# RMO102 — normal Alfvén speed as an additional constraint

## Result

**±20% preserves fast exclusion with margin; the limiting symmetric error for
this criterion is about ±24.4%, if the central normal Alfvén-speed estimate is
cAn1=w0.** The calculation already includes compression ±0.05 and normal
acoustic Mach ±10%. These are HARD error bounds, not 1σ statistical
uncertainties. They apply to normal Alfvén speed, not total Alfvén speed.

In the familiar error notation, cAn1=w0(1±20%) is exactly eta in [0.8,1.2].
The weaker tested interval [0.7,1.2] means −30%/+20% about the SAME centre;
it must not be labelled ±30%. The saved slow state's actual eta≈1.05409 is
compatible with the interval; it is not the chosen central estimate eta=1.

Within the same broad synthetic input set B used in R100–R101, the additional
normal-Alfvén interval eta in [0.8,1.2] excludes the whole evolutionary fast
family, including the stated switch-on boundary. The saved slow reference
remains compatible. The weaker interval [0.7,1.2] removes the particular fast
witness of R101, but neither the new bound nor its combination with the earlier
compression/Mach necessary condition excludes fast throughout the set.
No additional complete fast state is constructed for the weaker interval.

| Hard error about cAn1=w0 (interval) | Worst-case G | Saved R101 fast | Saved slow | Whole-set result |
|---|---:|---|---|---|
| −30%/+20% (eta in [0.7,1.2]) | -0.1722908739 | Outside | Inside | Not excluded by these bounds |
| ±20% (eta in [0.8,1.2]) | +0.1453751851 | Outside | Inside | Fast excluded |

These are predeclared MODEL bounds, not measured solar uncertainties or a derived
solar requirement. The result is conditional on their physical validity.

## How to read the errors

We report the central value, then ± an absolute amount (with units where
applicable) or ± a percentage of that stated centre. Unequal bounds are shown
as −lower/+upper. Here compression ±0.05 is an absolute dimensionless amount;
Mach ±10% and normal Alfvén speed ±20% are relative to their stated centres.
Each result specifies whether these are hard bounds or statistical errors.
For a robustness check, read which errors were tested, whether the conclusion
survives them and, when established, the limit of the stated criterion.
Input uncertainty is distinct from a solver residual or model limitations.
The labels 1σ, 2σ and 3σ are used only when the statistical model and the
meaning of sigma are supplied. Hard bounds are not converted into sigma.
Separate error bars also do not specify a joint dependence; the allowed
combinations must be stated. R102 uses all combinations in the declared box.

## Definitions

Use r=rho2/rho1, r0≈2.1677737267, r in r0±0.05 and
x=M_n/M0 in [0.9,1.1], M0²=3. The fixed upstream sound speed is 1/2,
so w0²=3/4 and x=w1/w0. Here w1 is the upstream normal plasma speed relative
to the front, not apparent image motion. Retain rho1=1, p1=3/20, gamma=5/3,
mu0=1 and the same fixed local normal as the earlier tests.

    cAn1 = |B1 dot n| / sqrt(mu0*rho1)
    eta = cAn1/w0

The quantity uses the NORMAL component of B. The total Alfvén speed
|B1|/sqrt(mu0*rho1), or a field component in the image plane, is not eta.
The reference w0 is fixed, whereas the actual w1 varies with x. Thus eta is
also not the reciprocal of the actual normal Alfvén Mach number: cAn1/w1=eta/x.

## Derivation and boundary handling

Mass conservation gives w2=w1/r. Continuity of normal B gives
cAn2=cAn1/sqrt(r). An ordinary evolutionary fast transition has
w2>cAn2; therefore

    w1² > r*cAn1², or x² > r*eta².

The switch-on fast limit can reach equality, so use the inclusive necessary
condition x²>=r*eta². A STRICTLY POSITIVE margin

    G = r*eta²-x²

excludes both the regular fast family and that boundary. G=0 is retained by
this necessary screen; it is not evidence that a switch-on state actually
exists at arbitrary remaining inputs. This is a standard characteristic
condition translated into bounded partial inputs, not a newly discovered law.
The normal-field bound itself does not need the gamma=5/3 energy relation;
gamma and thermal normalization are retained to compare with B consistently.

Other intermediate/nonregular jumps, rotational/contact structures, full fans,
smooth disturbances, material motion and emission effects are not excluded.
In particular, surviving slow compatibility is not unique slow identification.

## Certificate over the whole set

Each additional interval is combined with all points in B. For positive x,
r and nonnegative eta,

    dG/dr = eta² >= 0
    dG/deta = 2*r*eta >= 0
    dG/dx = -2*x < 0.

Consequently min G=(r0-0.05)*eta_min²-1.1². Exact Fraction arithmetic
propagates the saved r0 enclosure outward. No plotting grid, Monte Carlo,
branch scan or historical numerical audit is used. Direct substitution of
w2²-cAn2² at the same enclosed corner independently gives the opposite sign:

    (w2²-cAn2²)/w0² = (x²-r*eta²)/r².

The threshold for this sufficient whole-set exclusion is

    eta_min > 1.1/sqrt(r0-0.05)
            = 0.75588009172991968993679370293665378… .

The JSON stores a rational enclosure with width 1e-35; its lower/upper squares
are checked exactly against the rational threshold-squared enclosure. This
precision is numerical, not observational. At exact equality the sufficient
strict certificate no longer holds.

Equivalently, for symmetric fractional half-width epsilon about cAn1=w0,

    epsilon < 1 - 1.1/sqrt(r0-0.05)
            = 0.24411990827008031006320629706334621… .

Thus ±20% passes with margin; the limit of THIS sufficient exclusion criterion
is approximately ±24.4%. At or above the exact limit this criterion does not
certify exclusion; that fact alone does not prove fast existence. No new
scientific run is needed for this algebraic re-expression of the saved bound.
For a different central estimate c_est, the fractional half-width instead obeys
epsilon < 1 - 0.7558800917…*w0/c_est, provided the right side is positive.
No universal 24.4% instrumental precision is inferred.

For the weaker interval, the same low-r/high-x corner also satisfies the
earlier necessary acoustic condition r<=4*x²/(1+x²). Thus combining these
two necessary conditions does not repair that case. A full RH construction
would be needed to demonstrate a competing fast state inside this NEW weaker
magnetic interval; the saved R101 fast state is outside it.

## Reused witnesses and nonempty allowed sets

The saved slow reference has eta²=10/9; eta≈1.054092553, inside both
intervals. This comes from the saved R98 alpha=120 configuration, reused via
R99; it is not rediagnosed. It remains at the centre of B. The saved R101 fast
witness has eta²=(1.09)²/10; eta≈0.344688265, outside both intervals.
Removing this one example is therefore an inadequate argument for excluding
the whole family: the weaker and stronger intervals differ in their full-set
certificate even though both remove that example. The retained slow witness
shows that the stronger exclusion is not caused by an empty feasible set.

## Controls and observational use

The equality control r=2,x=1,eta²=1/2 returns NOT_EXCLUDED_BY_THIS_BOUND.
Negative or unordered eta limits are rejected. Requests using total or
sky-plane components are rejected as NORMAL_COMPONENT_REQUIRED.
Geometrically B perpendicular to n has positive total field and zero Bn;
this illustrates the component-substitution error, not a full shock solution.

For solar use, the field direction, front normal, density and relative plasma
speed must refer to the same upstream region. A bound obtained from a model
is conditional on that model. Shared density and geometric errors can link
these quantities. The Cartesian set here admits all combinations; it neither
assumes statistical independence nor estimates a covariance. A physically
justified smaller joint set may be used, but a convenient dependence cannot
be invented to force a family. No new solar measurements have been obtained.

## Reproducibility and article role

- Protocol: normal_alfven/PROTOCOL.md; SHA256 40782c77e9cbf0484385baf4cadb10c0287aece2d19baccebc57bf0eb4bfd212.
- One execution: normal_alfven/audit.py. Presentation reads its saved result.
- Saved R99 JSON: SHA256 34aa2f1d11bf10c6a5a5d9028486cdcf64737a1bed5129069241c84ae89f30c4.
- Saved R101 JSON: SHA256 9308a5849b12674fadb6c580ba39219320912b413f93e9c6ca6ca68d6bf816b6.
- Full bounds/signs/controls: RMO_normal_alfven.json and verification.json.
- Conservation/characteristic source context is preserved in the R99 report,
  including Fitzpatrick's original jump conditions:
  https://farside.ph.utexas.edu/teaching/plasma/Plasma/node105.html.
  No new source comparison or shock-law novelty is claimed.

R102 supports the ApJ argument that RMO should report what an additional
constraint actually excludes over the allowed inputs, rather than eliminate
only a chosen illustrative state. The existing production full-state API
is unchanged. No observational identification is made.

## Presentation and next proposed step


Native-browser acceptance remains open; controlled checks address these edits.

One next proposed test, not run: if only a total-Alfvén interval is supplied,
what bound on the field-to-normal angle is needed to guarantee the normal
component condition over B? This addresses the geometric dependency without
assuming an observed field direction or repeating the earlier geometry audit.