Reasoning
- In the weakly coupled heterotic string, the compactification volume V and coupling lambda drop out of the relation between the Planck mass and the string scale, forcing M_H = (alpha_G/8)^(1/2) M_p ~ 10^18 GeV - a TeV heterotic string scale is impossible.
- In the strongly coupled E8xE8 heterotic theory (11D M-theory on a line segment), identifying the M-theory scale M_11 with the electroweak scale forces the segment length R_11 = alpha_G^2 M_p^2 / (8 M_11^3) ~ 10^8 km - absurdly excluded.
- In type I theory the 4D gravitational and gauge kinetic terms give V^-1 = (alpha_G^2/2) M_p^-2 M_I^8 and lambda = 4 alpha_G after T-duality; choosing n of the six internal dimensions large with common radius R yields R^-1 = [(2/alpha_G^2)(M_I/M_p)^2]^(1/n) M_I, i.e. for M_I ~ 1 TeV and alpha_G ~ 0.1 - R^-1 ~ 10^-18 eV for n=1 (experimentally excluded), ~10^-3 eV (R ~ 100 microns) for n=2, up to ~10 MeV (R ~ 0.1 fermi) for n=6.
- Equivalently M_I ~ alpha_G^(2/(n+2)) M_p(4+n): the type I string scale sits at (slightly below) the higher-dimensional Planck mass. Matter localisation on the 3-brane is automatic (open strings end on D3-branes), removing the need for a field-theoretic trapping mechanism.
The calculation establishes internal consistency and uniqueness-of-route, not truth: it bears on the plausibility of the TeV-gravity hypothesis by showing the only perturbative string realization and fixing its parameter ranges.
Validity verdict (step 6)
Reconstruction: premises = the stated string kinetic-term / duality relations (heterotic weak, heterotic strong, type I); conclusion = a TeV string scale has exactly one perturbative realization, with the stated n and R ranges. Conditional on the relations, the rest is algebra: recomputed R^-1 = [(2/alpha_G^2)(M_I/M_p)^2]^(1/n) M_I at M_I = 1 TeV, alpha_G ~ 0.1 - n=1: ~1.4e-18 eV, n=2: ~1.2e-3 eV (R ~ 100 um), n=6: ~11 MeV (R ~ 0.1 fm) - all match. “Only in type I” rests on the charitable hidden premise that the three surveyed corners exhaust the 1998 perturbative options with gauge+gravity; conclusion is scoped to “concrete, calculable embedding”, not truth. Checked (algebra traced; the input relations are premises).