The second forecast was beautiful enough to make Yanchuan suspicious before he found a mistake.
Its shaded band was narrow. Its demand curve followed the expected afternoon pattern with only small uncertainty, and its recommendation would let the factory reduce contracted reserve capacity for the next quarter. Guo’s analyst called it the largest saving yet.
Ning asked where the weather assumptions came from.
“A commercial forecast feed,” the analyst said. “We use the same feed for temperature, humidity and expected cooling demand.”
“Do those count as three independent inputs?”
“They are three variables.”
“That wasn’t the question.”
The forecast combined variables produced by a single upstream weather model. Its uncertainty estimates treated errors in those variables as though they could vary separately. When the source underestimated a stagnant humid afternoon, it tended to underestimate all three together.
Yanchuan had seen the same comfort in two different spreadsheets that shared an excluded interval. He asked the analyst to show a week when temperature and humidity had both been higher than expected.
They found one. The predicted maximum stayed inside the shaded band until the actual measurements rose above it.
“That’s a rare event,” the analyst said.
“Possibly. How often does the source miss both conditions together?”
The analyst had no answer. The training set contained only a few comparable weeks. There was not enough evidence to turn the observed frequency into a reliable guarantee.
Guo wanted to know whether a conservative reserve margin would make the proposal uneconomic.
“First find what margin is needed,” Chen said. “Then price it.”
They ran two scenarios with coherent weather changes rather than independent random noise. The reserve requirement grew. The projected savings shrank and, in the most demanding scenario, disappeared.
The factory manager looked at the revised chart.
“So the model is worse?”
“It is less certain than the first chart claimed,” Yanchuan said. “That’s not the same as being less useful.”
He described a choice between keeping the present reserve and accepting a lower service commitment. The choice belonged to the people who would operate the factory and pay for its interruption, not to the mathematical convenience of a narrow band.
Gao asked whether the proposal could be tested for a month without selling the reserve capacity in advance. The analyst said the quarter’s pricing deadline would be missed.
“Then the price of learning is part of the choice,” Gao replied.
They kept the reserve for another cycle and gathered more measurements. Guo was displeased by the delay. He did not hide it. But he signed the revised study request rather than the earlier recommendation.
The following week the factory received its new tariff quotation. The price of holding reserve capacity had risen. Gao forwarded the invoice without commentary. The analyst called Yanchuan to ask whether a narrower confidence band could be justified with a different model. Yanchuan said the uncertainty came from the weather source, not the drawing software. The cost of caution had become an amount of money, not an argument he could win by being right.
Yanchuan went to Mei that night. She was comparing two fault-detection models whose errors appeared to agree. One had been trained partly on labels generated by the other. The apparent independent confirmation had a common source.
“We could build a museum of comfortable duplication,” she said.
“There’s a market for it.”
“Unfortunately.”
They cooked dinner and argued about how many measurements were needed before a model could reasonably learn a rare joint event. Mei refused to give him a universal number. The sample depended on the event, the consequence of being wrong and whether other physical knowledge constrained the estimate.
He liked that answer less than a number. He trusted it more.
The following day An called from Guangzhou. Their son had been learning weather signs at school and wanted to know why a forecast could say rain was unlikely and then rain could arrive anyway.
“Unlikely doesn’t mean impossible,” Yanchuan said.
“Then why do people get angry at forecasts?” the boy asked.
“Sometimes because they’re wrong. Sometimes because they thought a probability was a promise.”
“That’s silly.”
“Sometimes,” Yanchuan said. “People still do it.”
An laughed softly from the kitchen.
At work Chen asked Yanchuan to prepare a client memo that stated both the original projected saving and the revised bounded result. It would be tempting to replace the first figure and pretend nobody had considered it.
“Show the revision history,” Chen said. “Not because we enjoy being wrong. Because they need to understand why the answer changed.”
The memo explained the shared weather-source risk, the physical cooling constraints, the retained reserve and the limited evidence for extreme conditions. It recommended another measured cycle before reducing capacity. The client accepted the plan.
There was no launch announcement. No one published the impressive first chart.
At the family review Wanjiru said the investigators had not resolved the disputed visitor’s departure. Another witness was being consulted about the internal receiving-office corridor, but the account was not sufficiently reliable to share. An asked them to keep the family informed when an official assessment could be made.
Yanchuan noticed that he no longer tried to guess what the witness might say. He still wanted to. The desire and the decision had become separable.
His father called later. Jianming had looked at the weather forecast before repairing a neighbour’s exterior light and decided to wait. The rain never came.
“Waste of a morning,” he said.
“Did the work have to be done today?”
“No.”
“Then perhaps it wasn’t wasted.”
“Don’t sell me a model, son. I just wanted to complain.”
They talked about the neighbour instead.
Before closing the factory report Yanchuan added a note on dependent errors. Different columns in a table did not guarantee independent knowledge. A forecast could be honest in every calculation and still be too sure if all its assumptions leaned on one imperfect source.
He printed the revised band. It was wider, less graceful and more likely to contain the weather the factory would actually experience.
He sent the wider band to Gao with a note explaining why it had changed. Gao replied that he could work with a forecast that admitted when it might be wrong.