times 2 charts times 2 side conventions, 8 hit. R^s_5. The source states on page 3: 'If r_s(K) = 5, the set R^s_5 is too large to compute via our methods.' It is computed here: 145 Alexander polynomials, deposited as coefficient lists (R^s_4 has 27). Determinants occurring: 1, 9, 25, 49, 81, 121, 169. THE INDEPENDENT CHECK: Theorem 1.6 of the source proves by entirely different means - Kidwell's theorem on the z-degree of the Kauffman polynomial - that r_s(K) = 5 implies det(K) <= 169. The largest determinant in the computed set is exactly 169. Second check: every knot in Table 3 with r_s = 5 proven carries its Alexander polynomial in the set, 17 of 17. WHAT IS CALIBRATED AND WHAT IS PROVEN. Proven: that the boundary knot of each witness is the named knot (verified isometry of complements, Gordon-Luecke); Lemma 2.3 then gives the bound from the crossing number. Calibrated but NOT proven: that our constructor realises exactly the construction of Lemma 2.3 - evidence is exact reproduction of the source's own R^s_2 (2), R^s_3 (6) and R^s_4 (27 of 27, Table 1), plus 12/12 negative and 5/5 positive controls. The two results do not rest on that: they rest on objects whose boundary is verified. The residual risk is stated plainly - that a deposited object is not a valid labeled knotoid diagram in the source's sense. NOT CALIBRATED AT ALL: the completeness of the arc-diagram enumeration at c = 5 (26 diagrams); for c <= 4 the counts 1, 2, 8 match Lemma 3.1, Lemma 3.2 and Proposition 3.3 exactly, at c = 5 there is no source to check against. R^s_5 as deposited may therefore in principle be a subset of the true one. CONSEQUENTLY NOT CLAIMED: no exclusion argument of the form 'Delta not in R^s_5 implies r_s != 5', and in particular r_s(12a1202) = 6 is NOT claimed - it would follow from exactly that argument and remains a reasoned candidate. Note also the converse error, which an early draft of our own script printed for 13 cases before it was caught: Delta in R^s_n does NOT imply r_s = n, since R^s_n is a set of Alexander polynomials and Delta is not a complete invariant. The source uses these sets as an obstruction only, and so does this deposit. STILL OPEN: 12n23 is decidable with the same machinery and is not decided (the run died without a traceback inside SnapPea's C code). Twelve further knots were searched for c = 5 witnesses without success at |a| <= 5; absence of a witness in a bounded search is not an exclusion. A dependency-free reproduce.py re-checks the internal coherence of the data file - identity certificates, Gauss words, DT codes, the recomputed determinants and the 169 - and prints what it cannot verify without SnapPy. HONESTY NOTE: no invariant in this deposit is computed by an AI. Every number comes from established open tools - SnapPy/spherogram and SageMath - and is recomputable from the diagrams included here. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the questions, the decisions and the hardware are Stanggassinger's; the pipeline driving the tools is the joint contribution. The knot 12n706 is the knot of one of Stanggassinger's sculptures, Celtic-for-Playing (a rendered design, not a steel piece); that is how the case was chosen and it plays no role in the mathematics. Prior art was checked after the computation, not before, on 19 September 2026 and the version status again on 22 September 2026.