# Worked Demonstrations

**Version:** 1.0
**Date:** 2026-08-22
**Type:** Demonstration — the arithmetic the instruments produce

---

## ⚠ Every number on this page is invented

There is no property here. "Example Inn" is not a hotel, the counts are made up,
and the figures were chosen to make arithmetic legible rather than to resemble
any market. **Nothing here is a record of anything.**

Real filled sheets belong to the properties that fill them and are not published.
What publishes is the *mechanism* — and a mechanism is easier to check when you
can watch it run on numbers you know are fake.

Assume throughout: **Example Inn, 60 rooms, 36 king and 24 queen.**

---

## 1. Par, and why the denominator is the whole game

```
Par = (units served x consumption per unit x cycle length) + safety stock
```

Bath towels, at two per occupied room per day, on a ten-day supply cycle, with
20% safety stock:

| input | value |
|---|---|
| units served (rooms) | 60 |
| consumption per unit per day | 2.0 |
| cycle length (days) | 10 |
| subtotal | 60 × 2.0 × 10 = **1,200** |
| safety stock at 20% | 240 |
| **Par** | **1,440** |

Now run it with the wrong room count. Suppose someone takes 48 from a status
report instead of 60 from the room list:

| | correct | from a status report |
|---|---|---|
| units served | 60 | 48 |
| **Par** | **1,440** | **1,152** |

**A 20% error in one input becomes a 20% error in every par at once, in the same
direction, silently.** Nothing runs out immediately; the property simply operates
288 towels short of its own plan and discovers it on the worst day of the year.

This is why the room count is checked against the room list and never against a
report of work done.

---

## 2. What a cycle change does — the number that lives in two places

Same towels, same consumption, cycle shortened from ten days to seven:

| | 10-day cycle | 7-day cycle |
|---|---|---|
| Par | 1,440 | 1,008 |

If the runs move to weekly and par is not rebuilt, the property holds **432 towels
it is storing and paying for with no reason.** If the runs stretch to fourteen days
and par is not rebuilt, it runs 576 short and stocks out on the last days of every
cycle, repeatedly, with nobody writing down why.

**Change the cycle, rebuild the par.** They are one number in two places.

---

## 3. Cost per occupied room — the number a bank statement cannot give you

A statement shows what left the account. It has no denominator. Supply cost only
means something per occupied room:

| input | value |
|---|---|
| supply spend, month | $4,800 |
| occupied rooms, month | 1,100 |
| **supply cost per occupied room** | **$4.36** |

Now the same spend in a slower month:

| | busy month | slow month |
|---|---|---|
| supply spend | $4,800 | $4,200 |
| occupied rooms | 1,100 | 620 |
| **per occupied room** | **$4.36** | **$6.77** |

Spend fell by $600 and the cost per room rose by 55%. Read from the statement
alone, the slow month looks like successful cost control. It is the opposite: the
fixed part of the supply base is being spread over fewer rooms.

**This is the number the run sheet exists to produce, and it cannot be computed
without a room count and an occupancy figure.**

---

## 4. Rate position — reading a sell-out correctly

Six nights at Example Inn, with the compset median beside them:

| night | our rate | compset median | position | result |
|---|---|---|---|---|
| Mon | $119 | $104 | +14% | **sold out** |
| Tue | $119 | $113 | +5% | sold out |
| Wed | $119 | $118 | +1% | sold out |
| Thu | $159 | $116 | +37% | open |
| Fri | $139 | $133 | +5% | open |
| Sat | $174 | $134 | +30% | open |

The instinct is that the sold-out nights were the good ones. Read against the
compset they are the opposite: **the property sold out on its three lowest-priced
nights while every competitor kept selling.** It ran out of rooms before it ran out
of demand, and the last rooms went at the price that had been set rather than the
price the market would have paid.

A sell-out is not a result. It is a question: *what would these rooms have made at
a higher price?* And it is only answerable if the rate was recorded **before** the
night filled — which is why a single snapshot cannot answer it and two can.

---

## 5. Rotation coverage — finding the room nobody has entered

Preventative maintenance on a six-month cycle:

```
monthly round = keys / cycle months = 60 / 6 = 10 rooms per month
```

| month | rooms covered | required | debt |
|---|---|---|---|
| 1 | 10 | 10 | 0 |
| 2 | 7 | 10 | −3 |
| 3 | 10 | 10 | −3 |
| 4 | 6 | 10 | −7 |
| 5 | 10 | 10 | −7 |

Every month reads as "the round was done". The cumulative column is the only place
the property learns it is now **seven rooms behind**, and that at this rate some
rooms will go eight or nine months without anyone entering them.

**A log that shows only this month cannot show this.** The arithmetic has to be on
the page.

---

## 6. A rating that can be taken apart

One person, one period, weights set for the role beforehand:

| component | weight | score | weighted |
|---|---|---|---|
| Reliability | 25 | 9 | 2.25 |
| Output | 20 | 7 | 1.40 |
| Standard | 20 | 6 | 1.20 |
| Guest handling | 15 | 8 | 1.20 |
| Independence | 10 | 6 | 0.60 |
| Teamwork | 10 | 7 | 0.70 |
| **Total** | **100** | | **7.35 → 7** |

"A 7" tells the person nothing. **7 = reliability 9, standard 6** tells them exactly
where the number came from and exactly which component to move. The development
line must address the lowest scored component — here, standard and independence at
6 — not the strength.

And the arithmetic closes: weights sum to 100, weighted scores sum to the overall.
**An overall written first and fitted around afterwards rarely closes**, and the
failure is visible to anyone who adds up the column.

---

## 7. Review count as the leading indicator

Three months at Example Inn:

| month | score | count | new reviews |
|---|---|---|---|
| Jan | 4.2 | 412 | — |
| Feb | 4.2 | 419 | 7 |
| Mar | 4.2 | 421 | 2 |

The score does not move at all — and it cannot, because one bad review against 400
existing ones shifts an average by hundredths. **A property watching only the score
sees three identical months.**

The count is the signal. New reviews fell from 7 to 2 while nothing else changed.
That is either a collapse in volume or a collapse in asking, and both are findings
that arrive months before the score could ever show them.

**Scales are not interchangeable.** 4.2 out of 5 and 8.4 out of 10 are the same
standing. Averaging them without normalising produces a number that means nothing,
which is why the sheet records the scale beside every score.

---

## What a demonstration is not

It is not evidence. Every figure here was chosen to make a mechanism visible, and
choosing your own inputs proves only that arithmetic works.

What it does show is what each instrument computes, what inputs it needs, and which
error each one is built to catch — enough to decide whether the same machinery is
worth running on real numbers.

**The instruments publish blank. Every property's filled sheets are its own.**