# 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.
- 

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.