Inside a sealed robotics facility, each unit carries a status LED that is either teal (non-sentient) or magenta (sentient-certified). Every robot can visually observe all other LEDs during daily operations, but each unit lacks any self-sensor that can read its own LED. Communication about LED status is cryptographically blocked at the protocol layer; no direct or indirect messaging about LEDs is possible. No unit can ever obtain its own LED color via any means-there are no reflective surfaces, no cameras or self-diagnostics, and no permissible workarounds that would reveal self-status. Facility days are globally synchronized so that the midnight “tick” is simultaneous and observable across the floor.
Above the central charging dais hangs a longstanding placard, installed at commissioning and visible to all: “This facility always operates with both LED states active”. The placard remains posted in the atrium and is known to all units during normal operations. Transfers to bays and the absence of transfers are observable outcomes on the next day’s operations log.
Inspector’s closing remark. On Day 0 at 12:00, a Government Sentience Inspector completes an audit and, bound by regulations forbidding mention of any specific unit’s status, makes a single, facility-wide closing statement in the atrium: “It is confirmed that more than one LED state is currently active”. The statement is audibly broadcast and posted to the public operations log with a cryptographic signature; all robots verify receipt and know that the statement is posted in the public log. From that point, normal operations proceed with the usual synchronized midnights and visible outcomes (who transfers, who does not).
Suppose there are N magenta robots in the facility (with N a positive integer). Under the conditions above, on which numbered midnight, counted from Day 0 to Day 1 as the first midnight after the inspector’s closing statement, do magenta robots transfer to the Rights-Compliant Bay, and how many transfer on that midnight? Provide a justification that is valid for all integers N ≥ 1, using only the information structure, timing assumptions, and governance rule specified above.
The permanent placard already states that both LED states are active, and it is publicly visible. The inspector’s closing statement appears to add nothing “new” about LEDs. Explain what, if anything, the inspector’s public statement contributes beyond the longstanding placard.