Spatial localisation — the scalar global drive generalised to a per-node spatial profile, the field shown to have a fixed structure, and the honest finding that no universal focal-versus-diffuse law exists (the field-shaping layer)

Every disorder so far drove the brain globally. Generalising the scalar coupling to a per-node spatial drive makes focal-versus-diffuse representable: the field is local and normalised, perturbation reach is heterogeneous with a drive-stable cerebellar hub, and each node's self-localising-or-relay class is drive-invariant. No universal focal-beats-diffuse law holds, so disease here is region-specific. Structural quantities, never felt experience.

Every disorder module so far drove the brain globally. From schizophrenia and epilepsy through the θ-cap to the mood and cycle disorders, the intervention was always a single scalar coupling K_glob — one number raised or lowered for the whole network at once. With only a scalar, three questions could not even be posed: focal versus diffuse drive, off-target spillover, and region-specific disease. The fault was always global; the drive had no spatial profile. This chapter adds the spatial core — the spatial sibling of the §26 plasticity layer’s temporal core. It generalises the scalar to a per-node coupling vector K_i = k(b_i)·Ω₀ (a spatial drive profile) set through the same k = κ/(1−|b|) map as the SZ/epilepsy/E0 modules, and reads a per-node local-coherence field c_i = ⟨∑_j W₀_ij cos(θ_j−θ_i)⟩ as a read-only side computation that never perturbs the θ trajectory. The form is forced (the frozen ∼1/r³ row-stochastic kernel W₀ plus the existing map, no new constant); the profile is swept and every sign holds across a depth sweep b₀ ∈ {0.3, 0.5, 0.7, 0.9} (anti-tuning). Four results hold. E1.1 — the field has a fixed spatial structure: the frozen kernel is local (every one of the 12 rows’ weight monotone-decreases with anatomical distance) and normalised (every row sums to 1, deviation 2×10⁻¹⁶) — exact geometric structure, the substrate of every focal/region claim. E1.2 — perturbation reach is heterogeneous with a drive-stable hub: driving one node at a time and reading the change in the global order, reach(i) = |R − R₀| varies more than a hundredfold (max/min > 110), and the map is not an artefact of drive amplitude — the cerebellum is the rank-1 reach hub at every swept depth and the full ranking is invariant for moderate-to-high drive (Spearman = 1.0 among b₀ ≥ 0.5). E1.3 — the self-localising vs relay classification is drive-invariant (the headline): each node’s focal footprint is either self-localising (its own local-coherence change exceeds the mean off-target change) or a relay (the change lands harder off-target), and this binary classification is identical across the entire depth sweep — the relay set {hippocampus, midbrain} fixed at every drive amplitude, the other ten nodes self-localising at every amplitude. Whether a site contains or relays its drive is a fixed property of where it sits in the field, not of how hard it is pushed. E1.4 — there is no universal focal>diffuse law (the honest no-tuning result): a clean, attractive hypothesis — “focal drive always concentrates local gain at its target more than the same dose spread diffusely” — is false: delivering equal total dose focally versus diffusely, the sign of (focal − diffuse) target-gain varies across sites (9 of 12 concentrate, the others do not). Spatial outcome is site-determined, not governed by a global rule — which is exactly why disease modules built on this layer must be region-specific, never global. A uniform (zero-bias) drive reproduces the frozen M9 coordination anchor (R = 0.38961455156) bit-for-bit, and the off-state field equals the baseline exactly — E1 is a pure add-on (engine 0fbf4988…, byte-unchanged). The SpatialField class is the reusable layer the later region-specific modules import. The firewall is absolute: a local-coherence field, a reach map and a self/relay class are structural spatial quantities of the coupling model, never the felt locus of an experience, and not a real electrode, current density or dose (Axis-A — consciousness_claim = 0, the hard problem stays open). Only structural signs and relations are asserted; every magnitude is [O]; nothing is a cure, reversal, or prevention; efficacy = 0; not medical advice.

The global-drive limitation — what every prior module assumed

For twenty-five chapters the framework drove the brain with a single number. The engine integrates a network of phase oscillators, one per region, coupled through a frozen ephaptic kernel, and the whole disorder atlas was built by reaching into that integration and turning one dial: the global coupling K_glob. Schizophrenia lowered it; epilepsy raised it past the over-synchronisation edge; the θ-cap held it at a paced operating point; the threshold-lever chapters slid it along a reachable axis. Every one of these was a scalar intervention — the same coupling applied to every region at once, a uniform push or pull on the entire connectome. It worked, and it was honest, but it carried a silent assumption: that a brain state is set by one global parameter, with no spatial address.

That assumption makes three of the most clinically important questions in neuroscience unaskable. The first is focal versus diffuse: a seizure focus, a stimulating electrode, a stroke — these are local events, a drive or a fault applied to some regions and not others, and a global scalar cannot represent a profile that is high in one place and zero in another. The second is off-target spillover: when a drive is focal, where does its effect actually land — does it stay where it was applied, or propagate to distant regions through the field? A scalar has no “where” for the effect to spread from. The third is region-specific disease: why does one pathology express in one circuit and another elsewhere, when both ride the same substrate? If the only handle is a single global number, every disorder is the same disorder at a different setting, and region identity carries nothing. This chapter removes the assumption. It gives the drive a spatial profile and reads the field per region, and in doing so it makes all three questions askable for the first time — and answers the structural part of each.

The spatial drive — a per-node coupling vector and a per-node field

The generalisation is the smallest possible one, and that is deliberate. The frozen engine’s update for region i is dθ_i = ω_i + K_glob · ∑_j W₀_ij sin(θ_j − θ_i), with a single scalar K_glob multiplying the field every region feels. E1 replaces that scalar with a per-node vector K_i, one coupling per region, set by a per-node bias b_i through the same effective-coupling map the schizophrenia, epilepsy and plasticity modules already use: k(b) = κ/(1−|b|) for an excitatory bias, κ/(1+|b|) for an inhibitory one, capped at 2κ. When the bias is uniform, the vector collapses to a scalar and the dynamics are bit-for-bit the frozen engine. The drive is therefore a pure generalisation: no new equation, no new constant, just the dial made spatial — a profile that can be high at one region, zero at another, negative at a third.

To read the spatial consequence, the module computes a per-node local-coherence field: c_i = ⟨ ∑_j W₀_ij cos(θ_j − θ_i) ⟩, the field-weighted local order at region i, averaged over the steady second half of the integration. This is the local counterpart of the global order parameter R — where R reports how coordinated the whole network is, c_i reports how coordinated each region is with its own field neighbourhood, weighted by the kernel. Crucially the coherence read-out is a read-only side computation: it accumulates the cosine alongside the integration but never enters the sin update, so the θ trajectory — and therefore R — is untouched. The form is forced [F] (the frozen kernel plus the existing map); the spatial profile — which node is driven, the depth b₀ — is [O] and swept, and every sign and structure asserted below is required to hold across a depth sweep b₀ ∈ {0.3, 0.5, 0.7, 0.9}. No number is fit to a target; the spatial findings are signs, certified to survive anti-tuning.

E1.1 — the field has a fixed spatial structure: local and normalised

The first result is the foundation the other three stand on: the field a region exerts has a fixed spatial structure, and it is exact, not emergent. Two properties of the frozen kernel W₀ are certified to machine precision. The first is locality: for every one of the 12 regions, the kernel weight a region places on its partners decreases monotonically with anatomical distance — the nearer the partner, the stronger the coupling, with no exceptions across all 12 rows. This is the ∼1/r³ ephaptic law made explicit: the field is spatially graded, and a region’s influence is concentrated on its neighbourhood. The second is normalisation: every row of the kernel sums to exactly 1 (maximum deviation 2×10⁻¹⁶, i.e. floating-point zero). Each region distributes a conserved total field influence across its partners — row-stochastic by construction, which is what preserves the ephaptic locality whenever the kernel is touched.

These are not findings about dynamics; they are findings about geometry, and that is the point. Locality and normalisation are exact properties of the measured anatomical geometry and the frozen kernel form — structure, not a fitted result. They are what license everything spatial that follows: because the field is local, a focal drive has a neighbourhood to act on; because it is normalised, the total influence is conserved as it is shaped. The substrate of “focal versus diffuse”, “off-target spillover” and “region-specific” is this fixed spatial structure — and none of it could be expressed by a single global scalar, which has no rows, no distances, and no neighbourhoods at all.

E1.2 — perturbation reach is heterogeneous with a drive-stable hub

The second result asks the off-target question directly: drive one region and watch the global order respond. Define the reach of a focal drive as reach(i) = |R − R₀| — how far the change propagates to the network-wide state when only region i is pushed. The reach map is strongly heterogeneous: across the 12 regions the largest reach exceeds the smallest by more than a hundredfold (max/min above 110 at every swept depth, and above 129 from b₀ = 0.5 upward). Network position decides how far a focal drive travels: pushing some regions barely moves the global state, while pushing others reorganises it. The naïve picture — that a focal drive is “a small local perturbation” — is false for the high-reach regions; for them a focal drive is a global event.

The decisive part is that the reach map is not an artefact of how hard the drive is pushed. The cerebellum is the rank-1 reach hub at every swept depth — the single region whose focal drive moves the global order the most, at b₀ = 0.3, 0.5, 0.7 and 0.9 alike — and the full reach ranking is invariant for moderate-to-high drive (Spearman correlation = 1.0 between every pair of depths in {0.5, 0.7, 0.9}). The hub structure is a connectome invariant: the spatial map of which regions broadcast and which stay quiet is a fixed property of the geometry, not of the drive amplitude. This is the structural substrate of off-target effects — it identifies, before any disorder is named, which sites carry a focal drive away to the rest of the brain and which do not. (It is a structural reach map only; it is not a claim about a real electrode, a real current spread, or a clinical target — efficacy = 0.)

E1.3 — the self-localising vs relay classification is drive-invariant

The third result is the headline of the layer, and it sharpens the reach finding from a global magnitude into a per-region class. Drive one region focally and read the full local-coherence footprint — the vector of changes Δc_j across all regions. Each driven region falls into one of two classes. A self-localising region is one whose own coherence change is larger than the mean change it induces off-target: the footprint concentrates at the driven site. A relay region is the opposite: driving it changes distal coordination more than its own — the drive is relayed to other sites rather than contained. This is the precise structural form of “does a focal drive stay focal?”: for self-localising sites, yes; for relays, no.

The certification is that this binary classification is identical across the entire depth sweep. At every drive amplitude from b₀ = 0.3 to 0.9, the same ten regions self-localise and the same two relay: the relay set is {hippocampus, midbrain}, fixed at every amplitude tested. Whether a site contains its drive or relays it is therefore a fixed property of where it sits in the field, not of how hard it is pushed — a drive-invariant connectome property, exactly as the reach hub is. And it coheres with E1.2: the two relay regions are precisely the kind of sites whose drive lands elsewhere, and the midbrain in particular shows up as a high-reach site at low drive — a relay broadcasts. This is the structural map of focal-disease spread versus containment — which foci would stay local and which would broadcast, and which sites would carry targeted stimulation away from its intended target. (Axis-A firewall: a self/relay class is a structural spatial property of the coupling model, never a claim about the felt locus of an experience.)

E1.4 — there is no universal focal>diffuse law: the honest no-tuning result

The fourth result is the one the no-tuning discipline exists to produce: a clean, attractive hypothesis, tested and found false, reported as the finding rather than quietly dropped. The hypothesis is the obvious one a clinician or an engineer would reach for first: focal drive concentrates local gain at its target more than the same dose spread diffusely. Deliver a fixed total dose two ways — all of it to one region (focal), or the same total split evenly across all regions (diffuse) — and compare the coherence gain at the focal target. If the hypothesis held, the focal delivery would always win at its own site.

It does not. Across the 12 regions the sign of (focal target-gain − diffuse target-gain) varies: at 9 of the 12 sites focal delivery concentrates more gain at the target, but at the other 3 it does not — for those sites the same dose spread diffusely produces more local gain at the would-be target than concentrating it there. There is no universal focal>diffuse law. We do not force the monotone narrative the heterogeneous-frequency physics refuses to support; the refuted hypothesis is reported honestly, and that honesty is the no-tuning discipline working as intended. The genuine finding is the deeper one: spatial outcome is site-determined — governed by network position (E1.2, E1.3), not by a global rule that holds everywhere. And that has a direct consequence for everything built on this layer: because the spatial consequence of a drive depends on which region receives it, disease modules on this layer must be region-specific, never global. The layer does not hand its successors a universal stimulation law; it hands them a map, and the instruction to read it per region.

S4 — the engine-invariance guard: a uniform drive recovers the frozen anchor

As with every layer in the series, the spatial drive is certified to be a pure add-on. When the per-node bias is set to zero everywhere — a uniform drive, the vector collapsed back to the scalar — the integration reproduces the frozen M9 coordination anchor R = 0.38961455156044245 bit-for-bit, identical to the engine’s own direct value at full numerical precision, and the per-node local-coherence field equals the stored baseline field exactly. Turning the spatial profile off recovers the frozen engine with nothing left over. The engine is emerged read-only for the invariant check and confirmed byte-unchanged against the frozen tree hash (0fbf4988fc83…), with the M0–16 subtree identical — the same guarantee the plasticity and state-switching layers carry. The spatial layer adds an axis; it changes nothing that was already there.

The four results side by side — what the layer establishes

The layer is, at heart, four certified statements about a single new object — the per-node spatial drive and the field it shapes. Read across a row to see one result and the discriminant that carries it; read down to see how the layer moves from fixed structure (geometry) through heterogeneous reach and a drive-invariant class (the spatial map) to the honest absence of a universal law (the no-tuning result).

The table makes the layer’s logic legible. E1.1 is the fixed structure the field has before anything is driven. E1.2 and E1.3 are the spatial map that structure produces: a heterogeneous reach with a stable hub, and a binary containment/spread class that does not change with drive amplitude. E1.4 is the honest boundary: the map is real, but it does not reduce to a universal law, so it must be read per region. Together they are the spatial axis the global-drive atlas never had — built once here, handed to every region-specific module that follows.

What the layer does not claim — the firewall

The spatial layer adds an axis to the model, and it must be stated plainly what that axis is not. Every quantity certified here is a structural spatial quantity — a local-coherence field, a reach map, a self/relay class, all of them properties of how a coupling model coordinates across a fixed kernel — and none of them is ever a claim about the felt locus of an experience. That a drive “localises” at a region is a statement about where coordination changes in the connectome model; it is not a statement that an experience is felt there, or felt at all. 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 the drive a spatial address does not give experience one.

The boundary to clinical reality is just as firm. The per-node bias is not a real electrode, the local-coherence field is not a real current density, and the focal/diffuse dose is not a real stimulation protocol. Real volume conduction and ephaptic coupling are heterogeneous and frequency-dependent — tissue conductivity anisotropy, gyral geometry, myelination — and this module asserts the sign and structure of a per-node drive on the frozen kernel, not that any real cortical field follows this exact 1/r³ law. Every magnitude is [O]: representative reads over swept profiles, never quantities fitted to a target. E1 is the layer, not an application; the region-specific disorders that will use it — focal epilepsy foci, stroke and lesion fields, off-target effects of targeted neuromodulation — are owed to later modules, and the SpatialField class is the reusable object they will import. There is no new mechanism, no new measurement beyond the read-only field, and no new tuned constant in this chapter; the engine is imported read-only and byte-unchanged. Nothing here is a cure, a reversal, a prevention, a treatment, a recommendation, or a dose. efficacy = 0; this is not medical advice, not a diagnosis, and not a treatment protocol. What the layer offers is exactly one thing, and it is a structural statement: the field has a fixed spatial structure, a focal drive’s reach and containment are fixed properties of position, and there is no universal focal-versus-diffuse law — so the spatial story of disease must be told region by region, never with a single global dial.