# Encoded-baseline uniform ideal two-step phase tolerance Development — no registry entry. Physical promotion0. Source-conditioned mathematical/numerical results for the pinned PR195/197 synthetic stress trajectory. No independent-review or apparatus claim. ## Exact encoded-baseline anchor Let B0=Y0A/2^55 be the8191×321 exact sketch of the signed-integer raw record, and S=A/128. The saved actual binary64 baseline basis U0 has209 columns; every binary64 entry is treated as an exact dyadic. The coefficient C0 is an untrusted floating proposal. For encoded integers u,c,n and their denominators, compute G=uᵀu, H=uᵀn, b2=Σn². Then r²=||B0−U0C0||F²=||B0||F²−2tr(C0ᵀU0ᵀB0)+tr(C0ᵀU0ᵀU0C0) is an exact rational, evaluated by integer cross-Gram cancellation. No floating subtraction of nearly equal energies is used; a negative numerator rejects inconsistent evidence rather than being clipped. Its validated square root is <6.960e-13. A separate full raw projection reproduces G,H,b2 exactly. Portable replay validates the actual basis Gram, arithmetic and pinned raw receipt; it does not reread omitted raw records. The parent encoded-sketch certificate givesσ209(B0)≥s0 andσ210(B0)≤ε0. For Q0=U0C0, rank≤209 andσ209(Q0)≥s0−r=α. The certified positive α proves its rank is209. ## Uniform sketch amplification For arbitrary additive real errors with each of37×8191 separate62-vectors of norm≤δ, ||E||F≤√(37×8191)δ. Thus ||ES||2≤Cδ where C=√(37×8191)||S||2. Let Z=AᵀA. Since Z is positive semidefinite and symmetric, ||S||2≤||Z^k||∞^(1/(2k))/128, k=1,2,4. All three powers and absolute row sums are exact integers. Taking the minimum of independently outward-rounded bounds is safe. It yields C<707.294. The previous version used k=1 only, giving C<1080.819. Both versions and their outcomes are retained. Weyl gives exactly209 values above the unchanged source cutoffτ=2^-24 if Cδ2e. Align Uhat to W0 by a right orthogonal transform. Their distance is at most d=√(2−2√(1−t²)). This alignment changes the shift by similarity and preserves its eigenvalues. For η≥||U0ᵀU0−I||2 and L the last two rows of U0, β=√(1−||L||F²/(1−η))≤σmin(W0_minus). Consequently the new deleted-row matrix hasσmin(Uhat_minus)≥β−d; require β>d. ## Comparison matrix: floating is not exact least squares Saved T0 is a binary64 comparison matrix. It is NOT assumed to be the exact least-squares solution. With G0=U0ᵀU0, define Tbar=G0^1/2 T0 G0^−1/2. The validated saved shift residual yields σ=||U0_plus−U0_minus T0||2/√(1−η), ||W0_plus−W0_minus Tbar||2≤σ, ||Tbar||2≤(1+σ)/β=Mbar. The residual termσ is essential. For the exact least-squares operator That of the aligned ideal Uhat, ||That−Tbar||2≤b=[σ+d(1+Mbar)]/(β−d). This follows by multiplying Uhat_minus† into (Uhat_plus−W0_plus)−(Uhat_minus−W0_minus)Tbar+(W0_plus−W0_minus Tbar). No exact-LS claim about T0 is used. ## Normal source comparison and unique phase matching The pinned parent source reduction D0 has validated inverse bound hD=||K0||/(1−||D0K0−I||), norm dD and intertwining residual LJ≥||T0D0−D0J||2. The source J is the exact orthogonal two-step rotation matrix; its dyadic construction error was charged in the parent receipt. Conjugating That through G0^1/2 and D0 gives a perturbation of normal J bounded by R=hD·LJ+√((1+η)/(1−η))·hD·dD·b. Normal resolvent bounds confine eigenvalues to the target disks of radius R. Homotopy from J to the perturbation preserves one eigenvalue in each disk when2R is below the complete209-target minimum chord. For R<1, circular angle error≤asin(R); the additional strict2asin(R)4.620e-7; this is inconclusive, not a demonstrated failure or an optimal tolerance boundary. At δ≤2^-48, the integer-record lattice allows at most one coordinate to change by one raw quantum in each62-vector, with arbitrary coordinate/sign choices. Version1's2^-49 ball contains no distinct encoded record; this limitation is preserved explicitly. ## Actual-output control and remaining gap Before new fitting, a fresh deterministic record was frozen: at preparation a/time k, channel(a+k)mod62 changes by(-1)^(a+floor(k/62)) integer units, all other channels unchanged. Its exact vector noise norm is2^-48. All18,790,154 differences are checked, and separate generation reproduces its raw hash and gates. This constructed control is not independent or random data. Two fresh unchanged original/instrumented adapter runs learn209, with byte-identical emitted phase archives. The unchanged PR197 actual-output proof and its saved-array replay certify phase error<9.323e-10 radians per two-step sample. A supplemental source-aware comparison bounds actual assigned error<7.571e-15, without replacing the primary certificate. The uniform theorem above concerns EXACT top209 singular subspaces and EXACT least squares. It does not certify floating convergence for every member of the ball. Each actual floating output still needs its own basis/inverse/eigenpair/emission checks. Source-labeled frequency lifts remain conditional moduloπ. The old instrument-noise contract and204/205 HOLD, full-Hankel signature, apparatus source/clock/noise/metric mapping and nominal747-dimensional unobservable space are unchanged. Tolerances and grids remain O03 protocol controls, not theory constants. The v2 refinement was motivated after v1 evaluation, then separately frozen before its own evaluation. Only a deterministic exact upper bound on the SAME S was strengthened; no estimator, phase-separation gate or grid was retuned. Both versions and the same-team correction about saved T0 are retained. No blinded preregistration or independent governance review is asserted.