RMO QuickLook

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The angle is between the upstream magnetic field and the normal, the direction perpendicular to the front surface. At 90° the magnetic field lies along the surface. At 83.3° it tilts away from that configuration by 6.7°.

Every adopted combination of scalar inputs contains this checked angle range. It is a sufficient tested range. We have not shown that the type changes outside it, and we have not measured this angle on the Sun.

The adopted scalar limits are treated as simultaneous outer bounds. A measured covariance or confidence probability has not been supplied. Downstream field, pressure and velocity are linked by the equations and are not varied independently.

A fixed interval covering checks the entire selected model domain. Seventeen independent full-flux reconstructions and characteristic-speed calculations agree with the unchanged local diagnostic. The finite controls check the implementation; the continuous-domain argument supports the range statement.

For the actual solar front: the local field angle, plasma motion and association of radio, EUV and thermal measurements still need constraints. A compatible model is not yet an independent identification of the observed wave. Open the published evidence and remaining measurements.

Sources: Ma et al. (2011), Kozarev et al. (2011); MHD jump conditions and classes.

# RMO93 — input ranges and field tilt checked together

**Result in one sentence:** The fast-shock interpretation of this model survives the adopted input ranges together with a field-to-normal angle from 83.3° to 90°.

This is a robustness result for a constructed ideal-MHD model motivated by the 13 June 2010 event. It is not an independent determination of the observed EUV-front type. It joins the earlier perpendicular scalar-range check (RMO89) and the central-state field-tilt check (RMO92).

## What changed, in ordinary words

Previously we checked input ranges with an exactly perpendicular field, and separately tilted the field in one central model. We now allow both changes together. We change the input constraints, then solve for magnetic strength, pressure and velocity consistently on both sides of the front. We do not add unrelated noise to the calculated downstream quantities.

For all adopted scalar input ranges there is a common model angle interval from 83.3° to 90°. The plasma still enters faster than the fast-mode speed and leaves slower than it, with compression, magnetic amplification and entropy increase. Full conservation and the downstream super-Alfvénic normal-flow condition also hold. These combined checks support the fast-shock class for this constructed local family.

The angle is measured between the upstream magnetic field and the front normal. At 90° the field lies in the front surface. An angle of 83.3° is a departure of 6.7° from that case. This chosen tested range is not an observational uncertainty, not a maximum allowable tilt, and not a claim that the wave changes type at 83.3°.

## Inputs, provenance and units

| Quantity | Adopted range | Meaning |
|---|---|---|
| Density compression r | 22500/18769 to 32400/16129 (about 1.1988–2.0088) | Previously saved radio-based range under its lane-identification assumptions |
| Normal inflow U1 | 495.77–667.03 km/s | Previously saved speed envelope, treated as normal shock speed with upstream plasma at rest |
| Upstream sound speed c1 | 126–186 km/s | Previously saved thermal range with the same composition assumptions |
| gamma | 5/3 | Fixed model choice |
| Normal-field parameter h | 0–0.01 | Chosen model extension, not a measured magnetic angle |
| Common field-to-normal angle | 83.3°–90° | Contained in the certified domain for every adopted r and thermal ratio |

The source record and its SHA256 are saved in the JSON. The scalar bounds are treated as simultaneous hard outer limits. We do not assign a confidence level, assume a measured covariance, or claim that all combinations occurred on the Sun. Unknown association of the radio, EUV and thermal patches remains explicit.

## Algebraic reconstruction

In units rho1 = U1 = mu0 = 1, let b = c1²/(gamma U1²) and h = Bn²/(mu0 rho1 U1²). The enclosing b range is [1944000/90801841, 207576000/2457878929]. This ratio encloses all combinations in the adopted U1 and c1 ranges; it is not a new independent measurement.

For gamma = 5/3 define

```
M = 4-r-5br
d = 1-rh
W = r+5-2rh(4-r)
q = r(1-h)/d
A = 2M d²/(rW)
N = b r²[5r+1-2h(4r-1)] + (r-1)³
p2 = N/(rW)
```

Then rho2=r, un2=1/r, Bn1=Bn2=sqrt(h), Bt1=sqrt(A), Bt2=q sqrt(A), and ut2=sqrt(hA)(q-1). The upstream tangential velocity and second tangential components are zero. The reconstruction follows one continuous coplanar branch; the physical diagnostic itself receives no input family label.

Mass and normal-field continuity are built into these expressions. Tangential momentum determines ut2. Induction, normal momentum and total energy are checked as exact polynomial identities after clearing the positive denominators. These simplified expressions agree exactly with the earlier RMO92 algebra at the independent controls.

## Entropy and characteristic conditions throughout the domain

All 128 fixed cells of an 8 × 4 × 4 partition in (r,b,h) are checked by outward-rounded 50-digit interval arithmetic. This is a covering of a continuous domain, not a fraction of random trials. Positive M, d, W, A and p2 hold, with r>1 and q>1.

The entropy argument avoids treating correlated pressure and compression as unrelated intervals. Exact algebra gives

```
dp2/dh = 2 M (r-1)³/W² > 0
p2(0)(4-r) - b(4r-1) = A(0)(r-1)³/2 > 0.
```

Thus the entropy increase is at least the gas-Hugoniot value at the smallest compression. Its positive derivative in r has numerator 20(r-1)². The resulting lower enclosure for entropy change divided by cv is 0.0014871475049217420582322361124611737540351547578899. Energy conservation remains a separate requirement; entropy constancy is not substituted for it.

For the upstream state, the magnetosonic matrix trace is below un1²=1, which places its largest eigenvalue below the squared inflow speed. For the downstream state, its magnetosonic characteristic polynomial at un2² has a strictly negative value, placing the normal flow between the slow and fast speeds. The downstream normal flow exceeds the normal Alfvén speed because 1-rh>0. All signs are checked over every cell. Quantitative Mach bounds use the stable magnetosonic discriminant. Downstream scaling is simplified algebraically before interval evaluation to avoid repeated artificial dependence on r.

The resulting outer enclosures over the full (r,b,h) domain are:

| Quantity | Outward numerical enclosure |
|---|---|
| Upstream fast Mach | 1.1199803342954961148975377192067523747727135650837 to 1.9843537938201525349807875020541435216014529586962 |
| Downstream fast Mach | 0.58242253303646670785744819921850002723746603127782 to 0.94331002409022115466976821740450554779053545434143 |

The figure displays rounded labels; the full decimal enclosures are in JSON. These bars are mathematical bounds, not probability distributions. They also enclose the common 83.3°–90° subdomain.

## Why the common angle range is valid

A decreases with r, b and h in the certified domain. For r, the derivative of M is negative, d is nonincreasing, and W+r dW/dr is positive. For b, dA/db = -10d²/W. For h, the derivative sign follows from W-(4-r)d = 2r+1-rh(4-r)>0. These required signs are checked on the domain.

Consequently the largest terminal angle at h=0.01 occurs at the smallest r and b. Its outward interval is 83.231591939875001708461977142396395250102557688468 to 83.231591939875001708461977142396395250102557688573. For each r,b, the angle varies continuously and strictly from 90° down to a terminal angle no larger than that value. Every pair therefore contains the conservatively stated interval 83.3°–90°.

Arctangent bounds use an alternating power series with an explicit remainder. Pi is enclosed with Machin's identity, using the same outward arithmetic. The larger certified h domain contains angles down to about 75.87° for some inputs; this is not a common angle interval for all inputs, so it is not used as the headline result.

## Independent checks

Seventeen model controls comprise the 16 combinations of the endpoint r, U1, c1 and h values, plus one interior state. At each, an independent four-variable Newton solve uses the original full-flux equations and a perturbed perpendicular starting state, rather than the simplified answer formulas. It agrees with the reconstruction to better than 1e-55. Separate 80-digit full-flux residuals are below 1e-65, and independently calculated characteristic-matrix speeds agree with the existing diagnostic. Independent Mach values lie within the reported enclosures.

The unchanged classifier supports a fast shock at every control. A common Galilean boost preserves the result. An inconsistent 1% pressure perturbation is rejected. 660 recorded assertions pass, including the cell conditions and control checks. This count is not a number of independent observations. The continuous-domain proof, rather than the finite controls, supports the full-range statement.

## What remains unresolved

- The local solar field angle, front normal and upstream plasma velocity are not measured by this calculation; zero upstream flow remains an assumption.
- Radio-lane interpretation and the association of radio, EUV and thermal volumes still need matching at the selected patch and time.
- Error semantics, covariance and independent emission/non-wave comparisons are not supplied by this model certificate.
- The reconstruction does not enumerate all MHD families, determine a blast/piston driver, establish a complete Riemann fan or prove global uniqueness.
- No new raw observations, publication action, broad numerical campaign occurred.

The useful advance is specific: the conditional fast-shock model does not rely on holding either all scalar inputs exact or the field exactly perpendicular.

## Sources and reproduction

- [Takahashi & Yamada, ideal-MHD jump relations and local shock classes](https://arxiv.org/html/1310.2330v1#S2.SS2).
- [Fitzpatrick, Oblique MHD Shocks](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node105.html).
- [Ma et al. (2011), solar event and model context](https://arxiv.org/html/1106.6056v1).
- [Kozarev et al. (2011), event and magnetic-model context](https://arxiv.org/html/1406.2372v1).

Run `python3 joint_geometry_bounds/audit.py` for the certificate and `python3 joint_geometry_bounds/report.py` for this report and vector figure. Prior scientific modules and output records remain unchanged. Native browser acceptance is a separate interface check.

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The whole chosen one-parameter range passed conservation-linked and interval checks. Three separate reconstructions from the full flux equations and an independent wave-speed calculation agreed with the existing diagnostic. The plotted dots mark these three model controls.

There are 97 recorded checks, including the interval covering and negative controls. They are not 97 solar observations. Earlier scalar-error tests remain available; they have not yet been combined with this angular continuation.

What is still needed for the Sun? A field angle and plasma velocity matched to the selected front, with their uncertainties, and comparison with the observed emission. See the source and geometry audit.

RMO conserves total energy; entropy is checked separately. Total energy includes thermal, kinetic and magnetic energy. A shock can convert ordered flow into heat while conserving the total energy flux. Entropy increases in the checked model; it is not forced to stay constant.

We tested a deliberately wrong construction that keeps entropy constant. Mass, momentum and induction still matched, but total energy did not. RMO rejected it and assigned no wave type. This is a model control showing why the energy equation is necessary.

This distinction also matters in the history of shock theory. The modern MHD Riemann problem uses the correct energy jump condition: Takahashi & Yamada, jump relations. For the historical development, see Salas (2007). Our control is not a reconstruction of Riemann’s original gas calculation.

# RMO92 — a bounded oblique extension of the central solar model

**Result in one sentence:** In this model, allowing a small normal magnetic field preserves the fast shock; the downstream flow also turns.

This extends the constructed central model associated with the 13 June 2010 event. It does not determine the magnetic angle or wave type of the observed solar front. The three numerical states below are model results, not three observations.

## Physical question and chosen inputs

Does the previously checked fast shock disappear immediately when the field is no longer exactly parallel to the front surface? Here the field-to-normal angle decreases from 90° (perpendicular propagation) to about 81.09° (oblique propagation). This is a deliberately chosen model interval, not an observational error bar.

We keep compression r = 1.56, upstream normal speed relative to the front U1 = 600 km/s, upstream sound speed c1 = 156 km/s, and gamma = 5/3 fixed. We use the prior composition and an initially stationary upstream plasma in the adopted solar frame. Field amplitude and all downstream quantities are reconstructed jointly. This is not a rotation at fixed magnetic strength and not the full scalar-error domain checked in RMO89.

In normalized units rho1 = U1 = mu0 = 1, p1 = b = 507/12500, and the front is at rest. The continuation parameter is h = Bn²/(mu0 rho1 U1²), with 0 <= h <= 1/100. The upstream tangential velocity is zero in this frame. At h = 0 the reconstruction returns the saved perpendicular central state exactly.

## What the picture means

The plasma enters faster than the upstream fast-mode speed and leaves slower than the downstream fast-mode speed, while remaining faster than the downstream normal Alfvén speed. Compression, magnetic amplification, the conservation relations and entropy increase also pass. These combined conditions support the fast-shock class for this constructed local family. A large image speed alone would not be sufficient.

On the right, the downstream plasma acquires a velocity along the front surface. The magnetic field changes the direction of the flow as well as its compression. This is a model prediction for the selected family.

| h | Field-to-normal angle (degrees) | Upstream fast Mach | Downstream fast Mach | Tangential flow after front (km/s) |
|---|---:|---:|---:|---:|
| 0 | 90.000000 | 1.439436 | 0.737023 | 0.000 |
| 1/400 | 85.551185 | 1.439261 | 0.736994 | 10.839 |
| 1/100 | 81.090455 | 1.438728 | 0.736906 | 21.773 |

## Conservation-linked reconstruction

Let K = gamma/(gamma-1), q = Bt2/Bt1 and A = Bt1² in normalized units. The coplanar construction is

```
q = r(1-h)/(1-rh)
C = (h+K/r)(1-q²)/2 + q²/r
A = [1/(2r²) + (K/r)(b+1-1/r) - (1/2+Kb)] / (1-C)
rho2 = r; un2 = 1/r; Bn1 = Bn2 = sqrt(h)
Bt1 = sqrt(A); Bt2 = q sqrt(A)
ut2 = sqrt(h A)(q-1)
p2 = b+1-1/r + A(1-q²)/2
```

The tangential momentum equation gives ut2, tangential induction gives q, normal momentum gives p2, and total energy gives A. Mass and normal-field continuity are built into the parameterization. The code verifies the remaining rational conservation identities exactly, after clearing denominators whose positivity is checked on the chosen domain. Negative Bn with the corresponding reversed tangential flow is also checked at the three control states.

## Continuous-domain check and independent controls

All 32 adjacent intervals [j/3200,(j+1)/3200], j=0,...,31, are checked with outward-rounded 50-digit decimal interval arithmetic. They cover the entire chosen h domain. Reconstruction denominators, pressure, field strength and the fast-shock inequalities retain the required signs. The entropy increase divided by cv has a lower enclosure above 0.16277. Conservative rounded enclosures for the fast Mach numbers are 1.435–1.443 upstream and 0.733–0.741 downstream; the unrounded interval records are in the JSON.

The sign of C'(h) is checked throughout. With a positive constant numerator in A and 1-C positive, this implies A decreases; consequently h/A increases and the field-to-normal angle decreases continuously. The displayed endpoint angles are numerical evaluations. At exactly h=0 the separate perpendicular diagnostic applies; the nonzero-h states use the existing oblique diagnostic.

At h = 0, 1/400 and 1/100, an independent four-variable Newton reconstruction solves the full flux equations without using the continuation formulas. It agrees with the constructed states to better than 1e-60 in normalized units. Direct 80-digit flux differences are below 1e-65. A separate magnetosonic matrix calculation agrees with the diagnostic's characteristic speeds. Common Galilean boosts preserve classification, and a 1% inconsistent change of downstream pressure is rejected.

The diagnostic receives states and front speed without a family label. The construction intentionally follows the saved branch, so this is a local continuation check, not a blind search over all solution families. The 201 plotted samples visualize the curve; they do not replace the continuous-domain check. 97 recorded checks pass. Their count includes the covering intervals and controls, not independent solar events.

## Why energy cannot be replaced by constant entropy

RMO checks conservation of mass, momentum and **total energy**, together with induction and normal-field continuity. Total energy contains thermal, kinetic and magnetic terms. Entropy provides a separate admissibility condition: it need not remain constant through a shock and increases in this model.

As a deliberate negative control we impose p2 = p1 r^gamma, then reconstruct the field to satisfy normal momentum at h=0. Mass, momentum and induction still agree, and the entropy change is essentially zero. However the energy-flux difference is about -0.01390614 in the chosen normalized units. The unchanged diagnostic rejects these states and assigns no wave family. This is an illustration of an incorrect closure, not an admissible shock or a reconstruction of Riemann's historical gas calculation.

The historical distinction concerns an early shock treatment; the modern MHD Riemann problem uses total-energy conservation. Riemann's wave construction and the later correct shock relations are complementary parts of the modern framework.

## Limits and next dependency

The result does not establish global uniqueness, enumerate a full Riemann fan, determine the driver (blast/piston), fit EUV emission, or exclude non-wave image interpretations. It does not combine h uncertainty with all earlier scalar input ranges. A matched local field-to-normal angle, upstream plasma motion, radio/EUV/thermal association and joint observational uncertainty remain needed before treating this as a diagnosis of the selected solar front. No new raw data or new observed angle was used.

## Sources

- [Takahashi & Yamada, MHD jump relations and admissibility, Section 2.2](https://arxiv.org/html/1310.2330v1#S2.SS2).
- [Fitzpatrick, Oblique MHD Shocks](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node105.html).
- [Ma et al. (2011), event and perpendicular model context](https://arxiv.org/html/1106.6056v1).
- [Kozarev et al. (2011), event and magnetic-model context](https://arxiv.org/html/1406.2372v1).
- [Salas (2007), history of shock-wave theory](https://doi.org/10.1007/s00193-007-0084-z).

The current source audit and prior numerical records are preserved separately. RMO92 code and outputs are stored in `oblique_continuation/` and `results/oblique_continuation/`.

13 June 2010 · Ma et al. (2011) · Kozarev et al. (2011)

Published magnetic modelling makes a near-perpendicular shock nose plausible; the exact angle and upstream plasma motion remain assumptions of this calculation.

Kozarev et al., Section III.4 and Figure 4 use a potential-field model and discuss a possible quasi-perpendicular nose. This motivates exploring the model. It does not measure an angle of exactly 90°.

For the observer: the model has a reason to be tried, and it passes the existing conditional checks. To establish how closely it describes your selected solar front, constrain the local field angle and plasma flow as well as the shared region and time.

Next useful input: a matched front surface and magnetic-field estimate with an angle uncertainty. A model-derived angle is useful when identified as such. Choosing “perpendicular” in a menu supplies an assumption; it does not measure that angle.

The radio heights used by Ma et al., Section III.2 depend on a coronal density model. They are not a radio image locating the source. The existing upstream-flow check and EUV emission comparison remain available.

Save the evidence and explanation

# RMO91 — why use a near-perpendicular model for 13 June 2010?

**Result in one sentence:** published magnetic modelling makes a near-perpendicular shock nose plausible, while an exact 90° angle and stationary upstream plasma remain assumptions of the checked RMO model.

This continues the RMO89 source-dependency assessment. It adds evidence provenance; it does not change any numerical result or claim an independently identified solar MHD type.

## What the existing publications actually contribute

| Question | Available support | How RMO should use it |
| --- | --- | --- |
| Are the EUV and radio signatures related? | Ma associates their onset with the same eruption. | Event-level support; not a resolved radio/EUV patch match. |
| Where does the radio emission originate? | Heights are inferred through density models. | Keep model-dependent heights distinct from radio imaging. |
| Why investigate near-perpendicular geometry? | Kozarev's PFSS overlay motivates a possible quasi-perpendicular nose. | Qualitative magnetic-model support; no numerical angle bound. |
| Why is B_n exactly zero in the calculation? | Ma's thermal reconstruction adopts perpendicular jump equations. | A specified limiting model, not a measured angle. |
| Is the upstream plasma at rest? | No matched upstream normal flow is supplied in the reviewed records. | Retain zero flow as an assumption and preserve the RMO88 sensitivity result. |
| Do the thermal and kinematic inputs describe identical volumes? | Regional DEM and selected EUV traces have different spatial definitions. | Keep this association dependency in the observation model. |

Sources: [Ma et al. (2011), Sections II and III.1–III.3](https://arxiv.org/html/1106.6056v1); [Kozarev et al. (2011), Sections III.1, III.2, III.4 and Figure 4](https://arxiv.org/html/1406.2372v1). The structured record in `observer_evidence/sources.json` gives the individual locators and evidence status.

## The positive conclusion

The perpendicular reconstruction has a physical motivation in the literature. It is useful to explore whether a shock of this kind can reproduce the supplied constraints and whether its diagnosis survives errors. RMO89 already established conditional consistency across its adopted scalar bounds.

The new source reading adds the reason for choosing that model. It does not supply a second numerical angle measurement. A closed field configuration by itself is insufficient to calculate the angle: the magnetic vector and the local front normal must be paired at one position and time. A model-derived angle could still be useful, provided its origin and uncertainty are retained.

## Three levels of association

1. **Same eruption:** supported by the published timing and context.
2. **Same shock surface or nose:** an interpretation supported by the authors' comparisons and models.
3. **Same local patch at the diagnostic time:** not recovered as a direct radio-image/field/normal match in the reviewed material.

These levels should not be collapsed into a single yes/no flag. Event-level association is scientifically useful; a local jump inversion asks a more spatially specific question.

The radio height consistency and its density-derived speed also share a model dependency. They should not be counted as two independent position and velocity measurements. Likewise, magnetic strength or downstream temperature reconstructed from shock equations is an output of that interpretation.

## Consequence for the next calculation

The current B_n=0 result applies to its exact geometry. Qualitative support for a nearly perpendicular nose does not automatically extend that result to finite nonzero B_n. A future departure-from-perpendicular study would have to construct conservation-linked states, check all relevant characteristic and admissibility conditions, and clearly label any chosen angle range as a model scenario unless observational bounds become available.

Similarly, a model normal speed and an upstream plasma speed must be distinguished. The incoming speed in the front frame depends on their difference along the same normal. The previously checked upstream-flow sensitivity remains relevant; the present literature reading supplies no new numerical bound on that flow.

A useful next observational input would therefore be a local angle estimate from a matched front surface and magnetic model, with its uncertainty and provenance. This would constrain a scenario rather than silently replace unknown geometry by exactly 90°. Direct or explicitly model-based upstream-flow information and the matched emission comparison remain additional dependencies. This checkpoint starts no new data acquisition or numerical campaign.

## What changed in QuickLook?

The solar example now contains **Why this solar geometry?**, opening a compact evidence table and this explanation. Existing results, figures, model inputs, scripts and reports remain available. No unknown input is filled and no classification label is assigned by the new section.

Static preservation and navigation checks are recorded in `results/observer_evidence/verification.json`. They do not validate native Chrome rendering or downloads. No new physical solver execution was needed for this source audit.

Quoted errors are treated as simultaneous bounds for this test. Their statistical meaning and covariance remain uncertain; no confidence level is assigned. Derived pressure, field, flow and temperature are linked by the model and are not perturbed independently.

The geometry is prescribed. This calculation does not compare all oblique, slow or non-wave explanations of the actual solar event.

56 focused checks pass. Exact conservation identities and outward-rounded inequalities cover the full domain. Eight corner cases and the previous central model also pass separate high-precision flux and published compression-equation checks.

The existing diagnostic receives those nine complete state pairs without a family label and returns fast shock for each. The chosen perpendicular construction restricts the problem; this is an internal physical check, not an independent observational classification.

This is a saved checked calculation. Opening it does not start a new Python run. The existing solver and earlier scientific results are unchanged.

# RMO89: does the perpendicular solar model survive the input bounds?

**Result in one sentence:** the perpendicular magnetic model remains a fast
shock throughout the tested speed, sound-speed and radio bounds, if the upstream
plasma is at rest and the stated measurements describe the same front.

This is a conditional model calculation for the **13 June 2010 EUV event**.
Ma et al. (2011) already interpreted this event as a shock and used a perpendicular
magnetic model. RMO extends the model check across an explicit, linked input
domain; it does not claim a new solar discovery or an independent measurement of
the front's MHD family.

## Read the picture

The figure compares plasma flow relative to the front with the local fast-wave
speed. Each plotted number is a ratio, so 1 is the dividing line.

- **Before the front: 1.148–1.863.** The incoming plasma is faster than the local fast
  wave throughout the tested domain.
- **After the front: 0.630–0.879.** The outgoing plasma is slower than the local fast
  wave throughout the tested domain.
- The reconstructed density and tangential field both increase by the same
  factor, 1.199–2.009; entropy increases and conservation laws hold.

This super-fast to sub-fast transition supports a **fast shock in this model**.
It describes flow through one front, not how the front accelerates over time.
The ratio bars enclose all model outputs in the domain. They are not measured
Mach-number confidence intervals; upstream and downstream values remain linked.

## Published inputs and provenance

| Input | Value used | Source / condition |
| --- | --- | --- |
| Event and time | 13 June 2010, 05:40 UT | Ma et al. (2011), Section III.2 |
| Front speed | 495.77–667.03 km/s | RMO88 evaluation of every Kozarev Table 1 fit at 180 s from 05:37 UT |
| Upstream sound speed | 126–186 km/s | Ma: 156 ± 30 km/s; a supplied thermal-model constraint |
| Lower radio frequency | 127–137 MHz | Ma: 132 ± 5 MHz |
| Upper radio frequency | 150–180 MHz | Ma: 165 ± 15 MHz |
| Density ratio r | 22500/18769 to 32400/16129 | Derived from (upper/lower radio frequency)^2 |
| Ratio of specific heats | 5/3 | Fixed model choice |
| Magnetic direction | Bn = 0 exactly | Assumed perpendicular geometry, not observed here |
| Upstream plasma velocity | 0 in the frame of the adopted front speed | Assumed, not measured |

The plus/minus quantities are treated as simultaneous bounds for this calculation.
Their statistical definitions and joint covariances are not supplied by the
extracted records, so no confidence level is assigned. The outer set is deliberately
an enclosure of the scalar combinations; it need not describe the original fit's
attainable corners. Published profiles and their averages are not independent
pieces of evidence.

The sound speed is the supplied thermal input. A separate temperature value is
not varied independently alongside it. Derived density, pressure, field, flow
speed, Mach numbers and downstream temperature are not independent measurements.
The radio interpretation assumes paired upstream/downstream emission from the
same discontinuity and fixed composition. EUV is not radio emission.

## Linked construction, not independent noise on downstream states

Use one front-rest frame with positive flow from state 1 to state 2. Normalize
rho1 = U1 = mu0 = 1 separately for each model; the dimensional velocity scale is
the adopted U in km/s. Bn and tangential velocities vanish. Define

\[
b = p_1 = \frac{3}{5}\left(\frac{c_1}{U}\right)^2,\qquad
A = B_{t1}^2 = \frac{8-2r-10br}{r(r+5)}.
\]

Conservation fixes the other state:

\[
\rho_2=r,\quad U_2=1/r,\quad B_{t2}=r\sqrt A,\quad
p_2=b+1-1/r+\frac A2(1-r^2).
\]

The same b, r and A are used in every expression. None of the derived inputs is
perturbed independently. Magnetic-field strength is reconstructed under the shock
model, not independently measured. This is a restricted perpendicular family of
single discontinuities, not a search over full Riemann fans, oblique branches or
non-wave explanations.

## What was proved over the entire domain?

The following are algebraic identities or outward-rounded interval bounds over
the full scalar enclosure. The finite control states below are additional checks,
not a substitute for this whole-domain argument.

1. **Conservation:** mass and tangential induction follow from U2 = 1/r and
   Bt2 = r Bt1. With the prescribed zero components, the other tangential fluxes
   vanish. Exact rational polynomial cancellation proves normal momentum and
   energy conservation for all r, b with nonzero denominators.
2. **Physical states:** A > 0 everywhere. A positive expression for downstream
   pressure avoids a cancellation-sensitive sign test:

\[
p_2=\frac{br(5r+1)}{r+5}+\frac{(r-1)^3}{r(r+5)}>0.
\]

3. **Fast crossing:** cf1^2 = 5b/3 + A and
   cf2^2 = (5p2/3 + r^2 A)/r. The exact factorizations are

\[
1-c_{f1}^2=\frac{(r-1)[3(r+8)-5br]}{3r(r+5)}>0,
\]

\[
c_{f2}^2-1/r^2=
\frac{(r-1)[(10-r)(r+2)-5br^2]}{3r^2(r+5)}>0.
\]

Outward interval evaluation gives strictly positive lower bounds for both
expressions on the complete domain. No point is certified using only a central
state or a Monte Carlo fraction.

4. **Entropy:** put k = A/(2b) > 0. Conservation also gives

\[
\frac{p_2}{b}=\frac{4r-1+k(r-1)^3}{4-r}.
\]

Thus entropy/cv is bounded below by
ln[(4r−1)/(4−r)] − (5/3) ln r. Its derivative is
20(r−1)^2/[3r(4r−1)(4−r)] > 0 for 1 < r < 4.
At the lower compression bound its outward-rounded lower value is
**0.0014871**, strictly positive.

5. **Mach envelopes:** for cf1^2 the r derivative has numerator
(2+10b)r^2−16r−40 < 0; the b derivative is 5(r−1)/[3(r+5)] > 0.
For r^2 cf2^2 the r derivative has numerator
−2(1+5b)r^3−(6+70b)r^2+(90+50b)r+80 > 0,
and its b derivative is −5r^2(r−1)/[3(r+5)] < 0.
Every sign is interval-checked over the domain. These monotonicities give the
Mach endpoints in the figure, with outward rounding rather than an unproven
corner-extremum assumption.

## Independent controls and practical limits

**56 focused checks pass.** Eight scalar corners and the previously used
central model (r = 1.56, U = 600 km/s, c1 = 156 km/s) are separately evaluated
with 75-digit Decimal arithmetic. Direct mass, induction, momentum and energy
residuals are below 1e−65 in the normalized calculation. The separately evaluated
published compression relation, Ma Eq. 5, recovers r to the same tolerance.
These small residuals describe arithmetic, not observational precision.

The unchanged RMO77 perpendicular diagnostic receives the nine complete model
state pairs and front speed **without a family label**. Every pair passes its
existing conservation, entropy and characteristic-speed criteria and is classified
as a fast shock. Choosing the perpendicular construction itself already restricts
the candidate family; this is not a blind competition between solar fast and slow
interpretations. Controls at and beyond the gas boundary confirm that A becomes
zero or negative, rather than assigning a magnetic shock to arbitrary inputs.

The existing solver, uncertainty assessors, source audit and earlier results are
unchanged. This calculation runs separately and its checked result is embedded in
QuickLook. Loading the figure does not start a fresh browser-to-Python calculation.
Native browser acceptance remains open; PDF rendering and content preservation
are checked separately.

## What is still needed for a solar diagnosis?

The observed local front normal and field direction, upstream plasma motion, and
association of EUV, radio and thermal information remain unresolved dependencies.
The perpendicular geometry and zero flow are held fixed here. The RMO88 result
already shows why allowing a sufficient upstream flow can change the gas/magnetic
comparison. A positive result in the present slice does not certify those extra
dimensions of uncertainty.

The separate RMO84/85 EUV-delay comparison also remains open. This result does not
repair that comparison, fit an emission history, independently measure the field,
exclude oblique/slow/non-wave alternatives for the actual event, establish
stability or enumerate all Riemann solutions.

**Next useful step:** constrain the same-front geometry and plasma-flow assumption,
then test compatible observational constraints and competing interpretations. The
published shock interpretation is credited; RMO has verified conditional physical
consistency and robustness, not independently confirmed the solar type.

## Sources and reproducibility

- [Ma et al. (2011), Sections III.2–III.3, Eqs. 3 and 5](https://arxiv.org/html/1106.6056v1),
  [published article](https://doi.org/10.1088/0004-637X/738/2/160).
- [Kozarev et al. (2011), Table 1 and its note](https://arxiv.org/html/1406.2372v1),
  [published article](https://doi.org/10.1088/2041-8205/733/2/L25).
- [Fitzpatrick, Perpendicular MHD Shocks, Eqs. 7.269–7.277](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node104.html).

Source numbers are read from the unchanged saved RMO88 calculation and its source
record; no new observational data or atomic-rate files were acquired.
Run `python solar_magnetic_bounds/audit.py`, then `report.py` and `plot.py` from
that directory path in the project root. The JSON preserves exact domain fractions,
outward decimal bounds, source hash and the nine input/output control records.
The prior PDF/report and all previous QuickLook sections remain available.

At 05:40 UT we evaluate each published kinematic fit from its 05:37 origin: D = D0 + a × 180 s. The conditional speed envelope is 495.77–667.03 km/s and the adopted sound speed is 126–186 km/s. Gamma is fixed at 5/3.

The quoted errors are treated as simultaneous bounds for this test, not confidence intervals. Profiles and their averages are not counted as independent observations. Derived speed, Mach number and compression remain linked. The outer coefficient rectangle allows unknown covariance; its corners are not necessarily attainable in the original fit.

The fitted front speed is assumed to be its local normal speed. The radio lanes are assumed to sample the same upstream/downstream structure; composition is fixed. The source region and frame still need observational checks. Image motion is not plasma velocity.

29 focused checks passed: exact endpoint bounds, a separate general-gamma calculation, an inverse threshold check, and a constructed moving-plasma gas shock checked directly against conservation and entropy. These are arithmetic and model checks, not new observations.

The separate EUV-delay check retains its original result. This comparison does not replace it or independently establish a solar shock type.

# RMO88 — Does the magnetic comparison survive speed errors?

**Result in one sentence:** under the adopted normal-speed, sound-speed and radio bounds, a gas-only shock compresses too much if the upstream plasma is at rest; permitting an unmeasured outward plasma flow can change that conclusion.

Event: **13 June 2010, comparison at 05:40 UT**. This is a bounded extension of the RMO76 scalar comparison. It does not replace the RMO84/85 emission audit or classify a new solar front.

## Published result and RMO's added check

[Ma et al. (2011), Section III.3](https://arxiv.org/html/1106.6056v1) already compared gas-dynamic compression with the radio estimate and argued that magnetic effects matter. We retain that attribution. Our addition is to test this comparison across explicit bounds on the existing speed and sound-speed inputs, and to calculate how an unknown plasma velocity affects it. This is not claimed as a new solar discovery.

The kinematic coefficients are from [Kozarev et al. (2011), Table 1 and Section II.1, accessible arXiv v1](https://arxiv.org/html/1406.2372v1). They describe front edges under a spherical/radial deprojection. They are neither local plasma velocities nor independently measured shock normals. The two papers discuss association between EUV and radio features; exact spatial correspondence of every quantity is not established by this calculation.

## 1. Compare speeds at the radio time

We evaluate each published fit at 05:40 UT, 180 s after its 05:37 origin:

\[
D=D_0+a\Delta t.
\]

For this conditional test only, each quoted plus/minus is treated as a simultaneous hard bound. The resulting outer half-width is

\[
\delta D=\delta D_0+\Delta t\,\delta a.
\]

| Published profile | Central speed at 05:40 (km/s) | Conditional bounds (km/s) |
|---|---:|---:|
| 193/I | 599.69 | 548.39–650.99 |
| 193/II | 607.86 | 576.13–639.59 |
| 211/I | 584.89 | 516.39–653.39 |
| 211/II | 579.71 | 528.33–631.09 |
| 193/AVG | 603.77 | 543.62–663.92 |
| 211/AVG | 581.40 | 495.77–667.03 |

The union envelope is **495.77–667.03 km/s**. The profiles and their published averages are not six independent data sets. We do not average them again or combine their errors as independent evidence. All remain alternative source summaries within one conservative outer envelope.

This is not an observational confidence interval. If the published errors are marginal standard deviations instead, the variance is

\[
\sigma_D^2=\sigma_{D_0}^2+\Delta t^2\sigma_a^2+
2\Delta t\,\operatorname{Cov}(D_0,a).
\]

Without the covariance, the possible standard deviation spans
\(|\sigma_{D_0}-\Delta t\sigma_a|\) to \(\sigma_{D_0}+\Delta t\sigma_a\).
Our rectangle does not set covariance to zero: it encloses any joint set contained in the assumed marginal bounds. Some corners may be unreachable under the actual unrecovered fit covariance. Source error definitions are still needed before making a statistical claim.

## 2. Test gas-only compression with those bounds

We retain the supplied sound-speed range **126–186 km/s** and \(\gamma=5/3\). This does not rederive total gas pressure from electron thermometry. Composition, electron/ion temperatures and thermal closure remain assumptions.

In a frame where the front moves outward at normal speed \(D_n\), let the upstream plasma's outward normal velocity be \(v_n\). For the branch considered here,

\[
U_1=D_n-v_n>c_1,\qquad
X_g=\frac{4U_1^2}{U_1^2+3c_1^2}.
\]

The stationary-upstream comparison takes \(v_n=0\) and identifies the source front speed with \(D_n\). These are explicit physical assumptions, not additional measurements.

At rest, monotonicity gives **Xg = 2.8124–3.6132** over the full chosen speed/sound box. The radio-lane assignment gives

\[
X_r=(f_U/f_L)^2\in[(150/137)^2,(180/127)^2]
=[1.1988,2.0088].
\]

There is no overlap. The smallest gas compression exceeds the largest radio ratio by **0.8036**. We use the two original lane frequencies jointly; the derived density ratio is not varied independently of them. Electron and mass-density ratios are equated only for fixed composition.

**Physical interpretation:** with these assignments and assumptions, a gas-only adiabatic shock cannot explain the compression. Magnetic stresses remain a viable way to reduce it, consistent with the published argument. This test neither measures a field nor establishes its orientation or a fast/slow family. It also does not exclude non-wave explanations, other thermal closures, other feature associations or all parallel MHD structures.

## 3. A missing plasma velocity changes the inference

The greatest upstream speed relative to the front allowed by the gas model and the adopted radio/sound bounds is

\[
U_{1,\max}=c_{1,\max}\sqrt{\frac{3X_{r,\max}}{4-X_{r,\max}}}
=323.5827\ \mathrm{km\,s^{-1}}.
\]

Therefore the first possible contact between the scalar ranges, anywhere in the outer speed envelope, occurs at

\[
v_{n,\min}=D_{n,\min}-U_{1,\max}
=172.1873\ \mathrm{km\,s^{-1}}.
\]

This is a **threshold in the assumed model**, not a measured coronal flow. An outward flow below that threshold preserves the exclusion over this box; a flow at or above it is a necessary opening for overlap, not a guarantee that all observations can be fitted. Different normal geometry, sound-speed conventions or radio-lane assignments change the threshold.

As a falsification control, we construct one gas-shock endpoint with Dn = 495.77 km/s, c1 = 186 km/s, the radio upper ratio and the corresponding outward flow. Its states satisfy direct inertial-frame mass, momentum and energy balance and have positive entropy change. It demonstrates why gas-only exclusion is not unconditional when plasma velocity is unconstrained. It is not an observed state, a best fit, or a witness within an unknown narrower observational covariance region. No radio-emission or EUV image model is fitted by this control.

The plot's right panel varies the assumed upstream flow from 0 to 300 km/s, with the same linked gas relation throughout. This is a sensitivity curve, not a time history or measured flow distribution.

## What should be measured next?

**Constrain the upstream plasma velocity along the local front normal**, and verify the front, thermal region and radio source refer to the same relevant plasma and time. A Doppler component alone still needs its projection relative to that normal. Stereoscopy constrains geometry but does not itself supply plasma velocity.

The earlier reconstructed perpendicular fast-shock model remains a conditional example. The present scalar contrast cannot upgrade it to an independently identified solar shock. The separate EUV-emission comparison remains open with its original source and density-history limitations.

## Reproduction and checks

Run from the project root:

```sh
python3 solar_speed_check/audit.py
python3 solar_speed_check/plot.py
```

The audit uses exact fractions for the printed source decimals and monotonic endpoint bounds. A separate 75-digit Decimal calculation uses the general-gamma gas relation. Direct lab-frame Rankine–Hugoniot evaluation checks the constructed moving-plasma counterexample. **29 focused checks pass**, including endpoint construction, the inverse threshold, entropy, original nominal compression and retained/lost exclusion controls at 150 and 180 km/s. Arithmetic precision does not increase observational precision.

Files: `RMO_solar_speed_check.json`, `verification.json`, vector PDF/SVG, and this report. The PDF was rendered and visually inspected. All previous scientific scripts, results and images remain unchanged. No raw observations, new atomic database, global solver campaign, external contact or publication action occurred.

Each EUV delay constrains the density at a chosen temperature. Both must use the same density. Radio-derived density and compression also share a frequency input, so that dependence is kept in the calculation.

At 3.2 MK the two EUV ranges overlap near 4.5 × 10⁷ cm⁻³, below the radio range of about 7.0–10.0 × 10⁷ cm⁻³. The other four tabulated temperatures do not fit both EUV delays with one density. Untested temperatures and atomic-rate errors are not excluded.

A separate arithmetic check found that the table times correspond approximately to 6.0 × 10⁷ cm⁻³, while the preprint text states 9.4 × 10⁷ cm⁻³. The final publisher PDF has not been verified. The report keeps this source-version limitation explicit.

For independent RMO confirmation: verify the final source version and density convention; match the radio and EUV plasma and time; establish the error definitions; then test an emission calculation with the relevant thermal/density history, channel response and background. These are requirements for our verification, not a claim that the authors ignored them. No new observations or solar family classification are claimed here.