Priority deposit v3: 315 positive amphicheiral hyperbolic knots whose Conway polynomial does not split -- Part III, order 16 and a discrepancy in the rotation parameter

Version 3, superseding the deposit of 29 July 2026 (250 knots) on the same concept DOI. Here: 315 distinct hyperbolic knots that are positive amphicheiral but not strongly so -- each carries an orientation-reversing symmetry of order 2^a preserving the knot orientation, and carries no orientation-reversing involution, no negative amphicheiral symmetry and no inversion, so that neither Hartley's theorem nor Hartley-Kawauchi applies. For each, the Conway polynomial does not split as f(z)f(-z) with f in Z[z]. 18 lie at a=2 (rotary reflection of order 4), 261 at a=3 (order 8) and THIRTY-SIX AT a=4 (ORDER 16), the first of that order we have, across all four rotation parameters c in {3, 5, 11, 13}. NO NEW THEOREM IS CLAIMED. What is deposited is a generator for this class together with verified data. THREE CORRECTIONS TO VERSION 2, stated up front: (1) Version 2 reported c*l = 1 mod 2^a on all 250 as a confirmation of Conant's Lemma 4.2. That was empty. For a <= 3 every unit of Z/2^a is its own inverse, so c*l = 1 and l = c are THE SAME CONDITION there and no example in version 2 could separate them, at any c. (2) At order 16 they do separate, and our curves follow l = c, not c*l = 1: all 36 order-16 knots have l = c, so c*l = c^2 = 9 is not 1 mod 16, at every one of the four parameters. This is forced by the construction (the xy-modes are n = c mod 2^a and l is one of them). The natural reading is that our parameter c is his c^-1, i.e. our sigma_c realises his standard model with rotation parameter c^-1 -- and that reading has now passed its first test that could have failed: mod 16 the four parameters fall into the two inverse pairs 3*11 = 1 and 5*13 = 1, the corpus realises BOTH pairs, and each family carries l = c, exactly the pattern the inversion reading predicts. We still put the discrepancy on the table rather than choose the convention that makes our numbers agree with his; WE DO NOT CLAIM A COUNTEREXAMPLE TO LEMMA 4.2. (3) Crossing bounds. In versions 1 and 2 that field came from a simplification run that was not shipped; measured against the PD codes that were shipped it was too pessimistic in 233 of 250 cases and, in 2 cases, not reachable from the shipped diagram at all. In version 3 the field is by definition the crossing number of the diagram included with each knot, so a reader counts the crossings and holds the proof. Of the 250 carried-over bounds 234 fall, 2 rise, 14 are unchanged; the range over all 315 is 19 to 256. Also corrected: the construction-certificate bound. Version 2 quoted 3.99e-14, which is the bound for its 250 curves; over all 315 the worst defect is 7.51e-14. THE SHARP BRANCH: the constant-term congruence E(0) = (l^2-1)/8 mod 2 forbids something only when l = +-3 (mod 8), where E(0) must be odd. It now holds 7 knots at order 4, 78 at order 8 and 36 at order 16 -- 121 of 121 satisfied. All four order-16 values of l (3, 5, 11, 13) are +-3 mod 8, so the entire order-16 stock sits in the branch where the congruence genuinely forbids; before this deposit it had never been tested there, because no such example existed. NULL MEASUREMENT, because 315/315 is worthless without it: the same solver finds a presentation for 0 of 200 random polynomials with p(0)=1 and for 11 of 144 genuinely chiral hyperbolic knots (7.6 percent), against 315 of 315 here; both null rows re-measured on 3 August 2026 against the grown corpus, rates unchanged. ON THE OPEN QUESTIONS: for Conant-Manathunga Question 4.8 the 315 realise 66 distinct leading coefficients from -988869 to [phone]; for Question 4.9 we add nothing beyond their Table 1 -- the prime values realised are -7, -5, -3, 5, 7, 13 (absolute values 3, 5, 7, 13, all inside Table 1), the sole prime value the 65 new knots add is -3 (twice, both at order 8), and -3 itself stands in Table 1. A GUARD checks that the diagram shipped with a knot carries that knot's own volume (Mostow), with a self-test that swaps two diagrams and must break on exactly those two; run on all 65 new knots plus a 25-record sample of the carried-over 250: 90 of 90 hold. A MEASUREMENT records that the per-candidate deadline used during generation does not cut at random: among candidates it had previously killed and that were later completed, 225 of 1197 are counterexamples (18.8 percent), against 1602 of 41893 evaluated at first sight (3.8 percent). Runtime and genuine counterexample correlate, so deadline-limited counts understate in a biased direction. Each knot is a closed-form Fourier curve invariant under sigma_c = R_z(2 pi c / 2^a) composed with z -> -z, so positive amphicheirality is a property of the construction rather than of a computation; six numbers per knot reproduce the curve bit-exactly and a standalone numpy-only script is included, carrying a self-test that must fail against the wrong rotary reflection. Volumes 14.038 to 230.159. HONESTY NOTE: no invariant here is computed by an AI; every number comes from established open tools (Regina, SnapPy/spherogram, SageMath, SymPy, numpy) and is reproducible there from the planar diagrams included. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the geometry, the questions and the hardware are Stanggassinger's; the pipeline driving the tools and the curve-generating machines are the joint contribution. See README.md, NOTE.md, verification_log.md and SHA256SUMS in the archive.