math

fig. 1 — the bounty curve · line

derived from formula. bounty on y against fast blocks since the last mark on x, with the quiet window and the lapse limit marked. a pure function of one input, so it holds regardless of what the record contains.

fig. 2 — the band ladder · stepped bar

derived from formula. one bar per band, length proportional to its witnessed threshold. thresholds are written at deployment, so every bar draws now. crossed bands fill solid, bands ahead are outline, and the witnessed marker rests at the origin until the chain produces it.

fig. 3 — the quiet window · timeline strip

derived from formula. fast blocks since the last mark divided into bounty accruing, quiet window elapsed, and lapse callable. the strip is a constant of the contract; the position marker rests at the origin until the chain produces held.

fig. 4 — witnessed and dark · stacked area

derived from measurement. cumulative fast blocks against elapsed fast blocks, witnessed filled and dark left as ground, the diagonal marking the total both series sum to. the plot area stays empty until the chain produces the series.

fig. 5 — stall durations · histogram

derived from measurement. count of proven stalls against stall length, log scale in fast blocks with decade ticks. the dashed rule is the nominal slow interval, the solid rule is deepest. no bars until the register holds rows.

fig. 6 — observed clock ratio · multi-line

derived from measurement. fast blocks per slow block over time, observed solid and trailing median dashed. the flat dashed line is the nominal ratio and is a reference only.

— observed-- trailing median·· nominal reference

fig. 7 — marks against stall length · scatter

derived from measurement. one point per closed interval, stall length on x and fast blocks since the previous mark on y. the diagonal is where the two are equal, the boundary between an interval fully witnessed and one only bounded from below.

fig. 8 — coverage · step line

derived from measurement. witnessed over elapsed as a proportion, y fixed zero to one, drawn as steps because coverage changes only when a mark lands. no smoothing and no interpolation.

table 1 — every figure with its formula

figure	formula	contract	recomputed
held	fast − fast at last observed slow move
witnessed	Σ proven spans between marks inside a stall
dark	elapsed − witnessed
elapsed	fast − fast at record open
coverage	witnessed / elapsed
deepest	max proven stall, in fast blocks
bounty	f(fast − fast at last mark)
observed ratio	Δfast / Δslow
form index	count of crossed bands

where the two value columns disagree, both cells print in live

table 2 — invariants

check	left	right	status
witnessed is monotone non-decreasing
dark is monotone non-decreasing
lapses is monotone non-decreasing
witnessed + dark = fast blocks elapsed since the record opened
no interval is both closed and lapsed
deepest ≥ every proven stall in the register
form index = count of crossed bands

a failing row prints the whole line in live with the two disagreeing numbers beside it

table 3 — constants

constant	source	value
chain id	network
ArbSys precompile	0x0000000000000000000000000000000000000064
ArbGasInfo precompile	0x000000000000000000000000000000000000006c
quiet window	record, immutable
band count	record, bands array length
bounty coefficient	record, immutable