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Theories based upon strong real space (r-space) electron electroninteractions have long predicted that unidirectional charge density modulations(CDM) with four unit cell (4$a_0$) periodicity should occur in the hole dopedcuprate Mott insulator (MI). Experimentally, however, increasing the holedensity p is reported to cause the conventionally defined wavevector $Q_A$ ofthe CDM to evolve continuously as if driven primarily by momentum space(k-space) effects. Here we introduce phase resolved electronic structurevisualization for determination of the cuprate CDM wavevector. Remarkably, thisnew technique reveals a virtually doping independent locking of the local CDMwavevector at $|Q_0|=2\pi/4a_0$ throughout the underdoped phase diagram of thecanonical cuprate $Bi_2Sr_2CaCu_2O_8$. These observations have significantfundamental consequences because they are orthogonal to a k-space (Fermisurface) based picture of the cuprate CDM but are consistent with strongcoupling r-space based theories. Our findings imply that it is the latter thatprovide the intrinsic organizational principle for the cuprate CDM state.

Commensurate $4a_0$ period Charge Density Modulations throughout the $Bi_2Sr_2CaCu_2O_{8+x}$ Pseudogap Regime