mathematically, can we prove thermal efficiency of 2 stroke engine less that 4 stroke engine.

2 Stroke cycle may be considered inherently more efficient, as it reduces thermal and mechanical losses, but unless you make it very complex, 2-Strokes are tuned for a certain rpm and load range, and anything out of this frame would put it in the high emissions, poor economy range of operations.

From recent data publshed in SAE news, it's known that, among the different systems of scavenging for 2-Strokes, Loop scavenging is under any circumstance better than Uniflow scavenging, and there have been many types of Uniflow scavenging in the ICE history, starting from split pistons, and so on.
Recent attempts for a practical automobile 2-Stroke engine seem were conducted by Ford in the early 80's based in the low-pressure, direct fuel injection systems by the Australian Sarich, who recently won a local contract for a drone engine.

I have a pair of books on 2-Stroke engines purchased from SAE; as the interest for 2-Strokes today may be considered below zero, it can probably still be found at bargain prices: 'Emissions form Two-stroke engines' by Marco Nuti, ISBN: 0-7680-0215-X, 1998, and
'Two-Stroke Engines Technology and Emissions', Edited by Ben Sehaffer, Gordon Blair and George Lassanske. ISBN: 0-7680-0126-9, 1998.
There were some previous publications on the subject of 2-Stroke exhaust pipes design, but as these were UK publications, and there, the number of copies printed from certain highly-specialized technical issues having a low demand are also few, it makes books expensive, it can be also hard to find. Perhaps some can be found in second-hand book sites as abebooks.com, with the search word '2-Stroke' or 'two stroke engine'.

SAE papers 260039: 'Racing car development', by F S Duesenberg, and 920079: '2-Stroke Engine: The Lotus approach', by D W Blundell and M H Sandford contain surprisingly close designs, also in the website by Douglas Self, on peculiar distribution systems and valveless engines, you may have a look on a summary of several approaches to improving two-stroke engines.
There's also a site in French language dovoted to 'Moteurs non-conventionels' -Unconventional engines. The reference source for many of these renewed ICE attempts is the book: 'Some unusual engines' by LJK Setright, but I'd say these are more for hobbyists than for engineers.

A publication in Spanish language, available in PenDrive edition: 'El motor de Aviacion de la 'A' a la 'Z', by Ricard Miguel Vidal (Aeroteca, Barcelona) contains most Aviation Engines ever built, and it can be a good reference to start searching.

Patent databases as ESPACENET are a good place to search for ICEs, but just if you write: 'Rotary Engine' as search words, the number of patents stored and able to be retrieved would be much over the limit of patents that can be shown, fixed in 500.

Probably, cost of producing a 4-Stroke Wankel Cycle, air cooled housing and charge cooled rotor RCE is not much high than for a 2-Stroke, (See SAE paper 900035: 'The Introduction of a New Ultra-Lite Multipurpose Wankel Engine', by D Eiermann & R Nuber -Wankel R&D GmbH, and M Soimar -Rotec manufacturing and engineering, for some charts about what can be expected in terms of efficiency and production costs for the different combinations of housing and rotor cooling and construction in Wankel RCEs). For small engines, there's nothing like the research by Sachs (Fichtel und Sachs) and Yanmar Diesel in the 70's, and Norton in early 80's, also in SAE Papers. Kawasaki had also a patent on a charge cooled Wankel RCE.

With RCEs meeting fuel economy and emissions standards would be easier than with any kind of two-stroke engine, so, with the exception of emergent countries, were it contribute to a great extent to an unbearable environment, both from the point of view of smoke and of noises, you won't find today 2-Stroke engines, build, sold and on the streets.