Spatial-plasticity imprint — local delivery versus relayed consolidation (the first cross-axis coupling, marrying the spatial layer to the plasticity layer)

The first cross-axis coupling: where a focal drive leaves a lasting synaptic trace. Marrying the spatial layer to plasticity, targets split into local-imprint and relayed-imprint, fixed by location across intensity and rate. The new result: clean delivery does not imply clean imprint — plasticity delocalises it. Relayed imprint is the structural default. Structural quantities, never felt effect.

Two layers have been built and read separately. The §43 spatial-localisation layer built a fixed spatial map of the frozen ephaptic kernel and handed it forward to be read region by region — §44 (containment vs broadcast for a seizure), §45 (local vs diaschisis for a stroke), §46 (clean delivery vs off-target leak for neuromodulation) — every one of them a reading of the same map on a single axis, space: the instantaneous footprint of a focal drive. The §26 plasticity-consolidation layer built the orthogonal axis, time: a phase-correlation Hebbian update that lets the connectome retain a structural trace of the coordination it has been driven through (§37 gain, §39 decay, §41 stabilisation). This chapter takes the first cross-axis coupling in the series — it marries the two layers. A focal stimulation delivers to a place (the E1 question), and then the network consolidates what the drive provoked (the E0 question); so the question this chapter asks is the one neither axis could ask alone: where does a focal drive leave a lasting synaptic imprint — and is that place the same as where the instantaneous field landed? Both layers are imported, not re-derived: the per-node excitatory drive is §43’s SpatialField, the synaptic update is §26’s PlasticConnectome, and a new integrator steps the per-node drive on the evolving connectome while accumulating the pairwise phase coincidence the plasticity rule consolidates — no new constant. Every sign is required to hold across two sweeps at once: a stimulation-intensity sweep {0.3, 0.5, 0.7, 0.9} and a consolidation-rate sweep {0.03, 0.05, 0.08}. Four results hold. C1 — the local-imprint/relayed-imprint partition is drive-invariant across both sweeps: driving each of the 12 regions in turn and letting plasticity consolidate, the targets partition into local-imprint (the lasting trace concentrates on edges incident to the driven node) and relayed-imprint (the trace lands harder on off-target edges), and this partition is identical across the entire intensity×rate grid — the local set {brainstem, cerebellum, pallidum, striatum} fixed, the relayed set (eight) fixed. Whether a focal drive imprints locally or relays its trace is a fixed property of the driven site, stable under both how hard and how fast you drive. C2 — relayed imprint is off-target dominance: for a relayed target the mean off-target consolidated change |ΔW| exceeds the incident change (ratio above 1 — the neocortex at roughly 6.3×, the forebrain GABA interneuron ~5.7×, the field-relay hippocampus ~1.4× and midbrain ~1.4×) — the lasting trace bleeds off the driven node — while a local target concentrates the trace on its own edges (ratio below 1 — the brainstem at about 0.63×); the equivalence holds across the whole grid. C3 — delivery and imprint are decoupled (the genuinely new coupling result, and the honest negative): the trace-relay set strictly contains the §46 instantaneous-field-relay set {hippocampus, midbrain} — plasticity delocalises the imprint. The two field-relay sites are also trace-relayed (the subset is genuine), and six further sites — the neocortex, thalamus, hypothalamus, basal-forebrain cholinergic nucleus, forebrain GABA interneuron and olfactory bulb — deliver their instantaneous field cleanly (they self-localise in the instant, §46) yet imprint off-target once the trace consolidates. Clean delivery does not imply clean imprint: where a focal drive lands and where it leaves a lasting mark are two different places. And the refuted clean hypothesis is reported honestly — there is no universal “focal beats diffuse” trace law: only 3 of 12 sites imprint more under a focal drive than under a matched diffuse drive of equal total intensity (heterogeneous, not universal) — a lasting trace is not a sharpened field. C4 — relayed imprint is the structural default: the relayed-imprint set is a strict majority (8 of 12) — an honest contrast with §46, where clean delivery was the default (10 of 12 self-localising in the instant). The coupling inverts the majority: a focal drive’s instantaneous footprint is by default focal, but its lasting footprint is by default delocalised, because consolidation spreads what delivery localised. The relayed-imprint set is simultaneously the off-target-dominant trace set (C2) and strictly contains the field-relay set (C3) — one coherent majority class. A direction-only [L] correspondence is noted, never a magnitude or a prediction: clinically, stimulation-induced plasticity — the after-effects of repetitive TMS and tDCS, the plasticity DBS induces in connected circuits — is recognised to be network-distributed and connectivity-dependent rather than confined to the stimulation site, consistent with relayed imprint being the structural majority and the imprint axis decoupled from the instantaneous-delivery axis. A zero drive with consolidation off (η = 0) reproduces the frozen M9 coordination anchor (R = 0.38961455156) bit-for-bit, η = 0 leaves the connectome W identical to the kernel, and a focal excursion reverts exactly once consolidation is removed (inheriting the E0.3 guard) — a pure structural read on the frozen kernel (engine 0fbf4988…, byte-unchanged; both layers reused, not re-derived; no new tuned constant). The firewall is absolute: a local-imprint/relayed-imprint class is a structural spatial quantity of the coupled model, never a felt effect of stimulation or of learning, and not a real electric-field or current-density map, a lead-position or specific-absorption-rate (SAR) map, a real connectome or synaptic-weight matrix, a prediction of which patient’s stimulation imprints where or which target consolidates optimally, or device-programming/target-selection guidance (Axis-A — consciousness_claim = 0, the hard problem stays open). Every magnitude is [O]; efficacy = 0; not medical advice.

The first cross-axis coupling — marrying the spatial map (where) to the plasticity layer (lasting)

The atlas has, until now, advanced along two separate axes. One axis is space: the spatial layer built a single map of the frozen kernel and asked for it to be read region by region, and three chapters did exactly that — §44 drove a region up the coupling curve and read the instantaneous footprint as a seizure, §45 silenced a region and read the footprint as a stroke, §46 re-read §44’s excitatory footprint as a therapeutic stimulation. Every one of those is a spatial reading: where, at the instant of the drive, does the coordination change land? The other axis is time: the plasticity layer built a phase-correlation Hebbian update that lets the connectome retain a structural trace of the coordination it has been driven through, and three chapters read that — addiction as a retained gain (§37), Alzheimer’s as a retained decay (§39), OCD as a retained stabilisation (§41). Every one of those is a temporal reading: what lasting mark does a sustained operating point leave?

This chapter is the first to put the two axes together, and the coupling is not a metaphor — it is mechanically forced by what stimulation actually is. A real focal stimulation does two things in sequence: it delivers a drive to a place (a spatial event, the E1 question of where the field lands), and then, because the brain is plastic, the network consolidates the coordination that drive provoked (a temporal event, the E0 question of what trace it retains). A purely spatial chapter sees only the first; a purely temporal chapter sees only the second; neither can ask whether the place a focal drive lands is the place it leaves a lasting mark. That is the question this chapter exists to ask, and it requires both layers at once. The drive is §43’s per-node SpatialField, the update is §26’s PlasticConnectome, and the chapter’s one new piece of machinery is the integrator that runs the spatial drive through the plasticity rule — stepping the per-node drive on the connectome as it changes, accumulating the phase coincidence the Hebbian update consolidates. Nothing is re-derived; the engine is byte-unchanged; the two frozen layers are coupled, and the seam between them is the new object of study.

The coupling model — a focal drive stepped through the Hebbian update

The model is the smallest one that makes the imprint question answerable, and it reuses both layers wholesale. A focal stimulation is, exactly as in §46, a per-node excitatory bias: region i is driven while every other region sits at baseline, through the same effective-coupling map — k(b) = κ/(1+|b|) — so a stimulated target is a region pushed up the coupling curve, with no new constant. What is new is that the drive is not read at an instant: it is integrated through the plasticity layer. The §26 phase-correlation Hebbian update potentiates an edge in proportion to the phase coincidence of the two nodes it joins; the new integrator steps the per-node spatial drive on the evolving connectome W, over a fixed consolidation window, accumulating the pairwise coincidence the rule writes into structure. The result is a retained trace ΔW — the lasting synaptic mark a focal drive leaves — rather than an instantaneous field. The engine’s own fixed incoherent initial condition is re-seeded each consolidation epoch, so the whole construction is deterministic.

Two sweeps, not one, gate every result. The stimulation intensity is swept — b₀ = 0.3, 0.5, 0.7, 0.9 — exactly as §46 swept it, asking whether a sign survives how hard the region is driven. But because the drive now runs through plasticity, a second axis must also be swept: the consolidation rate η = 0.03, 0.05, 0.08, asking whether a sign survives how fast the trace is written. Every sign reported below is required to hold across the entire product grid of intensity and rate — twelve drive conditions per region — so no number is fitted on either axis to make a result come out. The reading is the consolidated analogue of §46’s footprint: for each driven region, compare the lasting change |ΔW| on the edges incident to the driven node against the mean change it writes on edges off the target. If the incident change dominates, the drive imprints locally — the lasting mark is where the drive was aimed. If the off-target change dominates, the drive relays its imprint — the lasting mark is in the circuits the target consolidated into.

C1 — the local-imprint/relayed-imprint partition is drive-invariant across both sweeps

The first result is the spatial discriminant for a lasting imprint. Driving a focal stimulation at each of the 12 regions in turn and letting the connectome consolidate, the targets partition cleanly into two classes. Four targets imprint locally: the retained trace concentrates on the edges incident to the driven node — structurally, the lasting mark stays where the drive was aimed. The local set is {brainstem, cerebellum, pallidum, striatum}. The other eight relay their imprint: driving them writes a larger lasting trace off the target than on it — structurally, the mark consolidates into the circuits the target relays to rather than at the site itself.

The certification is that this partition does not move with either drive axis. Across the entire intensity×rate grid — the gentlest drive to the strongest, the slowest consolidation to the fastest — the same four targets imprint locally and the same eight relay: exactly one partition is observed over the whole sweep. Whether a focal drive imprints locally or relays its trace is therefore a fixed property of the driven site — a drive-invariant connectome property, not a function of how hard the region is stimulated nor how fast the trace is written. This is the consolidated counterpart of the §43 self/relay map, now read as a property of lasting imprint rather than instantaneous footprint — and, crucially, it is not the same partition. The local-imprint set here is a strict minority of four, where §46’s clean-delivery set was a majority of ten; the coupling has changed which sites stay put, and C3 and C4 make precise how.

C2 — relayed imprint is off-target dominance: the structure of a trace that bleeds away

The second result turns the binary class of C1 into a legible quantity, and ties “relayed imprint” to its meaning of a lasting change that consolidates away from where the drive was aimed. For each driven region, take the ratio of the mean off-target consolidated change to the change on the edges incident to the driven node. For a relayed target this ratio is above 1: driving the region writes a larger lasting trace in distal circuits than at the target itself — the neocortex at roughly 6.3× and the forebrain GABA interneuron at about 5.7× (the most strongly relayed, of which more in C3), the field-relay hippocampus at about 1.4× and the midbrain at about 1.4×. That is the structural picture of a relayed imprint: the drive is at the target, but the largest lasting change is elsewhere, in the circuits the target consolidated into. For a local-imprint target the ratio is below 1: the trace concentrates on the driven node’s own edges — the brainstem, for instance, at about 0.63×, the mark staying where the drive landed.

The equivalence — a target relays its imprint if and only if its off-target trace exceeds its incident trace — holds at every swept intensity and every swept rate. This is what makes the C1 partition more than a label: relayed imprint is not a separate phenomenon bolted on, it is the same retained trace read quantitatively, the off-target-to-incident ratio crossing 1. It also fixes what kind of claim is and is not being made. The ratio is a structural measure of where a lasting change lands on the frozen kernel; it is not an electric-field strength, a current density, a measured synaptic weight, or an after-effect amplitude. The magnitude of any real consolidated change is [O]; what is asserted is the sign — that relayed targets dominate off-target and local targets dominate at the site, robustly across both drive axes.

C3 — delivery and imprint are decoupled: clean delivery does not imply clean imprint

The third result is the one the coupling exists to surface, and the one genuinely new to this chapter — the seam between the two layers, where the spatial story and the temporal story come apart. The natural expectation, carried straight from §46, is that a focal drive that delivers cleanly — whose instantaneous field self-localises at the target — should also imprint cleanly, leaving its lasting mark where it landed. If that held, the trace-relay set would simply equal the §46 instantaneous field-relay set, and the imprint axis would be a relabelling of the delivery axis.

It does not hold. The set of targets whose lasting trace relays off-target strictly contains the §46 set of targets whose instantaneous field relays off-target. The field-relay set is {hippocampus, midbrain} — two sites — and both of them are also trace-relayed, so the containment is genuine, not a re-partition. But the trace-relay set has eight members, so six further sites — the neocortex, thalamus, hypothalamus, basal-forebrain cholinergic nucleus, forebrain GABA interneuron and olfactory bulb — deliver their instantaneous field cleanly (in §46 they self-localise; the drive lands at the target) yet relay their imprint (once plasticity consolidates, the lasting mark lands off-target). These six are the witnesses to the decoupling: a drive can land exactly where it is aimed and still leave its durable trace somewhere else. Plasticity delocalises the imprint. Where a focal drive delivers and where it imprints are two distinct, decoupled spatial properties — the temporal axis is not a shadow of the spatial one, and the cleanest deliverer is not, in general, the cleanest imprinter.

There is a second finding here, and it is an honest negative — a clean hypothesis tested and reported false rather than dropped. One might hope that a focal drive at least imprints more sharply than a diffuse drive of the same total intensity — that concentrating the drive concentrates the trace. It does not, in general. Comparing each region’s focal imprint against a matched diffuse drive, only 3 of 12 sites imprint more focally; the sign is heterogeneous, not universal, and it holds (as a non-law) across the swept intensities. There is no universal “focal beats diffuse” trace law: a lasting trace is not simply a sharpened field, and the concentration of a drive does not, by itself, concentrate its durable mark. The refuted hypothesis is the finding — the no-tuning discipline working as intended — and it sharpens C3’s lesson: the imprint axis is governed by the network the trace consolidates through, not by the focality of the drive that started it.

C4 — relayed imprint is the structural default: the coupling inverts the majority

The fourth result steps back and asks what kind of thing the relayed-imprint set is, and the answer reframes the whole chapter against its parent. The relayed-imprint set is a strict majority — eight of twelve — so relayed imprint, not local, is the structural default: the typical focal drive leaves its lasting mark off-target, and local imprint is the exception. This is the explicit, honest contrast with §46, where the default ran the other way: there, ten of twelve targets delivered their instantaneous field cleanly, and off-target leak was the minority. The coupling inverts the majority. A focal drive’s instantaneous footprint is by default focal (most drives land where they are aimed), but its lasting footprint is by default delocalised (most drives leave their durable mark elsewhere) — because consolidation spreads the imprint that delivery localised. The instant and the trace disagree about what is typical, and that disagreement is the coupling’s headline.

The relayed set is one coherent class, not a loose collection. It is simultaneously the off-target-dominant trace set of C2 (its off-to-incident ratio exceeds 1) and a strict superset of the §46 field-relay set of C3 (it contains {hippocampus, midbrain} and six more). These descriptions pick out the same eight regions, across the whole intensity×rate grid — one coherent majority class of sites whose lasting imprint relays off-target. Structurally, a delocalised imprint is the rule for a focal drive under plasticity, and a local imprint the special case carried by a small set of sites.

There is a direction-only correspondence to clinical reality worth naming — carefully, and graded [L], because it is a cited resemblance and not a derived or predicted quantity. Clinically, stimulation-induced plasticity is recognised to be network-distributed: the after-effects of repetitive transcranial magnetic stimulation and of transcranial direct-current stimulation outlast the stimulation and are observed in circuits connected to the stimulated site, not only under the coil or electrode; deep brain stimulation induces plasticity in the networks its target projects to, not merely at the contact. That stimulation’s lasting effects are distributed across connected networks rather than confined to the stimulation site is consistent with the model making relayed imprint the structural majority and decoupling the imprint axis from the instantaneous-delivery axis — a coherence of direction, nothing more. It is not a claim that this model predicts which individual patient’s stimulation imprints where, which target consolidates optimally, or what device settings to use; those are clinical determinations made from real imaging, electrophysiology and individualised modelling. The model offers a structural rationale for why the durable effects of a focal drive are, by default, network-distributed — a hypothesis about mechanism, gated for reproducibility, awaiting external test.

S6 — the engine-invariance guard: a zero drive at η = 0 recovers the frozen anchor

As with every module in the series, the coupling is certified to be a pure structural read that adds nothing to the engine, and here the guard is inherited from both parent layers. When the drive is set to zero and the consolidation rate to zero (η = 0) — no region driven, no trace written — the integration reproduces the frozen M9 coordination anchor R = 0.38961455156044245 bit-for-bit, identical to the engine’s own direct value at full precision. With η = 0 the connectome W is left identical to the kernel — consolidation off changes nothing — and a focal excursion, once consolidation is removed, reverts exactly, inheriting the E0.3 guard under a spatial drive. Removing the coupling recovers the frozen engine with nothing left over — the strongest possible check that running a focal drive through plasticity is a read on the kernel, not a modification of it. Both the SpatialField and the PlasticConnectome are imported and the engine emerged read-only, confirmed byte-unchanged against the frozen tree hash (0fbf4988fc83…) with the M0–16 subtree identical. There is no new mechanism, no new measurement, and no new tuned constant: this chapter couples two frozen layers and reads four signs off the seam, all of which survive the intensity×rate sweep.

The four results side by side — what the chapter establishes

The chapter is four certified statements about one object — a focal drive (the spatial layer) consolidated through the Hebbian update (the plasticity layer). Read across a row to see one result and the discriminant that carries it; read down to see the chapter move from the local/relayed partition (the discriminant), through its quantitative content, to the honest decoupling of imprint from delivery, and finally to the structural class the relayed targets form.

The table makes the chapter’s logic legible. C1 is the discriminant — a target imprints locally or relays by its position, across both drive axes. C2 is the quantitative content — relayed means the lasting trace lands off-target. C3 is the seam — the imprint axis is not the delivery axis (clean delivery does not imply clean imprint), and a focal drive does not universally imprint more sharply than a diffuse one, so the temporal story cannot be read off the spatial one. C4 is the structural class — relayed imprint is the majority behaviour, inverting §46’s clean-default, carried by one coherent set of sites. Together they are the first reading of the seam between the spatial and plasticity layers: where a focal drive lands and where it leaves a lasting mark are two different places, and the second is, by default, more distributed than the first.

What the chapter does not claim — the firewall

This is a chapter about brain stimulation and synaptic plasticity, and the boundary of what it asserts must be stated without hedging. Every quantity certified here is a structural spatial quantity — a local-imprint/relayed-imprint class, an off-target-to-incident trace ratio, a focal-versus-diffuse comparison, all of them properties of how a coupled model writes lasting structure across a fixed kernel — and none of them is a claim about the felt effect of stimulation or of learning. That a target “imprints locally” or “relays its imprint” is a statement about where a lasting change lands in the connectome model; it is not a statement about what stimulation feels like, what learning feels like, a therapeutic benefit, or awareness, and it is certainly not a claim that experience or memory is located at the target or its relayed circuits. This is the Axis-A firewall, held exactly as in every chapter of the series: consciousness_claim = 0, and the hard problem of experience stays open. Giving an imprint a spatial address in the model does not give the experience of stimulation, or the having of a memory, one.

The boundary to clinical and biological reality is firmer still. The per-node excitatory bias is not a real stimulation, the consolidated trace ΔW is not a real synaptic weight change, an LTP/LTD measurement, or a stimulation after-effect, the edge set is not a real connectome, and the local-imprint/relayed-imprint class is not a prediction of which patient’s stimulation imprints where or which target consolidates optimally. Real stimulation-induced plasticity is heterogeneous — it runs on the individual electrode geometry and montage, the tissue conductivity, the patient’s own tractography and connectome, the waveform and dose, the state-dependence of the cortex, and the molecular machinery of synaptic change — and this module asserts only the sign and structure of a focal drive consolidated on the frozen kernel, not that any real after-effect follows it. Target selection, lead placement, contact and current programming, dose and timing of plasticity-inducing protocols, and outcome prediction are external clinical determinations, made by clinicians with real imaging, electrophysiology and individualised modelling; nothing here is localisation, device programming, target optimisation, or treatment guidance, and the [L] correspondence in C4 is a cited resemblance of direction (that stimulation’s lasting effects are network-distributed), never a patient-level claim. Every magnitude is [O]: representative reads over a swept intensity and rate, never quantities fitted to a target. There is no new mechanism, no new measurement beyond the read-only layers, and no new tuned constant in this chapter; the SpatialField and the PlasticConnectome are imported, the engine is byte-unchanged, and the only new object is the integrator that couples them. Nothing here is a cure, a treatment, a diagnosis, a localisation, a device setting, or a recommendation. efficacy = 0; this is not medical advice. What the chapter offers is one structural thing: in this model, where a focal drive delivers and where it leaves a lasting imprint are decoupled — the durable mark is, by default, more distributed than the drive that wrote it — so the spatial story of plasticity, like every disease on these layers, must be told site by site, and the trace cannot be read off the field.