Perception · W6 — Observer Moduli Space

W6: 2026-05-04 to 2026-05-08.
Canonical: CV-1.5.2 → CV-1.11 (+6 versions, +17 claims, 61 → 78 total).
The week's center of gravity: observer-dependence finally gets its own mathematics.

Start here#

W5 gave SCC its first multi-formation count bridge. W6 asked a harder question: what does it mean for two observers to see the same scene differently — and can that be made into a theorem?

The answer took the entire week, three distinct phases, and a complete reset of the theory's navigation structure.

This week in five lines#

• Six canonical versions in a single day (D4). P-F-A1 Package I went fully Cat A. K-selection got two new Cat B theorems. The stereo boundary theory was promoted to Cat A.
• Theory navigation was rebuilt around six epistemological questions. DECLARATION.md names the central question and T8 as the central theorem. hypothesis_tree.md HT-3.0 reorganizes all open hypotheses by Q1–Q6 rather than mathematical topic.
• Observer Moduli Space was built from scratch and fully closed in two days. OMS-2.0 went from nonexistent to Accepted — Full across eight sessions, eleven validation programs, and three major open problem closures.
• The symmetry group of SCC energy is trivial. $G_{\mathrm{cw}}=\{e\}$ computationally confirmed on static scenes; $G_{\mathrm{cw}}^{\mathrm{temp}}=\{e\}$ proved and confirmed for the temporal extension. This is the theorem behind why observer-dependent K-selection is non-trivial.
• Two real proof bugs were caught and corrected. C1.2 required $H_T\succ 0$ (not merely invertible); C1.4 needed an explicit vertex hypothesis. Honest restatements made both proofs tight.

Scoreboard#

The percentage drop is structural: nine new Cat B theorems increased the denominator faster than the numerator. Every Cat B here has a clear Cat A path.

The week's arc#

Day 1–2: clearing the ledger#

W5 left two items on the table. Both were resolved before W6's main work began.

T-L1-M canonical promotion (Day 1). The soft-count corollary to T-L1-F entered canonical as Cat A conditional. The promotion path was: four self-audit closures (R-0 through R-3) → external cold-review (~7 min, PASS) → same-day supervised authorization. This is the first time the audit → external review → supervised promotion pipeline ran end-to-end in a single session.

The theorem says: under the L1-J regime, the soft-count of the aggregate field tracks the active-slot count within an explicit two-term error bound controlled by admissible envelope functions in ${\Phi }_{\mathrm{res}}$.

Regime characterization (Day 2). NQ-G3-1 confirmed that the 22.9% feasibility anchor in T-L1-F is stable across four orders of magnitude of $\epsilon$. This is not a sensitivity finding; it is a robustness finding. The anchor is not an artifact of the production default.

The count after these two days: 47A / 5B / 5C / 5R = 62 claims. The remaining gap to CV-1.11 (78 claims) would be closed in a single day.

Day 3: debt paydown#

Day 3 was explicitly scoped as a redirection day. Five accumulated theory debts (from W1 through W5) were addressed in one session without touching canonical or working theory files.

The substantive output was the 8-axiom Langevin framework v0 (P-F-A1 through P-F-A8) and the OP-0009-Pre quotient formalism for the $S_K$-quotient structure on labeled K-field states. These were planning and design documents, not promotions. The canonical count did not change.

Day 4: CV-1.6 through CV-1.11#

This is the week's most unusual session. Fourteen sessions over one calendar day produced six canonical version increments. The sequence:

CV-1.6 (Sessions A–G): Stereo extension formalized. T-ST-5a Cat A (stereo-to-monocular consistency), T-OP6-B Cat B (boundary precision with stereo conditioning), D-ST-1..5 (stereo definitional layer). T-ST-5b signed off Cat B (narrow claim: GL-only regime).

CV-1.7 (Session I): P-F stochastic foundation. T-P-F-ε0 (Gibbs measure weak continuity at $\epsilon =0$) Cat A via dominated convergence. T-P-F-ε0-K (Kramers exponent stability) Cat B, conditional on H5 Morse stability.

CV-1.7 amended (Session K): T-OP6-B promoted Cat B → Cat A. Four sub-blockers closed: B1 topological separator, B2 curved Hausdorff ($d_H\le 2(\alpha /\beta {)}^{1/2}$ with $C<1.37$), B3 stereo conditioning, B4 ${\rho }_{bd}=1/(4\xi )$. OP-0006 (boundary precision) RESOLVED.

CV-1.8–1.9 (Sessions M–P): P-F-A1 Package I. Four theorems on the existence and ergodicity of the Gibbs measure under $T_{\ast }$:

T-PF-A1-GI uniqueness used Aronson 1968 Neumann heat kernel (any invariant $\nu \ll \text{Leb}\ll \pi$, then $L^2(\pi )$ kernel argument). T-PF-A1-PE used Payne-Weinberger 1960 on bounded convex polytopes (no smoothness needed) with Holley-Stroock perturbation. P-F-A1 Package I is fully Cat A.

CV-1.10 (Session R): T-K-Select-PF Cat B. Equilibrium K-selection via Gibbs sector mass: $K^{\ast }=\arg {\min }_{K}F(K;\mathcal{P})=\arg {\max }_K{p}_K$ where $\{p_K\}$ is the pushforward of ${\pi }_{T_{\ast }}$ under $K_{\mathrm{act}}$. OP-0005-EQ partially resolved.

CV-1.11 (Session Y): T-K-Select-OBS Cat B. Observation-conditioned K-selection via posterior sector mass: $p_K({\mathfrak{O}}_t)=Z_K^{\mathrm{obs}}/Z^{\mathrm{obs}}$. Canonical likelihood model §2.4 (LM1–LM3) verified. exp85 3/3 ALL PASSED. OP-0005-OBS partially resolved.

Alongside these promotions, T-Temporal-Identity and T-σ-Inherit were developed as working Cat B candidates with numerical anchors (exp83 4/4, exp84 5/5). They are not canonical; they are the targets CV-1.12 will address.

Day 5: rebuilding the map#

After six canonical versions in one day, the theory's navigation had become hard to orient in. Day 5 did not add theorems. It rebuilt the conceptual structure.

DECLARATION.md (DECL-1.0). A new 2-minute document at the canonical root. It states the theory's central question directly:

어떤 차이의 덩어리가 언제부터 하나의 객체가 되는가?
(When does a cluster of differences become an object?)

It names the primitive ($u_t:X_t\to [0,1]$), the central theorem (T8: phase transition at $\beta /\alpha >4{\lambda }_2/|W^{\prime \prime }(c)|$), and six epistemological questions (Q1–Q6) that organize all open work.

hypothesis_tree.md HT-3.0. The dependency tree was reorganized from mathematical topic groupings to epistemological question groupings (Q1: when does a boundary appear? Q2: can multiple formations coexist? Q3: how do formations evolve? Q4: how does discrete count emerge? Q5: how does temporal identity persist? Q6: how does observer dependence enter?). This makes the critical path visible: H-SINK (Phase 1, unblocked) closes Q5 Cat A.

Observer Moduli Space OMS-0.1 → OMS-1.0. The new theory framework was sketched: ${\mathcal{M}}_{\mathrm{obs}}={\Delta }^3\setminus \partial {\Delta }^3$ as the observer parameter space (weight $({\lambda }_{cl},{\lambda }_{sep},{\lambda }_{bd},{\lambda }_{tr})$), $V\in {\mathcal{V}}_{\mathrm{adm}}$ as an admissible observer landscape, and basin stratification as the mechanism linking parameter choice to percept. VP-1 ran: OP-OMS-009 (P-resolution injectivity) was RESOLVED-NEGATIVE — four counterexamples found with $\Vert d\Vert <0.15$ and $D_T>0.5$. Prop R1 proved.

Day 6: OMS-2.0 in five sessions#

The final day closed the Observer Moduli Space theory. Five sessions, eight validation experiments (VP-4 through VP-11), three open problems closed.

Sessions 4–5: VP-3 tested seven candidate symmetry transformations of the SCC energy. Six were rejected; transformation E (transport ablation on static scenes) was confirmed as a symmetry (Prop CW2). This established computationally that $G_{\mathrm{cw}}(P)=\{e\}$ for dynamic scenes — the core-weight symmetry group is trivial. OMS promoted 1.0 → 1.1 → 1.2.

Session 6: OMS-2.0 Conditional Accepted. Eight gates closed three hard blockers:

• OP-OMS-001 (core-weight symmetry): Proved conditional on H4 (existence of a non-degenerate Jacobian witness). H4 computationally confirmed via VP-8 (81% witnesses across three scenes). The proof uses a rank-obstruction theorem: $J_e=-G_T^{\top }{H}_T^{-1}{G}_T$ combined with a genericity dichotomy (G4/G5/G7/G8) establishes that the symmetry group has no non-trivial elements on dynamic scenes.

• OP-OMS-002+ (admissible landscape existence): An explicit non-trivial admissible landscape $V_{2,\tau }$ was defined (softened min of two observer readouts) and proved admissible. VP-9 confirmed basin nontriviality: 3 attractors on $P_{12}$, 4 on $S_3$, all with distinct readout pairs.

• OP-OMS-026 (branch structure): ${\Sigma }_{\mathrm{branch}}$ proved codim-1 with explicit decomposition ${\Sigma }_{ab}\cup {\Sigma }_{\mathrm{Hess}}\cup {\Sigma }_{\mathrm{AS}}\cup {\Sigma }_{\mathrm{SN}}$. VP-10 pseudo-${\Delta }^3$ branch map confirmed codim-1 consistency: 7 distinct branches, transition fraction 0.311 against 0.375 budget.

Session 7: OMS-2.0 Accepted — Static. Proof consolidation with no new experiments. Two real bugs corrected:

• C1.2 rank-equivalence: "$H_T$ invertible" changed to "$H_T\succ 0$" (positive definite). An indefinite invertible matrix does not give $\mathrm{rank}(A^{\top }{B}^{-1}A)=\mathrm{rank}(A)$.
• C1.4 rigidity: made explicit that the (Vertex) hypothesis is required (supplied independently by Prop CW1 and VP-3).

OP-OMS-032 closed: INTERVAL_CERTIFIED H4 witness, margin $4\times 10^{13}$ over IEEE bound (12 witnesses, 3 scenes). OP-OMS-033 proved: conditional fold theorem SN3 via Crandall-Rabinowitz applied to the SCC KKT system. OP-OMS-034 separated: temporal ${\Delta }^3$ is independent of static OMS-2.0. Appendix OMS (§A–§L, 20+ theorem items) added to canonical.md.

Session 8: OMS-2.0 Accepted — Full. VP-11 closed the temporal extension (OP-OMS-034).

Phase 1 (rank witness): 14 $\lambda$-points tested, 14/14 with rank 3 on the ${\Delta }^3$ tangent. The critical correction here: rank-3, not rank-4. Pre-Session-7 references to "4×4 minor" were tangent-dimension errors (documented as W28).

Phase 2 (${\Delta }^3$ branch map at $K=5$): 19 distinct branches, 7 ${\lambda }_{tr}$-unique. Two macro-regimes: static-cohesive (26.8%) and transport-coherent (17.9%). Theorem TS3 proved: temporal reduces to static exactly at ${\lambda }_{tr}=0$.

The final qualification: "Static (PROVED) + Full Temporal (COMPUTATIONALLY SUPPORTED on faithful reduced test)."

Deep dive: what OMS-2.0 actually says#

The central question of OMS is: given that two different weight vectors ${\Theta }_1,{\Theta }_2\in {\Delta }^3$ produce different minimizers $u^{\ast }({\Theta }_1)\ne u^{\ast }({\Theta }_2)$, what is the structure of the set of observers who would agree on the percept?

OMS-2.0 gives three answers:

1. The core-weight symmetry group is trivial. There is no non-identity transformation of $\Theta$ that leaves $u^{\ast }$ invariant for a generic dynamic scene. This means observer-space has no internal redundancy: different $\Theta$ genuinely correspond to different percepts.

2. The branch decomposition is codimension-1. The set ${\Sigma }_{\mathrm{branch}}\subset {\Delta }^3$ where the minimizer changes topologically is a codim-1 stratified manifold. It has four components: ${\Sigma }_{ab}$ (attractor-birth transitions), ${\Sigma }_{\mathrm{Hess}}$ (second-order degeneracy, identified with the SCC phase-transition surface ${\Sigma }_{T8}$), ${\Sigma }_{\mathrm{AS}}$ (asymmetric saddle), ${\Sigma }_{\mathrm{SN}}$ (saddle-node). Observers on the same side of ${\Sigma }_{\mathrm{branch}}$ agree on percept topology.

3. The moduli space admits non-trivial admissible landscapes. There exist legitimate observer functions $V\in {\mathcal{V}}_{\mathrm{adm}}$ with at least two distinct attractor basins. Observer-types (in the sense of basin-of-attraction) are a non-trivial partition of ${\Delta }^3$.

Together these say: observer-dependence in SCC is not a degeneracy or a gauge redundancy. It is a genuine structural feature, and its geometry is codimension-1 in parameter space.

What the week did not prove#

• T_ registration (OP-0021).* The stochastic temperature $T_{\ast }$ remains axiomatic. P-F-A1 Package I proves the Gibbs measure exists and is ergodic for any $T_{\ast }>0$; it does not prove which $T_{\ast }$ the SCC energy selects.
• Eyring-Kramers rates (Package II). The $k_{K\to K-1}$ transition rates remain conditional on H5 (Morse stability) and OP-0021. Package II is W9+ work.
• K-jump σ-inheritance (OP-0008). Post-merger ${\sigma }^A$ at a K-jump event is not determined by pre-merger ${\sigma }^A$ alone. This blocks T-MF-Synthesis and requires the Wigner-projection computation (W9+).
• T-Temporal-Identity / T-σ-Inherit canonical promotion. Both are working Cat B candidates with numerical anchors. They are the explicit targets of CV-1.12.
• OMS formal theorem rows in theorem_status.md. The Appendix OMS is in canonical.md. Formal row-by-row registration is the next bookkeeping task.

The epistemological frame (Q1–Q6)#

The new DECLARATION.md and HT-3.0 organize all open work around six questions. This is the best way to understand what remains:

Carry to W7#

The critical path is H-SINK.

The theorem to prove: in the SCC cost class, the Sinkhorn transport map $T_c(\mu ,\nu )$ satisfies a Lipschitz bound $L_g\le L_c$ using the Bigot-Cazelles-Papadakis (2019) framework.

Why it matters: H-SINK unlocks T-Temporal-Identity parts (a), (b), and (d) for Cat A promotion. These are the three constructive parts of the theorem — existence of the correspondence, uniqueness under the margin condition, and reduction to the K=1 case. Part (c) (interior persistence) remains Cat C pending OP-0011.

Closing H-SINK in W7 would produce CV-1.12 with +3A and bring the total to 57A / 14B / 5C / 5R = 81 claims.

Canonical impact#

W6 spans six canonical increments:

• CV-1.6: stereo extension (T-ST-5a Cat A, D-ST-1..5).
• CV-1.7: P-F stochastic foundation (T-P-F-ε0 Cat A, T-OP6-B Cat A, OP-0006 RESOLVED).
• CV-1.8–1.9: P-F-A1 Package I fully Cat A.
• CV-1.10: equilibrium K-selection (T-K-Select-PF Cat B).
• CV-1.11: observation-conditioned K-selection (T-K-Select-OBS Cat B).
• Appendix OMS (W6 D6): Observer Moduli Space §A–§M.

Aligned with Perception_theory canonical CV-1.11 (2026-05-06) + Appendix OMS (2026-05-08). The next milestone is CV-1.12, targeted for W7 via H-SINK.