What crossover and directional entry rules actually compare

Entry rules as inequalities

Technical entry procedures, including a single-average rise test and a dual-average crossover, can be written as inequalities that authorize a long stance when one computed quantity exceeds another.

A rule-based-entry is a testable procedure that maps a stated inequality, together with market inputs and execution constraints, onto enter, hold, or abstain.

Allowed operations

Algebraic reduction rewrites an entry inequality with allowed operations until only the prices that actually decide the signal remain.

Adding or subtracting the same quantity on both sides of a true inequality, or scaling both sides by a positive number, preserves the comparison. Scaling by a negative number requires reversing the inequality, and division by zero is disallowed.

Single-average and dual-average reductions

A moving-average-crossover is a rule-based entry that authorizes a long stance when a shorter simple average exceeds a longer one, or when a single simple average exceeds its own prior-session value.

A long-or-hold rule that requires a 10-session simple moving average to exceed its value from the prior session is algebraically identical to requiring today's close to exceed the close from 11 sessions earlier. After that 10-session reduction, the nine intervening closes cancel and therefore cannot change the moving-average entry decision.

A rule that goes long when a 4-session simple average exceeds an 8-session simple average on the same bar is equivalent to requiring today's 4-session average to exceed the 4-session average from five sessions earlier. That dual-average crossover can therefore be applied by tracking a single 4-session average rather than computing both averages on every bar.

What the directional long rule reduces to

The average-directional-index is a directional-movement construction in which a long signal is taken from a comparison of plus and minus directional indicators over a fixed lookback, after those indicators have been scaled by a true-range sum. A true-range sum is a non-negative aggregate of high-low range, including gap adjustments, that can cancel from both sides of a directional-indicator inequality.

Under the 14-period directional construction, the summed plus and minus directional moves collapse to a 15-session change in the high and in the low, respectively. When the long rule is plus directional indicator greater than minus directional indicator, and the 14-period true-range sum is treated as a shared positive divisor, the rule reduces to today's high-low range exceeding the high-low range from 15 sessions earlier.

The reduced range-versus-range comparison is offered as grounds for skepticism that the original directional long rule is a sound entry procedure.