# Two-step source-precision adapter and actual-output certificate Development — no registry entry. Physical promotion 0; independent review pending. All results concern two known synthetic PR195 records and their pinned learned modal source. They do not establish physical-frequency truth. ## 1. A different audit rule, preserving prior outcomes The original instrument-noise diagnostic retains its common/scaled cutoff counts205/204 on the wider stress sketch and its original proposer HOLD. This return defines a separate encoding-precision audit with frozen cutoff tau=2^-24. It does not change that older noise contract. The record-only adapter uses the same nested321-column sketch as PR196, but learns its order by its own SVD count above tau. It receives no source phases, rates, jitter or coefficients. Source hashes are checked for lineage only. Expected source orders105 and209 are used by the verifier as applicability conditions, never forced onto a different learned order. ## 2. Source-aware order and excitation Let X be the encoded time-by-coordinate record, quantum2^-48, and S=A/128 the dyadic sketch. Its mathematical value is Z=X_integer A/2^55. Integer products are guarded against overflow before execution, then FLINT forms exact G=Z^T Z at scale2^-110. All18,790,154 readings per record are projected and raw hashes checked before/after. Exact Grams and untrusted sorted dyadic eigenproposals are retained. STRUCTURAL_AUDIT.json checks symmetry, shapes and descending eigenproposal order for the actual hash-bound witnesses. As in PR196, exact orthogonality/eigenpair residuals and a polar/Weyl bound enclose each ordered eigenvalue of G. Let l_q be the qth lower enclosure. The declared source comparison has rank at most q, where q=105 for zero clocks and q=209 for the two-rate stress model. PR195's generator error is independently re-evaluated from the pinned coefficient numerators: \[ d_{\rm eval}=L\,2^{-120}+2^{-49},\qquad L=\max_c\sum_{r>0}|B_{{\rm integer},rc}|. \] Hence \[ \|Z-Z_{\rm model}\|_2 \le\sqrt{8191\cdot2294}\,d_{\rm eval}\|S\|_F =:\epsilon<1.034\,10^{-10}. \] If l_q>tau^2 and epsilon0.3585086 for the baseline and sigma209>0.0028804 for stress. Both exceed tau, while the next value is below epsilon. The floating counts105/209 agree with these source-aware exact encoded-sketch certificates. The source comparison itself therefore has exact rank105/209, because its qth singular value is still positive after subtracting epsilon. The exact-sketch count is not silently a certificate of every intermediate floating BLAS entry. The output proof below uses saved actual arrays directly and does not assume SVD or BLAS accuracy to transport phases. ## 3. Two-step closure and aliasing For stress, s=+1 or-1 labels the preparation rate class and \[ g_{js}(k)=e^{i(1+s\mu)\theta_j k} e^{is\theta_jj_0(-1)^k},\quad \mu=2^{-20},\ j_0=1/16. \] Preparation origins rotate coefficients without adding temporal functions. Since parity is unchanged by two steps, \[ g_{js}(k+2)=e^{2i(1+s\mu)\theta_j}g_{js}(k). \] The real comparison frame P has columns1,cos,sin for each j and rate class. It obeys P[2:]=P[:-2]J, where J is block diagonal with stationary block1 and orthogonal blocks \[ J_{js}=\begin{pmatrix}\cos(2\omega_{js})&\sin(2\omega_{js})\\ -\sin(2\omega_{js})&\cos(2\omega_{js})\end{pmatrix}, \quad\omega_{js}=(1+s\mu)\theta_j. \] Thus J is normal. Its complete target list is1 and exp(plus/minus2i*omega_js). Every pairwise unit-circle chord is recomputed with validated arithmetic; the minimum is bounded below by7.285423493e-7 for stress. This checks folded cross-pair collisions as well as same-pair rate-class splitting. Two-step eigenvalues identify omega only modulo pi. Primary results are circular two-step phase errors. Dividing by2 is only a conditional lift about the named source target; each target's lift integer is retained. No unrestricted continuous generator or SI frequency is identified. ## 4. Actual floating adapter and saved arrays For the learned order q, the actual adapter forms T=lstsq(U[:-2],U[2:]), calls the floating eigensolver, captures its arguments, and rounds all signed phase arguments onto the2^-60 grid. The original and instrumented variants share all proposal arithmetic; instrumentation runs only after proposal emission. Their proposal bytes match on both records. Saved witnesses include the actual U,T,eigenvalues,eigenvectors,inverse, angle outputs and emitted phase integers. The large U is split into two row archives without changing bytes. A verifier-only untrusted reduction D=U^T P_tilde and inverse witness K are proposed after the record-only fit. The source frame is generated with the validated96-bit recurrence from PR195; each nonstationary component is within2^-60 of its exact value. The basis orthogonality, two-step shift residual and source-frame span residual are checked and retained. The direct similarity proof does not require a guessed deleted-row or reference conditioning bound. ## 5. Direct similarity certificate Let T be the actual shift matrix, D an arbitrary square reduction witness, and K an inverse witness. Verify exact binary64 encodings and bounds \[ \rho_D\ge\|DK-I\|_2<1,\quad h_D=\|K\|_{2,\rm upper}/(1-\rho_D),\quad d\ge\|D\|_2,\quad L_J\ge\|TD-DJ\|_2. \] All residual products use exact integers. J is evaluated as a validated dyadic block rotation; its component errors below2^-60 contribute an operator error at most2^-59, so d*2^-59 is charged to L_J. For the actual returned diagonal Lambda and an eigenvector/inverse witness V,H, verify \[ \rho_V\ge\|VH-I\|_2<1,\qquad \delta_\lambda= \|TV-V\Lambda\|_{2,\rm upper} \frac{\|H\|_{2,\rm upper}}{1-\rho_V}. \] Then A=V Lambda V^-1 has exactly the returned eigenvalues and ||A-T||<=delta_lambda. Both V and D are certified invertible by their Neumann residual bounds. Consequently \[ \|D^{-1}AD-J\|_2 \le h_D\bigl(L_J+d\,\delta_\lambda\bigr)=:R. \] Because J is normal, every eigenvalue lies in a radius-R disk about its unit-circle spectrum. Along the homotopy from J to D^-1 A D, all spectra stay inside these disks. If2R