memory-meter

• The Mobius meter of a memory rule: M_W(x) = sum of mu(n) over the integers n <= x whose binary digit word the rule accepts, against that set's own mass A_W(x).
• Every width-1, width-2 and width-3 rule at dim 1, base 2 is read: 4, 16 and 256 codes, 276 rules in all, the width-1 codes being the memoryless row.
• An integer n >= 1 is its minimal base-2 word, coarsest digit first, no leading zero; 0 is excluded from every sum. The rule reading is mrlynum::memory::Rule: a window of k digits is w = sum_j c_j 2^(k - j), bit w of the code is set when that window is allowed, a word is accepted when every one of its k-windows is allowed, and a word shorter than k is accepted.
• No exponent is fitted. Printed per rule and phase are A_W(x), M_W(x), max abs M_W(t) over t <= x, the ratios M_W/sqrt(A_W) and max abs M_W/sqrt(A_W), with kappa, rho and card W from the crate.

THE METHOD

• One ascending pass over n <= 2^30 carrying the window profile of n, the bitmask of the 2^3 windows its word contains, by profile(n) = profile(n >> 1) | bit(n mod 8) for n >= 4; n is accepted by W exactly when profile(n) is a subset of W.
• The width-2 and width-1 profiles are induced from the width-3 one: for a word of three digits or more every 2-window is a prefix or a suffix of some 3-window and every digit lies in one, so one u8 per n carries all three widths. The words of one and two digits are entered by hand.
• The mass per profile is a 256-bucket count and A_W is its subset-sum transform at each phase; the meter is carried per rule so the running maximum is exact, and the subset-sum transform of the per-profile mu sums is asserted equal to it at every phase.
• Phases are x = floor(2^(level + j/4)) for level 8..30 and j = 0..3, 89 of them, 2^30 last.
• mu is mrlynum::factor::mobius_sieve, the crate's linear Mobius sieve; mrlynum::sieve carries the Sierpinski word and no Mobius, so nothing there is reused.
• rho and kappa come from mrlynum::memory::perron and kappa, which split the digraph into strongly connected components and make each component's Perron root exact against its integer characteristic polynomial, so every printed root is an algebraic integer of the right minimal polynomial.

THE CONTROLS

• The full line, k = 1 code 3, is the Mertens function: its meter reads -1, 1, 2, -23, -48, 212, 1037, 1928 at 10^1..10^8, asserted, which is A084237.
• The memoryless base-3 designs are enumerated directly from the same sieve and asserted against mobius-designs: digits {0,1} read (M, max abs M) = (11, 105) at level 14, (149, 173) at level 16 and (-30, 312) at level 18, digits {1,2} read (-1461, 1582) at level 18. Digits {0,2} at level 20 wants 3^20, past 2^30, so it is printed at level 14, 16, 18 and not pinned. The four pinned pairs are read at source in design-meter, which computes them and cites mobius-designs as their census.
• The profile recurrence is asserted against a direct digit recount on all 276 rules below 2^20, and profile containment against Rule::accepts on all 276 rules below 2^12.
• Code 7 at k = 2, the golden rule forbidding 11, opens 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, which is A003714 without its zero, and its mass is the Fibonacci number. Code 11, forbidding 10, opens exactly the Mersenne numbers A000225 without its zero, one per level.
• Leading zeros: a rule is zero-closed when prepending one zero to the word changes no membership below 2^20. Both masses are printed per rule and the criterion is asserted code for code.

RUN

• CARGO_BUILD_JOBS=4 cargo run --release -p memory-meter
• About thirty seconds on one thread, 23s of it the sieve; peak resident set about 2.15 GB, one i8 and one u8 per integer. Prints only, writes nothing.
• An optional depth argument runs a shallower table at the same phases: -p memory-meter -- 28 takes about seven seconds and 0.6 GB, and its rows are the first 81 phases of the deep run verbatim.

READS

• Every ratio below is a reading at a named phase, never a fit and never a sweep-wide claim unless it says so. rho and kappa are read off the transfer matrix of the word language while A and M are read off the integer set; on a rule that is not zero-closed those are different objects, and the columns sit side by side for that reason.
• The census is the 53 rules of all three widths holding at least 10^4 integers below 2^30; the floor is part of the census and is fixed before any reading. The same rules under a floor of 64 number 106.
• The full line at phase 30.00: A = 1073741824, M = -10374, max abs M = 11173, M/sqrt(A) = -0.316589, max abs M/sqrt(A) = 0.340973. Three codes carry it, k = 1 code 3, k = 2 code 15 and k = 3 code 255.
• The golden rule, k = 2 code 7, kappa = 0.098239, rho = 1.618033989: A = 2178309, M = 551, max abs M = 716, ratios 0.373329 and 0.485125 at phase 30.00. Code 14, forbidding 00, reads A = 3524576, M = -466, ratios -0.248218 and 0.454355.
• The three named k = 3 least codes at phase 30.00: 23 supergolden, kappa = 0.115204, A = 125492, ratios -0.448837 and 0.731125; 54 plastic, kappa = 0.260981, A = 13579, ratios -0.360425 and 0.677943; 127 tribonacci, kappa = 0.056639, A = 98950096, ratios -0.432878 and 1.239625.
• The normalised peak over the census at phase 30.00 spans [0.293624, 1.239625], least on k = 3 code 125 and largest on k = 3 code 127. Over all 89 phases the same quantity spans [0.500000, 2.169240], the top on k = 3 code 232 at phase 16.75; a last-phase span is not a sweep-wide span.
• Read against the full line at the same phase, the factor over the census is [0.861136, 3.635552] at phase 30.00 and reaches 6.375774 on k = 3 code 190 at phase 12.75 over all phases.
• Grouped by kappa over the census the means read 0.340973 on [0, 10^-9) with 3 rules, then 0.622171 on 6, 0.581446 on 14, 0.644840 on 12 and 0.645966 on 18 for the bands [10^-9, 0.1), [0.1, 0.2), [0.2, 0.3) and [0.3, 0.5). Every band of positive coupling sits above the kappa = 0 band, and among them the means are not monotone in kappa, so the coupling does not order the spread. Restricted to k = 3 the same bands read 0.340973, 0.698386, 0.581446, 0.644840, 0.645966 on 1, 4, 14, 12, 18.
• The full line's own normalised peak runs [0.272410, 0.500000] over the grid, the ceiling at phase 8.00, which is the grid's first point. That ceiling is a property of where the grid starts and not of the full line: below the grid the same ratio reads 1.000000 at x = 1, 0.894427 at 5, 0.832050 at 13, 0.718421 at 31 and 0.565685 at 200. Every band statement is therefore read beside the same-phase factor, which needs no grid.
• The falsification fires. Five of the 53 census rules never enter that band at any phase where they hold 10^4 elements, all of them above it: k = 3 codes 159, 182, 190, 218 and 250, holding 211116, 13607, 31535, 59860 and 4126645 integers. Under the floor of 64 the count is 32, the other 27 all holding under 500 elements.
• 16 of the 53 census rules attain their sweep-wide normalised peak in the last quarter of the phases, from 24.75 on. No rule at any width with at least 1000 elements has either ratio rise at every one of the last eight phases; that test asks max abs M to grow about 9% per quarter-level across two levels, so an empty answer carries little, and the late-peak count is the informative statistic.
• Zero-closed under one prepended zero: 3 of 4, 8 of 16 and 64 of 256, exactly the codes allowing the digit 0 together with the empty code, the codes allowing 01, and the codes allowing both 010 and 011. Under any number of prepended zeros the word language agrees with the integer set on 2 of 4, 4 of 16 and 16 of 256 codes, so at k = 3 the two readings part company on 240 of 256.
• Equal mass is not the same set. k = 2 code 14 and k = 3 code 126 both hold 28655 integers below 2^20 and both carry rho = 1.618033989, but they share only 1077: the symmetric difference is 55156, and 4 is the least integer in code 126 and not in code 14.

WITNESSES

• beneath, The memory meter - the meter table, the control band, the kappa bands and the zero-closed criterion.
• research/claims/ the memory-meter rows.

COLUMNS

• control the pinned lines; classes and reps the 88 orbit representatives at (1,3) under G_(1,3); rule one line per code with card W, rho, kappa, the zero-closed flag and both masses, the last-phase reading and a track of the normalised peak at level 8, 12, 16, 20, 24, 28, 30; row one line per code and phase; top, bottom, span, factor, latepeak, kappaband, band and rho the printed bands at two mass floors, gridstart the full line below the grid, pair the two equal-mass rules, reading the word language against the integer set; climbing the monotone scan; run the depth and the runtimes.