# RMO98 — one published slow-shock polar reproduced

## Result for an observer

RMO reproduces one regular slow-shock curve published by Urashima & Morioka
(1966). The curve describes possible plasma states behind one shock when its
orientation changes at fixed upstream conditions. It is not the shape of a
solar front. This result supports this tested model calculation and its local
classification; it does not identify a slow shock in an EUV image.

All 177 sampled states pass conservation and entropy checks and receive a
slow classification from the unchanged state-only diagnosis. The largest
deflection difference from seven preselected readings of the published curve
is 0.55512 degrees, within the predeclared 1.5-degree printed-curve tolerance.
The original publication supplies the model and graph; RMO reproduces them.
This is validation, not a claim to have discovered a new polar.

## Primary source and inherited audit

Shin-o Urashima & Shigeki Morioka, “Magnetohydrodynamic Shock Polar”,
Journal of the Physical Society of Japan 21, 1431–1439 (1966).
DOI: https://doi.org/10.1143/JPSJ.21.1431

The purchased nine-page PDF has SHA-256
31dae9cfeb93340e472153598b3de5f36280d82273b412b298299a9b771a0d56.
Earlier Checkpoint 10A was reused. The actual equations and figure were
visually checked from the primary pages, since the PDF text layer contains
only the publisher download header. Printed pp.1432–1433 give Eqs.(6)–(16);
Fig.3(a), p.1434, supplies the selected M1²=4, A1²=0.9, gamma=5/3 curve.
The source assumes flow and field parallel on both sides and distinguishes
the regular slow curve from the hatched nonevolutionary trans-Alfvénic curves.
Neither those other curves nor temporal stability is tested in RMO98.

## Matched inputs and conventions

The squared source numbers are M1²=4 and A1²=0.9, not M1=4 and A1=0.9.
Choose normalized rho1=1, v1=(1,0,0), p1=0.15, B1=(sqrt(10/9),0,0), mu0=1,
gamma=5/3. Their scale is arbitrary; no velocity in km/s or field in gauss
is inferred. Alpha is the shock-PLANE angle to upstream velocity. The normal
is n=(sin alpha,-cos alpha,0), with t=(cos alpha,sin alpha,0). All velocities
refer to one fixed frame in which every candidate plane is stationary.

The sample is alpha=91,91.5,...,179 degrees. Exactly parallel and zero-normal-
flux endpoints are excluded. Density and pressure are not held fixed downstream.

## Two calculations and physical checks

The RMO route uses its existing gamma=5/3 compression polynomial, followed by
magnetic and velocity jump reconstruction. A new wrapper uses the declared
1.9<rho2/rho1<2.4 bracket and permits the regular denominator to be negative.
The historical R95 wrapper was intentionally limited to its fast-sector
bracket and positive denominator; that code was not changed or rerun.

The source route separately transcribes Eq.(7) in
x=tan(alpha-theta)/tan(alpha), with its unique positive root in 1<x<10.
Eqs.(8)–(13) then give compression, field, pressure, total speed and deflection.
The two scalar formulations and recovery routes are separate; they describe
the same ideal-MHD physics and share elementary geometry/arithmetic utilities.

Both use 90-digit Decimal arithmetic. Original mass, momentum, induction,
total-energy and normal-field fluxes are evaluated for both state pairs.
Their maximum scaled residual is below 1.02e-69; maximum normalized state
disagreement is below 2.82e-69, inside the predeclared 1e-55 and 1e-50 limits.
These numbers describe arithmetic consistency, not accuracy of the source
plot or any solar observation.

Classification is applied after solving, using the unchanged local module,
with no input family field. Every sampled state has positive pressure,
compression, increasing entropy and the 3-to-4 characteristic transition:
upstream normal speed exceeds the slow speed but is below the normal Alfvén
speed; downstream normal speed is below the downstream slow speed. Tangential
field decreases. This is the source's regular slow family.

## Sample states

| Plane angle alpha | v2/v1 | Deflection (deg) | rho2/rho1 | Bt2/Bt1 | u_n1/c_s1 | u_n2/c_s2 |
|---:|---:|---:|---:|---:|---:|---:|
| 120 | 0.401442 | 24.361946 | 2.167774 | 0.170991 | 2.065739 | 0.652114 |
| 150 | 0.259547 | 42.641173 | 2.018359 | 0.180475 | 2.169993 | 0.678185 |
| 175 | 0.102799 | 20.446837 | 1.973183 | 0.183863 | 2.212332 | 0.686356 |

These table values are RMO calculations, not tabulated values from the paper.
The source figure does not contain independent numerical density measurements.
Across the sampled angles, compression is approximately 1.972–2.286,
tangential-field ratio 0.165–0.184, upstream slow Mach number 2.000–2.214,
and downstream slow Mach number 0.628–0.687. The upstream normal Alfvén Mach
number is sqrt(0.9)=0.94868 throughout. These are sampled ranges, not interval
certificates over measurement uncertainty.

## Comparison to the actual published numerical curve

Before predicting the polar, seven ink centres and four axes corners were
recorded from the 300-dpi rendering of Fig.3(a). An axes-only projective map
corrects the slight scan skew. No curve values or model parameters were used
to fit this map. The frozen coordinates and extraction method are in
external_polar/PROTOCOL.md and published_graph_readings.json.

Graphical bounds were declared as ±0.005 in v2/v1 and ±1.5 degrees in theta.
No statistical coverage is assigned. Each extracted central speed has one
resolved matching bracket in the 177-point angular grid. Its predicted
deflection is compared directly with the central printed reading.

| Printed v2/v1 | Printed theta (deg) | Calculated theta (deg) | Difference (deg) |
|---:|---:|---:|---:|
| 0.11976 | 31.023 | 31.162 | +0.139 |
| 0.15103 | 39.292 | 39.617 | +0.324 |
| 0.20226 | 43.705 | 43.678 | -0.027 |
| 0.25214 | 43.334 | 42.983 | -0.351 |
| 0.30186 | 40.019 | 39.725 | -0.294 |
| 0.35143 | 34.250 | 34.116 | -0.133 |
| 0.40187 | 24.797 | 24.242 | -0.555 |

All seven pass the fixed angular bound. The JSON also reports predictions at
the two speed-error-box edges; these are edge values, not a general interval
certificate. Digitizing an old printed graph provides a coarse external
numerical comparison. It cannot support the 69-digit accuracy of the internal
algebraic agreement. There is no claim of a continuous global root-count proof
from the sampled angular brackets.

## Negative control and reproducibility

At alpha=120 degrees, treating the plane angle as the normal angle for the
unchanged states fails normal-field continuity (INCONSISTENT_BN). Thus a common
geometric convention error is rejected before an ordinary family is assigned.

Commands from the project root:

    python3 external_polar/audit.py
    python3 external_polar/plot.py
    python3 external_polar/write_report.py
    python3 external_polar/build_view.py

Python, NumPy, SciPy and Matplotlib are required. The unchanged local diagnosis
and elementary R95 utilities are imported, but no historical audit main is run.
Full requests, states, reference reconstructions and diagnostics are in
RMO_external_polar.json. Numerical acceptance is recorded in verification.json.

## Scientific scope

Established: one regular aligned-field slow-polar example is reproduced using
the published formulas and agrees with extracted points from the published
graph. The existing diagnosis identifies its sampled exact states as slow.
This adds an external literature case to the earlier internal RMO examples.

Not established: comparison with an external executable solver, independent
expert validation, a blinded trial, full MHD-family coverage, degenerate limits,
measurement-error robustness of this slow case, a full Riemann fan, source-driver
history or identification of an observed solar slow shock. The original-field
alignment is an explicit source model assumption. R94–R97 are preserved.

## One next proposed scientific step, not run

Assess one incomplete-input diagnosis: at one selected R98 state, which of
the available compression, normal-flow and thermal constraints are sufficient
to reject a fast alternative when the magnetic field is not supplied? First
declare exactly which quantities remain known and which may vary. Either
construct an admissible competing state or give a bounded exclusion argument;
solver failure alone is not exclusion. This connects the validation result to
what an observer must actually constrain. No new error scan or event programme
is started automatically.

## Interface and preservation

QuickLook adds one folded MHD check with this figure and downloadable records.
One static grey summary under the title uses the footer's 13px font size and
names the four saved model types, explicitly marked Model tests and from the
supplied states and front speeds. A disclosure arrow opens a brief explanation
of the exact-input model checks. It has no version numbers, links or animation.
Solar observations and the full four-model evidence block remain in place.
The requested logo is 20% smaller and aligned with the title; earlier scientific
sections, controls, movie and branding pixels are preserved. Only structural,
asset and script-preservation checks are claimed; no native-browser acceptance.
The complete R98 project copy retains previous science and the source packet.