Binary-weighted measurement systems, such as successive-approximation converters built from discrete resistors, sets of reference weights or banks of binary thresholds, are usually specified with one tolerance class for all elements. Because an element's contribution to the conversion error scales with its binary weight, this over-buys precision for the less significant elements. This paper treats per-element tolerance as a cost-constrained allocation problem. For a power-law cost, in which price scales as t^−α with tolerance t, the error-variance-optimal allocation is t_i ∝ W_i^−2/(α+2), and the cost ratio to a uniform design of equal error variance, obtained in closed form, falls approximately as 1/N with the number of bits N. Monte Carlo simulation of a 10-bit converter confirms the closed form (cost ratios of 0.379 for α = 1 and 0.299 for α = 2 at equal RMSE). Because linearity yield is not a variance criterion, tolerance classes are also selected under a yield constraint and validated on 3000 independent units: at ≥ 99 % yield (|INL|, |DNL| ≤ 0.5 LSB) the allocated designs cost 48–74 % less than the cheapest uniform design for α = 0.5 to 2, and 64–67 % less with the catalogue prices of 120 commercial resistors (fitted α ≈ 1.2). SPICE Monte Carlo of a SKY130 capacitor array shows that integrated unit-cell arrays already realise the α = 2 optimum under area-only mismatch; a layout-level study shows that coarser cells for the significant bits help at equal area only when edge mismatch is significant and gradients are moderate.

Let s(n) be the side of the smallest square containing n non-overlapping unit squares. For sides just below an integer K, recent preprints show that s(n) < K for all n ≤ K² − O(√K) and beyond, using an L-shaped packing: a grid with two tilted bands along adjacent walls. Using this construction, found here independently, we prove that for every integer K ≥ 16 the wasted area is at most 8√K + 19 at every side in [K − 1/K, K). Thicker bands also beat the grid, with waste of order K^2/3. Optimized instances with rational angles and exact certificates, at sides in [K − 1/2, K), improve the best-known upper bound for 5308 values of n between 1182 and 9944. They beat the grid down to a fractional side that decreases from 0.62 at K = 35 to below 1/2 for K ≥ 78. A structural study of the best-known packings with n ≤ 1000 gives conjectures on tilted squares and tilt regimes, tested on 5077 packings with 1000 < n ≤ 10000.

Conventional reservoir-level measurement systems convert hydrostatic pressure into an electrical signal, which makes the entire sensing chain dependent on electronic components that are vulnerable to lightning-induced transients, power interruption, and long-term environmental degradation. This paper proposes a hydromechanical measurement architecture in which hydrostatic pressure is converted directly into an n-bit digital code through a purely mechanical successive-approximation procedure, with no electrical component anywhere in the sensing or comparison chain. A lever-balance comparator, pivoting on a knife-edge, is weighed against a reference column fed by a bank of individually calibrated, laminar-flow binary-pressure orifices; a fixed manual protocol, structurally identical to the classical single-pan binary-weight problem, halves the search interval at each of n steps, resolving the most significant bit first. A Monte Carlo simulation propagating a physically-derived, bit-dependent orifice calibration-retention error, differential fluid-density mismatch, and comparator friction through the full binary-search procedure identifies a practical operating point of n = 10 bits, giving an effective resolution of approximately 1.4 mm (7.7 effective bits) over a 1.024 m bench-scale range. The architecture extends our prior demonstration that the successive-approximation principle is independent of the specific physical quantity involved, showing here that it is also independent of the electrical domain altogether.

Let s(n) be the side of the smallest square that contains n non-overlapping unit squares. For large n a direct search over the 3n positions and angles is expensive, and the best-known packings for n < 1000 come from heavy GPU computations. We ask whether the structure of best-known packings is regular enough to guide a much cheaper search. In 531 certified best-known packings with n < 1000 we observe that 49% of tilted squares are at exactly 45^◦, that the dominant tilt is organised by the fractional part of the side, that neighbouring tilted squares share the same angle within 0.5^◦ in 90.6% of pairs, and that the waste per unit length of tilted boundary grows from about 0.04 at small tilts to 0.39 near 45^◦. A depth-3 decision tree that sees only n and is trained on odd n (or odd ⌈√n⌉) puts the structure class of the best-known packing of even n among its first two choices in 96–99% of cases, and it recovers the side-based regime rule from n alone. Used as a search prior above n = 1000, a structure family with a few parameters per n gives certified upper bounds for 5601 values of n in 1182 ≤ n ≤ 9946 that are below the entries of two public indices, by 0.084 on average and up to 0.207; for K = 100 the number of squares that must be removed from the K × K grid before the side drops below K falls from 87 to 54. For n < 1000 the same family does not beat the GPU records. All statements are labelled as proven, empirical or perspective; no optimality is claimed.

This paper presents a multi-sensor Binary Intensity-Search Measurement System (BRMS-M) that extends the binary-search, ADC-less sensing principle to N independent two-part sensors—laser, optical, and ultrasonic—each comprising a transmitter (Tx_i) and a receiver (Rx_i). Rather than replicating a measurement path per sensor, all N sensor pairs share a single binary-weighted resistor array that sets transmitter drive intensity, and a single central controller that performs the binary search. Each sensor pair is addressed by a shared selection signal (Sel_i), simultaneously activating its transmitter and enabling its receiver while isolating all other pairs. Measurement proceeds through an intensity-domain binary search: rather than comparing a resistance value against a threshold, the controller sweeps drive intensity from MSB to LSB and reads the receiver’s binary detection output directly, exploiting the sensor’s own detection threshold in place of a dedicated analog comparator. This confines the design to sensors with a monotonic intensity-to-detection response, qualified on the ground prior to flight. Each converged code is telemetered to the ground receiver once per hourly cycle, using a two-peak, pulse-interval encoding inspired by aircraft IFF (Identify Friend or Foe) systems, rather than a per-bit event-driven scheme. This architecture reduces N per-channel measurement paths to a single shared array and controller, positioning it as a scalable, low-power multi-sensor telemetry method suitable for distributed sensing on tethered platforms such as research balloons.