# Finite-window phase stability around a pinned fitted reference Development lane; physical promotion 0. Exact conditional matrix analysis plus constructor computations. Independent review is pending. The reference phases are learned from a numeric record; they are not nominal source modes, measured physical truth, or a uniquely identified continuous generator. ## 1. Source model and validated constants Let `n=8191`, `q=105` and `S=[W/64 | v/128]` be the pinned complementary sketch. Let `Z0=D0 S` be the pinned clean record's sketch. The previous exact161-by-161 Gram certificate gives `sigma105(Z0)>=L0`. Let \[ M_{k,:}=\sum_{j=1}^{q}e^{ik\theta_j}C_{j,:},\quad \theta_j\in\{0,\pm\widehat\theta_1,\ldots,\pm\widehat\theta_{52}\} \] be the real modal model obtained from the pinned clean learned proposal after applying the same sketch. The coefficient convention is the original stationary, cosine and sine convention. The full complex representation is only a change of basis. Every raw reading contributes to the exact integer sketch; no nominal 600-cell operator or nominal frequencies enter this calculation. `certify_phase.py` computes an exact unweighted squared residual `F` between the integer sketch and dyadic approximations to the modal phases. If `CA` is the integer product of the proposed coefficients with `[2W|v]`, its entries represent coefficients at scale2^-67. Every sine/cosine dyadic error is less than2^-60. With `Cmax` the sum of the largest absolute `CA` coefficient in each nonstationary row, \[ \|Z_0-M\|_2\le\|Z_0-M\|_F \le\varepsilon_0:=\sqrt F+ \sqrt{8191\cdot161}\,C_{\max}/2^{127}. \] The resulting bound is about2.639e-11. It is a newly computed fit bound, not an assumption that the quantized record has exact temporal rank105. Since `M` has rank at most105, Weyl's inequality and `alpha=L0-epsilon0>0` prove it has rank exactly105 and `sigma105(M)>=alpha`. Normalize the Vandermonde frame by `V[k,j]=exp(ik theta_j)/sqrt(n)` and set `G=V* V`. Off-diagonal entries use the finite geometric sum. Exact interval bounds and Gershgorin give `a I <= G <= b I`, with `a>0`. For every pair of distinct reference eigenvalues, the chord separation obeys \[ |e^{i\theta_j}-e^{i\theta_l}|\ge s_{\min}>0. \] All constants are outward dyadic bounds in PHASE_STABILITY_CERTIFICATE.json. The minimum reference chord separation is approximately0.00214408. ## 2. Conditional phase-matching theorem Suppose a new record is `D0+E`, with each62-vector error at most `delta` across 37 preparations and8191 times. Its sketch is `Z=M+R`, where \[ \|R\|_2\le e:=\varepsilon_0+ \sqrt{37\cdot8191\cdot371097/4096}\,\delta. \] Consider the **exact-arithmetic** estimator: take an orthonormal basis `U` of the leading105 left singular vectors of `Z`, delete its last/first row to form `U_-`, `U_+`, and calculate `T=U_-^dagger U_+` by exact least squares. This theorem concerns that estimator. It is not an implementation-error proof for the floating SVD/least-squares routine used in the proposer. Define \[ K=\sqrt{b/a},\quad \beta=\sqrt{1-q/(na)},\quad t=e/(\alpha-e),\quad d=\sqrt{2-2\sqrt{1-t^2}},\quad r=\frac{K d(1+K)}{\beta-d}. \] If `alpha>2e` and `beta>d`, every eigenvalue of `T` lies within distance `r` of some reference `exp(i theta_j)`. If also `2r=beta`. Projecting the leading left singular-vector equation for `Z=M+R` off the column space of `M` gives `||(I-QQ*)U||_2 <= e/sigma105(Z) <= e/(alpha-e)=t`. The leading subspace is separated because `alpha>2e`. A unitary alignment of `U` with `Q` has basis distance at most `d`. After that alignment, \[ \|U_+-U_-J\|_2\le d(1+K),\qquad \sigma_{\min}(U_-)\ge\beta-d. \] Least squares therefore bounds `||T-J||_2<=d(1+K)/(beta-d)`. Diagonalizing `J` bounds every perturbed eigenvalue within the stated radius (Bauer-Fike, directly by the diagonal resolvent/Neumann series). Along the homotopy `J+x(T-J)`, `0<=x<=1`, the eigenvalues stay inside these disks. If the full disks are disjoint, multiplicity cannot move between them, giving one-to-one matching. A disk of radius `r<1` about a unit-circle point subtends at most `arcsin(r)` from the origin. QED. ## 3. Certified grid boundary, not an algorithm-failure boundary The dyadic grid was frozen as `delta=2^-k`, `k=20,...,40`, before execution. The largest radius on this grid satisfying unique matching is `2^-27`. There `r<0.000646029` and phase error `<0.000646029` radians per sample. At the full `2^-20` vector budget the theorem gives only a one-sided enclosure: `r<0.088763` and angular error to some reference phase `<0.088880`; the disks overlap, so unique matching is not certified. This is a limitation of this sufficient bound, not evidence that actual phase recovery fails there. It does not establish that2^-27 is an optimal or necessary threshold. The previous temporal-order theorem remains valid at2^-20. Stable component count, retained conditioning, individual phase matching and convergence of a particular numerical fitter are distinct claims. ## 4. Two fitting controls Two new deterministic coherent phase-shift controls have declared vector-error budgets2^-28 and2^-20. They shift the lower member of the closest positive reference pair, using a source-bound coefficient L1 Lipschitz scale. A floating constructor rounds the additive perturbations to encoded integers; exact integer checks verify the actual vector/scalar error gates. Both remain known constructor records, not independent apparatus or blinded data. Returned phase numerators are compared exactly with the intended constructor phase numerators. These differences are empirical parameter discrepancies; the generated quantized record is not asserted to be exactly that modal model. Passing a control does not close a general numerical-implementation or optimizer convergence guarantee. One new full-record Hankel certificate separately tests the higher-budget record against the unchanged selection criteria. ## 5. Two identifiability limits **EXACT sampled aliasing.** For sampling interval `h=1/2` in normalized model units, every continuous frequency `omega+4*pi*m` has the same sampled complex phase as `omega`, for integer `m`. Even perfect sampled phase recovery cannot select a unique continuous generator without an alias restriction. **EXACT positive-noise obstruction.** For a nonzero one-mode amplitude vector `a`, `||a cos((theta+eta)k)-a cos(theta k)||<=||a|| k |eta|`. For any positive finite error budget there are distinct nearby phase models whose finite records differ by less than twice that budget at every sample; their error balls overlap. Thus exact real-valued phase identification is not uniformly possible under positive bounded error. Approximate/local bounds such as the theorem above remain meaningful. No result here identifies unobservable bulk dynamics, promotes an experimental bridge, supplies SI units, or changes the full-Hankel threshold/count/condition cap. O03 tolerances remain protocol controls, not theory constants.