# R135 E11: outer and inner front classification closure

**E11 R135 RESULT: FAST AND ANOTHER FAMILY BOTH REMAIN ADMISSIBLE.**

This is a conditional MHD result. **Neither visible E11 ridge has an independently selected observational MHD family.** The outer FAST reconstruction is more robust to magnetic obliquity than the old near-perpendicular test alone showed. A different unmeasured upstream-flow assumption also permits an outer SLOW completion. The independently tracked inner ridge retains checked conditional SLOW solutions when a specified outer-downstream state is transferred to the plasma between the fronts. None of these statements identifies two physical waves, a solar fast–slow pair or a complete Riemann fan.

## Exact structures and association

The event is 13 June 2010. The outer target is the selected AIA 193 Å crest near PA 116 degrees, not every point on the dome and not a radio image pixel. R109 gives the reference patch near X = 1137.700 arcsec and Y = -554.893 arcsec. Its projected normal is PA 120.277 degrees; the older 113–126 degree estimator spread is not a complete uncertainty interval. The effective pattern fit covers 05:38:32.06–05:40:08.06 UTC.

The inner 193 Å ridge was tracked separately in R120. Its common-interval projected radial speed is 409.1 km/s, with a finite method envelope 397.9–417.7 km/s. The matched outer estimator gives 687.5 km/s, with method envelope 686.2–784.5 km/s. These are different image tracks. The inner projected normal at the endpoint is PA 117.521 degrees, with method envelope 108.22–121.67 degrees. Neither normal PA is theta_Bn. Projection, state association and the full observational uncertainty remain outside those method envelopes.

The outer R110 model retains its original R107/R109 effective normal speed 705.535 km/s, derived from the 707.505 km/s radial fit and the G0 image-plane normal. It is not silently replaced by the newer matched 687.5 km/s comparison. The inner effective normal speed remains 408.968 km/s. These interval means and endpoint normals are conditional local-state proxies, not simultaneous 3-D plasma velocities.

| Association level | Supported conclusion |
| --- | --- |
| Same eruption | The retained Ma and Kozarev timing and morphology support studying EUV and radio together. |
| Same outer front/shock surface | The published comparisons support an interpretation; radio heights depend on density models. |
| Same local radio/EUV patch | Not independently established by radio imaging in the retained set. |
| Outer downstream equals inner upstream | A uniform intervening-state hypothesis, not an observed identity. |
| Inner ridge equals CME material boundary | A retained alternative, not an automatic classification. |

The inner front, CME/driver, outer EUV ridge, radio-emitting region and selected local nose/patch remain distinct. No outer radio compression, regional outer temperature or R119 slow exclusion is transferred to the inner jump as a measurement.

## Provenance and fixed dependencies

| Label | Quantity and dependency |
| --- | --- |
| MEASURED | Retained AIA detector signals; projected ridge positions and motion obtained from their saved reductions. These measure brightness patterns. |
| SOURCE_DERIVED | Radio lanes near 132 ± 5 and 165 ± 15 MHz, with compression and density derived under the lane/harmonic interpretation; regional temperature/sound-speed estimates; calibrated pointing; published magnetic/emission-model outputs with their stated dependencies. |
| ASSUMED | Paired upstream/downstream radio-lane interpretation, harmonic identification and radio-to-patch transfer; gamma = 5/3 and pressure/composition closure; G0 normals; assigned upstream motion; uniform intervening plasma for the inner test. |
| MODEL_DERIVED | Normalized model pressure, magnetic strength, downstream thermodynamics/velocity, characteristic speeds and model family boundaries. Radio compression and upstream density remain source-derived under explicit emission assumptions. |

R109 uses fL = 127–137 MHz, fU = 150–180 MHz and T = 1.4–2.2 MK as a declared scenario enclosure. At its centre, r = (165/132)^2 = 1.5625, rho1 = 9.0351e-14 kg/m^3 and T1 = 1.8 MK. Density, pressure and compression keep their shared radio/temperature dependence. This is not a measured independent covariance box or a probability distribution.

R93 instead retained speed 495.77–667.03 km/s and sound speed 126–186 km/s as its source inputs. These conventions are not silently equated with the R109 pure-hydrogen, equal-electron/proton-temperature closure or its 705.535 km/s image-normal proxy. The R93 proof is expressed in r and b = p1/(rho1 w1^2). Its mathematical applicability is checked in those normalized variables; it does not make the two observational reductions identical.

The published standoff-derived magnetic field discussed in R121 describes the outer upstream medium under its own geometry, radio-density and negligible-wind assumptions. It is neither an independent vector-field measurement nor a measurement of the plasma between the fronts. Shock-derived downstream temperature is not an independent thermometer. The saved R121/R122 emission audit did not establish one common local thermal state from the asynchronous AIA channels.

## What PFSS does and does not bound

Kozarev et al. (2011), Section 3.4 and Figure 4, retain an AIA/STEREO-A image comparison with a PFSS topology overlay. It motivates a possible quasi-perpendicular nose. It supplies no numerical field vector, local front-surface normal and uncertainty at the selected radio/EUV patch. No numerical theta_Bn bound is recovered from that overlay. A visual projected tangent cannot be promoted to an unmeasured 3-D angle.

**MEASURED theta_Bn interval: none. MODEL_BOUNDED theta_Bn interval: none.** The definition of an acute angle permits 0–90 degrees mathematically; that is not an observational bound. Exact 90 degrees remains an assumption. The old 83.3–90 degree interval was a tested model domain.

No new PFSS download is needed to establish the present conditional closure. A field model alone would not resolve the missing radio-to-EUV patch association and upstream plasma-relative speed. Obtaining only a global topology or a selected model nose would not supply those missing local observations.

## Actual ordinary-FAST geometry limit

The new calculation extends the R93 normalized input rectangle rather than rerunning its old certificate. Let r > 1, b = p1/(rho1 w1^2), h = Bn^2/(mu0 rho1 w1^2), t = r h and gamma = 5/3. Define M = 4-r-5br, d = 1-t and W = r+5-2t(4-r). The linked upstream tangential field satisfies A = Bt1^2 = 2 M d^2/(r W). The downstream field is obtained from the same state equations, not rotated or perturbed independently.

For the saved R93 domain, M and W remain strictly positive throughout 0 <= t <= 1. Exact rational conservation identities and a **256-cell, 8 × 4 × 8 Bernstein certificate** prove the positive-pressure and magnetosonic sign conditions over the continuous enlarged domain. The previous positive entropy lower bound carries over because dp2/dh = 2 M (r-1)^3/W^2 > 0. This is a continuous mathematical certificate, not a fraction of successful samples.

For 0 < t < 1 the flow crosses from region 1 to region 2: upstream super-fast, downstream sub-fast but super-Alfvenic. At t = 0 the separate perpendicular diagnostic applies. The field-angle mapping decreases continuously from 90 degrees toward zero because A decreases while h increases. Consequently the ordinary FAST branch covers **0 < theta_Bn <= 90 degrees under this model domain**. It does not terminate at 83.3 degrees.

At theta = 0 and t = 1, the switch-on/characteristic limit remains degenerate and is not assigned an ordinary family. Numerical guard neighborhoods also remain unresolved; the exact open-domain result does not override numerical classification guards. No observationally supported angle or universally unique solar class follows from this extension.

## Other outer roots and evolutionary checks

At each declared angle the complete inverse problem is quadratic in h: r h W sin(theta)^2 - 2 M (1-rh)^2 cos(theta)^2 = 0. Every algebraic root is retained. Stable 80-digit Decimal quadratic evaluation precedes the independent physical checks. There is no finite magnetic-field cutoff.

Under M > 0 and the retained 1 < r < 2.5 range, a regular SLOW state would require h > 1, where the required Bt1^2 is negative. The other finite real completions in 1/r < h < h_pole reverse the tangential field. They retain their conservation checks and historical R110 unresolved labels in the frozen baseline. R135 adds an explicit necessary evolutionary test; it does not alter those saved records.

For a coplanar base state, two linearized out-of-plane RH equations decouple: delta F = (rho u_n delta u_z - Bn delta B_z, u_n delta B_z - Bn delta u_z). The field-reversing candidates have three incoming Alfven amplitudes but only one outgoing amplitude for two independent jump conditions. Generic incoming perturbations cannot be matched. They therefore fail this **unrestricted ideal-MHD transverse evolutionary test**. Ordinary checked FAST/SLOW states have the appropriate two outgoing amplitudes and pass the local ordering and entropy conditions. This is not a claim about dissipative stabilization or a restricted coplanar model.

Contact states require zero mass flux, and an ideal rotational discontinuity preserves density and pressure. Neither can supply the imposed radio-derived r > 1 with finite mass crossing in this conditional single-jump problem. This does not observationally exclude a material boundary or emission pattern if the radio source does not belong to the visible patch. Switch limits, exact parallel degeneracies, smooth waves and full compound Riemann fans are not silently assigned an ordinary shock label.

## Why unknown upstream motion prevents a unique outer class

The relevant speed is w1 = Vn - u1n, not the image speed alone. At fixed r and p1/rho1, M changes sign at wcrit = sqrt[5 (p1/rho1) r/(4-r)]. The central R109 values give **wcrit = 308.615 km/s**, or **u1n = 396.921 km/s** at the fixed G0 pattern speed. The frozen R119 proof across its complete radio/T enclosure requires w1 > 428.022 km/s, equivalent to u1n < 277.514 km/s at that G0 speed. Neither threshold is a measured flow.

The predeclared stress probes use w1/wcrit = 0.75, 0.9, 1, 1.1 and 1.5, keeping each radio/temperature tuple linked. They are conditional probes around an analytical boundary, not observational error limits. At the equality boundary, candidates remain UNRESOLVED.

One checked central-input SLOW counterexample has assumed theta_Bn = 23.0 degrees and u1n = 427.782 km/s. It keeps the same radio frequencies, compression and upstream temperature. It has w1 = 277.753 km/s and w2 = 177.762 km/s, positive entropy production and scaled flux residual 4.44e-16. Its upstream/downstream field strengths are model outputs, 1.219 and 1.152 G. This is an existence counterexample, not a measurement, preferred state or claim that such a large upstream flow actually occurred.

Thus FAST is retained under the stated stationary/sufficient-inflow conditions; it is not the sole mathematical completion once the unmeasured flow condition is relaxed. Qualitative quasi-perpendicular context supports exploring FAST, but does not provide a numerical local angle bound with which to exclude every conditional SLOW completion.

## Which upstream flow must observations rule out?

The final flow-closure test goes beyond a necessary sign condition. It proves both sides of the conditional transition while retaining all numerical guard regions.

For the complete retained radio compression interval, write eta = b/bcrit, where bcrit = (4-r)/(5r). An exact Bernstein certificate covers **0 < eta < 1 and 0 <= r h < 1**. It proves ordinary FAST existence and excludes regular SLOW whenever w1 > wcrit, under the same local ideal-MHD/state-association conditions. One coefficient bound becomes zero only at an excluded boundary; nonnegative Bernstein coefficients and strictly positive interior anchor coefficients prove positivity on the specified open domain. The boundary is not forced into a class.

To establish actual SLOW onset, a second exact certificate uses the explicitly assumed coordinate h = 2 and b = bcrit(1+zeta), for **0 < zeta <= 1e-4** over the entire radio-compression interval. The linked fields are real, pressure and entropy are positive, and the characteristic ordering is 3 → 4. As zeta tends to zero, the angle tends to the parallel limit and u1n approaches ucrit from above. Thus **regular SLOW exists arbitrarily close above ucrit when the angle and field are not independently bounded**. This establishes the onset infimum; it does not assert SLOW at every larger flow or every fixed angle.

| Conditional planning question | Quantitative answer |
| --- | --- |
| Central R109 tuple at fixed G0 Vn = 705.535 km/s | wcrit = 308.615 km/s; ucrit = 396.921 km/s. |
| Conservative requirement for every retained radio/T tuple | Independently support **u1n < 277.514 km/s**, including its full uncertainty, at that fixed G0 normal speed. |
| What upstream flows must be ruled out for this whole-domain exclusion? | The supported interval must exclude u1n >= 277.514 km/s. This is sufficient for the stated envelope, not a universal solar threshold. |
| If normal front speed is uncertain too | Require **min_supported(Vn - u1n) > max_supported(wcrit)**, preserving joint dependence and local geometry. |
| What does this select? | FAST as the unique regular evolutionary compressive-shock family under the supplied radio jump, closure and nondegenerate geometry. It does not independently establish that the visible ridge is that shock. |

Across the declared radio/T enclosure, the conditional onset threshold ranges from 277.514 to 483.151 km/s. These are model-planning thresholds, not measured flows or confidence bounds. The maximum wcrit = 428.022 km/s occurs at the largest compression and temperature; monotonicity gives the exact enclosure extrema. The earlier R119 exclusion number is recovered as an analytical consequence, without rerunning its map or changing its result.

The additional 45 onset controls retain **27 checked SLOW states and 18 UNRESOLVED guard-limited states**. Numerical characteristic splitting becomes too small near the limiting parallel state; these controls are not labelled SLOW merely because an exact open-domain theorem exists. The finite controls supplement the rational proof and do not supply observational precision.

**The missing observation is now quantitative:** bound the same-patch plasma velocity relative to the local outer-front normal tightly enough to satisfy the inequality above, and establish that the radio compression belongs to that patch. Under G0 the normal lies in the image plane, so a Doppler LOS velocity alone is not this normal component. A material-motion diagnostic or sufficiently constrained vector-flow geometry is needed; brightness-pattern speed cannot substitute for plasma flow.

The inner-front task remains separate. Its relevant unknown is the plasma immediately ahead of the inner ridge, which may already have passed through the outer disturbance. The outer flow threshold must not be applied to that different state or to the inner jump.

## Inner front: independent track, conditional SLOW, unmeasured state

R120 already supplied a fully checked inner SLOW jump using the saved outer FAST state F1 as the uniform intervening plasma. That reference has inner inflow 155.269 km/s, compression 1.296476 and field-tangent ratio 0.909421. The required intermediate normal flow is about 254 km/s. These are conditional model quantities; the roughly 409 km/s observed pattern is not that plasma flow. All 108 saved F1 method/azimuth checks remain unchanged.

R135 extends the shared-state test across the central outer ordinary-FAST inverse family and azimuths 0–180 degrees in 5-degree steps, with outer angles 0.5–90 degrees in 0.5-degree steps. The same frozen inner speed and projected normals are used. Every nontrivial energy-polynomial root is recorded; the removed no-jump factor and complex, nonpositive, expansive, non-evolutionary and guarded candidates remain documented. Inner compression is solved independently. The outer radio compression is never imposed on the inner jump.

There are **6,660 inner model nodes and 19,980 algebraic root records**, including **2,470 accepted SLOW records**. No ordinary inner FAST root is accepted in this finite shared-state grid. The sampled SLOW examples span outer model angles 30.5–68 degrees. That extent is neither an observed angle range nor a certified continuous boundary. Four candidates remain numerically unresolved, including weak/characteristic and coalescing limits. Other unknown intervening states, 3-D normals and physical interpretations remain untested by this finite conditional construction.

The preserved R122 gap photometry shows channel-dependent emission changes. It does not isolate one fluid parcel, recover local vector flow or supply an inner density/field jump. A CME-related surface, material boundary or changing emission remains observationally possible. Two distinct tracks do not establish two MHD modes.

## Numerical safeguards and reproducibility

The outer extension contains **10,440 nodes and 20,880 algebraic root records**: 5,580 accepted FAST and 305 accepted SLOW records under their separately labelled scenarios. The largest accepted scaled flux residual across both front tests is **3.48e-15**, below 1e-10. Mass, normal B, all momentum and induction components, total energy, positive pressure, entropy and independently calculated characteristic speeds are checked at every accepted state. The classifier receives states and front speed without an input family label.

Predeclared guards include relative pole distance 1e-6, polynomial residual 1e-12, flux residual 1e-10, characteristic/coincidence and root-separation margins 1e-7, and positive entropy margin 1e-10. The 3,310 unresolved outer candidates comprise 3,240 candidates on the deliberate gas-boundary probes and 70 denominator-guard candidates. Guarded candidates are not forced into a class. Empty sampled regions are not certificates of absence.

A guard-order audit moved gas-boundary and coalescence tests ahead of sign/realness rejection. Thresholds and the tested domain were unchanged; every accepted physical state before and after that correction is hash-matched in the verification record. Targeted saved FAST/SLOW controls, a Galilean check, rejection of an inconsistent pressure perturbation, and separate scalar energy-root checks pass. These numerical checks do not convert source scenarios into observations or prove global stability.

## Final physical summary

**E11 R135 RESULT:** FAST AND ANOTHER FAMILY BOTH REMAIN ADMISSIBLE.

**Observed structure classified:** neither visible ridge has an independently selected observational MHD type. Outer and inner results are kept separate.

**Directly observed constraints:** saved AIA brightness structures, separately tracked projected ridges and their projected motion/tangents.

**Source-derived constraints:** outer radio frequencies and conditional compression/density, regional thermal inputs and calibrated imaging geometry, with their shared dependencies.

**Model-bounded geometry:** no numerical local solar theta_Bn bound is recovered. PFSS supplies qualitative topology only.

**Remaining assumptions:** radio-to-patch and state assignments, G0 geometry, upstream motion, thermal/composition closure and local ideal MHD; the inner test additionally assumes a uniform intervening state copied from the outer model.

**Supported theta_Bn interval:** no measured or independently model-bounded observational interval. The mathematical ordinary-FAST certificate covers 0 < theta <= 90 degrees under its stated input/flow conditions.

**Admissible MHD family/families:** outer conditional FAST and SLOW under different unmeasured flow/geometry choices; inner conditional SLOW under the tested shared-state hypothesis. These are not identifications of two solar waves.

**Why FAST is retained or rejected:** FAST remains physically admissible across the full regular angle continuation in the certified domain. It is not rejected, but observational uniqueness is not established.

**Why competing families are excluded or survive:** regular outer SLOW is excluded when the conditional inflow sign proof applies, but checked SLOW completions appear when unmeasured upstream flow is relaxed. Field-reversing coplanar candidates fail the stated full-MHD transverse evolutionary test. Other observational structures and degenerate limits are not erased.

**Single remaining bottleneck:** an independently established local plasma-state pair for each observed ridge. For the outer front this requires radio/EUV patch association and min(Vn-u1n)>max(wcrit), equivalent to an upper normal-flow bound strictly below 277.514 km/s for the frozen G0 speed and complete radio/T envelope. For the inner front it requires the intervening plasma and an independent jump across that ridge. One global normal or one assumed shared state does not close both problems.

The retained work of Ma, Kozarev, Gopalswamy, Lee and the earlier magnetic-model/theoretical sources keeps its scientific credit. RMO adds conditional consistency, completeness and exclusion checks; it does not replace those observations or methods. 

**MHD admissibility determines what observational precision is required; observational analysis determines whether that precision is achievable.**