How is antimatter distinct from negative mass? How is antimatter distinct from negative mass? I'm very interested in the concept of negative mass from a strictly mathematical perspective.  (i.e. this question should not be taken as relating to suppositions involving the harnessing anti-gravity.)  Rather, I am interested in the idea of "the opposite of something".  
My understanding is that anti-matter, although having opposite charge to normal matter, still has mass.  Is this correct?
How does negative mass work conceptually?  Is it related to "dark energy" or "dark matter".  Is is purely theoretical?
 A: At the moment, with a quantum theoretical standard model of particle physics where gravity is not included,  antimatter is defined as particles that have all the quantum numbers defining them with opposite sign, so that when particle hits antiparticle all quantum numbers add to zero.
Mass is not a quantum number and is always positive for individual particles, so the annihilation of particle on antiparticle gives energy which can split into completely different particle antiparticle pairs . The cleanest experiments were the LEP e+e- experiments at CERN and other laboratories, where the standard model has been very well validated.
As your link discusses, there are proposals where a negative mass may arise in General relativity but 

General relativity describes gravity and the laws of motion for both positive and negative energy particles, hence negative mass, but does not include the other fundamental forces. On the other hand, the Standard Model describes elementary particles and the other fundamental forces, but it does not include gravity. A unified theory that explicitly includes gravity along with the other fundamental forces may be needed for a better understanding of the concept of negative mass.

So antimatter in the standard model is well validated by the data and mathematically modeled and understood. Negative mass is an open research question.
Part of this research is the coupling of antimatter particles with gravity, and there are experiments studying this at CERN, with anti hydrogen and how it is affected by the earth's gravity.
A: Assuming the Einstein Equivalence Principle (inertial and gravitational masses are equal) holds for negative masses:

*

*a positive mass attracts all masses, either positive or negative.

*a negative mass repels all masses, either positive or negative.

*a negative mass behaves like a positive mass for an inertial force such as the gravitational force: a negative mass falls towards the Earth and does not exhibit anti-gravity behavior

*a negative mass behave oppositively to a positive mass for an non inertial force such as the Lorentz electromagnetic force or a pressure force (a negative mass accelerates in the opposite direction to the applied force)

So, when a positive and a negative mass are present together, the negative mass repels the positive mass which attracts the negative mass, leading to the controversial so-called runaway behavior.
if the two masses are equal and opposite, both masses would accelerate with the same constant rate and in the same direction (that of the positive mass), while keeping the same separation between each other.
When interacting with a gravitational field or an external electromagnetic field an positron (+e,+m) of positive mass could not be distinguished from a "electron" (-e,-m) of negative mass so that a hypothetic (-e,-m)---(+e,+m) particle pair could not annihilate contraryly to a classical (-e,+m)---(+e,+m) electron-positron pair and be used to transport an electric current.
According to the Dirac's description of the creation of a classical electron-positron pair, a negative energy electron (-e, -m) from the Dirac sea that gains an amount of energy (from a photon or an electric static field) equal to the energy gap width (2mc^2), can pass through this gap by tunnel effect to materialize into an electron (-e,+m) leaving a hole (+e,+mc2) equivalent to a positron. After this process, both particles are supposed to be at rest and non interacting.
For creating an exotic (-e,-m)---(+e,+m) pair, this very amount of energy would not be used to create a positive mass electron from a negative mass electron, but to strongly bind a positron to an negative energy electron from the Dirac sea through the energy gap (the negative energy electron stays in the Dirac sea). There would be no annihilation process as positive energy brought by radiation absorption would simply cancel the negative binding energy.
However, if such exotic particle pairs ever could exist, the existence of macroscopic dense objets of negative mass consisting of neutral negative mass particles is highly unlikely because negative mass particle repel each other and may not be separable from positive mass particles. Negative mass charged objects could not exist. Only neutral or charged null mass objects could exist due to the runaway behavior.
Interestingly, it has been shown, both theoretically and experimentally, that effective negative masses could be simulated by
using mass-spring oscillating systems (cf. J. Zhou et al, "Experimental study on interaction between a positive mass and a negative effective mass through a mass–spring system",
Theoretical and Applied Mechanics Letters 5 (2015) 196–199).
