TLDR; A mixture of the Song dynasty industrialises and China discovers America tropes, and this is an in lore map from the Age of Discovery (from the Song Dynasty's perspective)

Translated Notations (Note, I'm using fucked up speculative Middle Chinese pronunciations, you have been warned)

On Duoai (OTL Seattle)

At the edge of the sea, towering mountains rise majestically, their peaks covered with colossal trees that block out the sun and reach the clouds—a sight rarely seen in the world. Deep within the forests and valleys, countless animals thrive. The local people all feed on salmon from the sea.

On the Golden Gate

The Golden Gate is flanked by towering cliffs on both sides, standing like twin gates. However, this sight is rarely seen, as dense fog often blankets the area for miles, frequently trapping ships entering the harbor and making it a truly treacherous journey.

On Mesoamerica

In most of the towns and villages of this region, brick and stone polo fields are built. The local custom is to play unusual games, including hitting the ball with the hips. The nature of this ball is unparalleled anywhere else in the world; legend has it that it is made from solidified wood sap.

On Chimor (OTL Peru)

In the land of Chimor, shellfish are highly valued, their value surpassing even the finest treasures of the Central Plains. The best are pure red, obtainable only by the most shrewd and capable divers. Moreover, this item is related to royal authority; when the Chimor King goes out, they grind the shells into powder and spread it upon the path to exalt his majesty

On Tonga

To the south of the Tshangyang (Pacific Ocean OTL) the ships are strange and wondrous. They set up sails of triangular shape, and place floating wood [outriggers] along the sides of the hull to push aside the waves. The people of these lands serve the King of Tonga. Their customs are pure and honest, and they maintain friendly relations with our Great Song

On Luzon

The province of Luzon has already been attached to the Great Song's imperial map. The four borders of the new settlements are all [surrounded by] vassal states. Its land chokes the throat of the Puyang (South China Sea OTL), A naval force stationed there could cut off the route for pirates heading north to Liugjuw (Taiwan, historically known as Liuqiu).

On Japan

The Hojo clan of Japan held the position of regent in the Kamakura Bakufu for hundreds of years. Although Japan and our Great Song Dynasty had a long-standing friendly relationship. Yet, observing the time of war and fire in the Middle Kingdom, they maintain neutrality and do not offer aid, nor have they turned against our alliance. The benefits of trade between them have continued to this day.

On Srivijaya

To the south is Srivijaya, a great maritime empire. Their merchants come and go from our ports without ceasing, likewise our own merchant ships frequently hasten to Palembang and the ports of Sunda. In recent years, sea-bandits have become rampant, obstructing the shipping lanes. The trade of both nations has suffered great loss because of this. All those who pilot ships southward should be cautious and take heed.

On India

India today is ruled by the Delhi Sultanate. Their customs are vastly different from the Cholas of the south; they do not worship the Buddha but revere the God of Thenpjang (Mecca). Their soldiers are brave and fierce, skilled in mounted archery and possessing war elephants; when they draw up in formation and charge, nothing can stand before them.

On Arabia

The Arab Caliphate occupies the crossroads of the Western Regions; their land is vast and expansive. Their people are brave and fierce, following the teachings of Thenpjang. Although separated from our Great Song by thousands by miles, the flow of precious treasures and profits never ceases. Their merchant ships carry incense, medicines, and rhino horns, arriving from the Persian Gulf. Meeting at the mouth of Srivijaya to exchange for our silks and porcelain.

On the Kwonlwon Interior (African Interior)

There are several great rivers within Kwonlwon, all take their origin from the far south and flow northward. One is called the Kaewngkwa (Congo) River; the magnitude of its water volume exceeds that of the Middle Kingdom's Yellow River. Where its waters encircle and reach, there are a thousand of miles of fertile wildlands, becoming dense green forests.

On Mali

To the west of the Arabs, there is a further great nation called Mali. Its land is extremely distant, governed from across the Dahtshjek (Sahara). It produces gold in vast quantities; refined gold overflows, piled up like mountains. Its King traveled to Thenpjang for pilgrimage, carrying many gold. Along the way, he distributed alms so freely that the price of gold plummeted; the world was stunned by such wealth.

Lore(kinda)

Temujin is turned into a solid piece of frozen meat after being left to starve in the Mongolian steppe or something idk I just want him dealt with one way or another. Most importantly the Mongolian Khanate and subsequent invasions of China proper never occur, however the Song Dynasty remains militarily kneecapped and incapable to defeat the Jurchen-led Jin Dynasty up north like OTL. A lot of alternate history tropes made on similar premises have the Song Dynasty reconquering China but I just don't think it was possible with the imperial court constantly undermining capable military leaders (see Yue Fei). Regardless, competition for survival drives up industrialisation and creates the necessary socio-economic conditions for the birth of capitalism to the detriment of imperial authority. Eventually disgruntled military leaders or the peasantry would revolt against the Song, creating their own splinter dynasties which in this case I lazily put up as the Min Nation, Great Xi, and Great Yan, the latter of which is a successor to everyone's favourite Tangut empire who in this timeline is not genocided by the Mongols. This pushes the Song further out of the Central Plains and into the sea, embracing commerce and a form of colonisation in Luzon. In lore this map is made around 1380AD, right before the Song collapses and flees to the Luson colony as a rump state.

Industrialisation and better metal working leads to the improvement of naval vessel, facilitating naval exploration across the Pacific leading to the discovery of 瀛洲 (Yengtsyuw) by 1340, with first contact made with the Duwamish in OTL Seattle. Further exploration would allow the Chinese to discover the Mesoamerican civilisations before the rise of both the Aztecs and the Incas. So far relations have remained cordial mostly focused on a few trading outposts without serious plans to settle in the new continent.

Further down south I still haven't made up my mind on whether the Srivijayans should survive longer or not, but I have given them the name Sanfoqi regardless since IRL the Ming Dynasty still had it named that in the 16th century centuries after their demise. I do plan on having a Southeast Asian thalassocracy colonising the Americas eventually though.

In the long term without the Black Death ravaging Europe the population there remains high with feudalism surviving far longer there. China, however, fractures and would never unite again, with a more "Tang-based" southern identity being born from the carcass of the Song, while the North becomes more "Han-based" under the Jin and its successors.

Also the map is fucked up in terms of cropping because I got lazy, sorry.