feat(corpus): enrich CS-04 + CS-05 + CS-06 — Arc 2 Physics (manuscript-grounded)

corpus/construction-spine/identify-physical-carrier/index.md

@@ -91,28 +91,141 @@ construction_step_id: "CS-04"
 
 <div class="notice note"><strong>Status note.</strong> Build status reflects the current internal state of the Corpus. It does not imply external acceptance unless explicitly stated.</div>
 
-## What this step is required to do
+## 1. What this step must build
 
-The program must locate a structure inside the kernel capable of carrying physical semantics: locality, globality, entities, observables, interactions, dynamics, and lawfulness.
+The program must locate a structure inside the kernel capable of carrying physical semantics — locality, globality, entities, observables, interactions, dynamics, and lawfulness — *without* yet making any empirical claim.
 
-## What the corpus built
+By the end of this step:
 
-The Corpus identifies the physical-carrier problem through the relevant self-enrichment and unitary construction layer.
+- The **4+1 sector template** must be derived (not assumed) from the kernel's five generators.
+- **Boundary functoriality (Langlands<sub>0</sub>)** must be shown to *induce* the sector decomposition.
+- The **spectral algebra** must be earned at E<sub>0</sub> — the toolkit Books IV–V apply at E<sub>1</sub>.
+- The **Hinge Theorem** must establish that every result in Books IV–VII is a sector instantiation of Book III's enrichment structure.
+- The **No Knobs Principle** / **No Knobs Ledger** must establish that all inter-sector couplings are determined by `ι_τ` — no free parameters.
+- **Where physics lives**: local bulk projections must glue; the decompactification limit `τ³_R → ℝ³` must recover Euclidean space at human scales; the **3+1 spacetime signature** must be derived from the `τ¹ × T²` fibered product.
+- The **Eight Guarantees** — Spatial, Harmonic, Regular, Discrete, Legible, Codable, Coherent, Predictive — must be earned as the structural force names.
 
-## What this step builds
+What cannot yet be assumed: time, mass, energy, fields as physical primitives (CS-05); SI units (CS-06); life (CS-07).
 
-This step turns the Agenda question "where can physics live inside the kernel?" into a Corpus construction problem. It identifies the structural layer that can carry physical roles before measurements or empirical adequacy are discussed.
+## 2. The construction challenge
 
-## Why this step is required
+This step is hard for five interlocking reasons.
 
-Physics cannot be assumed as a finished external domain. The Corpus must first explain what kind of internal structure can support physical meaning at all.
+**2.1 Cannot assume spacetime, fields, particles, measurement, or physical law.** The kernel is mathematics (Books I–II). Physics must be *located* inside it without smuggling in physical primitives. The natural temptation — "let τ³ be a topological space" with all the implicit structure that brings — is exactly what must be avoided.
 
-## Key constructions
+**2.2 Must identify where locality and globality can live.** Local bulk projections must glue into a single, globally coherent three-dimensional space — but the gluing condition itself cannot be assumed; it must be derived as a theorem.
 
-- The carrier problem.
-- Locality and globality.
-- Entities, observables, and interactions.
-- Dynamics and lawfulness as internal roles.
+**2.3 Must not smuggle in physics as vocabulary.** Naming things "fields" or "particles" before the structure is in place is a category mistake. The 4+1 sector template must be a *consequence* of generator structure under boundary functoriality, not a label applied to a pre-conceived list.
+
+**2.4 Must distinguish carrier identification from physical content.** CS-04 identifies *where physics lives*. The actual physics — constants, laws, particles, cosmology — lives in CS-05 + CS-06. The boundary between "structural carrier" and "physical content" must be sharp; if it blurs, downstream empirical claims have nothing distinct to stand on.
+
+**2.5 Must earn the 3+1 spacetime signature.** The temporal/spatial split is normally postulated. It must be *derived* — from the `τ³ = τ¹ ×_f T²` fibered product structure inherited from Books I–II.
+
+## 3. What Panta Rhei builds
+
+The Corpus builds, at canonical scope, the structural physics layer E<sub>1</sub> via four manuscript-grounded constructions: the 4+1 sector template, the spectral algebra, the Hinge Theorem, and the Global Cartesian Gluing of "Where Physics Lives".
+
+Step 4 turns the Agenda question — *where can physics live inside the kernel?* — into a derivation, not a posit. The answer is that the same five generators that determine the τ-kernel determine, by functorial necessity at E<sub>1</sub>, both the sector decomposition and the spacetime signature. **E<sub>1</sub> is not a model of physics — it IS the structural layer where physics becomes definable.**
+
+### The 4+1 Sector Template (Book III Part II)
+
+The five generators of τ — `α`, `π`, `γ`, `η`, `ω` — induce **four primitive sectors plus one mixed sector** (the ω-coupling). The decomposition is *not chosen*: **Langlands<sub>0</sub> (boundary functoriality) induces it as a theorem**. The functor from boundary characters to interior holomorphic data preserves sector structure, so the 4+1 form is a functorial consequence.
+
+At enrichment level E<sub>1</sub>, the template instantiates as:
+
+| Generator | Sector at E₁ |
+|---|---|
+| `α` | Gravity (D-sector) |
+| `π` | Weak (A-sector) |
+| `γ` | Strong / computational |
+| `η` | Electromagnetic |
+| `ω` | Higgs / mass coupling |
+
+Two structural commitments accompany the template:
+
+- **Parity Bridge Theorem** — identifies the **weak sector** as the canonical carrier for the *computational bootstrap* that bridges to Book VI's life recovery.
+- **No Knobs Principle** — establishes that **all sector couplings are determined by `ι_τ`**, not as free parameters.
+
+### Spectral Algebra at E₀ (Book III Part III)
+
+The algebraic vocabulary for everything that follows is earned at E<sub>0</sub> — the same algebraic substrate as Books I–II. The spectral toolkit comprises:
+
+- **CRT Decomposition Theorem** — a τ-native Chinese Remainder Theorem via modular Bézout, *without signed arithmetic* — earned-language discipline in action.
+- **Hensel lifting** — performed constructively in residue carriers (III.T11, III.D21).
+- **Primorial ladder** `Prim(k) = p₁ · p₂ ⋯ p_k` — the canonical cofinal filtration unifying finite-level verification across all Millennium Problems.
+- **Spectral trichotomy III.T14** — every boundary character decomposes uniquely into B-supported, C-supported, and X-mixing components.
+- **Computable classifier `Label_n` (III.D23)** — replaces informal lobe language with computable predicates.
+
+### The Hinge Theorem (Book III Part VII)
+
+Part VII assembles the **Complete Dependency Chain** showing the full derivation path:
+
+> five generators → seven axioms → four orbits → ABCD coordinates → boundary ring → Central Theorem (Book II) → enrichment ladder (E<sub>0</sub> ⊊ E<sub>1</sub> ⊊ E<sub>2</sub> ⊊ E<sub>3</sub>) → 4+1 sector template → spectral algebra → Millennium clusters → enriched bi-square → computational collapse → Hinge.
+
+**Every link is earned; no postulates, no free parameters.**
+
+The **Hinge Theorem (III.T20)** itself states:
+
+> Every result in Books IV–VII is a sector instantiation of Book III's enrichment structure.
+
+The seven-book architecture is **derived, not postulated**. Export contracts to all four downstream books are formalized in Book III Chapter 62. The **No Knobs Ledger** (Book III Chapter 63) exhibits that every inter-sector coupling is canonically determined by `ι_τ`.
+
+### Where Physics Lives — Global Cartesian Gluing (Book III Part VIII)
+
+Local bulk projections glue (III.T21). The **decompactification limit** `τ³_R → ℝ³` recovers Euclidean space at human scales, and the Minkowski extension from the base `τ¹` provides 3+1-dimensional spacetime with the **correct signature**.
+
+The **Eight Guarantees Earned** revisit the gluing requirements with a theorem number and a structural force name attached to each:
+
+- **Spatial** — bulk projections glue locally
+- **Harmonic** — frequency / mode decomposition is internal
+- **Regular** — analytic discipline (τ-holomorphy)
+- **Discrete** — countable address structure
+- **Legible** — `Label_n` classifier is computable
+- **Codable** — sector-level encoding is well-defined
+- **Coherent** — boundary-character coherence holds
+- **Predictive** — closed-form prediction discipline (forward to CS-06)
+
+The **Temporal-Spatial Decomposition** unpacks the base `τ¹` direction as the temporal axis and the fiber `T²` directions as spatial — completing the 3+1 signature derivation.
+
+### E₁ Complete
+
+Book III Chapter 76 synthesizes: **`E_1` is not a model of physics — it IS the structural layer where physics becomes definable**. Export contracts to Books IV–V are formalized; the transition to E<sub>2</sub> (life layer, Book VI) is previewed. Book III closes the foundational arc; the τ-effective scope is honoured throughout.
+
+## 4. Why this matches the required answer-shape
+
+Step 4 identifies the physical carrier as a τ-internal derivation, not a presumption. Its admissibility is evaluated against the obligation to locate physics inside the kernel without importing physical primitives.
+
+**Gluing to previous steps.** CS-04 inherits CS-01's kernel + boundary algebra + holomorphy + τ-topos; CS-02's recovered mathematics; CS-03's enrichment ladder + Central Theorem + Categoricity. The 4+1 sector template is the structural reading of the five generators (CS-01 K1) under the Central Theorem's holographic principle (CS-03 II.T40). The spectral algebra at E<sub>0</sub> uses the boundary ring + CRT machinery already earned in Book I.
+
+**No-externalities discipline.**
+
+- **No assumed spacetime.** Spacetime emerges from the `τ¹ × T²` fibered product; the 3+1 signature is derived (III.T21).
+- **No assumed fields or particles.** Sectors are functorial consequences; particles, generations, and the constants ledger are CS-05 territory.
+- **No free parameters.** No Knobs Principle / Ledger commits all inter-sector couplings to `ι_τ`.
+- **No imported physics vocabulary.** "Sectors" is the structural vocabulary; "EM/weak/strong/gravity" are *instantiations* at E<sub>1</sub>, not primitives.
+
+**Earned language.** Every sector is a theorem (III.T03 — Langlands<sub>0</sub> induces decomposition). The carrier identification is *earned* via the Eight Guarantees. The 3+1 signature is *derived*.
+
+**Internal standpoint.** The carrier identification is τ-internal. E<sub>1</sub> is enriched-over-τ; "physics-as-structural-layer" is a τ-categorical claim, not a meta-physical assertion.
+
+**Step gluing — what later steps does it enable.**
+
+- **CS-05 Recover Internal Physical Grammar** instantiates the 4+1 template at E<sub>1</sub>: τ-time, τ-mass, τ-fields, τ-laws are defined inside the carrier.
+- **CS-06 Measurement Bridges** uses the No Knobs Ledger to calibrate every dimensionless ratio via `ι_τ` and the neutron-mass anchor `m_n`.
+- **CS-07 Recover Life** uses the **Parity Bridge Theorem**'s computational-bootstrap carrier as the route from physics to life recovery.
+- **CS-08 / CS-09 / CS-10** inherit E<sub>1</sub> as the layer above which higher enrichments (E<sub>2</sub> life, E<sub>3</sub> metaphysics) are constructed.
+
+**Bridge status.** CS-04 is *carrier identification*, not measurement. Bridges to standard physics begin at CS-05 (internal grammar) and consolidate at CS-06 (measurement / SI / prediction). Empirical claims have no place in CS-04 — that boundary is sharp and load-bearing.
+
+**Unresolved boundaries.** CS-04 does not by itself settle:
+
+- Specific physical content — constants, laws, particle content (CS-05 + Book IV).
+- Cosmological structure (CS-05 + Book V).
+- Empirical adequacy of any sector instantiation (CS-06).
+
+These are explicit handoffs, not concealed gaps.
+
+**This is an internal construction claim, not external acceptance.** Step 4 derives the physical carrier under τ-discipline; reviewer scrutiny is invited via Book III's full dependency chain, the No Knobs Ledger, the registry, and the TauLib formalization.
 
 ## Registry spine