Farey stack

• 2026-08-28 [Verified] The odd-carpet stack renders an RH-equivalent object and no route to a proof: the lit nodes are exactly the Farey fractions, each scale n contributing phi(n) new nodes, their discrepancy is the object of the Franel and Landau 1924 theorems, and the measured S2 * Q flattens near 0.656 (0.6560 at 2000, 0.6564 at 8000) with the local exponent walking to -1. Witness: mrlynum::lattice::new_nodes, lab/rs/farey-discrepancy.
• 2026-08-28 [Proved] The stack is an address, not a construction: the odd-carpet stack's brightness at x = (a_1/q, a_2/q) is the residue count B_N(x) = ceil(N/2) - sum_{r in S(x), r <= N} (floor((N - r)/(2q)) + 1) over the bad residues mod 2q, per-point cost independent of N, and the line stack's is floor(N/b); a stack of 5 * 10^17 layers, N = 10^18, evaluates exactly in a tenth of a second, and at N = 55 the Farey table holds 940 nodes summing to 1540 = N(N+1)/2; the closed form's proved boundaries are per-point only (an R x R raster costs R^2), exact representations only (a real-oracle input is undecidable on {n x integer}, irrationals with known continued fractions stay computable via Ostrowski), finite N only (infinite-depth membership is undecidable) and unweighted only. Witness: lab/py/carpet-stack-address, mrlynum::lattice::farey.
• 2026-08-28 [Verified] Immediacy buys no RH content: the Mobius-weighted node is the Mertens-type sum Sum_{k <= N/b} mu(kb), equal to M(N) at b = 1 and to M(floor(N/b)) at only 64 of 200 denominators at N = 200, with no polynomial-time algorithm for the Mertens function at binary input and the best known near x^(2/3); the rank closed form sum_d mu(d) sum_e floor(x e) re-imports Mobius, the meter's global readout collapses to sum_{n <= N} M(floor(N/n)) = 1 identically (checked exactly to N = 20000), the divisibility incidence array is the Redheffer matrix up to its first column, and Franel 1924 is already the symbolic all-Q reduction, so the route ends at Mertens. Witness: lab/py/mertens-meter.
• 2026-08-28 [Verified] The stack's complexity frontier is the sharing of its scales: per-pixel brightness with binary inputs is in P by fixed-dimension lattice-point counting (Barvinok 1994, two parity branches summed), destroying the shared scales makes "does any point reach maximum brightness" NP-complete (Simultaneous Incongruences, Garey and Johnson SP3), and making the ambient dimension part of the input makes "is any layer lit at this fixed point" NP-complete (Lagarias 1985) while polynomial at every fixed dimension, so a no-shortcut theorem for this stack could never separate P from NP; d(n) is not factoring-hard by the sigma route (sigma(pq) = pq + p + q + 1 recovers the factors while d(pq) = 4 carries nothing), and the O(q) residue sweep is polynomial in the denominator q, hence a unary-input algorithm. Witness: REFS.md.
• 2026-08-28 [Conjecture] The Baez-Duarte coefficients c_k = sum_n mu(n) n^{-2} (1 - n^{-2})^k = sum_{j=0}^k (-1)^j C(k,j)/zeta(2j+2) from an independent Mobius sieve to n = 10,000 read -0.316011506 at k = 1 to -1.68003e-5 at k = 1000 against a 450-digit reference -1.65958e-5, difference 2.04521e-7; the sieve agrees with the test vector [1,-1,-1,0,-1,1,-1,0,0,1] and with a second linear sieve at every integer through 10,000, counts 3053 minus-ones, 3917 zeros, 3030 plus-ones; consistent with the criterion and evidence for the Riemann hypothesis of exactly nothing.
• 2026-08-28 [Conjecture] S_N = sum_{k=1}^N (-1)^k C(N,k)/zeta(2k) is not a Riemann-hypothesis criterion tending to zero: it has the wrong zeta shift, omits the j = 0 term and tends to 2 - S_100 = 1.843329, S_500 = 1.967518, S_1000 = 1.983699; the sequential Baez-Duarte coefficient is c_k = sum_{j=0}^k (-1)^j C(k,j)/zeta(2j+2), the Nyman-Beurling distance d_N = inf ||1 - D_N||^2 is not the coefficient sequence and needs its own basis and Gram matrix, sum_{j | k} mu(j) is the Mobius inversion identity (1 at k = 1, 0 after), and direct binomial evaluation at 80 digits is nonsense by k = 500, so 450-digit arithmetic or the Mobius series is required.
• 2026-08-28 [Conjecture] Under the convention Q = 3^level the Landau discrepancy reads 0.166667, 0.549206, 1.150760, 2.118500, 3.187070 at Q = 3, 9, 27, 81, 243, computed in exact rationals by two routes that agree; D_Q/sqrt(Q) stays in [0.0962, 0.2354] and the last-three log-log slope is 0.464, consistent with O(Q^{1/2+eps}) and discriminating nothing, since five nested deterministic points cannot test a statement quantified over every positive epsilon.
• 2026-08-28 [Refuted] Stack brightness encodes the Mobius function, so a Baez-Duarte meter can replace the Franel table - neither brightness carries factorization data: the Farey stack gives B_Q(a/b) = floor(Q/b), 199 distinct values for the 10,000 denominators at Q = 10,000, every denominator from 5,001 to 10,000 sharing brightness 1 while mu runs -1, 0, +1; the gramstack gives 1 + K(a/b) with K = (-1)^a/b^2 for odd b and K = 0 for every even b, so all even denominators coincide; joining a node to a factorization through its denominator puts the arithmetic in the factorization. Witness: lab/py/carpet-stack-address.
• 2026-08-28 [Refuted] A design's Farey order is determined by its fill count - the stack of grid scales n = 1..Q lights exactly F_Q = {a/b : 1 <= a <= b <= Q, gcd(a,b) = 1} with brightness floor(Q/b), a boundary coordinate k/n reducing to a/b and recurring at every scale divisible by b, checked by literal stacking at Q = 30 on all 278 lit fractions; Farey order is Q, fill count plays no part, and every design gives the same sequence at fixed Q. Witness: lab/py/carpet-stack-address.
• 2026-09-11 [Proved] The Farey sequence restricted to a digit design counts without enumerating a fraction: with S_F the whole numbers whose every base digit lies in the digit set, card {a/b reduced : 0 < a <= b <= Q, b in S_F} = sum_{b in S_F, b <= Q} phi(b) and card {a/b reduced : 0 < a <= b <= Q, a and b in S_F} = sum_{b in S_F, b <= Q} sum over d dividing b of mu(d) #{multiples of d in S_F up to b}, the second by inclusion-exclusion on the divisors of b. Witness: lab/rs/farey-discrepancy.
• 2026-09-11 [Verified] Both restricted counts agree with a Stern-Brocot enumeration at every rung of both ladders, base 3 {0,1} to Q = 3^11 = 177147 and base 10 without the digit 9 to Q = 10^5, 45 checks and no failure, the largest being 9538759028 nodes on the full-set control. Witness: lab/rs/farey-discrepancy.
• 2026-09-11 [Verified] On the strict convention the discrepancy sums ride the node count instead of cancelling against it: the local exponents e_2 and e_1, single ratios between consecutive rungs, agree with the mass exponent to two decimals at both designs, +1.259 and +1.262 against +1.263 at base 3 {0,1} and +1.904 and +1.906 against +1.908 at base 10 without 9, while the same lane's S1/card holds two figures from Q = 2187 (card 4286 to 1080458) at base 3 and from Q = 10000 (card 11890654 to 963170938) at base 10, at 9.4e-2 and 5.2e-3, and S2/card likewise at 1.3e-2 and 3.6e-5, the base 10 rung below moving the first figure of S2/card from 4.1e-5. Witness: lab/rs/farey-discrepancy.
• 2026-09-11 [Proved] The strict digit-restricted Farey sequence at base 3 {0,1} misses the closed interval [1/2, 2/3] at every Q, so it does not equidistribute: a denominator with leading digit at position L satisfies 3^L <= b <= (3^(L+1) - 1)/2, a numerator with leading digit at the same position gives a/b >= 2*3^L/(3^(L+1) - 1) > 2/3, and one with leading digit at L - 1 or below gives a <= (3^L - 1)/2 < b/2, hence a/b < 1/2; the measured widest gap contains that interval at every finite Q and shrinks onto it from outside, 0.16827, 0.16720, 0.16684, 0.16673, 0.16669, 0.16667 at Q = 3^6 .. 3^11 from left endpoints 0.49931, 0.49977, 0.49992, 0.49997, 0.49999, 0.50000. Witness: lab/rs/farey-discrepancy.
• 2026-09-11 [Conjecture] The strict digit-restricted Farey sequence at base 10 without the digit 9 does not equidistribute either, on the settled constants alone and with no interval to argue from: its widest gap falls like 1/Q, 0.01136, 0.00113, 0.00011, 0.00001 at Q = 10^2 .. 10^5 against the control's 0.01000, 0.00100, 0.00010, 0.00001, so the base 3 emptiness argument does not transfer. Witness: lab/rs/farey-discrepancy.
• 2026-09-11 [Conjecture] Restricting only the denominator to a digit design keeps the square-root shape transplanted to that design: with D_Q = #{b in S_F, b <= Q} ~ Q^alpha and card ~ Q^(1+alpha), a node-count error of order sqrt(D_Q) puts e_2 at -1 and caps e_1 at alpha/2, and the measured e_2 reads -0.959 and -0.899 while S2*Q reads 0.8926 and 0.8536 against the control's 0.6782 and 0.6684 and S1/Q^(alpha/2) reads 0.243, 0.281, 0.267, 0.268, 0.274 at Q = 3^7 .. 3^11 and 0.213, 0.222, 0.207, 0.265 at Q = 10^2 .. 10^5, flat where the control's S1/sqrt(Q) falls, so the reading is S2 = O(Q^(-1+eps)) and S1 = O(Q^(alpha/2+eps)). Witness: lab/rs/farey-discrepancy.
• 2026-09-19 [Proved] The stack is the Farey resonance diagram up to the floor: normalising the brightness law gives the node a/b the height floor(Q/b)/Q, which lies in (1/b - 1/Q, 1/b] at every depth and equals 1/b exactly when b divides Q. Witness: lab/rs/farey-discrepancy.