# 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.