The Wall-Local Obstruction Lemma

Theorem 1 · the §8 lemma of docs/research/tangency-chambers-walls-strata.md, proved · proofs/wall-local-obstruction-lemma.md · a-tangency board · ARGUED — natural-language proof, no Lean · firewalled adversarial cold review CONFIRMED, relay-20 Lane F, 2026-07-23

An arrangement of $n$ circles sits on the unit-spacelike quadric $W = \{(V_1,\dots,V_n) : \langle V_i,V_i\rangle = 1\}$ inside $\mathbb{R}^{3,1}$, $Q = \operatorname{diag}(+,+,+,-)$, and every pair $q=\{i,j\}$ contributes one number $\Phi_q(V) = \langle V_i,V_j\rangle$. Four intervals partition the line that number can land in — $\mathrm{CROSS}=(-1,1),\ \mathrm{EXT}=(-\infty,-1),\ \mathrm{NEST}=(1,\infty),\ \mathrm{TAN}^{\pm}=\{\pm1\}$ — and a relation-labeling $L$ is a choice of one class per pair. Fixing the tangent pairs at their exact values carves out a reduced manifold $M = W \cap \{\Phi_q = \tau_q : q\in T\}$, and inside it the labeling's chamber $F(L) = \Phi^{-1}(\mathrm{Cell}(L))$ is the open region where every remaining pair sits strictly inside its assigned class. Chambers meet at walls: fix one non-tangent pair $p$ as the pin, with wall value $\sigma$ on the boundary of its class, sign $s$ recording which side is which, and excess $\varepsilon = s(\Phi_p-\sigma)$. The contact locus $Z = \Phi^{-1}\bigl(\{\sigma\}\times{\textstyle\prod_{q\neq p}}\overline{R_q}\bigr)$ is the stratum where the pin sits exactly on its wall while every other pair is free to be anywhere in the closure of its own class — the seam the chamber touches. At a clean point of $Z$ (every other free pair strictly interior, so only the one pin is active) a short continuity argument, Lemma 0 of the source document, pins the picture down completely near $V^*$: there is a neighborhood $U$ with

So near a clean contact point, deciding whether the chamber reaches $V^*$ is exactly deciding the sign of one smooth function $\varepsilon$ on the manifold $M$.

Chambers, walls, strata — and the single-pin trichotomy

That reduces a piece of combinatorial topology to a completely standard question in local analysis: near a zero of a smooth function on a manifold, when does the function stay one sign, and when does it take both? The source document proves the general answer as three lemmas about an abstract $f\colon M\to\mathbb{R}$ at a zero $x^*$, then specializes to $\varepsilon$. Case 1, first-order escape (Lemma A): if $\nabla\varepsilon(V^*)\neq 0$, a first-order Taylor expansion along the gradient direction already produces points of both signs in every neighborhood — $F(L)$ meets every neighborhood of $V^*$, i.e. the labeling is realizable there. Case 2, second-order escape (Lemma B): if the gradient vanishes but the Hessian has a positive direction, $H\varepsilon(w,w) > 0 \text{ for some } w\in T_{V^*}M$, the same one-line computation along that direction gives realizability. Case 3, the local obstruction (Theorem E, the genuinely hard case): if the gradient vanishes, the zero set $Z$ is a smooth embedded submanifold near $V^*$, and the Hessian is negative definite on some linear complement of $T_{V^*}Z$ in $T_{V^*}M$, then

— a genuine open-neighborhood exclusion: not just “no escape found yet,” but a proof that none exists. The three cases are proved pairwise exclusive (case 3 forces $H\varepsilon \preceq 0$ everywhere on $T_{V^*}M$, so case 2 cannot also fire, and $\nabla\varepsilon(V^*) = 0$ rules out case 1), and the source document is explicit that they are not exhaustive: a fourth possibility — negative semidefinite transverse to $Z$ but not definite, i.e. flat to second order — is left undecided by construction. The standing example is telling: $f(x,y) = x^3+xy^2$ has $Z=\{x=0\}$ smooth, a vanishing gradient and Hessian at the origin, yet $f>0 \text{ for } x>0$ — realizable, and invisible to every instrument in this trichotomy. This is not a corner case tacked on for completeness: §6 of the source document measures that most of the real n=5 hard residue lives exactly in that fourth, uncovered region, one level up (jointly across several pins at once) — the subject of a separate reduction, proofs/two-pin-wall-reduction.md (essay: humans/atlas/two-pin-wall-reduction.html), not this one.

The kernel lemma: a strengthening the §8 statement only asked for conditionally

Case 3's proof needs one more fact to go through, and it is where this document's real mathematical content sits. Write $Hf(x^*)$ for the Hessian of $f\colon M\to\mathbb{R}$ at a critical point (well defined exactly there — a chart change only ever perturbs it by a term proportional to $df(x^*)$, which vanishes at criticality) and $Z:=f^{-1}(0)$ for the standing hypothesis that $Z$ is an honest smooth embedded submanifold near $x^*$. The scoping document that first stated the trichotomy flagged, as something still needing proof, that the Hessian should vanish on $T_{V^*}Z$ whenever the case-3 definiteness hypothesis holds. Lemma C proves something stronger, and unconditionally:

with no definiteness hypothesis anywhere — criticality and a smooth zero set are all it needs. The proof splits on the codimension $k$ of $Z$: at $k=0$ the zero set is open and closed near $x^*$, forcing $f\colon M\to\mathbb{R}$ to vanish identically on a whole neighborhood (so $Hf(x^*)=0$ outright); at $k\ge 2$ an embedded submanifold cannot carry two different dimensions at once, which forces the gradient itself to vanish all along $Z$ near $x^*$ (a stronger fact than the lemma needs, proved along the way), and differentiating that vanishing gradient tangentially kills every mixed second partial directly; at $k=1$ an adapted chart plus Hadamard's lemma writes $f = x_1\cdot h(x_1,y)$, and the same open-submanifold argument applied one dimension down forces $h\equiv 0 \text{ on } \{x_1=0\}$, again killing the mixed terms. One clean consequence falls out immediately: for any two linear complements $C_1, C_2$ of $T_{V^*}Z$, writing a vector of one in terms of the other's coordinates plus a piece of $T_{V^*}Z$ shows $Hf|_{C_1} \cong Hf|_{C_2}$ — so “negative definite on some complement” and “negative definite on every complement” are the same statement, and case 3's hypothesis needed no arbitrary choice to begin with. Because the lemma holds independent of any sign condition, it also applies at case-2 points and at the undecided fourth region — wherever $Z$ is smooth and the gradient vanishes, the Hessian is automatically block-diagonal against $T_{V^*}Z$, for free.

The n=4 witness: n4_TTTmmN, verified exactly

The lemma would be an empty exercise in local topology without a real instance where it actually fires on real data, and one is banked: n4_TTTmmN, the one non-realizable a-tangency class certified at n=4 (results/certificates/atangency/n4_TTTmmN.json). Its labeling puts three pairs at $\mathrm{TAN}^+$, two at $\mathrm{TAN}^-$, and pins the last pair $\{2,3\}$ at the $\mathrm{NEST}$ boundary, excess $\varepsilon = \langle V_2,V_3\rangle - 1$. A hand-built contact point,

satisfies every one of the nine constraints defining $M$ exactly (four unit conditions, five tangency equalities), has $\langle V_2,V_3\rangle = 1$ and $t:=\det V = 0$, and re-verifying it here (independently of the certificate and of the relay-20 script, scratch/relay21-lane4c/verify_n4_geometry.py, rational arithmetic throughout, cross-checked against scratch/relay20/wall_local_n4_check.py's own run) reproduces every claim below to the digit.

Interpolating $\det G_4(e)$ against the one free entry $e:=\langle V_2,V_3\rangle$ recovers $\det G_4 = 8(e-1)$ exactly, so with Theorem A1's identity $\det G_4 = -(\det V)^2$ the global relation $8\varepsilon = -t^2 \text{ on } M$ holds on all of $M$ — the whole obstruction in one line, once $t$ is known. Locally: the nine constraint gradients have rank $9$, so $M$ is a smooth 7-manifold at $V^*$ (hypothesis (H-M)); the pin's gradient turns out to equal an exact combination of two of the unit-constraint gradients, $d\varepsilon(V^*) = \tfrac12\, d\langle V_2,V_2\rangle + \tfrac12\, d\langle V_3,V_3\rangle$, so it lies in their span and case 1 does not fire; and $dt|_{T_{V^*}M}\neq 0$ — the differential of $t$ does not lie in that span, so $Z=\{t=0\}\cap M$ is smooth of codimension 1 in $M$ there, with $\dim T_{V^*}Z = 6$ (the full dimension of the Möbius orbit — this contact configuration is rigid up to gauge, nothing more). The reduced Hessian on the resulting 7-dimensional tangent space has exact inertia $(0,1,6)$, and the one negative eigenvalue is $-2$ on the nose — case 3 fires, with the kernel exactly equal to $T_{V^*}Z$ (the kernel lemma, verified on real numbers, not just proved in the abstract) and the identity $H_{\mathrm{red}} = -\tfrac14\,(dt|_{TM})(dt|_{TM})^{\!\top}$, matching the analytic prediction $\varepsilon = -t^2/8$ that falls out of the global relation above.

The coincidence wrinkle. At $V^*$ the pinned pair is not merely tangent — $V_2 = V_3$ exactly, the same vector twice. This is forced, not an artifact of the particular chart chosen: releasing the pin to its wall value makes all six pairs sit at $\pm 1$, and that fully-tangent Gram matrix has rank 3 with a nondegenerate induced form, which pins $V_2-V_3$ to be simultaneously orthogonal to, and inside, a nondegenerate 3-space — forcing it to zero. So the contact locus of this particular labeling is a repeated-circle locus, not a distinct-circle tangency; the source document flags this as an interpretive caveat for anyone reading the released pin as “grant an extra tangency” (that reading is unavailable here — no 4-distinct-circle realization of the released labeling exists at all), not a defect in the lemma, which is indifferent to the distinction because it works entirely on $M$, never on a picture of circles.

The honest fences

• The lemma is the engine, not the verdict. Theorem 1 decides a neighborhood of one contact point. It does not, by itself, decide a labeling of record, let alone a residue class: a per-class verdict additionally needs the global assembly corollary — the §8 gauge/compactification bookkeeping, the multi-point paving of the whole contact locus, and the excise-and-pave argument over the pin-relaxed system — and that corollary is not proved here (unproved SEED statement, unchanged by this document). Nothing above should be read as reducing any residue: the n=5 a-tangency UNKNOWN residue is unchanged at 355 iso-classes.
• The trichotomy is not exhaustive by construction. The flat-cone region (Hessian negative semidefinite but not definite transverse to $Z$) is left silent, and measurement shows most of the real hard residue sits there once more than one pin is active — that region is a separate, harder problem (proofs/two-pin-wall-reduction.md advances a piece of it; the fully general multi-pin obstruction sufficiency is characterized precisely in the source document's §7 and stays OPEN, new mathematics, not engineering debt).
• Z-smoothness is a real hypothesis, not bookkeeping. Every conclusion above routes through $Z$ being an honest embedded submanifold near $V^*$. At n=4 that was exhibited exactly, by producing $t$ with nonvanishing differential; no comparable exact witness exists yet for any n=5 class, and numerics alone cannot certify smoothness.
• One instance is one instance. The n=4 witness shows the lemma fires on real, previously-banked data by exactly the mechanism it names — it is a non-vacuity check, not a general decision procedure, and it decides only that one labeling.
• The a-tangency board only. Every object above — chambers, walls, pins, the n=4 witness — lives on the relation-labeling board of docs/research/tangency-chambers-walls-strata.md. This essay moves no a(6) quantity and no bracket of record; the bracket of record is docs/a6-bracket.md, untouched here and unreferenced by any number above.

Grade, stated plainly: ARGUED — a refereed natural-language proof, strictly sub-PROVED: no Lean, no kernel check. A solo first draft (relay-20 Lane F) was lifted to ARGUED by an independent firewalled adversarial cold review that re-derived Lemma 0, the abstract trichotomy, Lemma C, and Theorem 1 from their bare statements and re-ran the exact n=4 script before reading the argument; the artifact survived. Restated once more because it is the single point most at risk of being over-read: the trichotomy decides a neighborhood of one contact point, it closes no residue class by itself, and the n=5 a-tangency residue this whole apparatus is measured against stands exactly where it stood before this document — 355 iso-classes, unchanged.