Best hypotheses so far along with how to test them:
1. Perhaps the parts of the CPI that companies in the stock market can produce are being outpaced by the components that are not produced, like real estate, and so can't contribute to an increase in share prices. (How to check: Look up changes in the CPI components, which are detailed in the reports.) (Edit: This does not hold up, almost all the increases are in energy, which is produced and held in reserve by public companies.)
2. Perhaps the consumer goods in the CPI are increasing in price far more rapidly than prices of goods and services overall, leading to an underestimate of the value of the dollar to anyone except a consumer. (How to check: Since capital goods are eventually used to make consumer goods, this should eventually correct itself. Also, is there a broader version of the CPI?) (Edit: This might be it, although energy is 8% of the CPI and GDP, it is only 4% of the S&P 500.)
3. Perhaps money is leaving the USD, US bonds, and US stocks and going somewhere else. (How to check: Make these same charts but with broader indices.)