How stable are horns and knife-edged ridges?

The Matterhorn in Switzerland -- the most famous example in the world of a "horn" or residual peak with three or more intersecting faces or rock walls.  Peaks such as these are often too steep for snow and ice to accumulate, although small impermanent ice masses and snow patches can present grave dangers to climbers since they are likely to break off without warning and go crashing down at high velocity.

Here is another splendid image of the Matterhorn -- the most photographed mountain in the world?

Further to this recent post:

https://brian-mountainman.blogspot.com/2020/02/residual-peaks-and-areal-scouring.html

there has been some discussion in the literature about the stability of steep pyramidal peaks (horns) and knife-edged ridges or arētes.  They occur in particular in glaciated alpine landscapes, but they are common too around the edges of ice caps and ice sheets where remnants of old interfluves stand well clear of the surfaces of outlet glaciers and where local cirque glaciation comes into play.

By the way, I'd like to see the word "tind" used instead of "horn" because it has a much longer association with the landscape.  According to Wikipedia:  From Middle English tind, tynd, from Old English tind (“tine, prong, tooth”), from Proto-Germanic tinduz, tindaz (“prong, pinnacle”), from Proto-Indo-European (e)dont- (“tooth, projection”). Cognate with Dutch tinne (“battlement”), German Zinne (“pinnacle, battlement”), Danish tinde (“pinnacle, battlement”), Dutch tinne (“tooth of a rake”), Icelandic tindur (“spike, tooth of a rake or harrow, pinnacle, peak, battlement”). Cf. the related tine. Also more distantly related to Dutch tand (“tooth, tine”), English tooth.  In Norwegian the word tind is also used specifically for a high and spectacular mountain pinnacle.

Ama Dablam, Nepal, Himalayas -- a mountain horn whose summit is at 6,812 m above sea level

Ulvetanna Peak, Queen Maud Land, Antarctica -- altitude 2,930 m.   Here we can see the remnants of ridges running out from the peak.  That in the foreground has almost been removed by coalescing cirque glaciers, and that in the middle distance is being whittled away, leaving very spectacular pinnacles and arete remnants behind.

Machapuchare, Himalayas, altitude 6,993m.  A sacred mountain which may or may not ever have been climbed.  Climber are asked not to set foot on the summit.  Really there are two summits connected by a vicious knife-edged ridge.

Fitz Roy in the Argentinian / Chilean Andes -- summit altitude 3,405 m.  This is the highest pinnacle in an area of jagged peaks and ridges affected by multiple small glaciers with many different orientations.

Mount Assiniboine in the Canadian Rockies.  Altitude 3,618 m.  Known as "the Canadian Matterhorn".

Stetind, Nordland, Norway, summit at 1392 m.  Unofficially Norway's "national mountain"..........  On the summit there is a remnant of an ancient plateau surface.

Aguille du Dru, in the French Alps.  Summit altitude 3754 m.  This very spectacular summit is the highest point on a long ridge of which much remains.

Tre Cime di Lavaredo -- three distinctive and famous peaks in the Italian Dolomite.  The highest summit lies at 2,999 m.  These are really the last remnants of a mountain ridge -- currently subject to intense frost weathering -- note the steep banks of scree, which do not accumulate in situations where flanking glaciers are still present.

I have not included Ketil (Greenland), Half Dome (United States), Mount Asgard (Baffin Island), and Mount Thor (Baffin Island) because although they are all located on ridges they are also flanked by glacier outlet troughs which have cut away, in each case, one particularly spectacular and vertical rock face.  So they are asymmetrical features, somewhat different in origin from the horns created at the intersections of mountain ridges affected by cirque and valley glacier erosional processes.

Back in the 1960's Professor David Linton had a theory that horns and knife-edged ridges are very stable features which can survive for many thousands of years in a state of equilibrium.  He never did explain this theory very well, and I doubt that it is very reliable.  He thought that horns formed at the intersections of ridges affected by cirque glaciers.  As cirques expand, their headwalls are pushed back further and further,  eating into the ridge until the ridge crest is breached, at which point glacial processes are reduced in effectiveness.  As noted above, many horns have three or four faces.  Linton thought that "dilatation of the rock parallel to each face" was an important mechanism of erosion, and he thought that this process would become less effective as the horn gets smaller, eventually reaching an equilibrium state.  In this state rock expansion and spalling will take place all over the horn rather than being concentrated on the flattish horn faces.  I don't buy into this hypothesis -- and it must surely be the case that exposed pyramidal peaks must be affected most of all by frost shattering on the edges or corners between the flattish faces -- and this process would increase as a horn gets smaller, rather than reducing.  If the flanking cirque glaciers are rendered ineffective as a result of ridges being breached or destroyed, it is possible that the mechanism for the glacial removal of rockfall debris might be reduced, and this might result in a buildup of scree, as we see in the case of the Dolomites.

I suspect, however, that theories of evolution and equilibrium states are all rendered redundant by the complex and unique histories of glacier / climate oscillations experienced by  all of the sites at which horns and knife-edged ridges are found.   Geology and structure play a part too.  No two sites have experienced exactly the same history -- so horns never have perfect asymmetry, and if you look at all of the examples illustrated above, the similarities are obvious -- but so are the deviations from the ideal.