Farey Discrepancy

Three verbs on one object: the node set of the scale stack, and the Franel-Landau discrepancy read on it. With rho_1 < ... < rho_m the nodes of a set in (0,1] ascending and delta_j = rho_j - j/m, the meter is S2 = sum delta_j^2 and S1 = sum |delta_j|.

VERBS

• stack - the lit nodes of the scale stack are the Farey nodes: the literal reduced lines of every scale n <= Q, the sorted window list and the mediant walk are one set, sized by sum_{k<=Q} phi(k), at Q = 10, 30, 60, 125. Each node a/b is drawn by floor(Q/b) of those scales, and scale n lights phi(n) nodes no earlier scale lit.
• meter - the Franel-Landau meter on F_Q, seven rungs Q = 125 .. 8000: S2*Q, S1/sqrt(Q) and the local exponents d ln S / d ln Q, with the sorted window route cross-checked against the mediant walk at the first four rungs. The top rung is Q = 8000 with 19455782 nodes.
• design - the same meter restricted to a digit design S_F, the whole numbers whose every base digit lies in a digit set. Two conventions, a and b both in S_F (strict) and b alone in S_F (denominator), against the unrestricted control, at base 3 {0,1} to Q = 3^11 and base 10 without the digit 9 to Q = 10^5. Per lane it prints S2 and S1 with their consecutive-rung ratios and local exponents e_2 and e_1, never a fit; the transplanted normalisers S2 Q^(e-alpha) and S1/Q^(alpha/2); the constants S1/card and S2/card; and the widest gap between consecutive nodes with its left endpoint.

HOW IT COUNTS

• One Stern-Brocot mediant walk per rung enumerates F_Q in ascending order; the walk itself carries no node list and the three lanes filter it as it runs, so each keeps its own rank and the restricted lanes ride the control for free. The verb's memory is the O(Q) sieves it builds per rung, a few megabytes at the top.
• The count is checked against a sieve that never enumerates a fraction: sum_{b in S_F} phi(b) for the denominator convention, and sum_{b in S_F} sum_{d | b} mu(d) #{multiples of d in S_F up to b} for the strict one. The chk column is that comparison, PASS at every rung of every table.
• Rungs are powers of the base so the design's set is self-similar at each one.
• The cut is the control's cost, one walk step per node of F_Q: 9538759028 steps at Q = 3^11, 3039650754 at Q = 10^5.

RUN

• bash scripts/cargo.sh cargo run --release -p farey-discrepancy from mrlyprod/ runs all three verbs; append -- stack, -- meter or -- design for one.
• stack and meter take under a second at 0.7 GB peak; design takes a minute at a few megabytes. Prints only, writes nothing.
• bash scripts/cargo.sh cargo test --release -p farey-discrepancy checks both restricted counts against brute-force enumeration at base 3, base 10 without 9 and the full set.

WITNESSES

• farey.md WHERE THE LINES LAND - brightness floor(Q/b) on every lit node, and the node table at N = 30.
• farey.md PRIMES ARE THE MAXIMALLY NOVEL SCALES - the new nodes of scale n number phi(n) for n = 2..30.
• farey.md FRANEL AND LANDAU, 1924 - card F_Q = sum phi(k) at Q = 10, 30, 60, 125.
• farey.md THE METER READS WHAT RH PREDICTS - nodes, S2*Q, S1/sqrt(Q) and the local exponents at the seven rungs.
• farey.md THE METER ON A DIGIT DESIGN - the two exponent tables, the count checks, the constants S1/card and S2/card, the widest gap between consecutive nodes, and the S2*Q and S1/Q^(alpha/2) readings of the denominator lane.
• farey.md FAREY ORDER IS THE STACK, NOT THE DESIGN - brightness by literal stacking at Q = 30 on all 278 lit fractions and up to Q = 125.