What are the Higgsless Theories that can explain the Higgs boson detection at the LHC? As many know, in 2012 the Higgs Boson was "detected" at the LHC. I have read that the Higgs boson was not actually directly observed, but the existence of the Higgs boson in the standard model helped to predict what was observed, and thus their existence was taken to be true. However, I have read in this stack exchange answer by Conifold (https://hsm.stackexchange.com/a/2128) that there are so-called "Higgless Theories" that include an effective version of the Higgs Boson, and are therefore "not ruled out by the LHC detection". My question is then, how do these Higgsless Theories (that are consistent with what was actually measured at the LHC) work?
 A: 
My question is then, how do these Higgless Theories (that are consistent with what was actually measured at the LHC) work?

Have you looked at the list your link gave for  theories without a Higgs field?
If you do you will see a long list of disparate theories that try to model the same data that the standard model,SM, of particle data fits,  with various successes as you will see reading the list and its references.
The SM can be thought as a data bank of 99% of measurements and observations at present, and its success is due to also being predictive for new data, as was the discovery of the Higgs boson.
The SM is a quantum field theory, and the introduction of the Higgs field in order to break the symmetry the model has at very high energies , to the mass spectrum we observe in the laboratory, carries with it the necessity of the existence of a Higgs boson, which was a prediction until the Higgs boson was discovered at the LHC.
Alternative theories attempting to model the same data have to embed the SM or show that the SM is derivable from the new theory, in order to fit the existing data.
Each alternate theory works with its own mathematics which cannot be described in a page on a question and answers site.
A: I speculated about this in a question I had about alternative mass generation. The Higgs mechanism is confusing. A Higgs potential is postulated. The potential is caused by the Higgs particles themselves an7d in the absence of this field the vacuum is said to have a positive potential energy (top of the hat). Very strange. When the field is present, the energy drops to a state on the rim of the Mexican hat. The energy ⁷of the field is zero then while the field is non-zero. There is a global U(1) symmetry between all these zero energy states. The field is a not a gauge field but interacts with massless particles to give mass. Mathematically sound but physically unsound, to say the least. Feynman once said that if it can't be explained in layman's terms the theory is off-line. I haven't seen one convincing explanation in these terms. A marble moving in molasses won't do. Light obtaining effective mass in media won't do. A famous person entering a room with people turning their attention to her, won't do. In fact, nothing will do. Does the field pull? No.
I proposed that the quarks and leptons are bound states of three massless particles. The W, Z, and the Higgs could be composites of six of these massless particles. Someone provided a link to the rison model. I don't see why this can't be the case. The Higgs mechanism hasn't been observed and there is a lot of space between $10^{-20}$ and $10^{-35}$ meter.
Quarks were once thought up to explain hadrons and mesons. These massless particles could explain quarks and leptons. A lot of fundamental problems would be solved. Matter/antimatter asymmetry  family generation, weak interaction, etc.
Here is the rishon model
Let me state the quark and lepton content. Say we have two fields, $C$ and $U$, charged and uncharged. Charge is $\frac{1}{3}$ Then:
Electron
$\overline{\rm CCC}$
Up quark
$CCU$
Down quark
$\overline{\rm CUU}$
Neutrino
$UUU$
Distribute the quarks over a proton and a neutron, and there are equal amounts of matter and anti matter!
Electron: $\overline{\rm CCC}$
Proton: $CCU$
$CCU$
$\overline{\rm CUU}$
Neutron: $CCU$
$\overline{\rm CUU}$
$\overline{\rm CUU}$
Neutrino:$UUU$
Equal amounts of particles and their anti's (6 of each)! Bye bye asymmetry!
A Higgs boson could be
$\overline{\rm UUU}UUU$
No charge, and the spin is zero. A $W$ and $Z$ could be similar and the masses are comparable. The $W^-$, for example:
$\overline{\rm CCC}UUU$
How simple can it be? The Weyl spinors could function as the basic two massless fields, and a gluon-like interaction could hold them together.
It seems that there comes a day the emblem of this PSE gotta be replaced...
What are the arguments against this model? In fact the model sits in the same position as where the quark model found itself in 1961, when it was introduced to explain the plethora of particle animals in the hadron and meson zoo. The situation is comparable with the quark and lepton zoo these days. The zoo is not that big but a model containing just two particles seems favorable to one with twelve. It can't get more economical. One basic particle is impossible. Twelve is way too much. Two is optimum. Apart from downvotes and negations of this model, I haven't seen a single argument against it. It's non mainstream, but it's what the OP asked for. And they got it. It's the model that can explain the Higgs particle. If this model was thought of before the recipe of Higgs, the particle would have been predicted just the same. Unluckily Higgs's recipe and the quark were introduced contemporary, and thinking already about composed quarks when just introduced would be too much. The W and Z bosons are not basic forces in this model, so there would be no need for a mass mechanism (like there is no need for such a mechanism in pion mediated strong interactions). Nor would there be such a need for leptons and quarks. One day people will point to this answer... Ain't it perfectly clear?
