**Verdict:** The idea is fine, but the core math is wrong as written, and the security claim is much weaker than stated.

**Fatal errors**

- **The transition is wrong.** T(x,y) = (x+y, x) doesn't map (F_n, L_n) to (F_{n+1}, L_{n+1}). From (1,1) it gives (2,1), but the next pair is (1,3).

- **Lemma 3.3 fails.** y² − 5x² = −4 at the seed (1,1), but at T(1,1) = (2,1) it equals −19. Part 2 of the lemma is false.

- **The ring story doesn't match.** F_n + L_nφ is not φ^n. The correct element is φ^n = F_{n−1} + F_nφ. Also, N(F_n + L_nφ) ≠ ±4, and (L² − 5F²)/4 = ±1 is the norm of (L + F√5)/2. "Up to units and identification" hides this mismatch.

**Fixes:** Use (F_n, F_{n+1}) with T(a,b) = (b, a+b) and the Cassini invariant b² − ab − a² = (−1)ⁿ. Alternatively, keep (F,L) with T(F,L) = ((F+L)/2, (5F+L)/2), which is always an integer since F and L have the same parity.

**Weaknesses**

- **Algorithm 4.1 is muddled.** Testing N(β − cα) = 0 is just an equality test, so the norm adds nothing. The "norm = ±4" alternative doesn't test orbit membership. The unit group {±φ^k} is infinite, so there is no "fixed finite set" of units.

- **Soundness is mislabeled.** The check accepts any point on the curve, so skipping steps or jumping to another orbit point passes. It only catches off-curve corruption.

- **The check is effectively redundant state.** If the program value must equal c·monitor_state, this is duplicate-and-compare with an extra invariant.

- **Overflow is fatal, not minor.** F_n exceeds 64 bits around n ≈ 93. Modular arithmetic makes the identity far weaker, since many wrong values satisfy it mod m.

- **There's no threat model.** The identity is public, so it gives no tamper resistance, and there's no comparison with CRC, MAC, or lockstep checks. "Constant-time" means O(1) here, not the cryptographic data-independent sense.

- **Naming and references.** "Pell–Lucas" is a different sequence. This is the Fibonacci–Lucas identity. "The framework" and "zero-drift axiom" are never defined.

- **The computability section is trivial.** Orbit membership is obviously decidable, so invoking undecidability adds nothing.

**Next steps:** Correct the recurrence, prove the lemma explicitly (or in Lean), define the threat model, handle overflow, and include a small implementation with benchmarks against a CRC or duplicate-state baseline.