# RMO97 — compression uncertainty and merging normal solutions **Result:** both R96 branches persist throughout the chosen narrow compression band 1.56 ± 0.0001. Their downstream y-velocity ranges remain separated by a certified gap greater than 1.63169 km/s, allowing independent compression choices on the two branches. The wider band 1.56 ± 0.0005 includes a single inverse fold where the branches meet. The transition remains fast at that point. **По-русски:** при небольшой выбранной ошибке сжатия две ветви сохраняются и их скорости по y различимы внутри модели. При более широкой ошибке допустимые ветви соединяются. Это слияние решений обратной задачи о нормали; fast-переход не превращается в slow и не исчезает как конечный скачок. Модельные границы ошибки не выданы за точность солнечных наблюдений. ## Fixed scope and meaning of the bands The pre-run protocol is `polar_uncertainty/PROTOCOL.md`. Keep the exact saved R95 upstream state: ρ₁=1, p₁=0.04056, u₁=(−1,0,0), B₁=(0.05618818561896245, 0.6422339004220036, 0), γ=5/3 and μ₀=1. Velocity unit is 600 km/s; density normalization is arbitrary. All candidate planes are stationary in one common frame, with normals in the same coplanar φ∈[−4°,4°] sector. Plotting axes stay fixed. Only compression is allowed to vary. The exact rational hard bands are [1.5599,1.5601] and [1.5595,1.5605]. They were selected using the saved R95 curve's approximate peak to test persistence and merging; they are not observed error intervals, confidence levels or blind predictions of the peak location. No earlier R95/R96 sweep or scientific audit was rerun. An error band allows a continuum of states. “Two roots” below refers to the two normal solutions for **each exact compression**. The narrow band's whole solution set consists of two separate continuous branch segments, not two points. ## Complete existence result in the declared domain There is one critical compression, enclosed by 1.56020295869730841751307358451 < r* < 1.56020295869730841751307358452. The exact stored rational enclosure has width below 10⁻³⁵. Its normal rotation is approximately φ*=0.9697644551°, or θBn≈84.0302355449° to the fixed upstream field. Numerical digits refer to the exact adopted model, not observed precision. | Exact compression within [1.5595,1.5605] | Normal roots in the declared sector | Physical meaning | |---|---|---| | r < r* | Two simple roots | Two distinct admissible fast shocks | | r = r* | One double orientation root | The branches meet at one admissible fast state | | r > r* | No roots | That exact compression has no solution under these fixed assumptions | The narrower band lies wholly below r*, so its branch segments remain separate. The wider band includes the fold, so its permitted segments join. A no-root subrange at its upper end does **not** reject the whole wider measurement band: that same band still includes compressions with admissible states. Neither result excludes solutions at other upstream states, outside this sector, or in other physical descriptions of a solar feature. At the fold representative, the upstream fast Mach number is approximately 1.439288 and the downstream value 0.736911. Compression remains 1.560203, pressure is positive and entropy increases. The fold is a singularity of normal recovery from compression; it is not a degeneracy of the MHD characteristic ordering, a weak-shock limit or a fast-to-slow transition. ## What additional velocity information retains its value? For the narrow compression band, rigorous interval recovery gives the following enclosures in the fixed Cartesian frame. Displayed endpoints are rounded outwards. | Branch | Normal rotation φ, approximate degrees | u₂x enclosure, km/s | u₂y enclosure, km/s | |---|---|---|---| | A | −0.215129 to 0.279113 | [−384.687215, −384.532626] | [−12.523789, −11.721841] | | B | 1.660081 to 2.153672 | [−384.305277, −384.284457] | [−10.090150, −9.283309] | The y enclosures have a certified gap greater than 1.6316919329 km/s before the displayed rounding. Each candidate may use its own compression anywhere inside the band; this is stronger than comparing only two roots at a shared central r. It is a conservative enclosure gap, not the proven exact minimum over all pairs of branch states. If each predicted y-velocity interval is broadened by an additional equal symmetric hard half-width ε, a sufficient disjointness condition is 2ε < 1.6316919329 km/s. For example, ε < 0.815 km/s is a conservative sufficient choice. This further error is separate from the compression uncertainty already included in the bars. It is not an instrument-resolution requirement, a one-sigma criterion, or a complete uncertainty model. A real measurement must also agree with a permitted prediction and trace the same plasma patch. The y direction is the fixed model axis. Its contrast must not be identified with an actual Doppler measurement unless the viewing geometry supports that projection. Upstream field/flow errors, line-of-sight geometry and emission weighting are not propagated here. These are plasma velocities, not an EUV brightness-pattern speed. At the fold the two branch states coincide. By continuity, their vector velocity difference tends to zero. Hence the wider band has no positive uniform branch-separation guarantee, even for a full vector-velocity measurement. Particular states away from the fold may still be distinguishable. This does not claim that all velocity information becomes useless throughout the band. ## Why the root-count statement covers an interval Use z=tan φ, a=B₁x, c=B₁y, h=(a+cz)², A=(c−az)² and b=p₁(1+z²). The exact bivariate polynomial is F(r,z)=2[4−r−5br](1−rh)² − rA[r+5−2rh(4−r)]. Its coefficients are rational values constructed from the saved input strings. The calculation uses no fit to the R95 plotting samples. Two exact rational rectangles cover r∈[1.5595,1.5605], z∈[−0.07,0.07]. On both, interval bounds certify F_r<0 and F_zz<0; conservative combined bounds include −F_r>5.68148 and −F_zz>7.84000. F_z is positive at both left enclosing boundaries and negative at both right boundaries. Thus F has one interior maximum in z. F is negative at z=±0.0699 throughout the band. The saved rational bound 0.06990, positive regular denominator, nonzero normal and tangential upstream fields, and tangential-field amplification. It also certifies the sufficient fast inequalities used in R96: upstream normal speed squared exceeds the magnetosonic trace; downstream normal speed squared exceeds cAn² and lies between the two magnetosonic roots. These bounds establish admissibility for the actual F=0 states, including the fold. Arbitrary off-curve recovered states in the rectangles are not claimed to conserve total energy. For the narrow velocity ranges, 64 equal r cells cover each branch. Exact endpoint root enclosures, root monotonicity and interval state recovery on each r/z cell produce the velocity hulls. The separation already passes at the declared 64 cells; no extra refinement or domain extension was needed. Four narrow-band endpoint states and one fold representative pass the original mass, momentum, induction, energy and normal-field checks in Decimal90. The maximum scaled residual is 3.278×10⁻³⁸, below the predeclared 10⁻²⁸ threshold. At each fixed normal, a separate original-flux Newton solve with six downstream unknowns agrees within 6.406×10⁻³⁸. A forward solve remains regular at this inverse fold. All five states receive `fast_shock` from the unchanged local classifier without a family label in the request. These are independent internal formulations and exact-domain sign checks, not independent solar observations or an external solver benchmark. Underlying ideal-MHD jump laws are the same as the R95/R96 formulation; the new contribution here is the bounded inverse-ambiguity and velocity-enclosure calculation. ## Figure, article consequence and preservation `RMO_polar_uncertainty.pdf` and `.svg` are vector figures. The left context curve reuses saved R95 points; the highlighted narrow segments and fold use R97 results. The right bars enclose the narrow-band y velocities, with the R96 exact-compression states marked. No panel is a front outline or trajectory. The results distinguish stable local type from stable inverse geometry and quantify when a particular additional constraint retains conditional discriminating power. This does not establish solar identifiability or robustness to untested upstream errors. The complete QuickLook keeps its R96 beginner entry and expandable groups. R97 is added as another saved MHD result; R94/R95/R96 figures, reports, calculators and tests are retained. No earlier scientific artifacts are overwritten. Native-browser, save-dialog and live-backend acceptance remain open. Reproduce only this stage from the project root with `python3 polar_uncertainty/audit.py` and `python3 polar_uncertainty/plot.py`. Saved results are readable without rerunning them. The JSON records exact polynomial coefficients, covering cells, fold brackets, endpoint counts, velocity enclosures, flux controls and input/code/protocol hashes.