| Single-harmonic spectra | statement | proposition 72 | arxiv |
| Convexity | statement | theorem 23 | arxiv |
| Finite support and inactive tails | statement | lemma 37 | arxiv |
| Farey parents and first violations | statement | theorem 22 | arxiv |
| Sharp convergence threshold | statement | theorem 2 | arxiv |
| One-coordinate initialization | statement | lemma 35 | arxiv |
| Slacks of the initial pairs | statement | lemma 56 | arxiv |
| Polyak's tuning and gradient descent | statement | corollary 20 | arxiv |
| The lag-one cubic | statement | lemma 67 | arxiv |
| Exact spectral interpolation | statement | lemma 7 | arxiv |
| 63487HSFAC | problem | problem 1 | arxiv |
| Functionals above the rate | statement | lemma 62 | arxiv |
| Central family near $\kappa_c$ | statement | theorem 54 | arxiv |
| Interpolation on arbitrary sets | statement | theorem 5 | arxiv |
| Dual nonnegativity at $\kappa_c$ | statement | proposition 51 | arxiv |
| Normalization | statement | lemma 4 | arxiv |
| Monotonicity and phase floors | statement | proposition 75 | arxiv |
| The special spiral | statement | proposition 10 | arxiv |
| Angle function | definition | definition 13 | arxiv |
| Periodic spectra | statement | proposition 76 | arxiv |
| Data of the arc | definition | definition 45 | arxiv |
| No strict certificate at the growth exponent | statement | lemma 34 | arxiv |
| The certificate as an integral quadratic constraint | statement | lemma 68 | arxiv |
| Dominant quadratic roots | statement | lemma 66 | arxiv |
| Feasible spectrum | definition | definition 6 | arxiv |
| One dip per line | statement | corollary 24 | arxiv |
| Homotopy | statement | lemma 69 | arxiv |
| Energy estimate | statement | lemma 33 | arxiv |
| Algebraic identities on the arc | statement | lemma 46 | arxiv |
| Quotients at the dominant real roots | statement | lemma 74 | arxiv |
| Growth envelope | statement | lemma 32 | arxiv |
| Closed form at $\kappa_c$ | statement | theorem 50 | arxiv |
| Fan-lag certificates | statement | theorem 27 | arxiv |
| Weighted shift | statement | lemma 70 | arxiv |
| The rational curve | statement | lemma 44 | arxiv |
| The central family | statement | proposition 55 | arxiv |
| Phase locking | statement | theorem 17 | arxiv |
| Large lags near the three-cycle | statement | lemma 53 | arxiv |
| A smaller rate on the kernel | statement | lemma 64 | arxiv |
| Bounded memory preserves the rate | statement | lemma 61 | arxiv |
| Linear gap | statement | lemma 57 | arxiv |
| Dual factorization | statement | lemma 11 | arxiv |
| Symmetries and monotonicity in the twist | statement | lemma 36 | arxiv |
| Reduction to one curve | statement | theorem 21 | arxiv |
| Spectral rate for Polyak's tuning | statement | theorem 9 | arxiv |
| Regularity and critical behavior | statement | theorem 3 | arxiv |
| Lag identities on the spiral | statement | lemma 38 | arxiv |
| One exponent suffices | statement | theorem 15 | arxiv |
| Near the unit circle | statement | lemma 73 | arxiv |
| A functional at the exact rate | statement | lemma 63 | arxiv |
| Pinning lags | statement | lemma 28 | arxiv |
| Contraction near the kernel | statement | lemma 65 | arxiv |
| Rates | statement | theorem 18 | arxiv |
| Explicit regions | statement | corollary 19 | arxiv |
| Complete spectral classification | statement | theorem 12 | arxiv |
| A two-harmonic thirteen-cycle | statement | proposition 29 | arxiv |
| Sign certificates | statement | lemma 48 | arxiv |
| The family $P_1$ | statement | proposition 43 | arxiv |
| The arc $P_2$ | statement | theorem 49 | arxiv |
| Positivity of the spiral kernel | statement | lemma 39 | arxiv |
| Exact rate and realization | statement | theorem 1 | arxiv |
| Convergence in a gap | statement | proposition 30 | arxiv |
| Multiplier criterion | statement | theorem 16 | arxiv |
| Endpoints | statement | lemma 47 | arxiv |
| Telescoping | statement | lemma 58 | arxiv |
| Threshold strip | statement | proposition 42 | arxiv |
| Spectral characterization | statement | theorem 8 | arxiv |
| Continuation | statement | lemma 60 | arxiv |
| Levels | definition | definition 25 | arxiv |
| First-order slacks | statement | proposition 52 | arxiv |
| Compactness | statement | lemma 31 | arxiv |
| Fan tongues | statement | theorem 26 | arxiv |
| Amplitude identity and tail bound | statement | lemma 40 | arxiv |
| The spiral outside its regime | statement | proposition 41 | arxiv |
| Plateau for classes of segments | statement | lemma 59 | arxiv |
| Fan geometry | statement | lemma 77 | arxiv |
| Symbols of one-sided certificates | statement | lemma 71 | arxiv |
| The angle at the threshold | statement | lemma 14 | arxiv |
