Longest path: positive float, and still holding the finish date

Filter a programme on total float equals zero and you expect back the chain that sets the finish. Three activities and two calendars are enough to break it. Close one day on the downstream calendar and the upstream activity picks up a working day of float while still sitting on the only chain there is, so the filter returns two of the three and leaves out the one holding the date. Put every activity on one calendar and the filter and the chain agree again.

Live result

The only typed chain has 3 activities and the zero float filter returns 2.

float still on activity A1 day

Activity A still holds the finish while carrying positive float.

The rule as commonly stated

The activities holding completion are exactly the activities whose total float is zero, so a zero float filter returns the complete finish setting chain.

Try a case

Close the downstream calendar

Consecutive closed dates1 day

Starting on 2026-01-02. At zero, the second calendar is declared but changes no placed date.

One scope and information cut

The activities, links, calendars, and dates must describe the same scope and information cut before the witness is shown.

The computable witness

A
Calendar: Daily calendar
Early start: 2026-01-01
Late start: 2026-01-02
Total float: 1

B
Calendar: Downstream restricted calendar
Early start: 2026-01-03
Late start: 2026-01-03
Total float: 0

C
Calendar: Daily calendar
Early start: 2026-01-04
Late start: 2026-01-04
Total float: 0

Activity	Calendar	Early start	Late start	Total float
A	Daily calendar	2026-01-01	2026-01-02	1
B	Downstream restricted calendar	2026-01-03	2026-01-03	0
C	Daily calendar	2026-01-04	2026-01-04	0

Only typed chain: A, B, C
Zero float filter: B, C
Omitted chain activity: A

The smallest breaker

Three one day activities form one serial chain. A and C work every civil date. B uses a calendar closed only on 2026-01-02. B moves to 2026-01-03, C finishes on 2026-01-04, and the finish boundary is 2026-01-05. A can move one working day without moving that finish.

The restoring condition

With one calendar throughout, zero lag, no overlap, no separate date constraint, and a finish computed from this serial chain, every early and late start coincides. The filter and the chain then both return A, B, and C.

What does not break the rule

Fixed reference with one daily calendar

All three activities use the daily calendar. The computed finish is 2026-01-04 and all three activities have zero float.

Finish boundary: 2026-01-04
Zero float filter: A, B, C

Fixed reference with one shared restricted calendar

All three activities use the same calendar closed on 2026-01-02. The finish moves to 2026-01-05, but all three activities still have zero float.

Finish boundary: 2026-01-05
Zero float filter: A, B, C

It tests a zero float filter on the typed three activity serial network, without deciding a programme, selecting a controlling set for use, allocating float, assigning delay, or deciding entitlement.

Here is the maths

The formula, in words

Place each activity forward on its own calendar from the prior finish boundary. Then place each activity backward from the computed finish on its own calendar. Total float is the signed count of that activity calendar's working dates from early start up to late start.

TF=LS-ES

where

TF

is total float,

LS

late start, and

ES

early start, counted in working dates on the activity calendar.

A worked example

$A$ works on 2026-01-01. $B$ cannot work on 2026-01-02, so it works on 2026-01-03. $C$ works on 2026-01-04 and the finish boundary is 2026-01-05.
$C$ is the third activity. Backward placement puts $B$ and $C$ on their early dates. It lets $A$ start on 2026-01-02 instead of 2026-01-01, so $A$ has 1 positive working day of float on its own calendar.
The only typed chain is $A$ , $B$ , $C$ . The zero float filter returns $B$ and $C$ , leaving $A$ out.

What this does not model

This is not a general scheduler. It accepts exactly three serial activities, zero lag, no overlap, one computed finish, and no more than two calendars. An imposed finish uses another sign convention and is refused.

Source basis

The United States Government Accountability Office Schedule Assessment Guide supplies public practice context for schedule logic, calendars, durations, and date calculation. It supplies no copied definition, method table, named test, or equivalence. Schedule Assessment Guide.

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This counterexample does not make a finding about a programme, decide entitlement, impose a finish, support activity overlap, allocate float, or say who may use it. Requests for a programme decision or an imposed date interpretation are refused.

Zero float filter witness