# RMO94 — Unknown upstream flow changes the inferred field **Result:** for the declared central model at a fixed 85° field-to-normal angle, the continued positive-field branch is a fast shock for **0 ≤ v_n < 383.950825… km/s**. Its field tends to zero at the upper endpoint, where a finite gas-dynamic shock remains. This is a conditional model domain, not a measured solar flow limit or an identification of the observed EUV front. ## Physical question and inputs R93 certified the joint adopted scalar and small-tilt model domain with upstream plasma at rest. R94 asks one further question: what changes when the front speed is held fixed but the upstream plasma has an outward normal flow? | Input | Fixed value | Status | |---|---:|---| | Front normal speed D | 600 km/s | Adopted central model value | | Upstream sound speed c1 | 156 km/s | Adopted thermal/composition model | | Density compression r | 1.56 | Adopted central radio-association model | | Gamma | 5/3 | Ideal-MHD thermodynamic closure | | Upstream field-to-normal angle | 85° | Chosen model angle, not a measurement | | Upstream density | Fixed normalization | No new physical density inferred | | Upstream tangential flow | 0 | Declared solar-frame model | | Outward upstream normal flow v_n | 0 to 444 km/s, upper endpoint excluded | Specified search envelope, not a coronal prior | The search envelope follows U = D − v_n > c1. It is a necessary supersonic-inflow envelope for this continued fast branch, not a domain for every MHD family. The actual branch ends before its proposed upper search limit. At each v_n we reconstruct magnetic amplitude, downstream pressure and velocity together. Thus the test compares different compatible upstream states. It is not an evolution of one parcel, a common Galilean boost, or an experiment with an independently measured field held fixed. Density is fixed, so the plotted Alfvén speed also measures the relative change of inferred field strength. ## A consistent solar frame Take the outward normal as +x and the front speed as +D. Upstream is the right state and downstream the left state. Then the signed front-frame velocities are \[ u_{n1}-D=-U,\qquad u_{n2}-D=-U/r,\qquad U=D-v_n>0. \] The calculation first constructs positive-inflow normalized states for convenient algebra, then explicitly maps them to this outward solar frame. For positive normal and tangential magnetic components, the mapped downstream tangential velocity is negative. Independent checks evaluate the moving-front condition \([F]-D[Q]=0\) for every conserved variable, including total energy. Merely changing the upstream velocity in an old state pair fails this condition. ## Fixed-angle closure and branch accounting Use rho1 = U = mu0 = 1 for the local reconstruction, and define \[ b=\frac{c_1^2}{\gamma U^2},\quad h=\frac{B_{n1}^2}{\mu_0\rho_1 U^2},\quad A=\frac{B_{t1}^2}{\mu_0\rho_1 U^2},\quad k=\cot^2(85^\circ). \] The angle condition is **h = kA**; h cannot be held fixed as the flow varies. With M = 4 − r − 5br, d = 1 − rh and W = r + 5 − 2rh(4 − r), the linked energy-conserving reconstruction is \[ A=\frac{2Md^2}{rW},\qquad q=\frac{B_{t2}}{B_{t1}}=\frac{r(1-h)}{d},\qquad p_2=\frac{br^2[5r+1-2h(4r-1)]+(r-1)^3}{rW}. \] Consequently the fixed-angle equation is \[ F(h)=hrW-2kM(1-rh)^2=0. \] For M > 0, F(0) < 0 and F(1/r) = 3(r − 1) > 0. The coefficient of h² in F is negative. There is one small root below 1/r and a second root above 1/r. The small root is the branch connected to the R93 central model. A stronger enclosure shows that this root remains below h = 0.01 throughout the continuation: F(0.01) is positive at the largest M, and a positive lower bound on F_h holds over the small-h enclosing interval. The small root tends continuously to zero as M tends to zero. Its cancellation-free quadratic expression is used in code. The other algebraic root is recorded, not silently discarded or classified. At v_n = 0 it has h ≈ 0.86133, beyond 1/r ≈ 0.64103. Its states and admissibility are outside this stage. Global uniqueness and exclusion of other families do not follow from uniqueness of the continued small root. ## Continuous physical checks This stage does not transfer the old R93 certificate beyond its domain. Along this fixed central curve the previous b envelope is left at v_n ≈ 184.19 km/s. The extended thermal interval is checked here with new algebra and bounds. Exact polynomial identities verify normal momentum and total energy on the extended interval; mass, normal-field continuity, tangential momentum and induction follow from the stated coupled reconstruction. A useful new expression for the entropy argument is \[ p_2-b\frac{4r-1}{4-r} =\frac{M(r-1)^3}{rW(4-r)}\ge0. \] Therefore throughout the positive-field branch and its endpoint, \[ \frac{\Delta s}{c_v}\ge \ln\frac{4r-1}{4-r}-\gamma\ln r \simeq0.02318042>0. \] A fixed 32-cell outward-rounded interval cover in M, with 0 ≤ h ≤ 0.01 as an enclosing proof rectangle, establishes positive pressure, d > 0 and q > 1. It bounds the upstream magnetosonic trace below the squared inflow and the downstream magnetosonic polynomial below zero at the squared downstream flow. Together with d > 0 this gives the fast characteristic crossing and downstream super-Alfvénic normal flow. The enclosing rectangle is a mathematical proof device, not a claim that h was independently varied in this experiment. | Quantity | Conservative enclosure over the proof rectangle | |---|---:| | Upstream fast Mach number | 1.35365 to 1.48696 | | Downstream fast Mach number | 0.71789 to 0.77899 | The enclosures include the fixed-angle curve and its gas endpoint. They are mathematical outer bounds, not statistical intervals or extrema attained along the displayed curve. In the strict positive-field interval the field components are nonzero; the degenerate endpoint is handled separately. ## The endpoint is a gas shock, not disappearance of all shocks The endpoint is M = 0, giving \[ U_* = c_1\sqrt{\frac{3r}{4-r}} =216.0491747\ldots\;\mathrm{km/s},\qquad v_* = D-U_*=383.9508253\ldots\;\mathrm{km/s}. \] At this point h = A = 0, so the field vanishes on both sides. The field direction is undefined exactly at the endpoint, although the approach has the fixed angle 85°. Compression remains 1.56, entropy increases, and the gas jump satisfies the full moving-front conservation equations. This is not a weak-wave limit and not a fast-to-slow conversion. For v_* < v_n < 444 km/s, M < 0. On the continued regular domain 0 ≤ h < 1/r, d and W are positive, so the reconstruction would require A < 0. This is an algebraic obstruction to this branch, not a failed root search. It does not exclude configurations outside that domain, other thermal closures, different source associations or non-wave explanations. The endpoint relation is the standard gas-Hugoniot inverse already used in R88. R88's ≈172.1873 km/s value was the first overlap possible anywhere in its adopted scalar box; it was not a central MHD endpoint. The two numbers refer to different questions and domains. R94 adds the explicitly fixed-angle coupled MHD continuation and its limiting behavior; no historical novelty claim is made for the gas relation or the general ambiguity caused by unknown flow. ## Observer-readable numerical examples | Assumed upstream flow, km/s | Front-relative inflow U, km/s | Inferred upstream Alfvén speed, km/s | Upstream fast Mach | |---:|---:|---:|---:| | 0 | 600 | 386.8 | 1.4392 | | 100 | 500 | 311.7 | 1.4355 | | 200 | 400 | 232.8 | 1.4287 | | 300 | 300 | 144.0 | 1.4143 | | 380 | 220 | 28.8 | 1.3870 | | 383.9508… | 216.0492… | 0, gas endpoint | 1.3849 | These are linked model states, not six observations. Even with the same front speed, compression, thermal input and chosen angle, the inferred field changes substantially with the assumed plasma flow. An independent flow or field constraint could restrict this family. Feasibility and error of such a measurement remain event-specific; neither is supplied by this calculation. Figure: [RMO_upstream_flow.pdf](RMO_upstream_flow.pdf). ## Independent implementation checks and their limits Seven declared controls use a separately written four-variable Newton solution of the original momentum, induction and energy equations, with Bn = cot(85°)Bt1. The initial guesses come from perturbed perpendicular states, not the new fixed-angle closed-form answer. A separate 90-digit conservative-flux evaluation tests both the front frame and the moving-front solar frame. A magnetosonic matrix calculation checks characteristic speeds. Relative state discrepancies are below 5.3 × 10⁻⁷⁰; scaled flux residuals below 1.0 × 10⁻⁸⁹. Arithmetic precision does not imply observational precision. Five ordinary controls receive a fast-shock label from the unchanged diagnostic, which receives no family label in its input. Two near-endpoint controls honestly return DEGENERATE_NOT_CLASSIFIED: at M = 10⁻⁶ the slow/normal-Alfvén separation is already below the frozen tolerance, and at M = 10⁻²⁰ the normal field is too small. The full mathematical curve has separate continuous support; the classifier is not claimed to resolve it arbitrarily close to the endpoint. At exactly zero field the magnetic extension returns UNMAGNETIZED_LIMIT. The analytic gas check is recorded separately, with no override of the classifier's status. An inconsistent 1% downstream pressure perturbation and an upstream-only velocity change are rejected by conservation. The actual first-run expectation failure and its correction are documented in `upstream_flow/execution_notes.md`; no tolerance or historical result was changed to obtain PASS. The final audit records **134 assertions**, with 32 covering cells and seven independent numerical reconstructions. This is independent formulation within the project, not an external solver, external expert review or held-out solar benchmark. Assertion counts are not counts of independent physical evidence. ## Scientific implication Supported wording: “For one central, fixed-angle model, nonzero assumed upstream flow changes the inferred magnetic strength while the continued positive-field branch retains a fast-shock characteristic crossing. The branch approaches a finite unmagnetized gas shock at the explicitly calculated central flow limit.” Do not upgrade this to a measured flow bound, a fixed-field robustness result, the joint R93 uncertainty domain, a unique solar diagnosis or a full fan. Matched radio/EUV/thermal association, local geometry, error semantics, emission predictions and observed alternatives remain separate scientific dependencies. ## Sources, reproduction and preservation - [Fitzpatrick, Oblique MHD Shocks](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node105.html), governing local jump relations and characteristic families. Re-read in this stage. - Saved R93 report and JSON in `results/joint_geometry_bounds/`: central anchor, earlier domain and assumptions. Previous calculation not re-run. - Saved R88 report in `results/solar_speed_check/`: different gas-only outer-box threshold and the upstream-flow distinction. Previous calculation not re-run. - Ma/Kozarev observational context remains that documented in R91 and R93; original observations and final journal source versions were not reanalyzed. From the extracted project root: ```bash python3 upstream_flow/audit.py python3 upstream_flow/plot.py ``` Python standard library provides the audit; plotting uses Matplotlib and NumPy. No numerical library installation or external data acquisition is needed. The full prior scientific code and results are preserved. The R93 QuickLook HTML remains the saved interface version; R94 does not claim new browser, Save As, backend or live-page acceptance. ## One next proposed step, not executed A proposed bounded comparison would specify one MHD shock-polar example anchored to the v_n = 0 R94 state. Fix the complete upstream state and a single stationary-front inertial frame, vary the front orientation in an explicitly bounded interval, and display downstream velocities in fixed global axes with linked RH, entropy and characteristic checks. The R94 reference transition must lie on the locus. Determine the angle domain and reference control before calculation. This would test what a polar adds to observer reasoning. It would not label the current variable-upstream curve a polar, require a full fan, or identify the separate PDS X/diamond observations.