One-shot binding of places to a fixed grid template: dead reckoning and content recall under motion noise

Preprint, version 0.22b (29 September 2026). Grid cells exist before an animal explores. We ask how little has to be learned for a fixed hexagonal phase template, advanced by self-motion, to localise sensory content that is arbitrary in what it is, provided it is unique to its site and stable across visits. The only learned quantity is an association between a site's sensory pattern and the template's phase, written in one shot at the first visit and thereafter rewritten at a small gain, with the site's identity supplied by the simulator to tell a first visit and to address the rewrite, a shortcut we declare; at every step the phase is nudged towards what is stored. Without binding the template drifts across the room; with it, the decoding error falls within the first tenth of the walk to about one arena cell and stays there under velocity noise of up to ten percent per step, and it remains bounded when the noise is tripled. Shuffling the velocity or the associations destroys the alignment. Since the sensory input alone identifies sites, what the bound template adds is measured directly: while the input is off the template path-integrates like any integrator, and when the input returns the bound template re-locks on the stored position within about ten steps, where an unbound template does not. None of this depends on the spatial smoothness of the input: with content that is white noise per site the binding anchors and re-locks the same way. What is stored is a memory of the room and not a re-fit to it: when the room is stretched after the map is written, decoded positions keep the old room's frame, compressed by the stretch, while a map bound from scratch in the stretched room follows the new room exactly. Kept anchors are fields that stretch with the room: in the metric of the deformation experiments the model's field spacing follows 93% of the room change without the rewrite and about two thirds with it, where rats follow about half (a Hebbian write at every step, which does not use the site identity, anchors as well with smooth content but, at the correction gain used throughout, keeps the old frame whole, so in the model as run the partial rescaling rests on the shortcut), and two rooms bound separately merge by a local reorganisation at the old wall that is an order of magnitude smaller than the animal's. A place stage of stored exemplars, never trained, aligns the template as well as a trained one, and the same binding on a continuous-attractor network reproduces the anchoring and the stretch. The regime is the easy one by design, unique content and a sensor that identifies every site on its own, and the result is reported as the measured floor of the design, with its controls and with what it costs. Data and code: per-seed result files, figures, the model code and the scripts that reproduce every figure are in the evidence pack, doi:10.5281/zenodo.22267940 (version 3; all versions: doi:10.5281/zenodo.22239562).Use of AI tools: Generative AI (Claude and Claude Code, Anthropic) assisted the author in writing the text, in implementing and running the simulation code, and in exploring and critically evaluating hypotheses and experiments. All scientific content, decisions and claims are the author's, who takes full responsibility for them; every number reported is an output of the deposited simulation code and is anchored to a versioned result file.