# RMO100 — can a partial-input fast exclusion survive bounded errors?

## Answer for an observer

Yes for the narrower declared model set. Around the R99 reference compression
r0=2.1677737267 and normal acoustic Mach M0=sqrt(3), hard errors of ±0.05 in
compression and ±5% in Mach preserve fast exclusion for every allowed input
combination. Increasing the Mach range to ±10%, while allowing every combination
with the same compression range, removes that uniform exclusion.

A third set has exactly the same marginal limits as the wider box but admits
only a specified strip of joint values. Fast remains excluded throughout that
strip. Thus marginal error sizes alone do not determine the answer; which
combinations are actually allowed also matters. The strip is an imposed model
relation, not a measured covariance or a relation inferred from this event.

| Case | Hard compression range | Hard Mach range | Allowed combinations | Minimum signed margin | Result |
|---|---|---|---|---|---|
| A | r0 ± 0.05 | M0 × [0.95,1.05] | Full Cartesian box | +0.02027075 | Fast excluded everywhere in the set |
| B | r0 ± 0.05 | M0 × [0.90,1.10] | Full Cartesian box | −0.07227152 | Set crosses the necessary boundary; no uniform exclusion |
| C | r0 ± 0.05 | M0 × [0.90,1.10] | Imposed dependent strip below | +0.00772848 | Fast excluded everywhere in the set |

All percentages refer to M_n, not its square. Compression error ±0.05 is
absolute. These are chosen hard model bounds, not estimated solar errors,
Gaussian standard deviations, confidence intervals or instrument requirements.
A box permits all combinations without asserting statistical independence.

## Reused basis and physical scope

R99 derived the necessary compression condition

    r ≤ 4 M_n²/(M_n²+3),   γ=5/3,

for a local evolutionary fast shock in ideal MHD with scalar pressure, including
the stated perpendicular, parallel/gas and switch-on limits. R100 uses that
saved proof; it does not re-prove every jump relation or rerun R98/R99 audits.
The saved exact-rational compression enclosure is propagated outward here.
Its width below 1e-65 encloses the known model number and is distinct from
the deliberately much wider ±0.05 input-error range.

M_n is upstream normal PLASMA speed relative to the front divided by upstream
acoustic speed. It is not an image-front Mach inferred from a bright moving
edge without plasma-flow and thermodynamic assumptions. The normal, equation
of state and meaning of total pressure are unchanged from R99. Magnetic field
is unspecified. The known slow reference lies in all three sets and remains
an existing compatibility witness; no new solar type is identified.

With x=M_n/M0 and M0²=3, define

    F(x)=4x²/(1+x²),      G(r,x)=r−F(x).

Strict G>0 throughout the allowed set excludes fast over that set. A point with
G≤0 is merely not excluded by this necessary condition. The presence of such
a point does not prove that a complete admissible fast state exists. R100 does
not construct a new full state in set B, nor identify slow uniquely or exclude
all intermediate, compound, non-MHD or emission alternatives.

## Exact certificates for the two boxes

For x>0, dF/dx=8x/(1+x²)²>0, and dG/dr=1. Therefore the minimum margin in
a Cartesian box is at the lowest compression and highest Mach. Its maximum
is at the highest compression and lowest Mach. No grid search is required.

If [L,U] is the saved rational enclosure of r0, the minimum-margin enclosure
is [L−e_r−F(1+e_M), U−e_r−F(1+e_M)]. For A, e_r=0.05 and e_M=0.05:

    F(1.05)=2.0975029727…,  min G=0.02027075409… >0.

For B, e_r=0.05 and e_M=0.10:

    F(1.10)=2.1900452489…,  min G=−0.07227152212… <0.

Set B also contains the reference, where G>0, so it has certified overlap
with both sides of the boundary. This is loss of a uniform exclusion, not a
fast-shock identification. All endpoint expressions and outward bounds are
stored as exact fractions and decimal displays in RMO_partial_bounds.json.

A second arithmetic route clears the strictly positive denominator:

    H(r,x)=r(1+x²)−4x²=(1+x²)G.

For these ranges r<4, dH/dr=1+x²>0 and dH/dx=2x(r−4)<0. Direct polynomial
evaluation at the corners gives the same strict signs, independently of the
rational division used in F. It is an internal complementary formulation,
not an external executable validation.

## Same marginal limits, different allowed set

For C impose exactly

    −1≤t≤1,  −0.01≤δ≤0.01,
    x=1+0.1t,  r=r0+0.04t+δ.

Its marginal compression extrema are r0−0.05 and r0+0.05, while x ranges
from 0.9 to 1.1, exactly as for B. But low r and high x are no longer an
allowed arbitrary combination. This is additional joint information. It
must be justified independently before applying it to observations.

The general set-theoretic rule is that restricting an allowed set cannot
introduce a point outside the larger set. It does not guarantee that all
nonexcluded points disappear. Another dependence could leave them present.
Physical conservation laws relate states, but they do not establish this
particular strip for incompletely measured inputs. This relation was chosen
as a declared illustration and is not derived from new solar data or a new
MHD solution. Each set is nonempty and includes the saved reference.

Along this strip

    ∂G/∂δ=1,
    ∂G/∂t=0.04−0.8x/(1+x²)².

For x∈[0.9,1.1], the subtracted term is at least
0.8×0.9/(1+1.1²)², which exceeds 0.04. The derivative is therefore strictly
negative on the full interval. The minimum occurs at t=1, δ=−0.01:

    min G = r0+0.03−F(1.1) = 0.007728477878… >0.

The exact upper derivative bound and both rational endpoint margins are saved.
This argument covers the full strip, not just its centre line or sampled points.

The complementary proof expands the lower envelope of H at δ=−0.01 and r0=L
as a cubic in t, then sets t=2s−1, 0≤s≤1. The saved Bernstein coefficients
are all strictly positive. Because Bernstein basis functions are nonnegative
and sum to one on this interval, the smallest coefficient is a rigorous lower
bound for H. Reconstructing the power coefficients from the Bernstein form
agrees exactly. No subdivision or parameter retuning was needed.

## Convention and equality controls

The squared-Mach limits for ±10% in Mach are [2.43,3.63]. Treating the same
percentage as an error in M_n² instead would give [2.7,3.3]. Applying the
bound to that narrower, incorrect interval would misleadingly preserve the
fast exclusion for B. The explicit endpoint-convention guard rejects it.
The wrong result is retained as a rejected control, not a scientific option.

At the equality control r=2, x=1, G=0 and the screen correctly does not exclude
fast. Nonpositive Mach and noncompressive r≤1 inputs are rejected by this
screen's domain guard. These controls address concrete interpretation and
boundary risks. They do not claim completeness of the general MHD classifier.

## What this establishes for RMO and the ApJ paper

The exact-input exclusion in R99 can be extended to selected hard-bounded
uncertainty sets. The new evidence establishes both a robust case and a
controlled loss of uniform exclusion, together with a same-marginals example
showing the role of joint information. The mathematical certificates are
exact for these model sets; this does not establish the validity of their
error bounds for a solar observation.

The existing full-state API and its missing-B response are unchanged. R100
adds a saved bounded analytical screen and its explanatory figure, not a
general observational uncertainty solver. Full initial Riemann problems,
source driving and spatial front morphology remain separate questions.

Figure: results/partial_bounds/RMO_partial_bounds.pdf. 

## Reproducibility and one next proposal

Protocol frozen before the one R100 run: partial_bounds/PROTOCOL.md.
Reproduction commands, only for an intentional separate reproduction:

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

The proof source is the saved R99 report, results/partial_inputs/
RMO_partial_inputs_report.md, based on the original ideal-MHD jump relations
in Fitzpatrick, https://farside.ph.utexas.edu/teaching/plasma/Plasma/node105.html.
The reference state comes from the previously checked Urashima & Morioka
(1966) polar, https://doi.org/10.1143/JPSJ.21.1431. Neither earlier benchmark
was rerun. Source and protocol hashes are retained in verification.json.

One next proposed stage, NOT EXECUTED: determine whether the wider box B
actually contains a complete admissible fast state, using a bounded
construction with original conservation, entropy and characteristic checks.
This would distinguish loss of the exclusion certificate from demonstrated
physical coexistence of fast and the saved slow alternative. No automatic
all-event programme, new observations, full fan campaign or book is started.