# RMO101 — a complete fast witness inside R100 box B

## Checked answer

**A physically admissible fast transition exists inside the wider synthetic
input set B.** It has r=2.15 and M_n/M_0=1.09. Together with the previously
checked slow reference at r0=2.1677737267, M_n/M_0=1, this establishes two
different admissible families within the same bounded partial inputs.
It does not establish two solutions for identical complete states, classify
a solar front, or prove feasibility of every point in B.

R100 found loss of uniform exclusion. R101 adds the complete-state witness
needed to turn that loss into demonstrated ambiguity under these partial bounds.

## Inputs and units

All states use normalized mu0=1, gamma=5/3, rho1=1 and p1=0.15. The common
front-rest frame has S=0, normal n=(sqrt(3)/2,1/2,0) and tangent
t=(-1/2,sqrt(3)/2,0). These velocities are plasma velocities, not image speeds.
The new upstream tangential velocity is zero, whereas the saved slow state has
a different tangential velocity and magnetic field; both were withheld in the
partial input specification. No hidden equality of full upstream states is used.

B: r in r0 ± 0.05 and x=M_n/sqrt(3) in [0.9,1.1]. These are illustrative
hard limits, not observational errors or a probability distribution.
The candidate is strictly inside B (x clearance 0.19 and 0.01; compression
clearances approximately 0.03222627 and 0.06777373).
It is outside A because x>1.05 and outside C: at t=0.9 its delta is
approximately -0.05377373, below -0.01. This agrees with R100's exclusions.

## One predeclared construction

The protocol fixed r=43/20, x=109/100, h=1/10 before numerical execution.
No search, tuning or historical numerical test was performed.
Write W=w1²=3*x²/4, b=p1/W, h=Bn1²/W and a=Bt1²/W.

    a = 2(4-r-5*b*r)(1-r*h)^2 / [r*(r+5-2*r*h*(4-r))]
    q = Bt2/Bt1 = r*(1-h)/(1-r*h)
    rho2 = r, w2 = w1/r, ut2 = Bn1*(Bt2-Bt1)/w1
    p2 = p1 + W*(1-1/r) + (Bt1²-Bt2²)/2

The scalar energy reduction constructs a candidate; it does not assign its type.
The complete states are then substituted into the original fluxes and diagnosed.

| Quantity | New fast witness |
|---|---:|
| r | 2.15 |
| M_n² | 3.5643 |
| w1² | 35643/40000 |
| h | 0.1 |
| a | 0.00364335901436 |
| Bt2/Bt1 | 2.46496815287 |
| p2 | 0.618381768688 |
| p2/p1 | 4.12254512459 |
| entropy increase / cv | 0.14069098442 |

Exact rational parameters, Cartesian states and the family-free request are
in RMO_fast_witness.json. The magnetic field is nonzero and oblique, with
Bn and Bt both nonzero; this is a regular fast shock, not a gas-only endpoint.

## Checks independent of the construction equation

Original local mass, momentum, induction, energy and normal-field fluxes were
evaluated with exact Fraction arithmetic. All residuals are exactly zero after
factoring known nonzero dimensional/radical factors. In particular, the
independent normalized energy fluxes are

    F1 = gamma/(gamma-1)*b + 1/2 + a
    F2 = gamma/(gamma-1)*(p2/W)/r
         + [1/r²+h*a*(q-1)²]/2 + q²*a/r - h*a*q*(q-1).

Direct Cartesian vector substitution at 90 decimal digits separately gives
maximum scaled RH residual 3E-90.
Both routes check the original flux laws; agreement with the construction
polynomial alone would not suffice. The classifier uses the existing numerical
implementation; its result is checked against these exact and vector routes.

Pressure and density are positive. For gamma=5/3, entropy growth is certified
exactly by (p2/p1)^3/r^5>1, equivalent to log(p2/p1)-(5/3)log(r)>0.

For each side, P(z)=z²-(cs²+vA²)z+cs²*cAn² has the two magnetosonic
speed-squared roots. Exact P(cAn²)<0 orders slow < normal Alfvén < fast.
Upstream P(w1²)>0 and w1² lies above the quadratic vertex, hence w1>cf1.
Downstream P(w2²)<0 and w2²>cAn2², hence cf2>w2>cAn2>cslow2.
All inequalities are strict; none relies on a rounded plotted value.

| Normal speed (normalized units) | Upstream | Downstream |
|---|---:|---:|
| Plasma w | 0.943967690 | 0.439054740 |
| Fast cf | 0.504965776 | 0.699563781 |
| Normal Alfvén cAn | 0.298508794 | 0.203581308 |
| Slow cslow | 0.295573292 | 0.201485625 |
| w/cf | 1.869369638 | 0.627612165 |

The unchanged RMO full-state diagnostic receives states, geometry and front
speed without a family label and returns fast_shock. The saved slow witness
is copied from the prior checked result, not newly recalculated. The full-state
API still requires its documented inputs; no general missing-data solver was added.

## What an observer can conclude

If only these bounded r and normal acoustic Mach constraints are supplied,
they do not distinguish fast from slow in this model. The ambiguity is now
demonstrated by two complete, different states. Other independent measurements
are needed to distinguish them. This result does not make every EUV wave
ambiguous and does not identify either family in the displayed solar movie.

## Evidence and reproducibility

- Protocol: fast_witness/PROTOCOL.md, SHA256 a7e912f22c2b82a053fd9b00fd701ed0286592cf80fdac7f8a306c2e0b707ad6.
- One execution: fast_witness/audit.py; presentation reads its saved JSON only.
- Saved source: results/partial_inputs/RMO_partial_inputs.json, SHA256 34aa2f1d11bf10c6a5a5d9028486cdcf64737a1bed5129069241c84ae89f30c4.
- R99's proof and R100's three-set certificates are reused unchanged.
- The saved R98 comparison uses Urashima & Morioka (1966),
  https://doi.org/10.1143/JPSJ.21.1431; its full scientific provenance remains
  in the earlier report. No new published-polar comparison is claimed here.
- Static integration/asset checks apply to QuickLook. Native-browser acceptance
  remains open; it is separate from the scientific transition checks.

## One next proposed test — not run

Test whether a declared independent bound on upstream normal Alfvén speed can
exclude fast over all of B. The criterion must apply over the full allowed set,
not merely distinguish the two displayed witnesses; observational provenance
would be needed before using such a bound for a solar event.