Apologies for the long post, but hopefully it makes sense & also apologies if I'm on the wrong track.
I guess you have seen the census returns for Arthur for 1901, 1891 & 1881 which indicates somewhere between 1881 & the 1891 census Arthurs father Charles dies (Sarah a widow in 1891). Arthurs brother Walter born in Hull ~1884 family permanently moved by then?
Yorkshire BMD have a record of a death of a Charles Scarborough in 1890 age 44, St Marys Hull - could this be your Charles and his age on 1881 census is wrong? (for me it's not 100% clear).
Searching on Familysearch for baptisms around 1846, there looks to be a likely candidate as on 19/10/1846 Charles Scarborough was baptised in Tatteshall Lincoln - parents John & Harriet Scarborough.
Although strangely there is another baptism for a Charles Scarborough on 6/8/1854 at Tattershall, parents John & Harriet?
There is another baptism for a John Scarborough, parents John & Harriet on 6/4/1854 in Tattershall.
The 1861 census for Tattershall seems to show that the first baptism date for Charles is correct, along with brother John.
John Scarborough 57 Waterman or Boatman born Tattershall.
Harriet " 50
Charles " 14
Ann " 11
John " 8
Have not been able to find Charles in 1871, but father John, Mother Harriet & brother John are still in Tattershall - John still a boatman.
If this John Scarborough turns out to be in your line then on the 12/5/1843 there's a marriage for John Scarborough to a Harriet Paulson in Glentworth Lincoln - which may warrent a closer look.
Phew, hopefully it helps, would love to find Charles in 1871 to see what age he is listed as.
Kev.
