T-PreObj-1 — Pre-Objective Multi-Peak Formation Mechanism (W4 Capstone)

Hero · W4 Capstone group · Cat A. Source: C-0700 / P-0700 (T-PreObj-1) + C-0701 / P-0701 (T-PreObj-1G graph-class independent) + C-0702 / P-0702 (Lemma 4 quadratic form PD). Verification: E-0090 (L=12, 3-digit theory–experiment agreement), E-0091 (L=32 dichotomy). Canonical version: CV-1.3 (W4 close, 2026-04-24). Full proof: Canonical Spec — Part 5 · §13 W4 close additions. Narrative: Week 4 Extended weekly post.

Statement#

Let $G$ be a finite connected graph satisfying hypotheses (G1)–(G4) (graph-class regularity conditions; see canonical §12). Let $u_0^*$ be the F=1 single-disk minimizer of pure $\mathcal{E}_{\mathrm{bd}}|_{\Sigma_m}$. Under full SCC parameters $\lambda_{\mathrm{cl}}, \lambda_{\mathrm{sep}}, \lambda_{\mathrm{bd}} > 0$, the canonical statement has five sub-statements:

(i) Disk non-criticality. $u_0^*$ is NOT a critical point of full $\mathcal{E}$ on $\Sigma_m$. Equivalently: under full SCC, gradient flow leaves the F=1 disk.

(ii) Multi-peak attractor. Gradient flow from $u_0^*$ converges to $u^*_{\mathrm{end}}$ with $\mathcal{F}(u^*_{\mathrm{end}}) > \mathcal{F}(u_0^*)$, where $\mathcal{F}(u) = \#\{x \in X : u(x) > u(y)\;\forall y \sim x\}$ is the local-maxima count (threshold-independent topological invariant).

(iii) Lemma 4 (Quadratic form positive definite). The matrix $M \in \mathbb{R}^{2\times 2}$ defined by $M_{11} = \|g_{\mathrm{cl}}\|^2$, $M_{22} = \|g_{\mathrm{sep}}\|^2$, $M_{12} = \langle g_{\mathrm{cl}}, g_{\mathrm{sep}}\rangle$ is positive definite under linear independence of the gradients $g_{\mathrm{cl}} = \nabla \mathcal{E}_{\mathrm{cl}}|_{u_0^*}$ and $g_{\mathrm{sep}} = \nabla \mathcal{E}_{\mathrm{sep}}|_{u_0^*}$, with destabilization magnitude $\|\nabla\mathcal{E}(u_0^*)\|^2 = \lambda^\top M \lambda > 0$ for any $\lambda \in \mathbb{R}^2_{>0}$.

(iv) IC sensitivity. Basin attraction from $u_0^*$ depends sensitively on initial-condition eigenmode alignment.

(v) IC-protocol dichotomy (thermodynamic limit). The asymptotic endpoint of gradient flow under different initialization protocols satisfies:

• Adaptive bounded protocol → bounded F endpoint, $\mathcal{F}_* \leq F^{\mathrm{first-pitchfork}} + O(1)$.
• Random initialization → endpoint scales as $\mathcal{F}_*^{\mathrm{random}}(L) \sim L^{2.8}$ (where $L$ is the graph linear scale).

T-PreObj-1G (graph-class independent). Conclusions (i) and (ii) hold on any finite connected graph satisfying (G1)–(G4) — not just on the test grid families.

Category breakdown (canonical lines 1120–1123).

• (i), (iii), (iv), (v) dichotomy form: Category A (rigorous proof + 3-digit numerical confirmation).
• (ii) qualitative existence of multi-peak attractor: Category A; exact $\Delta\mathcal{F}$ magnitude: Category B.
• (v) precise exponent $k \approx 2.8$ for random IC: Category B (empirical fit, theoretical derivation open).

Proof idea (sketch)#

Lemma 4 (quadratic form PD). Define $M \in \mathbb{R}^{2 \times 2}$ as the inner-product matrix of the gradients $g_{\mathrm{cl}} = \nabla \mathcal{E}_{\mathrm{cl}}|_{\hat u_F}$ and $g_{\mathrm{sep}} = \nabla \mathcal{E}_{\mathrm{sep}}|_{\hat u_F}$ at the F=1 candidate $\hat u_F$. Under linear independence of $g_{\mathrm{cl}}$ and $g_{\mathrm{sep}}$ (generic, holds in (G1)–(G4)), $M$ is positive definite, and the destabilization magnitude $\Lambda^T M \Lambda > 0$ for any $\Lambda \neq 0$.

(i) Non-criticality. At the pure-$\mathcal{E}_{\mathrm{bd}}$ minimizer $\hat u_F$ with F=1, $\nabla \mathcal{E}_{\mathrm{bd}} = 0$ on the constraint surface. The full energy gradient is then $\lambda_{\mathrm{cl}} g_{\mathrm{cl}} + \lambda_{\mathrm{sep}} g_{\mathrm{sep}}$; by Lemma 4, this is non-zero on the constraint tangent unless $\lambda_{\mathrm{cl}} = \lambda_{\mathrm{sep}} = 0$. So $\hat u_F$ is non-critical.

(ii) Multi-peak attractor. From non-criticality, gradient flow leaves $\hat u_F$ with destabilization magnitude $\geq \Lambda^T M \Lambda > 0$. The descent direction $-\nabla \mathcal{E}$ has positive overlap with the F=2 destabilization mode (verified via second-variation analysis). Compactness + monotone decrease (T14) + Łojasiewicz forces convergence to some critical point; phase-space analysis shows the attractor is multi-peak F≥2.

(iii) IC-protocol dichotomy. Adaptive bounded protocols stabilize F by closing the iteration on a bounded scale; random initialization explores F space uniformly until energy minimization concentrates the field — this exploration phase scales with graph size, giving the empirical $\sim L^{2.8}$ scaling on tested grid families.

T-PreObj-1G. The argument above uses only abstract spectral / isoperimetric properties guaranteed by (G1)–(G4); no specific graph features needed. Therefore the conclusion lifts to any such graph. $\square$

Why this is a hero#

T-PreObj-1 is the W4 capstone — the conceptual centerpiece of nearly a year of work. It establishes that SCC's pre-objective character is a theorem, not a modeling choice.

The pre-objective claim is at the heart of SCC's identity: "objects are not the starting point; they are derivative; what is primitive is a graded cohesion field where multi-peak structure emerges." Before W4, this was a design intent — the theory was organized to express pre-objective character, but it was not forced to. T-PreObj-1 makes it forced: under full SCC parameters, the F=1 single-disk (the closest analog to "a single object") is not even a critical point, while F≥2 multi-peak configurations are the default attractor. Single-object thinking is energetically inconsistent with the full SCC framework.

The graph-class independence (T-PreObj-1G) lifts this from "true on grids" to "true on any finite connected graph satisfying (G1)–(G4)." The pre-objective character is therefore graph-class independent — it is a property of the SCC framework itself, not of any particular spatial substrate.

T-PreObj-1 also resolves F-1 (the K=2 vacuity Critical open problem) via the F-1 Resolution Corollary: F-1 splits into a pure-$\mathcal{E}_{\mathrm{bd}}$ portion (the proved theorem T-Merge (b)) and a full-SCC portion (T-PreObj-1 (i) — F=1 is non-critical). The dichotomy "K=1 cheaper vs observed K>1" is dissolved because the static and dynamic layers refer to different objects (commitment CN15 Static/Dynamic Separation).

The IC-protocol dichotomy (iii) is a separate result of independent interest: it identifies which initialization protocols are well-posed versus which produce arbitrarily-large endpoints. The $\sim L^{2.8}$ scaling under random initialization is a quantitative empirical signature awaiting a closed-form derivation (NQ-148 family).

Logical dependencies#

• Builds on: T1 (existence), T14 (gradient flow convergence), T-Merge (b) (isoperimetric ordering — the pure-$\mathcal{E}_{\mathrm{bd}}$ piece of F-1 split-resolution), spectral analysis of (G1)–(G4) graphs.
• Builds into: F-1 Resolution Corollary, CN15 Static/Dynamic Separation, T-V5b-T (translation-invariant graphs are a special class within (G1)–(G4) and admit Goldstone-mode analysis on top of T-PreObj-1), σ-framework on $\Sigma_m$ (single-formation Hessian signature).

See also#

• Full proof in canonical: Canonical Spec §13 T-PreObj-1, T-PreObj-1G, Lemma 4
• F-1 Resolution Corollary (the OP-0001 closure): Canonical Spec Part 4 §12 W4 Resolution Banner
• CN15 Static/Dynamic Separation (the conceptual key): Canonical Spec Part 5 §14 CN15
• W4-extended Goldstone refinement: T-V5b-T hero page
• Narrative of how this came together: Week 4 Extended weekly post