My understanding of the India/Pakistan split is that it was primarily motivated on religious terms.
Sikh majority Punjab doesn't really fit into the ideological split of Muslim majority Pakistan and Hindu majority India.
Firstly as a side note to the rest of this post I would just like to briefly clarify what a "Hindu" actually is since it isn't exactly a very accurate term to refer to a person's beliefs and also so I can follow up with the relationship between Sikhs and Hindus at the time. A Hindu is, in literal terms, a denizen of the area bounded by Burma, the Himalayas and the Sindhu (Indus) river and was never a term used by Indians themselves until the Islamic conquests; at which point any practitioner of the various beliefs of India at the time would be referred to as a "Hindu" and be given Zimmi status accordingly by the nascent Islamic governments. Speaking on basis of belief, there isn't a monolithic belief system called "Hinduism" - the beliefs of Hindus varied and still varies wildly depending on the individual (although there is a common-ish culture, which is, along with the sake of brevity, the two main reasons why the term "Hinduism" is often used): there are atheistic schools (shakhas) like Purva Mimansa and Sangkhya and theistic ones as well (for example, various sub-schools of Vedanta, Yoga and the majority of belief systems that modern Hindus follow such as Vaishnavism and Shaivism). Many of them are not based on strict dogma and many often use fables or mythology as a medium for communicating ideas and philosophies. Many of them are also ritualistic, but many of them also totally reject ritualism. Over time, presumably as a result of cultural contrast against Islam in the subcontinent and the decline of autochthonous philosophical thought in India (which is apparent by the dearth of innovative philosophical texts and fables being created in the Mediaeval and Early Modern periods with the same impact as say, the Upanishads or the Mahabharata), Hindus began referring to themselves as "Hindus" and the qawm (a people or nation in a more complex, gestalt political entity) of Hindus thus emerged.
Sikhs and most Hindus (most Hindus being Vaishnavites or Shaivites come the turn of the 20th century) were and still are culturally and religiously alike: similarities such as various common beliefs (like enlightenment, Moksha; belief in the soul, Atman; belief in reincarnation, Sansara; belief in karma; etc...), frequent marriages between the two groups and acceptance of the caste system (yes, even by many Sikhs, despite it being contrary to their Gurus' teachings) led to an overall familiarity and symbiosis between the two groups. The saving of many Kashmiri Pandits by Shri Guru Tegh Bahadur Ji and the reverence many Hindus have for the Sikh Gurus also contributed to this. The percentage of Sikhs in the Punjab region was always low (14.9% in all Punjab in 1941, six years before partition, and less than 8.6% before the 20th century): this low percentage and the familiarity between Sikhs and Hindus led to Sikhs generally not bothering with claiming an independent state in the partition era (although later movements, like the Khalistan movement of the 1980s did, which I'm not going to get into) because the Sikhs were not likely to get their own state and if the Hindus got their own state, the Sikhs would likely benefit from it too. There were exceptions to this mutualism though like Dayanand Saraswati's criticism of Sikhism and propagation of the monotheistic Hindu reformist movement Arya Samaj which became somewhat popular in Punjab.
The Punjab region, overall, was and is majority Muslim (53.2% Muslim in 1941) and before partition the western 60%-ish of Punjab generally had an absolute majority of Muslims, excluding many urban areas (nearly 40% of the total urban population was comprised of Hindus in 1941). As stated before, only 14.9% of the population of the Punjab region + internal princely states like Patiala were actually Sikh in 1941, the remainder being mostly Hindus with some Christians mixed in. The eastern remainder of the Punjab region was majority Hindu. Analysing all of this, it is clear that the British needed to partition Punjab if they were to fulfil the demands of the popular Pakistan movement and amicably leave the subcontinent quickly with good terms on both sides (or so they thought), which was in their best interest because upkeep for India was expensive after the Second World War and the British simply couldn't afford that (it was also in their best interest for trade purposes). The diverse region could obviously not have been given to one country or another due to said trade purposes and also because of the strife that such a move would cause relative to simply partitioning the region. As an addendum, the post-partition Punjab region was divided into West and East Punjab and in the 1960s East Punjab was partitioned again, into the majority Punjabi-speaking new Punjab region (which is a third of the size of the former East Punjab), Haryana and Himachal Pradesh.
I don't know much about the internal politics leading up to partition, but the partition demarcation line was drawn by Cyril Radcliffe, a man who had never been to India, and was told to leave as many Hindus in India and as many Muslims in Pakistan as possible. However, because Sikhs were instrumental in assisting the British in various frontiers, from fronts in the Second World War to the famous Battle of Saragarhi, the British felt compelled to grant more land to the Sikh qawm (whom sort-of allied with the Hindu qawm when it comes to the desire for an independent state, as explained previously), so important cities like Amritsar (which was actually majority Muslim) were given to India instead, even though they should have technically gone to Pakistan. Also, as stated before, urban areas and many cities (like Gujranwala, Rawalpindi, etc...) deep in West Punjab generally had more Sikhs and Hindus than rural areas in the same region, so many very diverse cities actually went to Islamic Pakistan rather than India, which is constitutionally secular.
Overall, Sikhs were generally closely aligned with Hindus socially and often politically too. This compounded with the renown of Sikhs in the eyes of the British, the religious distribution of the Punjab region and the British's financial interests led to the British partitioning the region between India and Pakistan, with no real comparable demand for a third Sikh state.
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