# RMO99 — excluding a fast alternative with partial inputs

## Result for an observer

For one saved R98 state, density compression and normal acoustic Mach number
exclude an evolutionary fast shock without prescribing magnetic-field strength
or direction. The compression is 2.1677737; the normal acoustic Mach squared is
3. At that Mach number the necessary upper compression bound for a fast shock
is 2. This is an exclusion of one alternative within the stated ideal-MHD
model. It is not a unique slow-shock identification or an observed solar result.

Two constructive controls show why the missing inputs matter. If the normal
flow is unknown, or if the upstream thermal constraint is unknown, an admissible
fast state can have exactly the same compression. Both controls satisfy the
original conservation laws, entropy increase and a fast characteristic crossing.

## Inputs and scope

Reference: the saved R98 alpha=120-degree state from the Urashima–Morioka
benchmark, with gamma=5/3, rho1=1, p1=3/20 and w1²=3/4. Here w1 is the normal
upstream PLASMA velocity relative to the front. It is not an image-front speed.
The upstream acoustic speed a1 satisfies a1²=gamma*p1/rho1=1/4; M_n²=w1²/a1²=3.

All magnetic components, tangential velocities and downstream pressure are
withheld from the partial-data test. A competitor is not required to retain
the reference field/flow alignment. The normal and rest-frame interpretation,
one planar steady transition, isotropic scalar pressure and ideal-MHD equation
of state remain assumptions. Only the combination p1/rho1 is needed thermally;
absolute density sets the chosen normalization.

The exact compression r_star is the root enclosed by the rational calculation
in the JSON. Its initial interval [2.1677,2.1678] is an enclosure of a known
model number, not an observational error bar. It agrees with the saved R98
compression to better than 1e-60. The R98 polar audit was not rerun.

## Derivation of the necessary fast bound

Work in a stationary-front frame and remove a common tangential velocity so
that u_t1=0. Let m=rho1*w1=rho2*w2, r=rho2/rho1>1, and normalize magnetic
stresses by rho1*w1² (mu0=1 in the saved states). Define

    b = p1/(rho1*w1²),   h = Bn²/(rho1*w1²),
    K1 = |Bt1|²/(rho1*w1²),   K2 = |Bt2|²/(rho1*w1²).

Tangential momentum, induction and normal momentum give, respectively,

    u_t2 = Bn(Bt2-Bt1)/m,
    (1-rh) Bt2 = r(1-h) Bt1,
    p2 = p1 + rho1*w1²(1-1/r) + (|Bt1|²-|Bt2|²)/2.

The tangential quantities can be two-component vectors. These identities do
not assume a particular field direction. Substituting into the ORIGINAL
total-energy flux, for gamma=5/3, gives the undivided relation

    (r-1)(4-r-5br)/(2r²)
      = [(1+2rh)K2 - (5+2rh-4r)K1]/(4r).                 (1)

An ordinary evolutionary fast shock has a downstream normal speed exceeding
the downstream normal Alfvén speed: w2²>Bn²/rho2. Thus rh<1. Since h>=0,
the induction relation gives Bt2=q*Bt1 with

    q = r(1-h)/(1-rh) >= r.

Consequently K2>=r²*K1 and the numerator on the right of (1) is at least

    K1[(1+2rh)r²-(5+2rh-4r)]
      = K1(r-1)[r+5+2rh(r+1)] >= 0.

It follows that 4-r-5br>=0 and therefore

    r <= 4/(1+5b) = 4 M_n²/(M_n²+3),    gamma=5/3.     (2)

For h=0 the perpendicular fast case is covered directly. If both tangential
fields vanish, (1) gives the gas compression relation. At the switch-on
boundary rh=1, the undivided induction relation forces Bt1=0 because r>1;
the right-hand numerator in (1) becomes 3*K2>=0. Therefore (2) also excludes
that fast boundary for the present inputs. We do not infer boundary behaviour
from a polynomial multiplied by a vanishing denominator.

Equation (2) is a necessary condition, not a sufficient fast diagnostic.
Falling below its curve does not prove that any fast solution exists. The
two controls below establish existence by constructing actual states.
Other intermediate/nonregular jumps, compound structures, full Riemann fans,
smooth fronts and non-MHD or emission explanations are not excluded here.

Source context: Richard Fitzpatrick, original ideal-MHD jump relations,
[Oblique MHD Shocks](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node105.html),
Eqs.7.280–7.286 and the switch-on/off discussion, accessed 2026-09-08.
The inequality is derived explicitly above from those conservation laws; no
claim to have discovered a new shock law is intended. Reference model:
[Urashima & Morioka (1966)](https://doi.org/10.1143/JPSJ.21.1431), Fig.3(a),
already checked in R98.

## Applying the bound and constructing omission controls

With M_n²=3, equation (2) requires r<=2. The reference has r_star>2, so no
evolutionary fast shock, including the stated limits, can satisfy that same
compression and normal acoustic Mach number. The proof covers unknown magnetic
strengths and directions; it is not the failure of a finite magnetic grid.

For the two controls, retain rho1=1, gamma=5/3 and r=r_star. Choose h=1/10 and
A=K1=1/20 as counterexample parameters. They are not measured values or priors
used by the exclusion. Recover b_c directly from the compression relation:

    b_c = [4-r-A*r*(r+5-2rh(4-r))/(2(1-rh)²)]/(5r)
        = 0.117093952756… .

| Inputs retained | Missing scalar constraint | Constructed values | Result |
|---|---|---|---|
| r_star, rho1, p1=0.15 | Normal flow w1 | w1=1.131822690 | An ordinary fast state exists |
| r_star, rho1, w1²=0.75 | Thermal p1/rho1 | p1=0.08782046457 | An ordinary fast state exists |
| r_star, rho1, p1=0.15, w1²=0.75 | Magnetic field only; downstream pressure/tangential velocities also unspecified | M_n²=3, fast bound=2 | Fast excluded |

For both constructive controls M_n²=5.1240904067, p2/p1=4.489266436,
Bt2/Bt1=2.490985687 and entropy increase divided by c_v=0.2121881302.
Their different dimensional scalings place them at the same point in the
dimensionless figure. The unchanged diagnostic returns fast_shock for both
without receiving a family label. Their original scaled flux residuals are
below 1e-55. Complete states and actual residuals are in the JSON; these
arithmetic tolerances are not observational precision.

The reference M_n²=3 is below the necessary boundary
M_n²=3*r_star/(4-r_star)=3.5494094126. This scalar boundary describes the
condition, not an instrument requirement or a full uncertainty analysis.

## Checks and implementation limits

The pre-run protocol fixed the reference, omitted inputs, two counterexample
parameters and acceptance criteria. Exact Fraction arithmetic brackets the
reference compression and checks the polynomial/reduced-energy numerator
identity coefficient by coefficient. Decimal reconstruction is compared with
the original energy flux and all other conservation fluxes. The controls
retain the appropriate known quantities and have positive pressure, entropy
increase, field amplification and a 1-to-2 characteristic transition.

The existing full-state API still returns INSUFFICIENT_DATA when B is omitted.
It has not been converted into a general missing-data solver. R99 adds a
separate analytical necessary-condition screen and its saved demonstration.
That screen returns FAST_EXCLUDED or a weaker outcome; it never infers slow
from absence of a fast candidate. The source R98 slow solution remains an
existing compatible witness rather than a newly unique identification.

## What a solar observer must constrain

The useful pair is density compression and normal acoustic Mach number.
Estimating it requires the plasma velocity relative to the front, its normal
and a justified upstream pressure-to-density ratio. EUV intensity contrast
is not automatically density compression. An electron temperature alone does
not supply total pressure without composition/ion-temperature assumptions.
The present example uses exact model inputs; no solar measurement uncertainties,
covariance, projection/emission ambiguity or same-patch association is propagated.

## Reproducibility and article role

From the project root, for a deliberate reproduction in a separate copy:

    python3 partial_inputs/audit.py
    python3 partial_inputs/plot.py
    python3 partial_inputs/write_report.py
    python3 partial_inputs/build_view.py

Closing R99 reused the saved audit; no historical numerical run was repeated.
R99 supports a limited observer-oriented claim:
some incomplete but physically matched scalar inputs can exclude a fast
alternative without prescribing B. It does not establish unique solar diagnosis.

## One next proposed stage, not run

Test finite hard bounds on the compression and normal acoustic Mach number
for this same case: does the entire allowed joint set remain above the fast
bound, or does it overlap the non-excluded region? Declare the set and its
dependence assumptions first. Completion requires a certified exclusion or
an explicit unresolved overlap; satisfaction of this necessary condition is
not proof of fast existence. No further event or family programme is started.

## Interface and preservation

The accepted R97 beginner layout and saved R98 result are retained. R99 adds
one folded MHD check and a concise update to the small grey model-result line.
The logo is 7% larger than the accepted R98 display size and appears at the
left, with title and summary at the right. SVG lettering and fan are vector
paths; the accepted solar illustration is embedded as a lossless raster crop.
This is a hybrid logo, not a vector reconstruction of observational solar data.
Prior scientific scripts and the movie are preserved. Native-browser acceptance
is not claimed. A distinct RMO_QuickLook_R99.html avoids ambiguous download names.