# R103 — which angle and total-speed errors preserve fast exclusion?

## Result for the observer

**A TOTAL Alfvén-speed error of ±20% together with a field-to-normal angle
30° ±5° preserves fast exclusion throughout the declared input set.** The
angular reserve is small: the limiting symmetric angle half-width is about
5.09°, and it must be strictly below the exact boundary. At 30° ±10° the same
±20% speed interval no longer certifies exclusion. This does not prove fast
existence. The previously checked slow state remains inside both sets.

The central total-speed estimate is cA0=w0/cos30° (equal to 1 in the saved
velocity normalization). Thus the central NORMAL speed remains w0, exactly
as in R102. These central estimates are model inputs, not fitted solar values
or the actual saved slow speed. All cases simultaneously include absolute
compression bounds r=r0±0.05 and normal acoustic M_n=M0(1±10%), M0²=3.
These are hard bounds, not 1σ, 2σ or 3σ statistical errors.

| Quantity | Central value | Hard error | Full interval |
|---|---|---|---|
| Total Alfvén speed | cA0=w0/cos30° | ±20% | [0.8,1.2] cA0 |
| Field-to-normal angle, A | 30° | ±5° | [25°,35°] |
| Field-to-normal angle, B | 30° | ±10° | [20°,40°] |
| Compression | r0≈2.1677737267 | ±0.05 absolute | [r0−0.05,r0+0.05] |
| Normal acoustic Mach | M0=√3 | ±10% | [0.9,1.1] M0 |

| Angle bounds with total speed ±20% | Worst-case exclusion margin G | Conclusion |
|---|---:|---|
| 30° ±5° | +0.00262720013 | Fast excluded throughout; small reserve |
| 30° ±10° | −0.14951092262 | Fast exclusion not certified |

This is why the R102 normal-speed error cannot simply be relabelled as a
total-speed error. Direction must also be accounted for. The mathematical
limit depends on the declared central values and all other input bounds.

## What is new, and what is reused

R102 supplied a whole-set necessary fast condition for a NORMAL Alfvén-speed
constraint. R103 introduces only the projection of a bounded TOTAL speed using
a bounded relative field angle. It is not a repeat of the earlier geometry
audit, a new shock-family diagnostic or an inferred field direction from images.

Keep the same fixed local normal, rho1=1, p1=3/20, gamma=5/3, mu0=1,
sound speed 1/2 and w0²=3/4. Let x=M_n/M0=w1/w0, where w1 is normal plasma
speed relative to the front. The front normal itself is held fixed here.
If its uncertainty changes x as well as the field angle, that joint uncertainty
must be propagated for the actual observation; this test has not done that.

No observed solar uncertainties, statistical independence or covariance are
assumed. Each declared box allows all combinations. A justified smaller joint
set could strengthen a result, but a convenient dependence cannot be invented.

## Projection and exact whole-set bound

For directed theta in [0°,180°],

    cAn1 = cA1*abs(cos(theta))
    a = cA1/cA0
    cAn1/w0 = a*abs(cos(theta))/cos(theta0), theta0=30°.

The saved mass/normal-field and downstream characteristic argument requires
x² >= r*(cAn1/w0)² for the fast family including its switch-on boundary.
Therefore a strictly positive minimum of

    G = r*a²*cos²(theta)/cos²(theta0) - x²

excludes that family throughout the allowed inputs. On these acute angle
intervals the worst corner is lowest r, lowest a, largest theta and highest x:

    min G = (r0−0.05)*0.8²*cos²(theta_max)/cos²30° − 1.1².

Monotonicity, not sampling, proves this is the global minimum. If the allowed
directed interval crosses 90°, the minimum normal component is zero. Field
reversal is handled through the absolute cosine, not by assigning negative
Alfvén speeds. Equality G=0 is retained by this necessary condition; it does
not establish that a complete switch-on state exists at those scalar inputs.

## Read the boundary as an allowable error

Let T=0.7558800917299197… be the saved R102 lower-bound threshold. For symmetric
total-speed error epsilon_A about cA0 and an acute angle interval,

    (1−epsilon_A)*cos(theta_max)/cos(theta0) > T.

For total speed ±20%, the exact critical angle is bracketed by
35.08859068431775° and 35.08859068497259°. Thus the symmetric angle half-width
about 30° must be below approximately 5.088590685°. The ±5° case passes;
±10° does not. The many digits certify numerical resolution, not observational
precision. Observer-facing summaries use about 5.09°.

Equivalently, at a specified maximum angle,

    epsilon_A < 1 − T*cos(theta0)/cos(theta_max).

At theta_max=35°, the total-speed half-width must be below about 20.0867%.
At theta_max=40°, it must be below about 14.5466%. At zero angular width about
30°, the expression recovers the saved R102 limit of about 24.412%.
These are strict limits of this sufficient exclusion criterion; losing it
neither constructs fast nor proves every other test would be inconclusive.

## Checks and nonempty allowed sets

The protocol was fixed before one new audit. Exact Fraction arithmetic uses
the saved compression enclosure. A rational pi enclosure follows from
Machin's arctangent identity with alternating-series remainder bounds; cosine
enclosures use 18 terms and the next-term remainder, rounded outwards. The
critical angle is bracketed by certified signs to better than 1e−9 degree.
Ordinary floating-point cosine values agree but are not the proof.

A separate substitution of the downstream squared speeds uses

    (w2²−cAn2²)/w0² = x²/r² − a²*cos²(theta)/(r*cos²(theta0)).

Its enclosed sign is opposite to G in both cases. Zero angular width at 30°
reproduces the saved R102 margin exactly. The reflected interval [145°,155°]
has the same projection as [25°,35°]. Intervals crossing 90° and completely
unknown direction do not yield a positive lower normal bound. Exact equality
is retained; invalid, nonfinite, unordered or out-of-range bounds and missing
or radian angle units are rejected rather than read as degrees.

The saved R98 slow reference, carried through R99, has total cA1²=10/9 and
theta=30°, with r=r0 and x=1. It belongs to both new boxes. Its previously
checked RH/entropy/classification results are reused, not rerun. Thus the
positive exclusion is not an artifact of an empty physically compatible set.
No new full fast state is constructed for the wider-angle box.

## Reproducibility, scope and paper role

- Frozen protocol: field_angle/PROTOCOL.md.
- One new execution: field_angle/audit.py.
- Exact bounds, source hashes and controls: RMO_field_angle.json; summary: verification.json.
- The figure/report are produced by field_angle/present.py from saved results.
- Source condition and limitations: ../normal_alfven/RMO_normal_alfven_report.md,
  with its preserved R99 derivation and primary jump-condition reference.

The ApJ result is a concrete example of specifying which combination of field
magnitude and direction errors preserves exclusion of an alternative. A robust
partial-input exclusion is conditional on a physically valid joint input set.
It is not a unique slow identification, a full Riemann fan, an observational
classification or a universal requirement on a solar instrument.

QuickLook retains all earlier science, the accepted header/logo, embedded and
standalone image viewer with frame 23 and Original: GOES/SUVI credit. No images
or processing are changed. Native-browser acceptance remains open.

One next proposed bounded stage: consolidate the saved R98–R103 evidence into
an observer-facing claim/assumption/missing-measurement table for the ApJ paper,
to close this local-exclusion block before selecting further scientific tests.
No further numerical programme is launched by this R103 stage.