Tsumura calculates a Galactic listening threshold of 1.32 and a 5.5-contact cap

A preprint posted to arXiv by Kohji Tsumura calculates how often Galactic civilizations could actually be in contact, treating appearance times, light-travel delay, and finite lifetimes as part of the network rather than as background. The paper separates four kinds of link: listening, audibility, mutual listening, and relay, plus a stricter reply-closing condition in which a return signal must arrive within the sender's lifetime. Contactability is defined as eta equals pi times the surface density of active civilizations times the square of the maximum detection range.

This is a mathematical model, not a sky survey: Tsumura derives expected contact counts for a uniform population, estimates percolation thresholds, then applies the formulas to a structured Galactic disc. A dimensionless ratio, chi, compares detection range with the light horizon set by a civilization's lifetime. Averaged over appearance times, the lifetime listening count is 2 eta for every chi and, at fixed range, for any lifetime distribution. Audibility and mutual-listening counts together sum to 4 eta. As chi approaches zero, the listening connectivity threshold falls from the coeval value of 4.51 to 1.32 plus or minus 0.01.

The Fermi question is usually framed as distance. The sharper issue is causal connectivity: two societies can share a galaxy and still fail to close a reply inside either lifetime. An ideal complete survey plus our non-detection gives a reference bound, eta at most 2.75 at the Sun's Galactocentric radius. That bound sits above the listening, audibility, and relay thresholds studied here, so a sparse disc can still percolate. Against a coeval snapshot, spacetime contact moves the local-contact frontier out by 3.2 kiloparsecs, while a typical civilization hears at most 5.5 others.

The mean reply-closing count drops below one once chi is about 1 or larger, so one-way contact can be common while completed conversations stay rare. That is a prediction about network topology, not a detection. The work is an unrefereed arXiv preprint: analytic counts plus threshold estimates, with no new telescope data and no measured civilization density. The 2.75 bound assumes an ideal survey; real searches are incomplete, patchy, and frequency-limited. Independent recalculation of the 1.32 threshold, and a version that folds in actual survey completeness, would be the test that strengthens or breaks the result.