memory-zeta

• The Dirichlet series of a memory rule, zeta_W(s) = sum_(n in S_W) n^(-s) over the positive integers whose minimal base-2 string the rule accepts, and the zeros of its matrix Lyndon cofactor Z_W(s) = det(I - 2^(-s) T) zeta_W(s).
• peel.py is the arbitrary-precision control on the double-precision matrix ladder of mrlynum::automaton; census.py is the double-precision engine that carries the zero census and reads its own control back.
• The peel: with E_j(w) the vector whose entry u sums n^(-w) over the accepted words of exactly j digits ending in state u, and G_P = sum_(j >= P) E_j, splitting a word on its last digit gives (I - 2^(-w) T) G_P(w) = E_P(w) + sum_(l >= 1) binom(-w,l) 2^(-w-l) Gamma_l G_P(w+l), and zeta_W(w) = 1^T D_(P-1)(w) + 1^T G_P(w) for the Dirichlet polynomial over the accepted words of at most P-1 digits.
• The cofactor never divides by the vanishing determinant: Z_W(s) = det(I - 2^(-s) T) D_(P-1)(s) + 1^T adj(I - 2^(-s) T) N(s) with N the right side of that identity, so it reads on the whole m = 0 pole comb, and the residue at a simple root x_0 of the determinant is that same lift over -x_0 log 2 det'(x_0). No eigenvector is solved for.
• det(I - x T) and adj(I - x T) are polynomials in x with integer matrix coefficients, both taken by Faddeev-LeVerrier from T, so the engine runs at any width with no eigen-decomposition anywhere in the loop.

THE CENSUS

• Z_W is meromorphic with one simple pole at each level-m tooth for m >= 1 and none on the m = 0 combs, which the determinant strips all at once, one factor for every eigenvalue. The census counts by the argument principle per cell and adds back the level-m poles the cell holds, each named by its exact position, so every printed cell count is a zero count.
• Each added-back pole is read and not assumed: its residue is a 48-point circle mean of Z_W at radius --ring and again at 0.4 of it, printed with the gap between the two radii, with (s - s_0) Z_W at 1e-5 beside it to catch a double pole and a blank point on the same line as the null.
• The clearance from every zero to the nearest outer box edge and to the nearest internal cell edge is printed; an internal edge is sampled at the same points and bisected the same way from both sides, so the two cells' phase contributions cancel and only the split between neighbours could move.
• The grid is derived and not chosen: the column edges are the midpoints of the comb and pole lines inside the box, subdivided so no column is wider than 0.75, and the row edges are the midpoints of the tooth and pole heights, so every cell holds at most one tooth.
• The phase is accumulated on a contour seeded at step 0.05 and bisected on any step above one radian, and the largest surviving step is printed beside the count. A count is resolved and not certified: nothing here bounds Z_W'/Z_W on the contour.
• A box of winding one is split along its longer side until it is 0.01 across, then polished by secant, and the largest abs(Z_W) at a located zero is printed.
• Occupancy at a tooth is read at the radius 0.45 of lab/py/zeta-family, with the distance from every tooth to the nearest located zero printed whatever the radius says.
• The nonzero eigenvalues are read off the degree of det(I - x T), whose coefficients are integers from the same Faddeev pass, and a root with a relative imaginary part below 1e-9 is snapped real. Reading them off the characteristic polynomial with a magnitude cutoff instead admits numerical debris: the full rule at width 3 has det = 1 - 2x, one eigenvalue and one comb, and the cutoff sold it three more near zero and a spurious tooth on the real axis.
• The first-order tooth law is u_1 = -r/R, r the residue of zeta_W at the tooth and R the regular part, R taken as the mean of Z_W/det on a circle of radius 0.3, which is exact for the Laurent tail and needs no fit.
• Both verbs print two errors at every tooth and name each one: the modulus miss abs(d - abs(u_1)), which is what the occupancy law is stated on, and the vector miss abs(z - t - u_1), which tests the direction too. One word for two quantities is how the weaker of them gets published.
• Occupancy is a per-tooth Boolean, so every tooth also prints the number of located zeros inside the radius. At width 3 one tooth of the 50 holds two, and 34 occupied teeth hold 35 zeros.
• The truncation bound is carried entrywise as a nonnegative vector through the majorant sum_k abs(adj_k) abs(x)^k / abs(det) of the resolvent, which is the adjugate's own coefficients and needs no norm and no primitivity, plus a measured relative allowance of 1e-14 against the largest quantity the ladder carries. The majorant is not certified by a residual, which is the one step short of the crate's left-of-abscissa branch, and the control is what closes it.
• The right edge is a proved zero-free wall and not a choice: the least element of S_W is 1, the coefficients are nonnegative, and zeta_W(2) < 2, so abs(zeta_W(s) - 1) < 1 on Re s >= 2 and the determinant has no root right of the abscissa.

THE OCCUPANCY CENSUS

• teeth walks the width-k rule classes under G_(1,k) at dim 1, keeps the classes whose nonzero eigenvalues carry exactly two distinct moduli and no repeated root, and runs the zero census of census on one box for all of them, printing one occupancy row per pole line.
• A line is a distinct abs(lambda) and not a distinct eigenvalue: a conjugate pair puts two combs on one line, offset by +-arg lambda / log base, and the line's teeth are their union.
• The box is cut to fit the budget and the cut is printed: -1.15 < Re s < 2, 0.02 < Im s < 20, against the 43.1 of the single-rule census. The left edge holds every radius 0.45 disc of every second line at width 3, the deepest reaching Re s = -1.144241913631, and the height holds two teeth of the first line and two or four of the second.
• The left edge is chosen by the contour guard and not by the discs alone. The guard measures every level-m pole of the rule against the contour, inside the box and outside it, since a pole just outside is never added back and is the worse neighbour; at -1.2 codes 54 and 62 carry a pole line 0.002842615688 outside the contour and code 223 one 0.011370462752 inside it, and at -1.15 every pole clears 0.02. The floor is exactly 0.02, the cut Im s > 0.02 against the level-m pole on the real axis. Both boxes read the same occupancy on all nine rules.
• Each row prints the rule, its class size, abs(lambda_2)/rho, the arguments on the second line, the teeth on each line, the teeth occupied at radius 0.45, every tooth-to-zero distance, the zeros lying off every tooth, and the zeros and poles within 0.02 of a contour with the least pole clearance. Each row is the argument-principle count of census, cell by cell, with the level-m poles added back, resolved and not certified.
• bridge is the control on a new rule. The width-3 rule 55 forbids exactly 011, 110 and 111, which bans an adjacent pair of ones inside every 3-window, so it accepts the width-2 rule 7's set with the single integer 3 adjoined, the word 11 carrying no window. The verb checks that membership over 1 .. 2^span and then checks Z_55(s) = Z_7(s) + det(I - 2^(-s) T) 3^(-s) at seven points, each against the sum of the two bounds.
• That identity crosses widths: the 4-state ladder, adjugate and peel meet the 2-state ones that carry control.json. It also says the pole data is the same on both rules at every m >= 0, orders and residues included, since zeta_55 - zeta_7 = 3^(-s) is entire; the determinant vanishing at every m = 0 tooth is how the census reads that back, and the zero sets are not the same.

THE DIAL

• dial is the finite perturbation probe. For a finite F of positive integers disjoint from S_W the perturbed set S_W + F has zeta_(W+F) = zeta_W + P_F with P_F(s) = sum_(n in F) n^(-s), so the engine carries it by adding F to the Dirichlet polynomial over the short words and changing nothing else: no new ladder level, no new truncation and no new majorant, which is why the perturbation needs no new control. The added term is a finite exact sum and travels inside the same relative allowance as the rest of the polynomial.
• The perturbation is entire, so every pole, order and residue of zeta_W is shared by zeta_(W+F) and only the zero set can move. Two of the three probes that check it are blind by construction and the verb says so: a 48-point circle mean annihilates every Taylor order below 47, so an entire addition of any size passes the residue probe, and the determinant vanishes at a tooth, so the tooth probe prints its own floating-point residual. The probe that measures the added part is the identity Z_(W+F)(s) - Z_W(s) = det(I - base^(-s) A) P_F(s) read off the teeth, at t + --rho, printed beside the modulus of the added part there.
• Occupancy is read by the argument principle on the occupancy circle itself and not on a box: the winding of Z_(W+F) on a 40-point circle of radius --rho about the tooth, bisected on any step above one radian, plus the level-m poles inside the circle, which are read from a pole list padded one unit past the box in both coordinates so that no pole a circle reaches is missed. The verb prints the deepest reach of any circle against the nearest pole line left of the box.
• The 0.02 edge rule of the census becomes an annulus: the count is taken at --rho minus the guard and again at plus it, and a cell whose two counts differ is printed as a seam, the reported occupancy being the inner one. Both baselines read no seam at all. On the perturbed grid a seam whose inner count is 0 is an undetermined occupancy, and a minimum over a tooth's candidate row is then convention-dependent even when the winning cell is itself off the seam, so each line prints its undetermined count and each tooth prints its smallest occupying singleton and its largest empty candidate on both readings.
• The first-order law of the occupancy census moves under the knob through its constant term alone: the principal part is fixed, R becomes R + P_F(t), and u_1(F) = -r/(R + P_F(t)). The higher Taylor coefficients of the regular part move too, the linear one by P_F'(t), which the first-order law does not read. The verb prints the measured occupancy and that prediction as two bit strings over the printed candidate list, one row per tooth, with the mismatching candidates named.
• The scoreboard is printed against a baseline and not alone: each line prints its cells, how many read occupied, how often the first-order law calls a cell right and how often the constant occupied predictor does. A law that scores below the constant predictor on a line is not a selector on that line.
• --deep runs a greedy search for the F that drives abs(R + P_F(t)) to zero at each occupied tooth, one candidate at a time, measuring the disc after every step, which is the attempt to empty a tooth rather than to fill one. A greedy chain is one path, so the verb also runs the exact meet-in-the-middle minimisation over every subset of size at most --deep whenever the candidate list is 24 long or less, printing the minimiser, the minimum, the emptying threshold abs(r)/rho and the disc measured at the minimiser.

RUN

• uv run python research/lab/py/memory-zeta/peel.py, 3.2 seconds, prints the control values.
• uv run python research/lab/py/memory-zeta/peel.py control, 4.6 seconds, writes control.json, the only arbitrary-precision run in the study; the rewrite is byte-identical to the stored file.
• uv run python research/lab/py/memory-zeta/census.py control, 0.05 seconds, reads control.json back through the double-precision engine.
• uv run python research/lab/py/memory-zeta/census.py census, 8.3 seconds, the golden rule, code 7 at width 2.
• uv run python research/lab/py/memory-zeta/census.py census --width 3 --code 23 --left -0.75, 16.3 seconds, the supergolden rule.
• uv run python research/lab/py/memory-zeta/census.py census --left -1.2, 8.3 seconds, the widened box that holds every occupancy disc of the second comb and reads the same 20 zeros.
• uv run python research/lab/py/memory-zeta/census.py teeth --width 2 --left -1.2 --height 20.0, 3.3 seconds, the one two-line class at width 2.
• uv run python research/lab/py/memory-zeta/census.py teeth --width 3 --left -1.15 --height 20.0, 75.5 seconds, the nine two-line classes of the 88 at width 3, the occupancy table and the first-order law.
• uv run python research/lab/py/memory-zeta/census.py bridge, 0.25 seconds, the width-3 rule 55 against the width-2 rule 7.
• uv run python research/lab/py/memory-zeta/census.py census --width 3 --code 55 --left -1.15 --height 20.0, 7.6 seconds, the four-state ladder's Laurent data against the two-state one.
• uv run python research/lab/py/memory-zeta/census.py dial --width 2 --code 7 --top 40 --height 43.1 --left -1.2 --deep 16, 16.0 seconds, the nine teeth of the golden rule against the 22 integers of 2 .. 40 outside S_W.
• uv run python research/lab/py/memory-zeta/census.py dial --width 3 --code 23 --top 40 --height 43.1 --left -0.75 --deep 0, 47.0 seconds, the fourteen teeth of the supergolden rule against its 27 candidates.
• --seed, --rho, --eps, --hug, --ring, --span, --top, --guard, --deep, --peel, --shift, --cut, --left, --right and --height are the dials.
• Halving the seed costs more and not less: 0.025 raises the contour evaluations from 11777 to 21599 and the run from 8.3 to 14.9 seconds, doubling it to 0.1 drops them to 7298 and 5.2, and the 45 cell rows and the 20 zero rows are identical at all three.
• Prints only, writes nothing except under peel.py control.

WITNESSES

• The page lines of beneath, ### The memory zeta.
• zeta_W(3) = 1.154012963277642016659466, zeta_W(2) = 1.415825532884777929125692, zeta_W(0.8) = 9.536379694275011510923898 and zeta_W(1.2 + 9i) = 1.906409024243906069557735 - 0.453243424778265833646527i, each met by mrlynum::automaton inside its own bound, the largest gap 3.0e-15 against a bound of 9.54e-13.
• The six residues at m = 0, j = 0, 1, 2 on both combs. On the comb at Re s = log_2 phi the gaps against the crate are 6.0e-16, 2.6e-15 and 9.3e-16 against bounds near 1.3e-13; on the comb at Re s = -log_2 phi, where the crate reads through its left-of-abscissa branch and bounds near 4.1e-9, the gaps are 1.1e-12, 3.7e-12 and 4.3e-12.
• That second comb is the reason this study exists: every crate number on it is produced by one branch and is met here from outside it.
• control.json carries fourteen rows at dps 40, peel 14, shift 60, cut 30, against the census engine at peel 8, shift 12, cut 24: two zeta_W values, ten Z_W values including five located zeros and the census contour points -0.95 + 20i and 30i, and two residues, one on each comb. The largest gap is 1.134e-11, on the row Z at -0.95 + 20i against its bound 7.348e-11, and no row falls outside its printed bound.
• Five of those rows evaluate Z_W at a located zero printed to twelve decimals and read abs(Z_W) below 9.81e-12 in the other lane. Both lanes run the same peel, the same Faddeev adjugate and the same l-cut, so this is a precision control and not a second method; the independent lane is the crate.
• The new rule's control is bridge: 3 is the only integer of 1 .. 262143 on which the width-3 rule 55 and the width-2 rule 7 disagree, and Z_55 - Z_7 - det(I - 2^(-s) T) 3^(-s) reads at most 1.168e-13 over seven points including three teeth and one located zero, every point inside its bound.
• The occupancy table at width 3 reads 9 two-line classes of 88, 50 teeth, 34 occupied at radius 0.45 holding 35 zeros, every zero located on every rule, largest residual 3.236e-11, largest phase step 0.999909 radians, largest bound 9.110e-09, and no pole within 0.02 of a contour. Two zeros are, -1.134547677+3.580553251i on code 127 and -1.143621954+17.814806003i on code 63; the box at -1.2 reads the same occupancy with the failure on the poles instead.
• The first-order law at width 3 reads largest modulus miss 0.739013203 and largest vector miss 1.287895060, both at code 63, line 1, Im s = 13.597080, abs(u_1) = 0.520202113 against a nearest zero at 1.259215316.
• The dial's baseline is the census read back through a different contour: the circle probe reads 4 of 4 and 0 of 5 on code 7 and 3 of 4 and 7 of 10 on code 23, the same occupancy the cell census prints, with no seam on either baseline and the deepest circle reaching Re s = -1.164241913631 on code 7 against a nearest excluded pole line at -1.305758086369.
• The falsification that measures something: the off-tooth identity misses by at most 2.384e-15 at code 7 and 4.003e-16 at code 23 against an added part of up to 1.912203 and 1.708983 in modulus, over three probes. The residue and tooth probes beside it read 1.776e-15 and 1.332e-15 over the 4 level-one poles of each box and 1.250e-13 and 1.789e-14 over the teeth, and both are blind: a 48-point circle mean annihilates an entire addition, and the determinant vanishes at a tooth, so those two numbers are an aliasing floor and a determinant residual and would print at that magnitude however the addition were implemented.
• The dial meets the one cross-width control that exists: S_7 + {3} is S_55, and the dial's grid row for the added element 3 reads the second comb occupied at Im s = 4.532360 and empty at 13.597080, which is the 1 of 2 the four-state ladder prints for code 55.
• The same control measures the Laurent shift on a four-state ladder against a two-state one: at Im s = 9.064720 code 55 reads r = 0.210170579-0.581938843i, digit for digit code 7's, and R = 1.313430833+1.119663028i against 1.714940435+0.882338583i, a difference of -0.401509602+0.237324445i, which meets 3^(-t) = -0.401509601+0.237324445i up to one unit in the last place of the two nine-decimal prints it is read from.
• Every empty tooth of code 7 is occupied by one added integer, {3}, {6}, {7}, {11}, {11} at Im s = 4.532360, 13.597080, 22.661801, 31.726521, 40.791241, and on code 23 the empty teeth are occupied by {7}, {5}, {7}, {15} while two occupied teeth are emptied, Im s = 2.678332 by {6} and 42.645269 by {6} and by {7}.
• Those least singletons are read on the inner convention and two of each list move on the outer one: code 7 reads {3}, {6}, {3}, {11}, {6} and code 23 reads {6} at 9.064720 and {11} at 38.937214. What does not move is the bound, 11 at code 7 on both readings, the four fillings and the two emptyings, both emptied teeth printing an empty seam list, and so the count 6 of 14.
• Occupancy under the knob is undetermined on 9 of the 110 cells of code 7's second comb, 10 of the 108 of code 23's abscissa comb and 9 of the 270 of its second line, each a cell whose inner count is 0 and outer count positive.
• The exact minimum of abs(R + P_F) over all 4158861 subsets of size at most 16 of the 22 candidates is 1.095277075, 0.784350607, 1.295268436 and 1.662786204 at code 7's four abscissa-comb teeth, the greedy chain attains every one of them, and the disc at each exact minimiser keeps its zero off the seam, 1/1, 2/2, 1/1, 1/1, the one at Im s = 18.129441 gaining a second.
• That search bounds the first-order proxy and not the occupancy, and at Im s = 9.064720 the two part company: the minimum 1.095277075 falls below the emptying threshold abs(r)/rho = 1.374951382, the law predicts abs(u_1) = 0.564905571 and an empty disc, and the disc reads 1/1 with no seam.
• The scoreboard: code 7 line 1 holds 110 cells, 77 occupied, the first-order law right 75 times against the constant predictor's 77; code 23 line 1 holds 270 cells, 222 occupied, the law right 218 against 222; code 23 line 0 holds 108 cells, 84 occupied, the law right 105 against 84; code 7 line 0 is 88 of 88 on both.
• On the outer reading of the seam the subdominant deficits widen and the abscissa-comb margin does not survive: code 7 line 1 reads 80 against 86, code 23 line 1 219 against 231, and code 23 line 0 95 against 94, three of whose four teeth are 27 of 27 occupied and score alike, so the whole margin sits on the one tooth carrying 10 undetermined cells.
• The four level-one poles of code 7 read residues -1.990368154340-0.795661945868i, -0.350975872907-0.436714265460i, -3.135030562964-2.032376530959i and -1.028888122837+2.036528502776i, the two radii agreeing to 7.3e-14 or better, each simple to 5.1e-05 against (s - s_0) Z_W at 1e-5, with a blank point on the same line at 4.6e-16.