# R134 E05 robustness and alternative-discriminator closure

**E05 R134 RESULT: FAST AND SLOW BOTH REMAIN ADMISSIBLE**

**Two conditional MHD families are physically resolved, but observational family selection remains unresolved.** Here, physically resolved means separately constructed and checked under stated conditions, not an observed shock identification or a proof that every possible MHD interpretation has been enumerated.

R134 tests whether the retained non-geometric observations, their shared reductions and defensible conditional assumptions exclude either family. They do not. Both families survive every retained source-scenario set. Both also contain physically checked examples consistent with an additional conditional fit to the ten EIS reference spectra. No new observation, preferred geometry or independent error box was introduced to select a winner.

There is a useful additional diagnostic: the **sign of the change in total magnetic-field strength between the same upstream and downstream states**. Within the ordinary ideal-MHD shock construction, FAST increases the field strength and SLOW decreases it. E05 does not presently provide that independent local state-to-state field measurement.

## What the observations constrain

The retained EIS target remains exposure 25, central rows 334–338, with the R133 shutter interval 2011-02-16 14:28:40.646484–14:29:25.645943 UTC. The original R132 and R133 results and every original accepted root remain unchanged.

The Fe XIII relative centroid response is +16.559600881 km/s for reference exposures 1–10. Its saved formal noise-only uncertainty is approximately 1.136 km/s. References 1–5 and 6–10 give +14.592077 and +18.527124 km/s for the same target aperture. These share the target spectrum and are alternative reference choices, not independent measurements or a confidence range.

R133 establishes that the local AIA aperture is brighter during the EIS exposure than in the early reference under all **16 retained registration alternatives, by 2.4–8.5%**. This is a range across registration choices and sampled exposures, **not a confidence interval**. No unique local propagating ridge, independent AIA–STEREO correspondence or observationally bounded 3-D alpha has been established.

The AIA early reference at approximately 14:25:32.840 UTC is not the EIS spectral reference interval. Neither is independently demonstrated to represent the upstream parcel of a particular local shock. A brightness ratio and a Doppler reference difference therefore cannot automatically be combined as two measurements of one state jump.

## Provenance and dependencies

| Label | Retained quantity or interpretation |
|---|---|
| MEASURED | EIS detector spectra and relative centroid response; AIA detector samples and projected brightness evolution; exposure records. |
| SOURCE_DERIVED | Spectral reductions, conditional Fe XIII effective density, calibration/pointing products, retained registration alternatives and published speed proxies. |
| ASSUMED | Temperature scenario; reference-is-upstream and emitting-state-is-downstream assignments; Doppler-to-velocity-jump mapping; plasma/front association; selected normal/field geometry; assigned upstream motion; ideal-MHD closure. |
| MODEL_DERIVED | Normal front and plasma-relative speeds, compression, upstream density/temperature, pressures, magnetic fields, velocity jumps and all conditional separators. |

Density and temperature are paired through the same saved R130 atomic-response fit. At the central aperture, the 1.0, 1.584893 and 2.5 MK temperature scenarios give effective densities approximately 1.92926, 2.35054 and 2.84208 × 10^8 cm^-3. These temperatures are assumptions for the response calculation, not measured temperature bounds. The saved density profile limits are conditional noise-only delta-chi-square = 1 intervals with nuisance reoptimization; they are not a joint observational error region.

The density-sensitive 203/202 information shares the 202 intensity measurement and calibration with other spectroscopic reductions. R134 does not multiply formal Doppler and density error intervals into an invented independent box or infer an unmeasured covariance ellipse. At fixed temperature, Doppler constraint and geometry, absolute density cancels from the normalized MHD admissibility equations. Positive density rescaling changes dimensional densities and fields, with B proportional to sqrt(ne), while leaving the normalized root and characteristic ordering unchanged. This scaling is a model property, not an independently adjustable density prior.

The retained 336 km/s AIA and 371 km/s EIS published means are tested as alternative speed-proxy assignments. They are not simultaneous exact measurements of the same instantaneous local normal speed and are not endpoints of a measured speed error interval. No independent absolute upstream normal-velocity interval is supplied by the relative EIS centroid. Assigning zero upstream flow remains a condition of R132, not an observation.

The 16 R133 registrations relocate the same detector aperture relative to imaging; they do not produce 16 new spectral measurements. The older lower/upper EIS apertures are retained as separate spatial-association sensitivity cases and are paired with their own density, Doppler and geometry records. They do not replace the selected central EIS aperture or its R133 result.

## Bounded calculation and retained roots

The pre-run specification defines 54 paired source scenarios: three retained apertures, three shared-target reference choices, three temperature/density response scenarios and two alternative published speed proxies. Eighteen refer to the primary central aperture. Temperature and density always stay paired; Doppler and density always refer to the same aperture.

Each source tuple is checked at 703 declared angular locations: alpha from -85 to +85 degrees in 5-degree steps plus -89 and +89 degrees; theta from 5 to 85 degrees in 5-degree steps plus 1 and 89 degrees. The 703 matching nodes of the unchanged central R132 tuple are reused directly from saved records. They are not a rerun of the R132 map.

A separate normal-flow stress test uses the 18 central tuples, alpha = 45, 54.2269698 and 60 degrees, theta = 1–89 degrees in 2-degree steps, and S/cs2 = 0.5–2.0 in 0.1 steps. Here S is the front speed relative to upstream plasma and cs2 is the downstream sound speed. This is an explicitly ASSUMED conditional stress domain around a physical sonic scale, not a measured or exhaustive upstream-flow range. Velocities outside it remain unknown, not excluded.

The calculation contains **76,842 nodes: 76,139 newly solved and 703 reused**, with **112,644 algebraic root records**. There are **36,937 accepted records: 33,933 FAST and 3,004 SLOW**. These counts include the reused subset. All 15,151 accepted roots in the complete inherited R132 result remain preserved separately. Counts of overlapping model/grid alternatives are not probabilities, independent samples or fractions of physical parameter-space volume.

Every one of the 54 stationary-upstream source-scenario sets and all 18 central normal-flow sets retains accepted examples of both families. This is a conditional survival result. It does not assert that every point in a scenario domain has a solution, that both exist at one fixed complete state, or that two fronts were observed.

## Conservation, characteristics and numerical safeguards

The unchanged R131 diagnostic and the R132 complete quadratic root construction remain the physical basis. New R134 code supplies paired inputs and an explicitly variable upstream-relative normal speed. It checks an independent dimensional flux calculation and an observer-frame Galilean transformation. The retained model-code diff identifies every change.

All algebraic roots at every declared node are recorded, including complex roots, nonpositive field-squared roots, nonpositive-pressure states and other rejected candidates. There is no finite field-strength search cutoff. Accepted ordinary roots satisfy compression, positive pressures, entropy increase, mass/momentum/induction/energy conservation and the characteristic transition 1 → 2 for FAST or 3 → 4 for SLOW.

The R132 guard values are retained: minimum |mu| = 10^-3; relative rational-pole distance = 10^-6; maximum normalized polynomial residual = 10^-12; maximum scaled flux residual = 10^-10; characteristic boundary/coincidence margins = 10^-7; and the other explicitly saved pressure, entropy, field and root-separation guards. Near cancellation, the complete stable quadratic formula uses the existing 70-digit Decimal fallback without overriding the guards.

The largest accepted independent scaled flux residual is approximately **3.17 × 10^-12**, below 10^-10. The largest constructed tangential LOS residual is approximately **1.37 × 10^-18**, below 10^-12. The observer-frame and upstream-rest-frame physical classifications agree. The saved velocity construction still has tangential velocity along e1 parallel to n × LOS; its zero LOS projection remains an ASSUMED geometry explicitly checked at each accepted state.

No new guarded candidate was encountered on this finite stress grid. This does **not** certify guard-free cell interiors or a complete continuous map. All frozen R132 unresolved nodes and boundary cells remain unresolved and unchanged. They are not silently removed by the coarser R134 sampling. No family-exclusion or universal precision claim is based on a sampled empty region.

## Conditional reference-spectrum test

The retained reference spectra provide more information than a scalar downstream density. R134 therefore tests the additional assumption that reference exposures 1–10 represent one upstream plasma state. The reference-is-upstream association and a single-temperature intra-ion model remain ASSUMED.

The test jointly fits all ten retained 203 Å spectral windows and their 202 Å integrated-area constraints. Six nuisance parameters per exposure describe the 202 area, Fe XII amplitude, centroid, width, background and slope. The Fe XIII energy ratio and self-blend ratio are calculated for a model's upstream temperature and density with the retained ChiantiPy/CHIANTI atomic data. The older fixed self-blend ratio 0.40 is not treated as an exact physical measurement.

For each model self-blend ratio, a comparator fits one shared free energy ratio and all 60 nuisance parameters. This is a conditional noise-only spectral comparison, not a full joint Doppler/density confidence distribution, a Bayes factor or proof that the reference is physically upstream.

The original selected FAST (45,80 degrees) and SLOW (60,30 degrees) examples have delta chi-square values approximately 3.924 and 2.522 against their respective free-ratio comparators. This difference between two examples does not exclude either family.

To test family survival, the same reference-data criterion was applied to both complete sets of accepted R132 roots. Approximate atomic interpolation only ranked candidates inside the saved response grid. The four closest candidates of each family were then checked with exact atomic response; the best exact-ratio candidate of each family received the joint spectral fit. No other root was deleted or relabelled, and no sampled minimum was promoted to a global likelihood optimum.

| Family | Conditional alpha, theta (degrees) | Predicted reference 203/202 ratio | Delta chi-square | Reduced chi-square |
|---|---:|---:|---:|---:|
| FAST | 14, 15 | 0.227772 | 0.01720 | 1.2958 |
| SLOW | 73.875, 51 | 0.227812 | 0.01574 | 1.2957 |

Both selected family witnesses have freshly rechecked conservation, entropy and characteristic admissibility. They retain the same R132 selected Doppler, downstream density, temperature and speed-proxy constraints and lie close to the conditional free-ratio optimum of the additional reference-spectrum test. Their angles are model coordinates, not measured E05 angles. This is an existence demonstration for both families, not observational family selection or a completed covariance analysis.

## Why upstream motion matters for alpha

Let D_lab = V cos(alpha), mu = -l·n and delta = d/mu. If v1n is the upstream normal velocity in the observer frame, the relevant MHD speed is **S = D_lab − v1n**, and compression is **X = S/(S − delta)**. The downstream normalized pressure is P2 = 2 X kB T2/(mp S^2), using SI speeds in that expression. The observer-frame state is obtained through a common normal Galilean boost; the relative Doppler jump is unchanged.

For the retained energy polynomial, the ordinary-family necessary sign changes where `A = 2(5 P2 X − 4X + 1)/X^2 = 0`. With downstream sound speed cs2, the positive threshold is `Scrit = (delta + sqrt(4 delta^2 + 9 cs2^2))/3`. FAST requires S > Scrit and SLOW requires S < Scrit within the compressed ordinary construction. These are necessary conditions; the full root checks are still required.

At the central R132 inputs, alpha = 45 degrees gives Scrit approximately 218.072 km/s and v1n at the separator approximately +44.264 km/s. At alpha = 60 degrees, these are approximately 216.372 and -30.872 km/s. They are conditional model quantities, not inferred velocities.

The completed stress grid retains a checked SLOW root at alpha = 45 degrees with v1n approximately +53.512 km/s, and a checked FAST root at alpha = 60 degrees with v1n approximately -44.207 km/s. These are counterexamples to treating alpha alone as a universal discriminator after relaxing upstream rest. They are not adopted flows or preferred E05 geometries.

R132's 45 ± 5 degree and 60 ± 5 degree planning bounds, and its separator near 54.227 degrees, remain valid under their frozen assumptions. R134 does not change that map or its precision result. It establishes that applying those bounds to a broader physical claim also requires justification of the upstream-flow and state-assignment conditions. Alpha remains the useful geometric discriminator within R132; it is not sufficient by itself over the expanded unknown-flow domain.

## Alternative discriminator: signed magnetic-strength jump

The standard tangential ideal-MHD jump relations give `q = Bt2/Bt1 = X(1 − bn^2)/(1 − X bn^2)`, where X > 1 and bn uses the upstream density and upstream-relative speed normalization. In an ordinary FAST transition, bn^2 < 1/X and q > 1. In an ordinary SLOW transition, bn^2 > 1 and 0 < q < 1. Continuity of Bn means that the sign of `|B2|^2 − |B1|^2` is the sign of `|Bt2|^2 − |Bt1|^2`.

Thus a reliably associated **increase in total field strength** can reject the checked ordinary SLOW interpretation, while a **decrease** can reject the checked ordinary FAST interpretation. This statement follows from established MHD jump relations and characteristic ordering; it is not a new law or a claim to replace existing methods. The sign was checked for every inherited accepted R132 root and every accepted R134 record.

This observable is a state-to-state change, not an absolute field value. Sampled absolute fields, upstream density and upstream temperature overlap between the families; source-derived quantities that both model constructions share cannot independently select a family. A field-change sign also does not by itself prove a shock or exclude smooth waves, intermediate/compound structures or emission alternatives outside this two-family construction.

For bounded field-strength errors, a sufficient sign determination requires `epsilon_B1 + epsilon_B2 < abs(B2 − B1)`, including calibration and association errors. The entire observational difference interval must remain on one side of zero. Equal per-state absolute errors must each be less than half the predicted jump. Equal relative bounded errors must be below `abs(B2 − B1)/(B2 + B1)`.

| Saved conditional example | B1 (G) | B2 (G) | B2 − B1 (G) | Equal absolute error per state | Equal relative error per state |
|---|---:|---:|---:|---:|---:|
| FAST (45,80 degrees) | 0.925217 | 1.023508 | +0.098291 | < 0.049146 G | < 5.0439% |
| SLOW (60,30 degrees) | 3.733122 | 3.715630 | -0.017492 | < 0.008746 G | < 0.2348% |

These are **witness-specific conditional planning values**, not measured uncertainties or a universal requirement for the full families. No uniform finite precision is certified over the continuous uncertain domain, including weak/degenerate and numerically unresolved limits. The finite sample's smallest jump is not a lower bound on every surviving physical solution. E05 has no retained independent local upstream/downstream field-strength pair to apply this test now.

## AIA brightening and physical association

For an illustrative optically thin component, a brightness ratio can be written `I2/I1 = 1 + f [X^2 (G2/G1)(L2/L1) − 1]`, where f is the upstream component's fractional contribution, G is the temperature/instrument response and L is its emitting depth. Those quantities and the identity of the component are not independently measured here. This equation explains why 2.4–8.5% brightening does not uniquely determine X or a wave mode; no values were fitted to force a preferred family.

The idea of one evolving disturbance remains **WORKING HYPOTHESIS — NOT AN OBSERVATIONAL RESULT**. R134 does not turn smoothing, SVD, brightness correlation, a reference spectrum or a selected geometry into an independently identified shock surface. The R133 EIS/AIA/STEREO assessment and its absence of a bounded alpha remain unchanged.

## Final physical summary

**E05 R134 RESULT:** FAST AND SLOW BOTH REMAIN ADMISSIBLE.

**What is directly observed:** relative Fe XIII spectral response and local AIA brightness evolution. The local aperture brightens by 2.4–8.5% across the retained registration alternatives; this is not a confidence interval or mode identification.

**What is source-derived:** conditional spectral density and line ratios, calibration/registration/time products and alternative published speed proxies. The conditional reference-spectrum fits add a reproducible consistency test, not a measured upstream state.

**What remains assumed:** the connection between spectroscopic plasma, reference and imaged front; upstream/downstream assignment; temperature/atomic and ideal-MHD closures; and any selected upstream motion, normal or field azimuth.

**Admissible MHD family/families:** separately checked conditional FAST and SLOW families. All inherited R132 accepted roots remain intact.

**Why a competing family is excluded or survives:** no retained independent jointly incompatible observation excludes either family. Both survive the paired source scenarios, explicit motion tests and additional conditional reference-spectrum existence check. Widening a domain cannot exclude its included witnesses.

**Best remaining discriminator:** bounded local 3-D geometry remains useful under the justified R132 flow/closure conditions. A separately measured signed magnetic-strength jump across the same state pair is an alternative. Neither is supplied adequately by the retained E05 observations.

**Minimum observation required for final classification:** independently identify the same local front and its upstream/downstream emitting states, then obtain either a bounded local 3-D normal together with sufficient upstream-flow and thermal constraints, or a bounded magnetic-strength difference with a definite sign and compatible MHD state checks. Geometry or magnetic change alone must not be promoted to an observed shock without the required physical association and admissibility evidence.

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

The result is useful because it identifies both what the existing spectra can test and what remains missing. It does not rank groups, catalogues or methods. The retained observational studies, EIS/AIA/SECCHI data, CHIANTI/ChiantiPy atomic response, numerical libraries and standard MHD theory keep their provenance and scientific credit.