small language model

Firing Shrinkage Has Two Values Depending on the Denominator, and Volume Shrinkage Is Not Three Times the Linear ── linear shrinkage is 0.150868 on the green size and 0.177674 on the fired size, and a 100 mm fired part needs a 117.767361 mm mould ── the separator is whether shrinkage is measured against the green size or the fired size ── [Paper 754]

Ceramics shrink on firing, and the shrinkage is often quoted as a single "shrinkage" figure. What that figure is divided by, how volume and length differ, and how density scatter appears in size are counted. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──that sinte...

Radar Range Resolution Comes from Bandwidth, Velocity Resolution from Time, and a One-Wrap Velocity Error Shifts Range One Cell ── with 1 GHz of bandwidth the range resolution is c/2B=0.149896 m, and 30 m/s wrapping to -8.935373 m/s shifts the corrected range by 1.000033 range resolutions ── the separator is whether the velocity has been mistaken by one wrap ── [Paper 802]

A continuous-wave radar that sweeps its frequency measures a target's range and velocity with one device. What sets the range resolution, the velocity resolution and the velocity limit is laid out, and it is counted how range and velocity mix within one chirp and what happens to the range reading when the velocity exce...

The Earliest Sunset Comes Half a Month Before the Winter Solstice: the Gap Between Sundial and Clock Is the Sum of Two Causes ── the equation of time is the sum of an eccentricity part (amplitude 7.6550 minutes) and an obliquity part (amplitude 9.8664 minutes) ── changing by 0.4975 minutes a day around the winter solstice, it brings the earliest sunset at latitude 35.7 degrees north 15.61 days before the solstice ── [Paper 596]

The earliest sunset of the year is easily taken to fall on the winter solstice, the shortest day. What this paper shows is that the gap between sundial and clock (the equation of time) is the sum of two causes, the eccentricity of the orbit and the tilt of the ecliptic, and that around the solstice it changes by about...

Condensation Happens Without Any Attraction: What Decides Is the Wavelength Overtaking the Spacing ── the condensation temperature of an ideal Bose gas contains only constants, the mass and the number density, with no interparticle potential at all ── the separator is whether switching off the interaction makes it disappear ── [Paper 663]

It is natural to think that a phase changes because particles attract one another. But the condensation of bosons happens without any attraction at all. The condensation temperature of an ideal Bose gas is computed in three cases, and what fixes that temperature is examined. No new theorem or law is claimed. Scope of t...

Errors Not Yet Found Can Be Counted from Two Readers' Overlap, but Come Out Too Few If Both Miss the Same Ones ── If two readers find 20 and 25 errors with 10 in common, the total is estimated at 50.000000 and the errors not yet found at 15.000000 ── if the readers' blind spots are opposite, 27 are reported as 155.571429 ── [Paper 585]

One way to estimate how many errors remain in a manuscript is to have two people read it separately and use the overlap of what they found. What this paper shows is that the estimate is right only when the two readers' misses are unrelated; if they miss the same errors it comes out too few, if they miss opposite ones t...

A convergence model of bioelectric, gap-junctional, and Hippo-YAP signalling in oral cancer stem cell maintenance knowledge graph: evidence-tiered snapshot and pipeline (2026-09-28)

Evidence-tiered knowledge graph of the proposed convergence between bioelectric signalling, gap-junctional communication and Hippo-YAP signalling in oral cancer stem cells, together with the pipeline that builds it and the complete validation record. WHAT THIS IS A narrative review proposes a mechanistic model. A model...

The Number of Ways to Place Non-Attacking Queens Does Not Always Grow with the Board ── Place n queens on an nx n board so that none attacks another along a row, column or diagonal ── 12x12 has 14200 ── [Paper 624]

A chess queen moves any distance along rows, columns and diagonals. The ways to place n queens on an nx n board with no two attacking each other are counted (the n-queens problem). No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the 92 placements of eight queens and...

A Node-by-Node White-Box Walkthrough of a Quantized Hybrid Gated DeltaNet Language Model: Instrument Calibration and the Information Trajectory from Embedding to Output

Widening Helps Either Fully or Not at All, With Nothing In Between ── The five stage capacities sum to 650 per hour, but the line delivers the smallest of them, 80, which is 0.1231 of the sum ── averaged from 80 to 250 it is only 0.1176 ── [Paper 550]

The capacity of a line follows from the capacities of its stages. What this paper shows is that it follows as a minimum rather than a sum or a mean, and that widening the minimum helps in a step from 1 to 0. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──that a ser...

Most of the Water Taken Out of Wood Changes Neither Its Size Nor Its Strength ── of the water removed from 80 % to 12 %, a share of 0.735294 changes neither size nor strength, and the remaining 0.264706 brings the shrinkage and the gain in strength ── the separator is whether the water leaving is free water in the cell cavities or bound water held in the cell walls ── [Paper 748]

Drying wood makes it shrink and grow stronger. Yet it does not shrink in proportion to the water removed. Which water changes what is counted through shrinkage and strength at each moisture content. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the fibre saturatio...

In Anisotropic Etching, the Depth of the Pit Is Set by Crystal Planes, Not by Etching Time ── the depth is 71.143691 μm at 100 minutes and 71.576704 μm at 200, so doubling the time gives 1.006086 times ── the separator is whether the rate ratio between the two planes is large ── [Paper 758]

When a crystal is dissolved to dig a pit, one readily supposes that longer immersion digs deeper. But when the dissolution rate differs between crystal planes, the planes decide the shape. Whether time or crystal planes set the depth is counted, divided by the ratio of plane rates. No new theorem or law is claimed. Sco...

Most of the Smoke Is Not What Burned but Air the Plume Pulled In, So What Sets the Volume of Smoke Is the Ceiling Height ── A 1000 kW fire burns only 0.076923 kg/s of fuel, yet the plume at a height of 5 m carries 9.216677 kg/s ── the separator is whether the volume of smoke is set by the fuel or by the ceiling height ── [Paper 728]

The smoke in a fire is taken to be what burned. Counted by mass, most of it never burned at all. What burned and what became smoke are set side by side in the same units. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──plume entrainment, smoke layer descent and the...