What is the origin and significance of karahafu (唐破風)?

I believe that the karahafu most likely developed in Japan, but I have also been wondering if there are any architectural precursors from China and/or Korea. I recently came across this Song-Dynasty painting which depicts a building with something resembling a karahafu: https://www.clevelandart.org/art/1915.705

However, there's a possibility the pavilion in this image has nothing to do with the karahafu, since the karahafu-like gable depicted in the pavilion is situated on the side as a balcony, as opposed to at the entrance.

The Wikipedia article on this feature emphasizes that it began as a symbol of spiritual auspiciousness and then evolved into a more secular symbol of high status before wider diffusion.

It was named thus because the word kara could also mean "noble" or "elegant", and was often added to names of objects considered grand or intricate regardless of origin. [...] The first known depiction of a karahafu appears on a miniature shrine [...] Initially, the karahafu was used only in temples and aristocratic gateways, but starting from the beginning of the Azuchi–Momoyama period, it became an important architectural element in the construction of a daimyō's mansions and castles. The daimyō's gateway with a karahafu roof was reserved for the shōgun during his onari visits to the retainer, or for the reception of the emperor at shogunate establishments. A structure associated with these social connections naturally imparted special meaning. Gates with a karahafu roof, the karamon (mon meaning "gate"), became a means to proclaim the prestige of a building and functioned as a symbol of both religious and secular architecture. In the Tokugawa shogunate, the karamon gates were a powerful symbol of authority reflected in architecture.

This strikes me as a clear example of how something like the European "civilizing process" may have also taken place in Japan. In other words, it is logical that as the secular state expanded its power, the rituals and symbols that were first developed as markers of nobility were gradually emulated by the society as a whole.

Kara-Ha-Fu, is, for example, according to this page, like this,( the encircled "style" of the building in yellow ).

The usage in palatial and religious architecture, particularly prior to the Edo period, places the karahafu closer to the Sinicised (or, perhaps, Koreanised) traditions of Japanese architecture.

Yes, according to the author of the page,

唐とは中国の意味ですが、どうやら中国にはないらしい

Translation

唐(Kara in Japanese, Tang in Chinese respectively) means China, but it seems these styles can not be found in China.

The picture you see above is the 東大寺大仏殿(I am sorry there is no English Wiki available(But you can see at least the picture), which is,

758年（天平宝字2年）に完成した。

Translation

It was finished to be built in AD 758.

To your first question

Although the design of the karahafu is quite different from anything that can be found on the continent, it is also atypical pre-Buddhist architecture, suggesting that it doesn't derive from vernacular or "purely indigenous" Japanese styles (e.g. taisha-zukuri (大社造)). The usage in palatial and religious architecture, particularly prior to the Edo period, places the karahafu closer to the Sinicised (or, perhaps, Koreanised) traditions of Japanese architecture.

According to this page(sorry not in English)

Quote

反曲カーブが印象的な唐破風は、「唐」と付くあたりから中国から入ってきたと想像される方もおられるかも知れませんが、日本独自のもので、鎌倉時代に生まれ、安土桃山時代に最も隆盛を迎えた様式だそうです。破風の形式としては、最も格が高いとされ社寺建築や城郭に主に使われてきました。

Translation

Many people would guess that from the name 唐破風, the impressive recurved 唐破風 style is supposed to be imported from China, but it is indigenously Japanese style, and the origin is from Kamakura period,（English) and it made the full debut in, Azuchi-Momoyama period(English). ---> The status of the style was considered at that time by the people was most ex graded so that the style was used for many castles and Shinto shrines/Buddhist shirines. ---

Architectural structure and the origin

I will try to be as simple as possible.

This is the anatomy of wooden house in English.

In Japanese architecture of the roof, the each part of the roof is categorized into these parts (please look at the below picture) and 破風 is there (yellow part).

This page says,

It corresponds with the upper half of the king post part in English.

And then what is unique about 唐破風?.

As you mention,

There are also peculiar features of the karahafu indicative of conscious design choices. The ibara-hire (茨鰭) on the barge board and rafters, for example, can be seen in even the earliest depictions of Karahafu despite being relatively difficult to create (compared to a simple, arched gable) and serving no apparent function.

According to the same page,

唐破風は曲線を連ねた形状の破風板を、屋根に付けたもの。

Translation

唐破風邪 is the special type of 破風 made from curved barge (The ibara-hire (茨鰭))

What then is 茨鰭？

It is the collection of the curved wooden parts that support 破風.

In my opinion, what enabled these kind of structure is Japanese buildings were mostly made by woods, using such as Japanese cypress, whereas the Westen buildings were mostly made from stones. (Same might go with Chinese ones)

Why is the style called 唐破風?

As @Brian Z suggests, and this page also says,

初期の唐物屋は今日で言う高級美術商で、茶室で使う絵画や茶器などの名物を扱う商人を広く唐物屋と呼んでいた8　唐物風の品は国内産であっても唐物と呼ばれていた。

Translation

The early 唐物屋(hard to translate:Would be "Chinese products dealers") are in reality high price antique dealers, dealing with wealthy scrolls and cups for tea in the tea room. 唐物風("Chinese style") goods were called as 唐物("Chinese goods") even though they were perfectly 100% Japanese.

So around 15th-16th century, the name "唐" become not to denote "China" but instead become to mean pricy things and did not correspond at all with "China".

How old does the structure date back?

鴨都破神社（Ka-Mo-Tsu-Ha Shrine)

This shrine dates back to Asuka period. And the shrine has an entering with 唐破風. (But I highly suspect that this entering was added in later dates).

What is the significance of the structure with respect of the viewpoint from architecture?

This thesis, luckily written in English says,

This study considers the process of a value and the application of the traditionalcurve of inflexion ‘‘Kara−hahu” to the architectural design after the Meiji era ．First，the curve had a gentle slope in the Heian era and itwas valued beautiful.

Conclusion.

How old does the style date back?

Well, it looks like the style dates back to at least Heian period, which coincidentally went the same period with Tang dynasty in China.

Why is the significance of the name?

Well, as the time descends, the name 唐 denotes "beauty, pricy" things so that it was applied to many castles and shrines from the viewpoint of the beauty.

Thank you.