Chapter 10: The Necessary Height

## Chapter 10: The Seventh Head

Halen was halfway through his broth when Mara placed the candidate register beside it.

Steam blurred the lower edge of the screen. Halen covered his bowl with both hands.

“You people have learned to threaten food with paperwork.”

“Only food with inadequate witnesses,” Mara said.

Renn made a sound that might have been a laugh before remembering they were still objecting to the investigation.

They were in the fabrication records room, not at any candidate bench. Vale stood at the second terminal with an untouched copy of the routing archive. Mara had generated her query from the spacer’s authenticated dimensions, laminate class, bridge elevation, boss spacing, and the twelve-day fabrication interval preceding the A-17 access window.

She had not marked a preferred result.

The screen separated the search into three sections.

The first showed one initially queried station: D-1.

The second showed four stations that survived dimensional review: precision router B-2, mills C-3 and C-4, and portable cradle P-6.

The third contained eleven station-and-head compatibility rows. Beneath them, a deduplicated ledger listed exactly seven physical heads:

R-8. R-11. R-14. T-12. T-19. T-22. T-27.

Renn leaned over the screen. “Eleven or seven?”

“Eleven possible pairings,” Mara said. “Seven pieces of metal. We establish the tool by pieces of metal.”

“And D-1?”

“Queried, then excluded before the plausible station pool. It remains visible so no one can accuse us of deleting an inconvenient result.”

She turned to Vale. “Start without my annotations.”

Vale read the dimensional requirements and rebuilt the query. D-1 appeared with the other initial results. Its certified jaw travel fell eleven millimeters short of the required bridge cut. He compared the certificate with the latest physical inspection rather than accepting the archive field alone.

“D-1 excluded by insufficient jaw travel,” he said. “It does not enter the four-station pool.”

Mara entered his finding beside hers.

The four surviving stations produced the same eleven compatibility rows. Vale ordered the system to deduplicate their head serials.

The ledger returned seven.

He read them aloud.

“R-8, R-11, R-14, T-12, T-19, T-22, T-27.”

“Now exclude stations,” Mara said. “Then heads. Separately.”

B-2 could accept the laminate and reach every spacer surface. Its compatibility rows named R-8, R-11, and R-14. But no sheet from the numbered pump-seal allotment had entered its controlled rack. Vale checked raw intake, retained offcuts, completed parts, and witnessed waste rather than relying on the job queue.

Every compatible sheet assigned to B-2 remained physically accounted for.

“B-2 excluded by material routing and complete retained accounting,” Vale said.

“Marks remain relevant,” Mara said. “They belong in the head findings.”

R-8 had eight cutting teeth. Its calibration strip repeated eight uninterrupted passages, not the spacer’s ten clean passages followed by an eleventh-position skip.

“R-8 excluded by tooth count and periodicity,” Vale recorded.

R-11 had eleven teeth, but its left-hand helix lifted withdrawal material toward the opposite face. Its retained strip showed the predicted reversed burr.

“R-11 excluded by helix and burr direction.”

R-14 carried fourteen teeth at a tighter pitch. A chipped third tooth left a paired witness mark on every retained strip, neither feature appearing on the spacer images.

“R-14 excluded by fourteen-tooth spacing and its distinct third-tooth witness pair.”

Three physical heads were gone. Mara did not allow the display to hide them.

C-3 had received laminate from the same material family, though not the numbered pump-seal sheet. Its compatibility rows named R-14 and T-12. The outer profile lay within its envelope, but the boss recesses did not fit any fixture certified during the relevant interval. A suitable narrow fixture had arrived later, five days after A-17’s access window.

Vale checked the fixture receipt clock and the receiving witness signature.

“C-3 excluded by the period-specific fixture limit.”

T-12 had remained mounted there through most of the interval. Its retained offcuts showed nine cutting passages and a straight withdrawal scar without the crescent lift visible on the spacer.

“T-12 excluded by nine-tooth rhythm and straight withdrawal.”

Portable cradle P-6 carried compatibility rows for T-12, T-22, and T-27. Its movement log placed it aboard the tug Nacre Resolve throughout the twelve-day interval.

“Logs can travel without machines,” Renn said.

Vale opened the tug’s departure image. P-6’s frame serial stood beneath the seal strap. A recent weld repair crossed the rear cradle support like a pale crooked finger. The arrival image showed the same repair and the same strap unbroken. Berth pressure records and cradle isolation telemetry supported the images.

“Authenticated physical continuity aboard the tug,” Vale said. “P-6 is excluded.”

He did not use that exclusion to dispose of its heads.

T-22’s sealed calibration pieces preserved eleven intact passages. Its grind geometry raised a withdrawal burr opposite the spacer’s crescent.

“T-22 excluded by intact eleven-tooth sequence and opposite burr direction.”

T-27 used a variable pitch. Its retained test pieces alternated narrow and wide tooth spacing, producing a paired cadence. The spacer and the C-4 tabs showed uniform spacing up to the repeated skip.

“T-27 excluded by variable-pitch cadence,” Vale said.

Renn counted the ledger with one finger, touching no evidence.

“R-8, R-11, R-14, T-12, T-22, T-27. Six exclusions.”

“One survivor,” Mara said.

Vale returned to his untouched archive copy and rebuilt the entire sequence again: one queried station excluded before the pool, four plausible stations, eleven compatibility pairings, seven unique physical heads, six individually excluded.

His result matched Mara’s.

Only then did she close the register.

“Now we go to C-4.”

Halen looked at his covered bowl. “It had a promising childhood.”

At C-4, Renn verified the breaker against the local diagram and opened the motor bus under Vale’s observation. Vale tested for stored power. No one mounted a head or energized the spindle.

The fabrication bay smelled of cured laminate, cold lubricant, and dust the extraction system had failed to remove from corners. Halen set three retained C-4 calibration tabs in a padded manual cradle. Worn metal clicked softly against polymer.

Mara displayed authenticated images of the missing spacer.

Ten clean passages crossed the first tab. At the eleventh position, the cut broke into a shallow stutter. A crescent of lifted laminate marked withdrawal.

The defect repeated on the second tab, the third, the spacer edge, and both photographed screw seats.

Halen measured spacing and burr direction twice. Vale repeated the alignment without using Halen’s scale settings.

“Uniform eleven-tooth interval,” Vale said. “Repeated interruption at the same rotational position. Same withdrawal direction.”