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From the SDC we have (\Delta\phi\lesssim \frac1\lambdaM_!P). For a monomial potential (V\propto\phi^p), the field excursion during inflation is
[ \Delta\phi \simeq \sqrt2p,N,M_!P, ]
implying that models with (p\gtrsim 1) and (N\sim 50!-!60) typically violate the SDC unless (\lambda) is substantially below unity.

The refined condition imposes a limit on the second slow‑roll parameter:
[ \eta_V \equiv M_!P^2\fracV''V;\leq;-c'. ]
A tachyonic direction (negative curvature) is therefore allowed, but its magnitude must be order‑one in Planck units.

Combining the TCC with the Friedmann equation yields an upper limit on the inflationary Hubble parameter:
[ H_\rm inf;\lesssim; 10^-20,M_!P;\approx; 10^4,\rm GeV, ]
corresponding to an energy scale well below that of typical Grand Unified Theories. This dramatically lowers the expected amplitude of primordial tensor modes. ampland com

Statement: No sub‑Planckian quantum fluctuation should ever become larger than the Hubble horizon. In an expanding universe with Hubble parameter (H), this leads to the bound
[ a(t_\rm end),\ell_\rm P;<; H^-1(t_\rm end)\quad\Rightarrow\quad N_\rm tot;<;\ln!\Bigl(\fracM_!PH_\rm i\Bigr), ]
where (N_\rm tot) is the total number of e‑folds and (H_\rm i) the Hubble scale at the beginning of inflation.

Original form: For any scalar potential arising from a quantum‑gravity UV completion, the gradient satisfies
[ M_!P,\fracV;\geq;c;\sim;\mathcalO(1). ] From the SDC we have (\Delta\phi\lesssim \frac1\lambdaM_

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Proposal (2023): The computational complexity of a low‑energy EFT grows exponentially with the geodesic distance traversed in moduli space, providing an information‑theoretic complement to the SDC. Original form: For any scalar potential arising from

Statement: In any consistent quantum‑gravity theory, traversing a geodesic distance (\Delta\phi) in field space larger than a critical value (\mathcalO(1),M_!P) triggers an infinite tower of states whose masses scale as
[ m ;\sim; m_0,e^-\lambda,\Delta\phi/M_!P, \qquad \lambda\sim\mathcalO(1). ]

The appearance of a light tower invalidates the EFT description beyond (\Delta\phi\sim M_!P).