LUMOs are delocalized over their entire length, and therefore their fundamental gap corresponds to ΔΕac. Such edge-modified AGNRs are obtained by eliminating the empty (non-aromatic) endrings of the standard AGNRs, which also eliminates end-states and zigzag end-bonds. This is a clear manifestation of the importance of aromaticity for AGNRs (and graphene itself). Comparing the behavior of the "bulk gap" in Figs. ) and 2(b), we can see that due to quantum confinement (both lateral and longitudinal) the (HOMO-1) and (LUMO+1) states defining the "onebody" ΔΕac are also affected by the abrupt appearance of the edge states, in sharp contrast to the ("one-body") open singlet gap which seems to be practically insensitive to the appearance of the end-states, contrary to what is expected from its original definition. This in fact emphasizes the "many-body" nature of the end states through their connection with inversion symmetry conflict, which is further supported from Figs. ) and S1. The "correct" behavior (with length variation) of the "one-body" ΔΕac is given by the (delocalized) HOMO-LUMO gap of the edge-modified AGNRs in Fig. ). As we can see in Fig. ) the value of ΔΕac (HOMO-LUMO gap of the edgemodified AGNRs) as a function of length, as L →∞, seems to saturate to the value of 1.22 eV. This could be misleading since only lengths up to about 140 Å have been considered. To remedy this problem we have recently suggested to fit the calculated ΔΕac as a function of L efficiently and transparently to a polynomial of the form ΔΕac(L)=A+B×L -C , where the value A corresponds to the gap at infinity, ΔΕac(∞)=A, and the constant C to some short of effective ("fractal") dimensionality (here equal to 1.20). As we can see in the inset in Fig. ), the projected ΔΕac value is 1.07 eV, which is also verified by the TDDFT result ΔΕac =1.01 eV (see Fig. ). The TDDFT value (1.01 eV) is clearly closer to the value of 0.85 eV measured by Lawrence et al. , assuming a very reasonable substrate screening (of about 0.15 eV), as we have suggested recently. However, further correct information is given in Fig. , showing the spectra of the 2x22 and 2x23 AGNRs immediately before and after transition, respectively (see Fig. ). As is illustrated in Fig. ), in the 2x23 AGNR (immediately after the transition) there is a strong peak value at 0.87 eV, very close to the recently measured STS gap of 0.85 eV. Detailed analysis of the TDDFT results shows that this peak includes transitions involving end-states to a large percentage (about 60%). Thus, the calculated value of 0.87 eV and the measured 8 gap should be assigned to ΔΕzz. This, contrary to the "one-body" gap, restores the expected correct behavior of ΔΕzz at (and after) Lc. Even more interesting is the fact that extrapolating to longer AGNRs gives a gap of 0.85 eV (exactly), which is an unexpected full agreement with experiment, as is shown in Fig. . Fig. ) also shows that, contrary to the "one-body" (open-singlet) ΔΕzz gap of Fig. , both "manybody" gaps, ΔΕzz, and Δεζζ (the latter corresponding to the "real energetical separation of the endstates), and the one-body HOMO-LUMO gap, which involve end-states, change discontinuously at the critical length (~100 Å), where Δεζζ and HOMO-LUMO gaps drop, while ΔΕzz increases. Thus, the observed 8 gap opening (of about 0.30 eV) is due to the increased aromaticity at the critical length, and the mixing of bulk and end-states at an almost equal amount. Lawrence et al. have attributed such gap opening to the different electrostatic potential felt by valence electrons at different regions of the ribbon due to the positive partial charge on the hydrogen atoms along the sides of the AGNR. However, the paradigm of edge-modified AGNRs contradict such interpretation. Our present work reveals that the gap opening is a many-body effect related with the aromatic transition and the change from bulk-like (ΔΕac) to coupled "surface-bulk" end-states (ΔΕzz). On the other hand, the calculated Δεζζ gap of 0.1 eV in Fig. ) is in full agreement with the results of Kimouche et al. Thus, Kimouche et al. , and Lawrence et al. , have apparently ("correctly") measured different kinds of gaps. Moreover, the same could be true for the value of 2.8 eV measured by Zhang et al. , which could be assigned as a tentative ΔΕac value, either for very short AGNRs (without end-states), or for longer AGNRs with a strong "bulk" transition from deep occupied states (well below HOMO-1 orbital) higher unoccupied states (well above the LUMO+1), and thus much larger than the real ΔΕac (which is technically determined by the HOMO-1, LUMO+1 difference). We can also observe in the 5-AGNRs that differences between the "onebody" and "many-body" (TDDFT) methods for assigning ΔΕac, ΔΕzz, and Δεζζ are relatively large (or even unusual) compared to the 7-and 9-AGNRS, discussed below, where the corresponding differences are of the order of 0.1-0.2 eV. This could be related to the fact that the 5-AGNRs (contrary to 7-and 9-AGNRs) are topological and aromatic mixtures. Thus, the three seemingly conflicting measurements for the 5-AGNRs could be attributed to different length samples (and/or different positions of the STS tip). Yet, alternatively, one could claim, based on the GW results , that there is a substrate interaction of equal magnitude (0.85 eV) and the "real gap" is 1.7