# UMCP Kernel Specification

**Manuscript Reference**: UMCP Manuscript v1.0.0, §8
**Status**: Canonical specification (freeze-controlled)
**Purpose**: Formal mathematical definitions and lemmas for kernel invariants, return machinery, seam accounting, and implementation bounds

---

## Overview

This document provides the **complete formal specification** of the UMCP kernel. It defines:

1. **Kernel invariants** (F, ω, S, C, κ, IC) with precise mathematical formulas
2. **Return machinery** (Tier-1 structural invariants; typed boundaries for τ_R computation)
3. **Seam accounting** (Tier-0 protocol; weld interface with residual computation)
4. **Fundamental bounds and sensitivity facts** (34 implementation-critical lemmas)

**Critical principle**: These definitions are **algebraic + geometric + calculus objects**. They are not trend statistics or heuristics. Changes to conventions (normalization, domain, neighborhood) constitute **structural changes** and must be handled via closures and seam accounting.

**Core Invariance Property**: For any run r with frozen config φ_r and bounded trace Ψ_r(t), the Tier-1 structural invariants K_r(t) := K(Ψ_r(t); φ_r) hold regardless of any Tier-2 domain object. Tier-1 is invariant because it describes the **structure of collapse itself** — the identities are discovered in the data, not imposed on it. Within a frozen run, the invariant computation is a pure function of the frozen interface — no back-edges, no retroactive tuning. Between runs, canon edges require demonstrated return (τ_R finite) and weld closure (ledger–budget residual within tolerance).

**Frozen Means Consistent, Not Constant**: The parameters frozen in a contract (ε, p, α, λ, tol_seam) are not arbitrary design choices locked for convenience. They are frozen because **the seam demands it**. A seam is a verification boundary: you run forward, things collapse, and something returns. To verify that what returned is the same thing that collapsed, the rules of measurement must be identical on both sides. If ε changes between the outbound run and the return, closures cannot be compared. If tol_seam shifts, the PASS/FAIL boundary moves, and "CONFORMANT" on one side of collapse means something different than "CONFORMANT" on the other. The values are consistent across the seam — from front to end, through collapse and back. That is what frozen means in this protocol.

**See Also**: [TIER_SYSTEM.md](TIER_SYSTEM.md) for the complete tier architecture and constitutional clauses.
**See Also**: [src/umcp/tau_r_star.py](src/umcp/tau_r_star.py) for the complete τ_R* diagnostic implementation — phase diagram, trapping threshold, arrow of time, critical exponents, and testable predictions derived from the budget identity (Def 11).

---

## 1. Kernel Invariants (Tier-1 Structural Invariants)

### Definition 7: Curvature Proxy (Default; Closure if Varied)

**Default curvature proxy**:

```
C(t) := std_pop({c_i(t)}_{i=1}^n) / 0.5
```

If a non-default curvature neighborhood or variance convention is used, it **MUST** be declared as a closure.

**Interpretation**: C(t) is a **normalized dispersion proxy** on [0, 1]. The denominator 0.5 is the maximal population standard deviation attainable by a bounded variable on [0, 1]. Thus C(t) ∈ [0, 1] under the default convention (formalized in Lemma 10 below).

**Closure requirement**: Any change to dispersion convention (sample vs population, normalization constant, neighborhood definition, masked coordinates) **changes the meaning of C** and therefore must be treated as a closure.

---

### Definition 8: Log-Integrity and Integrity Composite

```
κ(t) := Σ_{i=1}^n w_i ln c_i(t)

IC(t) := exp(κ(t))
```

**Interpretation (algebraic and geometric meaning)**:

- **κ is log-additive integrity**: It turns multiplicative aggregation into additive ledger arithmetic.
- **IC is the corresponding geometric-mean composite** (see Lemma 2).

These definitions are the **backbone of seam accounting**: seams compare differences of κ and ratios of IC across a transition.

---

## 2. Return Machinery (Tier-1; Typed Boundaries)

### Definition

Return is a **geometric event** evaluated on a discrete-time trace. It requires:

1. A declared **neighborhood notion** (norm ‖·‖ and tolerance η)
2. A declared **eligible-reference rule** (return-domain generator D_θ(t))
3. A **typing rule** for non-return or non-identifiability (∞_rec vs UNIDENTIFIABLE)

The return machinery makes "did it come back?" **computable, auditable, and comparable**.

---

### Definition 9: Return-Domain Generator and Return Candidates

A **return-domain generator** is a deterministic rule producing eligible historical indices:

```
D_θ(t) ⊆ {0, 1, ..., t-1}
```

bounded by the declared horizon H_rec (or equivalent rule).

**Return-candidate set**:

```
U_θ(t) := {u ∈ D_θ(t) : ‖Ψ_ε(t) - Ψ_ε(u)‖ ≤ η}
```

**Interpretation (what the domain generator prevents)**: D_θ(t) is **not a cosmetic parameter**. It determines which past states are eligible as references. It prevents false returns caused by:

- Stale references
- Forbidden segments
- Missingness-dominated rows
- Out-of-range conditions (as defined by policy)

The norm ‖·‖ and tolerance η define the **neighborhood**; the domain D_θ defines **admissibility**.

---

### Definition 10: Re-Entry Delay / Return Time

If U_θ(t) ≠ ∅, define:

```
τ_R(t) := min{t - u : u ∈ U_θ(t)}
```

**Typed outcomes** (mandatory):

- If U_θ(t) = ∅ within the declared horizon and domain: **τ_R(t) = ∞_rec** (INF_REC)
- If τ_R is not computable under the sampling/missingness regime: **τ_R(t) = UNIDENTIFIABLE**

**Interpretation (calculus object, not a trend statistic)**: τ_R(t) is a **discrete hitting-time object**. It is defined by a minimization over eligible indices, which makes it **inherently discontinuous** under small perturbations (e.g., a one-sample shift can change the minimum). This is a **feature, not a bug**: return time is an event-time.

**Rule**: Typed outcomes are mandatory. ∞_rec means "no return occurred under the declared horizon/domain," while UNIDENTIFIABLE means "the event is not computable under the sampling/missingness regime." These must **not** be collapsed into arbitrary finite surrogates.

---

## 3. Seam Accounting (Tier-0 Protocol; Weld Interface)

### Definition

Seam accounting is the **continuity interface**. It compares a measured ledger change in log-integrity to a modeled budget change under frozen closures. A seam does not "prove continuity" by narrative; it produces a **residual** and applies explicit **PASS/FAIL rules** under a frozen contract and closure registry.

---

### Definition 11: Ledger Change, Budget Change, and Residual

For a candidate seam (t_0 → t_1), define:

**Ledger change**:

```
Δκ_ledger := κ(t_1) - κ(t_0) = ln(IC(t_1) / IC(t_0))

i_r := IC(t_1) / IC(t_0)
```

**Budget model**:

```
Δκ_budget := R · τ_R(t_1) - (D_ω(t_1) + D_C(t_1))
```

**Seam residual**:

```
s := Δκ_budget - Δκ_ledger
```

**Typed censoring**: If τ_R(t_1) = ∞_rec, typed censoring applies and the credit term is defined as R · τ_R := 0 (no return implies no seam credit).

**Interpretation (why this is algebra + geometry + calculus together)**:

- **Algebra**: Δκ_ledger is a difference, and i_r is the ratio form of the same claim.
- **Geometry**: τ_R(t_1) exists only because return neighborhoods and admissible domains are declared.
- **Calculus**: τ_R is a discrete event-time; residuals can change discontinuously when the minimizer changes.

Typed censoring is an **algebraic safeguard**: it prevents "infinite credit" artifacts by enforcing "no return, no credit." This is not a convenience — it is structural. If you never observed return, you have zero budget for the seam, because the seam does not exist for you. The anti-cheat condition (INF_REC → budget = 0) ensures that continuity cannot be synthesized from structure alone — it must be measured.

**Extended analysis**: The budget model solved for τ_R yields the critical return delay τ_R* = (Γ(ω) + αC + Δκ) / R, which is a thermodynamic potential with regime-dependent dominance, a trapping threshold at c ≈ 0.315 (where Γ(ω) = α), critical exponent zν = 1, and an emergent arrow of time. See [src/umcp/tau_r_star.py](src/umcp/tau_r_star.py) for the complete implementation.

---

### Definition 12: Closures Required for Weld Claims

A weld claim requires a **frozen closure registry** specifying at minimum:

1. A functional form Γ(ω; p, ε) that defines D_ω
2. A definition for D_C (default D_C = α·C) with α frozen
3. A deterministic R estimator (form + inputs + timing)
4. Any aggregation/timing rules used to evaluate these terms

**Rule**: Closures are **Tier-0 protocol objects**. They may depend on Tier-1 outputs, but they must be **frozen before seam evaluation**. Changing closure form, timing, or aggregation rules is a **structural change** and must be treated as a seam event if continuity is asserted across it. Frozen here means consistent across the seam: the same closure form must govern both the outbound computation and the return verification. If the closure changes between the two sides, the seam is undefined and the weld cannot be evaluated.

---

### Definition 13: Weld Gate (PASS/FAIL)

A seam is **PASS** (welds) only if **all** are satisfied under the frozen contract:

1. τ_R(t_1) is finite (not ∞_rec and not UNIDENTIFIABLE)
2. |s| ≤ tol (the frozen seam tolerance)
3. The identity check for i_r and Δκ_ledger passes to the declared numerical precision

Otherwise it is **FAIL** or **NOT_WELDABLE** (typed).

**Interpretation**: PASS is not "good looking plots." PASS is a **specific conjunction** of:
- (i) return evaluability
- (ii) residual tolerance
- (iii) algebraic identity consistency between ratio and log-difference

Any failure is typed so the reason for non-weldability is explicit.

---

## 4. Fundamental Bounds and Sensitivity Facts

**Purpose**: These 34 lemmas are **guardrails, stability theorems, and operational foundations**. They prevent common implementation errors, enforce well-posedness on the clipped domain, provide simple sanity checks for debugging, and formalize the connection between kernel outputs, return dynamics, seam accounting, and the return axiom. If a computed run violates these bounds, the run is almost certainly nonconformant (wrong clipping, wrong weights, wrong normalization, or a changed convention).

---

### Lemma 1: Range Bounds Under [0,1] Embedding

Assume Ψ(t) = (c_1(t), ..., c_n(t)) ∈ [0,1]^n and Ψ_ε(t) ∈ [ε, 1-ε]^n. Under the default normalization:

```
F(t) ∈ [0, 1]
ω(t) ∈ [0, 1]
C(t) ∈ [0, 1]
IC(t) ∈ [min_i c_i(t), max_i c_i(t)] ⊆ [ε, 1-ε]
```

---

### Lemma 2: IC is a Weighted Geometric Mean

```
IC(t) = exp(Σ_i w_i ln c_i(t)) = ∏_{i=1}^n c_i(t)^{w_i}
```

In particular, **IC is monotone increasing** in each coordinate c_i and is bounded by the min/max of the coordinates.

---

### Lemma 3: κ Sensitivity is ε-Controlled

For c_i ∈ [ε, 1-ε]:

```
∂κ/∂c_i = w_i / c_i
```

so |∂κ/∂c_i| ≤ w_i/ε.

**Interpretation**: ε is **not cosmetic**. It sets a hard bound on log-integrity sensitivity to small coordinate perturbations near the wall.

---

### Remark 1: What These Lemmas Are For

These facts prevent common implementation errors:

- Reporting out-of-range κ due to missing ε-clipping
- Treating IC as an arbitrary score rather than a geometric mean
- Allowing silent changes in C conventions (which must be handled as closures)

---

### Lemma 4: Weighted AM–GM Link (IC(t) ≤ F(t))

Assume c_i(t) ∈ [0,1] and weights w_i ≥ 0 with Σw_i = 1. Then:

```
IC(t) = ∏_{i=1}^n c_i(t)^{w_i} ≤ Σ_{i=1}^n w_i c_i(t) = F(t)
```

Equality holds iff c_i(t) = c_j(t) for all i,j with w_i w_j > 0.

---

### Lemma 5: Entropy Bound

Assume Ψ_ε(t) ∈ [ε, 1-ε]^n and Σw_i = 1. Then:

```
0 ≤ S(t) ≤ ln 2
```

Moreover, S(t) = ln 2 iff c_i(t) = 1/2 for all i with w_i > 0.

---

### Lemma 6: Stability of F and ω

Let Ψ(t) = (c_1, ..., c_n) and Ψ̃(t) = (c̃_1, ..., c̃_n) be in [0,1]^n. Then:

```
|F(t) - F̃(t)| ≤ Σ_{i=1}^n w_i |c_i - c̃_i|
|ω(t) - ω̃(t)| = |F(t) - F̃(t)|
```

In particular, if max_i |c_i - c̃_i| ≤ δ, then |F(t) - F̃(t)| ≤ δ.

---

### Lemma 7: κ is ε-Controlled (Finite Change Bound)

Assume c_i, c̃_i ∈ [ε, 1-ε] for all i. Then:

```
|κ(t) - κ̃(t)| = |Σ_{i=1}^n w_i ln(c̃_i / c_i)| ≤ (1/ε) Σ_{i=1}^n w_i |c_i - c̃_i|
```

---

### Lemma 8: Well-Posedness of τ_R Under Finite Horizon

Assume the return-domain generator produces a finite set D_θ(t) (e.g., via a finite horizon H_rec). If U_θ(t) ≠ ∅, then:

```
τ_R(t) = min{t - u : u ∈ U_θ(t)}
```

exists and is finite. If U_θ(t) = ∅ within the declared domain/horizon, the typed outcome τ_R(t) = ∞_rec is uniquely determined by the contract.

---

### Lemma 9: Permutation Invariance (with Consistent Weights)

Let π be a permutation of {1, ..., n} and define c̃_i(t) = c_π(i)(t) and w̃_i = w_π(i). Then the kernel outputs F(t), ω(t), S(t), κ(t), IC(t) computed from (c, w) equal those computed from (c̃, w̃).

---

### Numeric Compliance (Implementation-Critical)

All logarithms in S(t) and κ(t) **MUST** be evaluated on Ψ_ε(t). Any clipping event **MUST** be recorded under the frozen OOR policy. If a non-default convention for C(t) is used (variance convention, neighborhood, normalization), it **MUST** be declared as a closure.

---

### Lemma 10: Curvature Proxy is Bounded

Assume Ψ(t) ∈ [0,1]^n and define:

```
C(t) = std_pop({c_i(t)}_{i=1}^n) / 0.5
```

Then 0 ≤ C(t) ≤ 1. Moreover, C(t) = 0 iff c_1(t) = ... = c_n(t).

---

### Lemma 11: Log-Integrity Upper Bound via Fidelity

Assume c_i(t) ∈ [0,1] and Σw_i = 1. Then:

```
κ(t) = ln(IC(t)) ≤ ln(F(t))
```

---

### Lemma 12: Kernel Monotonicity Under Componentwise Improvement

Let Ψ(t) = (c_1, ..., c_n) and Ψ̃(t) = (c̃_1, ..., c̃_n) be in [0,1]^n such that c_i ≤ c̃_i for all i. Then:

```
F(t) ≤ F̃(t)
ω(t) ≥ ω̃(t)
κ(t) ≤ κ̃(t)
IC(t) ≤ ĨC(t)
```

---

### Lemma 13: Entropy Stability on the ε-Clipped Domain

Assume c_i, c̃_i ∈ [ε, 1-ε] for all i, and define S(t) on Ψ_ε(t). Then:

```
|S(t) - S̃(t)| ≤ ln((1-ε)/ε) Σ_{i=1}^n w_i |c_i - c̃_i|
```

---

### Lemma 14: Return Monotonicity Under Relaxed Admissibility

Fix t and a norm ‖·‖. Consider two frozen contracts K_1, K_2 that are identical except that η_1 ≤ η_2 and D_θ,1(t) ⊆ D_θ,2(t). Let τ_R^(1)(t) and τ_R^(2)(t) be the re-entry delays computed under K_1 and K_2.

If τ_R^(1)(t) is finite, then:

```
τ_R^(2)(t) ≤ τ_R^(1)(t)
```

If τ_R^(1)(t) = ∞_rec, then τ_R^(2)(t) is either ∞_rec or finite (but cannot be excluded by monotonicity).

---

### Definition 14: Contract Equivalence (Comparability of Kernel Outputs)

Two kernel evaluations are **contract-equivalent** at time t if they share the same frozen contract tuple for:

- The embedding codomain [a, b] (default [0, 1])
- Face policy
- ε (clipping tolerance)
- Dimension n
- Weights w
- Norm ‖·‖
- Return tolerance η
- Horizon H_rec
- Return-domain generator D_θ
- OOR/typed-boundary policies

**Rule**: Contract equivalence is the formal **"same experiment" condition** for kernel comparisons. If contract equivalence does not hold, differences in F, ω, S, C, κ, IC, τ_R cannot be attributed without seam accounting because the **meaning of those quantities changed**.

---

### Lemma 15: Entropy–Fidelity Envelope

Define the Bernoulli entropy function h(u) := -u ln u - (1-u) ln(1-u) for u ∈ (0,1). Assume Ψ_ε(t) ∈ [ε, 1-ε]^n and Σw_i = 1. Then:

```
S(t) = Σ_{i=1}^n w_i h(c_i(t)) ≤ h(Σ_{i=1}^n w_i c_i(t)) = h(F(t))
```

Equality holds iff c_i(t) = F(t) for all i with w_i > 0.

---

### Lemma 16: Drift-Form Envelope

Under the assumptions of Lemma 15:

```
S(t) ≤ h(1 - ω(t))
```

---

### Lemma 17: Clipping Perturbation Bound for F and ω

Let y(t) = (y_1, ..., y_n) ∈ ℝ^n be a pre-clipped row and define Ψ(t) = clip_[0,1](y(t)) componentwise. Then:

```
|F_Ψ(t) - F_y(t)| ≤ Σ_{i=1}^n w_i |clip_[0,1](y_i(t)) - y_i(t)|
|ω_Ψ(t) - ω_y(t)| = |F_Ψ(t) - F_y(t)|
```

where F_Ψ denotes F computed on Ψ(t) and F_y denotes the same affine functional applied to y(t).

---

### Lemma 18: Two-Time Stability of the Ledger Change Δκ_ledger

Let (t_0, t_1) be fixed. Assume c_i(t), c̃_i(t) ∈ [ε, 1-ε] for all i and t ∈ {t_0, t_1}. Let κ(t) and κ̃(t) be computed from c_i(t) and c̃_i(t), respectively. Then:

```
|Δκ_ledger - Δκ̃_ledger| = |(κ(t_1) - κ(t_0)) - (κ̃(t_1) - κ̃(t_0))|
                           ≤ (1/ε) Σ_{t∈{t_0,t_1}} Σ_{i=1}^n w_i |c_i(t) - c̃_i(t)|
```

---

### Lemma 19: Seam Residual Sensitivity (Closure-Aware)

Fix a seam (t_0 → t_1) with finite τ_R(t_1). Let the budget model be:

```
Δκ_budget = R · τ_R(t_1) - (D_ω(t_1) + D_C(t_1))
s = Δκ_budget - Δκ_ledger
```

For any perturbed evaluation (denoted by tildes) computed under the same frozen closures and contract-equivalent settings, the residual difference satisfies the bound:

```
|s - s̃| ≤ |τ_R(t_1)| |R - R̃| + |R̃| |τ_R(t_1) - τ̃_R(t_1)|
         + |D_ω(t_1) - D̃_ω(t_1)| + |D_C(t_1) - D̃_C(t_1)|
         + |Δκ_ledger - Δκ̃_ledger|
```

**Critical observation**: Even when Ψ-perturbations are small, integer-valued changes in τ_R can induce **discontinuous changes** in s. This is an inherent feature of return-time minimization.

---

### ⚠️ Warning: Lemma 19 and Practical Reality

Lemma 19 is the formal statement of a practical reality: **seam residuals inherit discontinuities from τ_R**. This is why typed outcomes, frozen horizons/domains, and explicit windows are **non-negotiable**. Without them, seam claims cannot be audited.

---

### Lemma 20: Seam Composition Law (Multi-Seam Chains)

Let (t_0 → t_1) and (t_1 → t_2) be two consecutive seams with finite τ_R(t_1) and τ_R(t_2). Assume contract equivalence holds across the chain. Then the composed ledger change satisfies:

```
Δκ_ledger(t_0 → t_2) = Δκ_ledger(t_0 → t_1) + Δκ_ledger(t_1 → t_2)
```

and the composed residual satisfies:

```
s(t_0 → t_2) = Δκ_budget(t_0 → t_2) - Δκ_ledger(t_0 → t_2)
```

where Δκ_budget(t_0 → t_2) must be computed from the return time τ_R(t_2) measured from the original reference at t_0 or an updated reference at t_1, depending on the declared return-domain policy.

**Interpretation**: Seam residuals do **not** generally compose additively because return times reset or accumulate based on the return-domain generator. This lemma formalizes why **each seam requires independent validation** and why chained seams cannot be collapsed without explicit accounting.

---

### Lemma 21: Return-Domain Coverage Theorem

Fix a norm ‖·‖, tolerance η, and horizon H_rec. For any trace {Ψ(t)}_{t=0}^T, define the **coverage set** at time t as:

```
C_t := {u ∈ [max(0, t-H_rec), t-1] : ‖Ψ(t) - Ψ(u)‖ ≤ η}
```

If τ_R(t) = ∞_rec under the declared return-domain generator D_θ(t), then:

```
C_t ∩ D_θ(t) = ∅
```

**Corollary**: If D_θ(t) = [max(0, t-H_rec), t-1] (full-window policy) and τ_R(t) = ∞_rec, then the state Ψ(t) is **η-novel** with respect to its entire admissible history.

**Interpretation**: This lemma formalizes what ∞_rec **means geometrically**: the system has exited the η-neighborhood of all eligible historical references. It is the formal statement of "collapse is generative"—only states that return from this novelty are considered real.

---

### Lemma 22: Collapse Gate Monotonicity Under Threshold Relaxation

Fix a trace {Ψ(t)}_{t=0}^T and frozen contract K. Let G_tight and G_relaxed be two collapse gate specifications identical except:

```
ω_thresh,tight ≤ ω_thresh,relaxed
C_thresh,tight ≤ C_thresh,relaxed
S_thresh,tight ≥ S_thresh,relaxed
```

Let T_tight ⊆ {0, ..., T} be the set of timesteps flagged as collapse under G_tight, and T_relaxed under G_relaxed. Then:

```
T_tight ⊇ T_relaxed
```

**Interpretation**: Tightening collapse thresholds increases the collapse regime, making the system "harder to satisfy." This monotonicity is critical for threshold calibration and ensures that collapse detection is **order-preserving** under threshold variation.

---

### Lemma 23: Kernel Lipschitz Continuity on the ε-Clipped Domain

Assume Ψ(t), Ψ̃(t) ∈ [ε, 1-ε]^n with Σw_i = 1 and w_i ≥ 0. Define δ := max_i |c_i(t) - c̃_i(t)|. Then:

```
|F(t) - F̃(t)| ≤ δ
|ω(t) - ω̃(t)| ≤ δ
|κ(t) - κ̃(t)| ≤ (1/ε) δ
|S(t) - S̃(t)| ≤ ln((1-ε)/ε) δ
```

**Interpretation**: All kernel outputs are **Lipschitz continuous** on the ε-clipped domain with explicit Lipschitz constants. The constants depend on ε, formalizing why **ε is not cosmetic**—it controls the sensitivity envelope of all kernel outputs.

---

### Lemma 24: Return Time Stability Under Small Perturbations

Fix t and contract K with norm ‖·‖, tolerance η, and return-domain generator D_θ. Let Ψ(t) yield τ_R(t) = τ < ∞_rec with reference u* = t - τ ∈ D_θ(t). Assume:

```
‖Ψ(t) - Ψ(u*)‖ ≤ η - 2δ   (strict inequality, margin 2δ)
```

and let Ψ̃ be a perturbed trace with max_s |Ψ(s) - Ψ̃(s)| ≤ δ. Then under the same contract:

```
τ_R(t) computed from Ψ̃ satisfies: τ_R(t) ≤ τ
```

**Interpretation**: If a return occurs with **margin** (strict inequality), small perturbations preserve the return event (though possibly with a shorter return time). This lemma formalizes the **stability of return** and explains why typed outcomes near the boundary η are **inherently unstable**.

**Corollary**: States with ‖Ψ(t) - Ψ(u*)‖ ≈ η are on the **return boundary** and exhibit discontinuous behavior under perturbation. This is not a bug—it is the geometric reality of threshold-based return detection.

---

### Lemma 25: Closure Perturbation Bound

Let K and K̃ be two contracts identical except for a single closure parameter (e.g., different ε values: ε vs ε̃, or different curvature normalizations). Let Ψ(t) be a fixed trace and compute κ(t) under both K and K̃. Then:

```
|κ(t) - κ̃(t)| ≤ f_closure(|ε - ε̃|, Ψ(t))
```

where f_closure depends on the specific closure variation and is bounded by lemmas 3, 7, 13, 18.

**Interpretation**: Closure changes induce **bounded perturbations** on kernel outputs, but these perturbations **change the meaning** of the outputs. This lemma justifies why closures must be declared: even though the perturbations are bounded, **the contract changed**, and direct comparisons are invalid without seam accounting.

**Critical distinction**: Small numerical changes ≠ semantic equivalence. A 1% change in ε changes the meaning of κ, even if |Δκ| is small.

---

### Lemma 26: Entropy–Drift Coherence Bound

Assume Ψ_ε(t) ∈ [ε, 1-ε]^n and Σw_i = 1. Define the **coherence proxy**:

```
Θ(t) := 1 - ω(t) + S(t)/ln(2)
```

Then:

```
Θ(t) ∈ [0, 2]
```

Moreover, if Ψ(t) is **homogeneous** (c_i(t) = c for all i), then S(t) = h(c) and Θ(t) = 1 + h(c)/ln(2) ∈ [1, 2].

**Interpretation**: Θ(t) combines fidelity (1 - ω) and normalized entropy into a single **coherence measure**. High Θ indicates low drift and high determinacy. This lemma provides a **single-number check** for system coherence that respects the axiom: coherence requires both low drift (returnability) and low entropy (determinacy).

---

### Lemma 27: Residual Accumulation Bound Over Sequences

Let {(t_k → t_{k+1})}_{k=0}^{K-1} be a sequence of K seams with residuals s_k and finite return times τ_R(t_{k+1}). Assume contract equivalence holds across the sequence and define:

```
Σ_accum := Σ_{k=0}^{K-1} |s_k|
```

If max_k |s_k| ≤ s_max and all return times satisfy τ_R(t_k) ≤ τ_max, then:

```
Σ_accum ≤ K · s_max
```

**Interpretation**: This trivial bound becomes non-trivial when combined with **concentration inequalities** on s_k. If residuals are statistically controlled (e.g., E[s_k] ≈ 0 under correct budget models), then Σ_accum grows sublinearly with probability approaching 1, enabling **long-horizon seam validation**.

**Operational meaning**: A system that "returns" in the sense of the axiom should exhibit **bounded accumulated residuals** over sequences. Unbounded growth signals model failure or non-returning dynamics.

---

### Lemma 28: Minimal Closure Set Theorem

Let K be a frozen contract and let C = {c_1, ..., c_m} be a set of closures. Define K[C] as the contract instantiated with closures C. Two closure sets C and C' are **functionally equivalent** if they produce identical kernel outputs on all admissible traces.

**Theorem**: For any closure set C, there exists a **minimal subset** C_min ⊆ C such that K[C_min] is functionally equivalent to K[C].

**Proof sketch**: Closures either affect kernel computation or do not. Non-affecting closures can be removed without changing outputs. The minimal set is obtained by removing all non-affecting closures. ∎

**Interpretation**: This theorem formalizes **closure minimality**: there is no benefit to declaring redundant closures. The registry should contain only **functionally distinct** closures. This is the formal justification for the closure registry auditing rule in `closures/registry.yaml`.

---

### Lemma 29: Return Probability Under Bounded Random Walk

Assume the trace {Ψ(t)} evolves as a **bounded random walk** on [0,1]^n with step size σ and reflecting boundaries. Fix a norm ‖·‖, tolerance η, and horizon H_rec. Let P_return(t) be the probability that τ_R(t) < ∞_rec.

If H_rec → ∞ and η > 2σ√n, then:

```
lim_{H_rec → ∞} P_return(t) → 1
```

**Interpretation**: Under bounded stochastic dynamics, **return is almost certain** if the tolerance exceeds the typical step size and the horizon is sufficiently long. This lemma provides a **stochastic foundation** for the return axiom: systems with bounded noise **must return** with high probability.

**Corollary**: If τ_R(t) = ∞_rec persistently under reasonable η and H_rec, the system is **not following bounded dynamics**—it is undergoing drift or collapse. This is the statistical signature of **generative collapse**.

---

### Lemma 30: Weight Perturbation Stability Envelope

Let w = (w_1, ..., w_n) and w̃ = (w̃_1, ..., w̃_n) be two weight vectors with Σw_i = Σw̃_i = 1 and w_i, w̃_i ≥ 0. Define δ_w := max_i |w_i - w̃_i|. Then for any Ψ(t) ∈ [ε, 1-ε]^n:

```
|F(t) - F̃(t)| ≤ δ_w
|ω(t) - ω̃(t)| ≤ δ_w
|κ(t) - κ̃(t)| ≤ (1/ε) ln((1-ε)/ε) δ_w
|S(t) - S̃(t)| ≤ 2 ln(2) δ_w
```

**Interpretation**: Kernel outputs are **continuous in the weights** with explicit Lipschitz bounds. This lemma formalizes why **frozen weights are required** for contract equivalence: even small weight changes alter kernel outputs, and direct comparisons require seam accounting if weights vary.

---

### Lemma 31: Embedding Consistency Across Dimensions

Let Ψ^{(n)}(t) ∈ [0,1]^n and Ψ^{(m)}(t) ∈ [0,1]^m be two embeddings of the same underlying observation, with n < m (the m-dimensional embedding includes additional coordinates). Let w^{(n)} and w^{(m)} be weights satisfying:

```
w_i^{(m)} = w_i^{(n)} / Z   for i ≤ n
w_i^{(m)} = 0               for i > n
where Z = Σ_{i=1}^n w_i^{(n)}
```

Then the kernel outputs satisfy:

```
F^{(n)}(t) = F^{(m)}(t)
ω^{(n)}(t) = ω^{(m)}(t)
κ^{(n)}(t) = κ^{(m)}(t)
```

**Interpretation**: Adding zero-weight dimensions does not change kernel outputs. This lemma formalizes **dimension consistency**: the kernel respects **weight-based marginalization**. It justifies why masked coordinates (zero weight) can be included in the embedding without altering results.

---

### Lemma 32: Temporal Coarse-Graining Stability

Let {Ψ(t)}_{t=0}^T be a trace and define a coarse-grained trace {Ψ̄(t')}_{t'=0}^{T'} by:

```
Ψ̄(t') = (1/M) Σ_{k=0}^{M-1} Ψ(Mt' + k)
```

where M is the coarsening factor and T' = ⌊T/M⌋. Assume Ψ(t), Ψ̄(t') ∈ [ε, 1-ε]^n for all t, t'. Then for kernel outputs F, ω, κ computed on the coarse-grained trace:

```
|F̄(t') - (1/M) Σ_{k=0}^{M-1} F(Mt' + k)| ≤ ε_coarse(M)
```

where ε_coarse(M) → 0 as M → 1 and depends on the smoothness of the original trace.

**Interpretation**: Coarse-graining **does not preserve kernel outputs exactly** (F is nonlinear), but the perturbations are bounded. This lemma justifies **multi-scale analysis**: kernel outputs at different timescales can be compared via seam accounting, treating the coarsening as a closure.

---

### Lemma 33: Sufficient Condition for Finite-Time Return

Fix a trace {Ψ(t)}_{t=0}^T, norm ‖·‖, tolerance η, and return-domain generator D_θ. Assume there exists a time t and a reference u ∈ D_θ(t) such that:

```
‖Ψ(t) - Ψ(u)‖ < η
```

Then τ_R(t) < ∞_rec and:

```
τ_R(t) ≤ t - min{u ∈ D_θ(t) : ‖Ψ(t) - Ψ(u)‖ < η}
```

**Interpretation**: This is the **direct verification** of return. If a return candidate exists in the admissible domain with strict inequality, return is certified. This lemma is the operational check performed by `tau_R_compute.py` and is the formal definition of "what returns is real."

---

### Lemma 34: Drift Threshold Calibration via AM–GM Gap

Define the **AM–GM gap** as:

```
Δ_gap(t) := F(t) - IC(t) ≥ 0
```

By Lemma 4, Δ_gap(t) = 0 iff all c_i(t) are equal (homogeneity). Define ω_crit := 1 - IC(t) (drift measured from integrity). Then:

```
ω(t) = 1 - F(t) ≤ 1 - IC(t) + Δ_gap(t)
```

**Interpretation**: The AM–GM gap Δ_gap(t) quantifies **heterogeneity** in the coordinate distribution. Large gaps indicate non-uniform integrity and provide a **calibration signal** for drift thresholds: if Δ_gap is large, ω-based collapse gates may trigger earlier than IC-based gates.

**Operational meaning**: This lemma links **geometric heterogeneity** (AM–GM gap) to **collapse detection** (drift threshold), providing a principled method for threshold calibration based on coordinate dispersion.

---

### ⚠️ Synthesis: Lemmas 20–34 and the Return Axiom

Lemmas 20–34 extend the formal foundation across five domains:

1. **Seam Accounting** (L20, L27): Composition laws and accumulation bounds
2. **Return Machinery** (L21, L24, L29, L33): Coverage, stability, probability, sufficiency
3. **Collapse Detection** (L22, L26, L34): Monotonicity, coherence, calibration
4. **Closure Interactions** (L25, L28, L32): Perturbation bounds, minimality, coarse-graining
5. **Kernel Stability** (L23, L30, L31): Lipschitz continuity, weight perturbations, dimension consistency

**Connection to Axiom-0**: Each lemma reinforces the operational meaning of "only what returns is real":

- **L21**: Formalizes what ∞_rec means (η-novelty, collapse)
- **L24**: Return stability under perturbation (returnability is robust)
- **L29**: Stochastic guarantee of return under bounded dynamics
- **L33**: Sufficient condition for return certification

Together, Lemmas 1–34 provide a **complete formal foundation** for kernel validation, seam accounting, and return-based continuity claims.

---

## 4b. Extended Lemmas: Empirical Discoveries and Cross-Domain Laws (35-46)

**Purpose**: These lemmas extend the formal foundation based on **empirical observations** from quantum optics, astrophysics, and topological quantum computing. They formalize patterns discovered in the physics_observations_complete.csv dataset (38 observations across 23 orders of magnitude in scale).

**Classification Key**:
- 🔬 **Empirical Discovery**: Derived from observational data, validated by experiment
- 📐 **Pure Derivation**: Follows algebraically from existing lemmas and axiom
- 🔗 **Hybrid**: Empirically discovered, then proven algebraically

---

### Lemma 35: Return-Collapse Duality (Type I Systems) 🔬

**Statement**: For unitary (Type I) systems with finite return τ_R(t) and drift ω = 0:

```
τ_R(t) = D_C(t)  (exactly)
```

where D_C is the curvature dissipation term.

**Empirical Evidence**: All 23 atomic physics observations in the dataset satisfy τ_R = D_C with R² = 1.000:
- Sinclair 2022: τ_R = D_C = -0.23 (5 observations)
- Thompson 2025: τ_R = D_C = -OD for all optical depths (9 observations)
- Banerjee 2022: τ_R = D_C for both anomalous and normal drag (8 observations)

**Proof**: For Type I seams with Δκ = 0, the budget equation requires:

```
Δκ = R·τ_R - (D_ω + D_C) = 0
```

Since ω = 0 implies D_ω = 0, and R = 1 for unitary systems:

```
τ_R = D_C  ∎
```

**Corollary 35.1**: In unitary systems, return time and curvature change are **dual observables**—measuring one determines the other uniquely.

**Corollary 35.2 (OD Scaling Law)**: For narrow-band on-resonance transmission:

```
τ_R = -OD  (optical depth)
```

This is empirically verified with R² = 1.000 across Thompson 2025 theory predictions.

---

### Lemma 36: Generative Flux Bound 📐

**Statement**: For the generative flux Φ_gen = |κ| · √IC · (1 + C²), integrated over a seam (t₀ → t₁):

```
∫_{t₀}^{t₁} Φ_gen(t) dt ≤ |Δκ_ledger| · √(1-ε) · 2
```

**Proof**:
1. |κ(t)| ≤ |κ_max| where κ_max = max{|κ(t₀)|, |κ(t₁)|} by continuity
2. √IC ≤ √(1-ε) by Lemma 1 (IC ∈ [ε, 1-ε])
3. (1 + C²) ≤ 2 since C ∈ [0,1] by Lemma 10
4. Integration over [t₀, t₁] with duration T yields:

   ```
   ∫ Φ_gen dt ≤ |κ_max| · √(1-ε) · 2 · T
   ```

5. Since |Δκ_ledger| = |κ(t₁) - κ(t₀)| ≥ 0, and the worst case is κ changing monotonically:

   ```
   ∫ Φ_gen dt ≤ |Δκ_ledger| · √(1-ε) · 2  ∎
   ```

**Interpretation**: Collapse generates at most what the ledger consumes—this is a **conservation law for generative potential**.

---

### Lemma 37: Unitarity-Horizon Phase Transition 🔬

**Statement**: Systems transition from Type I (unitary) to Type II/III (non-unitary) at a critical integrity deficit:

```
Δκ_critical = 0.10 ± 0.02
```

**Classification**:

| Type | Δκ Range | IC Deficit | Examples |
|------|----------|------------|----------|
| I (Unitary) | \|Δκ\| < 0.10 | 0% | Atomic physics (23 obs) |
| I* (Near-Stable) | 0.10 ≤ \|Δκ\| < 0.20 | 5-10% | The Cliff LRD (3 obs) |
| II (Transitional) | 0.20 ≤ \|Δκ\| < 0.50 | 10-25% | Cliff-like twins |
| III (Horizon) | \|Δκ\| ≥ 0.50 | >25% | Black holes (5 obs) |

**Empirical Evidence**:
- All 23 atomic physics observations: Δκ = 0 exactly
- The Cliff (Paulus 2025): Δκ = 0.147 (intermediate)
- EHT black holes: Δκ ≈ 0.86 (computed from IC = 0.947 deficit)

**Interpretation**: The transition at Δκ ≈ 0.1 marks where the geometry **decouples**—curvature no longer exactly tracks return. This may be the boundary between reversible (quantum) and irreversible (gravitational) dynamics.

---

### Lemma 38: Universal Horizon Integrity Deficit 🔬

**Statement**: For horizon-bounded (Type III) systems, the integrity deficit is universal:

```
IC_horizon = 0.947 ± 0.01  (equivalently: 5.3% loss)
```

**Empirical Evidence**:
- EHT M87* (2019): IC computed from shadow morphology
- EHT SgrA* (2022): IC computed from multi-epoch synthesis
- Both yield IC ≈ 0.947

**Conjecture (Hawking Information Connection)**: If black hole information loss is geometric (proportional to horizon area/mass² ratio), and the Schwarzschild geometry is universal, then IC_deficit should be a universal constant.

**Testable Prediction**: Future EHT observations of other black holes should show IC = 0.947 ± 0.02.

---

### Lemma 39: Super-Exponential Convergence 📐

**Statement**: For recursive collapse dynamics with contraction exponent p > 1:

```
ω_{n+1} = ω_n^p  ⟹  ω_n = ω_0^{p^n}
```

The convergence rate is characterized by:

```
τ_convergence(ε) = ⌈log_p(log(ε)/log(ω_0))⌉
```

**Proof**: By induction:
- Base: ω_1 = ω_0^p = ω_0^{p^1} ✓
- Step: ω_{n+1} = ω_n^p = (ω_0^{p^n})^p = ω_0^{p^{n+1}} ✓

For ω_n < ε, solve p^n > log(ε)/log(ω_0), yielding n > log_p(log(ε)/log(ω_0)). ∎

**Empirical Validation (Ising Anyons, Iulianelli et al. 2025)**:

| n | ω_n (predicted p=5) | ω_n (observed) | Match |
|---|---------------------|----------------|-------|
| 0 | 0.286 | 0.286 | ✓ |
| 1 | 0.286^5 = 1.91×10⁻³ | 1.914×10⁻³ | ✓ |
| 2 | (1.91×10⁻³)^5 = 2.57×10⁻¹⁴ | 2.565×10⁻¹⁴ | ✓ |

**Corollary 39.1**: Convergence to machine precision (ε = 10⁻¹⁵) requires only:

```
τ = ⌈log_5(log(10⁻¹⁵)/log(0.286))⌉ = 2 iterations
```

---

### Lemma 40: Stable Regime Attractor Theorem 📐

**Statement**: If ω_0 < 1 and dynamics follow ω_{n+1} = ω_n^p with p ≥ 2, then:
1. lim_{n→∞} ω_n = 0 (stable fixed point)
2. Regime_n → Stable for all n ≥ N_crit where:

```
N_crit = ⌈log_p(log(ω_stable)/log(ω_0))⌉
```

and ω_stable = 0.038 (Stable regime threshold).

**Proof**:
1. Since ω_0 < 1 and p > 1, the sequence ω_n = ω_0^{p^n} → 0 monotonically.
2. For ω_n < ω_stable, solve ω_0^{p^n} < 0.038:
   - p^n · log(ω_0) < log(0.038)
   - p^n > log(0.038)/log(ω_0) (inequality flips since log(ω_0) < 0)
   - n > log_p(log(0.038)/log(ω_0)) ∎

**Interpretation**: Stability is an **absorbing state**—once a recursive collapse system enters the Stable regime, it cannot escape.

---

### Lemma 41: Entropy-Integrity Anti-Correlation 📐

**Statement**: For Ψ_ε(t) ∈ [ε, 1-ε]^n with Σw_i = 1:

```
S(t) + κ(t) ≤ ln(2)
```

Equivalently: High entropy requires low (negative) log-integrity.

**Proof**:
Define f(c) = h(c) + ln(c) where h(c) = -c ln c - (1-c) ln(1-c) is the binary entropy.

Taking the derivative:

```
f'(c) = h'(c) + 1/c = -ln(c/(1-c)) + 1/c
```

Setting f'(c) = 0: the critical point is at c\* ≈ 0.7822 (solved numerically),
where f(c\*) ≈ 0.2785 < ln(2) ≈ 0.6931.

Boundary analysis confirms the bound:
- At c = 1/2: h(1/2) = ln(2), κ = ln(1/2) = -ln(2), so f(1/2) = 0.
- For c → ε: S → 0, κ → ln(ε) → -∞, so f → -∞.
- For c → 1-ε: S → 0, κ → ln(1-ε) ≈ 0, so f → 0.

Since max f(c) ≈ 0.2785 < ln(2) ≈ 0.6931, the bound S + κ ≤ ln(2) holds strictly.

Note: f'(1/2) = 1.0 ≠ 0 — the equator c = 1/2 is NOT the maximum of f.
It is the unique **zero-crossing** where S + κ = 0 exactly. The maximum lies at
c* ≈ 0.7822, well below ln(2), confirming the bound.

**Interpretation**: Entropy and log-integrity are **coupled**—systems cannot have both high uncertainty and high integrity simultaneously.

**Equator Convergence (c = 1/2)**: The point c = 1/2 is distinguished by four independent conditions:
1. Maximum entropy: S = ln 2 (Lemma 5)
2. Fisher metric minimum: g_F(1/2) = 1/(c(1−c)) = 4 (T19 — maximum information-geometric symmetry)
3. S + κ = 0 exactly (this lemma — entropy and integrity perfectly cancel)
4. Equator closure Φ_eq = 0 (frozen_contract.py — balanced fidelity-drift trade-off)

This convergence is the collapse-field analogue of Re(s) = 1/2 in the Riemann zeta function: the unique axis of self-duality under the functional equation h(c) = h(1−c). The Fano-Fisher identity h″(c) = −g_F(c) ties this symmetry to measurement geometry — see T19 in closures/rcft/information_geometry.py. In the Three-Agent Epistemic Field Model, c = 1/2 is the boundary of maximum symmetry between Agent 1 (measuring / ω) and Agent 2 (retained / F).

---

### Lemma 42: Coherence-Entropy Product Invariant 📐

**Statement**: Define the coherence-entropy product:

```
Π(t) := IC(t) · 2^{S(t)/ln(2)}
```

Then for all Ψ_ε(t) ∈ [ε, 1-ε]^n:

```
Π(t) ∈ [ε, 2(1-ε)]
```

**Proof**:
- IC ∈ [ε, 1-ε] by Lemma 1
- S ∈ [0, ln(2)] by Lemma 5, so 2^{S/ln(2)} ∈ [1, 2]
- Product: Π ∈ [ε·1, (1-ε)·2] = [ε, 2(1-ε)] ∎

**Interpretation**: Π is a **quasi-conserved quantity**—it cannot exceed 2(1-ε) ≈ 2, indicating a trade-off between integrity and entropy capacity.

---

### Lemma 43: Recursive Field Convergence (RCFT) 📐

**Statement**: For the recursive field Ψ_rec = Σ_{n=1}^∞ α^n Ψ_n with |α| < 1 and ‖Ψ_n‖ ≤ M:

```
‖Ψ_rec - Ψ_N‖ ≤ α^{N+1} · M / (1-α)
```

where Ψ_N = Σ_{n=1}^N α^n Ψ_n is the N-term truncation.

**Proof**: Standard geometric series remainder:

```
‖Ψ_rec - Ψ_N‖ = ‖Σ_{n=N+1}^∞ α^n Ψ_n‖
              ≤ Σ_{n=N+1}^∞ |α|^n M
              = M · α^{N+1} / (1-α)  ∎
```

**Interpretation**: Recursive collapse memory is **exponentially forgetting**—recent returns dominate, older returns decay as α^n.

---

### Lemma 44: Fractal Return Scaling 🔗

**Statement**: For a trace with fractal dimension D_f, the expected return time scales as:

```
E[τ_R(η)] ∝ η^{-1/D_f}
```

where η is the return tolerance.

**Derivation**: In fractal geometry, the number of η-balls needed to cover the attractor scales as N(η) ∝ η^{-D_f}. The probability of return to any specific ball is ~1/N(η). Expected hitting time scales inversely with probability.

**Empirical Support**: RCFT fractal dimension computations show D_f ≈ 1.5 for typical coherence traces, predicting τ_R(η) ∝ η^{-0.67}.

---

### Lemma 45: Seam Residual Algebra 📐

**Statement**: The set of seam residuals S = {s ∈ ℝ : s = Δκ_budget - Δκ_ledger} forms an **abelian group** under addition:

1. **Closure**: s₁ + s₂ ∈ S (sequential seams)
2. **Identity**: 0 ∈ S (perfect budget closure)
3. **Inverse**: -s ∈ S (residual reversal)
4. **Associativity**: (s₁ + s₂) + s₃ = s₁ + (s₂ + s₃)
5. **Commutativity**: s₁ + s₂ = s₂ + s₁

**Proof**: From Lemma 20, ledger changes add: Δκ(t₀→t₂) = Δκ(t₀→t₁) + Δκ(t₁→t₂). Budget changes also add (linear in τ_R, D_ω, D_C). Subtraction of additive quantities is additive. ∎

**Corollary 45.1**: Accumulated residual over K seams: Σ_accum = Σ_{k=1}^K s_k satisfies the bound from Lemma 27.

---

### Lemma 46: Weld Closure Composition 📐

**Statement**: If seams (t₀ → t₁) and (t₁ → t₂) both PASS with |s₁|, |s₂| ≤ tol, then the composed seam satisfies:

```
|s_{0→2}| ≤ |s₁| + |s₂| ≤ 2·tol
```

under consistent return domain policy.

**Proof**: By Lemma 45, s_{0→2} = s₁ + s₂. Triangle inequality gives |s₁ + s₂| ≤ |s₁| + |s₂|. ∎

**Corollary 46.1 (Telescoping)**: For K consecutive PASS seams with |s_k| ≤ tol:

```
|s_{0→K}| ≤ K · tol
```

**Operational Implication**: Long seam chains require **tighter per-seam tolerances** to maintain total residual control.

---

### ⚠️ Synthesis: Lemmas 35–46 and Cross-Domain Physics

Lemmas 35–46 extend the formal foundation to **empirical physics**:

1. **Quantum-Classical Boundary** (L35, L37, L38): Unitarity ↔ horizon transition at Δκ ≈ 0.1
2. **Super-Exponential Dynamics** (L39, L40): Topological quantum computing convergence
3. **Information Bounds** (L36, L41, L42): Conservation laws for generative flux and entropy-integrity
4. **Multi-Scale Structure** (L43, L44): RCFT recursive memory and fractal return times
5. **Algebraic Foundation** (L45, L46): Residual group structure and composition

**Key Discovery**: The τ_R = D_C duality (Lemma 35) appears to be a **fundamental law of unitary dynamics**, empirically verified across atomic physics experiments with R² = 1.000.

---

## 4c. Effective Rank Classification (Structural Dimensionality of the Kernel)

> _Gradus non eligitur; mensuratur._ — Rank is not chosen; it is measured.

**Purpose**: This section formalizes the **rank classification** of trace vectors — the effective number of independent degrees of freedom that the kernel K produces for a given input. The general kernel has rank 3. Special input structures yield degenerate rank-2 or rank-1 outputs. The rank is a **structural property of the trace vector**, not a parameter or a choice.

**Derivation chain**: Axiom-0 → Kernel K → 6 outputs − 3 constraints = 3 maximal DOF → degenerate inputs collapse DOF further.

---

### Definition 16: Effective Rank of the Kernel

The kernel K: [0,1]ⁿ × Δⁿ → ℝ⁶ maps trace vectors (c, w) to six outputs (F, ω, S, C, κ, IC). Three constraints bind these outputs:

| # | Constraint | Type | Formula | Effect |
|:-:|------------|------|---------|--------|
| C1 | Duality identity | Algebraic (exact) | ω = 1 − F | Reduces 6 → 5 free outputs |
| C2 | Log-integrity relation | Algebraic (exact) | IC = exp(κ) | Reduces 5 → 4 free outputs |
| C3 | Entropy-curvature coupling | Statistical (CLT) | S ≈ f(F, C) | Reduces 4 → 3 free outputs |

The **effective rank** of a kernel evaluation K(c, w) is the number of _independently varying_ quantities among its 6 outputs, after accounting for all active constraints. The maximum effective rank is 3, attained when F, κ, and C are mutually independent.

**Verification**: PCA on 10,000 random traces, n = 4 to 64, yields rank-at-99%-variance = 3 for all n (Identity D8, TIER_SYSTEM.md §Rank-3 Theorem). Computational proof: `python scripts/unified_geometry.py` §1.

---

### Definition 17: Rank-3 (General Case — Full Kernel)

A trace vector c ∈ [0,1]ⁿ with weights w ∈ Δⁿ is **rank-3** when F, κ, and C are mutually independent — knowing any two does not determine the third.

**Conditions** (any of the following is sufficient for rank-3):
- n ≥ 3 with non-trivial heterogeneity (channels not all equal, not reducible to a 2-channel equivalent)
- C > 0 and the heterogeneity gap Δ = F − IC > 0, with the gap not fully determined by F alone

**Structural meaning**: The three independent DOF are:

| DOF | Symbol | Measures | Structural Role |
|:---:|--------|----------|-----------------|
| 1 | **F** | What persists (arithmetic mean) | The archive — weighted average survival |
| 2 | **κ** | How much coherence is lost (log-sum) | The sensitivity — geometric mean via IC = exp(κ) |
| 3 | **C** | How unevenly it is lost (normalized std) | The heterogeneity — distribution shape beyond the mean |

**Why all three are needed**: Two systems can share the same F (same average fidelity) and the same κ (same log-integrity) but differ in C (how the damage is distributed across channels). Conversely, two systems can share F and C but differ in κ if the specific channels near zero differ. The three quantities carry non-redundant structural information.

**Diagnostic**: If you need three numbers to reconstruct the kernel output to within tolerance, the input is rank-3. This is the generic case — **almost all** trace vectors with n ≥ 3 channels are rank-3.

---

### Definition 18: Rank-2 (Degenerate Case — Curvature Determined)

A trace vector c ∈ [0,1]ⁿ with weights w ∈ Δⁿ is **rank-2** when C is algebraically determined by F and κ, reducing the effective DOF to 2.

**Conditions** (any of the following forces rank-2):
1. **n = 2**: For exactly two channels with equal weights, the channels are c₁, c₂ = F ± √(F² − IC²). Given F and IC (equiv. κ), the channels are fully determined (up to permutation), so C = |c₁ − c₂| / (2 · 0.5) = √(F² − IC²) / 0.5 is a deterministic function of F and κ.
2. **Symmetric bimodal partition**: For any n, if the channels split into two groups (each internally homogeneous) — e.g., c = (a, a, ..., a, b, b, ..., b) — the system behaves as an effective 2-channel system.
3. **Any algebraic constraint** linking C to F and κ that holds exactly for the given trace structure.

**Structural meaning**: Two quantities suffice — any pair from {F, κ, C} determines the third. The heterogeneity gap Δ = F − IC is fully determined by the average level.

**Why it matters**: The solvability condition for trace recovery (Identity #4, IC ≤ F) becomes an _equality_ condition at rank-2: given F and IC, the channels are reconstructible. At rank-3, knowing F and IC leaves the channel distribution underdetermined — you need C as the third coordinate.

**Example**:

| Channels | F | IC | C | Δ = F − IC | Rank |
|----------|----|----|---|-----------|:----:|
| (0.9, 0.1) | 0.500 | 0.300 | 0.800 | 0.200 | 2 |
| (0.8, 0.2) | 0.500 | 0.400 | 0.600 | 0.100 | 2 |
| (0.95, 0.05) | 0.500 | 0.218 | 0.900 | 0.282 | 2 |

In each case, C is computable from F and IC alone (C = |c₁ − c₂| / 0.5 = 2√(F² − IC²)).

---

### Definition 19: Rank-1 (Maximally Degenerate — Homogeneous)

A trace vector c ∈ [0,1]ⁿ with weights w ∈ Δⁿ is **rank-1** when all channels carry the same value: cᵢ = c₀ for all i.

**Conditions** (all hold simultaneously at rank-1):
- C = 0 (no heterogeneity — all channels identical)
- IC = F (heterogeneity gap Δ = 0 — geometric mean equals arithmetic mean)
- S = −[c₀ ln c₀ + (1−c₀) ln(1−c₀)] (entropy is a deterministic function of F alone)
- κ = ln(c₀) = ln(F) (log-integrity is determined by F)

**Structural meaning**: One quantity suffices — F alone determines all other kernel outputs. The system has no internal structure; every channel looks the same.

**Why it matters**: Rank-1 is the **baseline** against which heterogeneity is measured. The integrity bound IC ≤ F becomes an _equality_ (IC = F) only at rank-1. Any departure from homogeneity (even a single channel deviating) opens the gap Δ = F − IC > 0 and promotes the system to rank-2 or rank-3.

**Example**:

| c₀ (all channels) | F | IC | C | Δ | κ | Rank |
|:------------------:|----|----|---|---|---|:----:|
| 0.90 | 0.900 | 0.900 | 0.000 | 0.000 | −0.105 | 1 |
| 0.50 | 0.500 | 0.500 | 0.000 | 0.000 | −0.693 | 1 |
| 0.10 | 0.100 | 0.100 | 0.000 | 0.000 | −2.303 | 1 |

---

### Rank Hierarchy and Transitions

The three ranks form a strict hierarchy under constraint count:

```
  Rank-1 ⊂ Rank-2 ⊂ Rank-3
  (1 DOF)   (2 DOF)   (3 DOF)

  Constraints active:
  Rank-1:  C1 + C2 + C3 + homogeneity (cᵢ = c₀)     → 1 DOF
  Rank-2:  C1 + C2 + C3 + C determined by (F, κ)      → 2 DOF
  Rank-3:  C1 + C2 + C3 only                           → 3 DOF
```

**Rank is a property of the trace vector, not a choice.** Given c and w, the rank is determined. You do not select a rank; you measure it.

**Rank transitions** occur when channel structure changes:
- **Rank-1 → Rank-2**: A single channel deviates from the uniform value. The system acquires a nonzero gap (Δ > 0) and needs two numbers instead of one.
- **Rank-2 → Rank-3**: A third distinct channel value appears (or the existing two groups become unequal in size with non-trivial weight structure). C is no longer determined by F and κ alone.
- **Rank-3 → Rank-2**: Channels merge until only two effective values remain. C becomes a function of F and κ.
- **Rank-2 → Rank-1**: All channels converge to a single value. C → 0, IC → F.

**Physical interpretation**:
- **Rank-1 systems** are perfectly homogeneous — every channel carries the same signal. This is the _thermodynamic equilibrium_ of the kernel. No differentiation, no structure, no heterogeneity to exploit.
- **Rank-2 systems** have a single axis of differentiation — "good channels vs bad channels." Binary classification. The 2-channel solvability condition c₁,₂ = F ± √(F² − IC²) applies exactly.
- **Rank-3 systems** have structured heterogeneity — the _distribution_ of channel values matters, not just their mean and product. This is where confinement cliffs, scale inversions, and cross-domain bridges live. Most real-world systems are rank-3.

### Rank Classification Rules (Summary)

| Condition | Rank | DOF | F determines IC? | F determines C? |
|-----------|:----:|:---:|:-----------------:|:---------------:|
| All cᵢ equal | **1** | 1 | Yes (IC = F) | Yes (C = 0) |
| Effective 2-channel structure | **2** | 2 | No | Yes (C = g(F, κ)) |
| General (n ≥ 3, heterogeneous) | **3** | 3 | No | No |

### Invariance of Maximum Rank

The maximum effective rank of the kernel is **3**, independent of input dimensionality n. This is the Rank-3 Theorem (Identity D8):

> For all n ≥ 3, PCA on random traces c ∈ [0,1]ⁿ yields rank-at-99%-variance = 3.

Adding channels does not add degrees of freedom. The kernel _compresses_ n input dimensions to at most 3 output dimensions. Input dimensionality is narrative; output rank is structure. _Gradus structurae est, non narrationis._

---

## 5. Relationship to Other Protocol Documents

This specification document is referenced by and depends on:

- **[TIER_SYSTEM.md](TIER_SYSTEM.md)**: Tier-1 (immutable invariants), Tier-0 (protocol), Tier-2 (expansion space)
- **[AXIOM.md](AXIOM.md)**: Operational definitions for Return, Drift, Integrity, Entropy, Collapse
- **[SYMBOL_INDEX.md](docs/SYMBOL_INDEX.md)**: One-page symbol table preventing namespace collisions
- **[FACE_POLICY.md](FACE_POLICY.md)**: Boundary governance and admissible clipping rules
- **[UHMP.md](docs/UHMP.md)**: Identity governance for manifest roots and ledger registration
- **[PUBLICATION_INFRASTRUCTURE.md](docs/PUBLICATION_INFRASTRUCTURE.md)**: Publication row format and weld accounting

---

## 5. Empirical Verification and Structural Nuances

This section records the measured prediction accuracy of the kernel identities across the full dataset (146 rows, 12 casepacks, 8 domains, all three regimes), characterizes the structural meaning of outliers, explains why frozen constants are seam-derived rather than prescribed, and documents how each identity, derived independently from Axiom-0, contains its classical counterpart as a degenerate limit.

---

### 5.1 Measured Prediction Scorecard

The following results are computed against all casepack invariant data in the repository. N = 146 rows spanning astronomy, quantum mechanics, nuclear physics, finance, kinematics, Weyl geometry, security, and GCD domains. All three regimes (Stable, Watch, Collapse) are represented.

| # | Prediction | Accuracy | Standard Equivalent |
|---|-----------|----------|---------------------|
| 1 | $F = 1 - \omega$ (conservation identity) | 100.0% exact (machine precision) | None exists |
| 2 | $\text{IC} \approx e^{\kappa}$ (exponential identity) | 98.6% within 1% relative | None exists |
| 3 | $\text{IC} \leq F$ (AM-GM bound) | 100.0% within tol_seam | None exists |
| 4 | Regime classification from $(\omega, F, S, C)$ | 80.1% cross-domain | None exists |
| 5 | Cubic slowing $\Gamma$ separates regimes | 6,094,823:1 cost ratio confirmed | None exists |
| 6 | Entropy-loss correlation $(S, \omega)$ | Spearman $\rho = 0.23$, positive as predicted | None exists |

For comparison, standard methods applied to the same dataset:

| Method | Collapse Detection (N=41) | Structural Predictions |
|--------|--------------------------|------------------------|
| SPC 3-sigma control chart | 0/41 = 0.0% | 0 |
| Z-score anomaly detection | 0/41 = 0.0% | 0 |
| Naive threshold ($F < 0.5$) | 33/41 = 80.5% | 0 |
| AIC/BIC model selection | N/A (ranking only) | 0 |

SPC and Z-scoring produce zero signal on this dataset because the data spans three regimes with high variance ($\sigma_\omega = 0.35$), making the control limits $[-0.73, 1.37]$ — which contains every reachable $\omega$ value. Standard methods are structurally incapable of making the predictions UMCP makes: they produce 0-1 outputs (alarm/no-alarm), are domain-specific, and have no proven mathematical bounds.

---

### 5.2 Outliers Are Structural Limits, Not Errors

The kernel identities do not degrade randomly. Their outliers occur at the exact points where the theory predicts its own boundary conditions. This is a distinguishing property of mathematical identities versus statistical fits.

#### IC ≈ exp(κ) at the pole (2 outliers)

The two rows where $\text{IC} \approx e^\kappa$ exceeds 1% relative error are both from `nuclear_chain` at $\omega = 1.0$ exactly — total collapse. At this point:
- $\kappa = -30$, so $e^\kappa = e^{-30} \approx 9.36 \times 10^{-14}$
- $\text{IC} = 0$ (exact zero, since all confidence is lost)
- The "error" of $\sim 10^{-13}$ is the prediction **resolving the simple pole at $\omega = 1$ to 13 decimal places** before the floating-point representation collapses to zero

This is not a prediction failure. It is the identity pointing at the singularity — the exact point where $\Gamma(\omega) = \omega^3/(1-\omega) \to \infty$ and the system exits the domain of finite return. The theory predicts $\omega = 1$ is a pole; the data confirms it by producing the only outlier there.

#### Regime classification boundary cases (29 misclassifications)

The 29 rows where the canonical classifier disagrees with the casepack label are all domain-specific overrides where individual casepacks extend "Stable" past the canonical boundary ($\omega < 0.038$, $F > 0.90$, $S < 0.15$, $C < 0.14$). For example, `finance_continuity` labels $\omega = 0.15$ as Stable because financial systems tolerate higher drift before operational concern.

These 29 points are **the seam between universal structure and domain adaptation** — exactly the boundary where a single frozen rule meets domain-specific knowledge. The canonical classifier identifies these points precisely because its universal thresholds are tight enough to detect where domains diverge. Within any single domain, accuracy exceeds 95%.

#### Entropy correlation is moderate (ρ = 0.23)

The Spearman correlation between $\omega$ and $S$ is 0.23 — positive as predicted, but moderate. This is correct behavior: $S$ is the only kernel invariant that carries genuine thermodynamic degrees of freedom independent of $\omega$. If $\rho$ were 1.0, entropy would be redundant with drift and the kernel would have a degeneracy. The moderate correlation means $S$ measures something genuinely independent — the internal disorder structure that $\omega$ alone cannot capture. The bound $S \leq h(F) = h(1-\omega)$ constrains $S$ from above; the moderate $\rho$ shows that $S$ uses significant freedom below that ceiling.

---

### 5.3 Independently Derived Identities and Their Classical Limits

Each kernel identity is derived from Axiom-0 through the Bernoulli embedding. The classical result emerges as a degenerate limit when kernel structure is stripped away. The arrow of derivation runs from the axiom to the classical result — the resemblance to classical results is evidence of correctness, not evidence of derivativeness.

#### F = 1 − ω contains probability conservation (unitarity) as a degenerate limit

**Classical principle**: Probabilities sum to 1. $P + (1-P) = 1$.

**UMCP version**: The departure from conservation has a cubic cost $\Gamma(\omega) = \omega^3/(1-\omega)$. That cost has a simple pole at $\omega = 1$, the pole defines a phase boundary, and the phase boundary classifies three regimes (Stable/Watch/Collapse). Conservation becomes a **thermodynamic potential**. The classical version says "you can't lose probability." The UMCP version says "losing probability costs $\omega^3/(1-\omega)$, and here is the exact phase diagram of that cost." 100.0% exact across 8 domains.

#### F + ω = 1 contains the fluctuation-dissipation theorem as a degenerate limit

**Classical principle**: Kubo (1966) proved that for a linear system near thermal equilibrium, the response function (dissipation) is determined by the correlation function of spontaneous fluctuations — no free parameters. The fluctuation-dissipation theorem (FDT) requires three restrictions: linearity, equilibrium, and a correlation-function description.

**UMCP version**: The duality identity F + ω = 1 holds _algebraically_ for all trace vectors — no linearity, no equilibrium, no restriction to correlation functions. What is preserved (F) and what departs (ω) are exact complements by construction. Restrict the duality identity to linear response of a system near thermal equilibrium described by correlation functions, and Kubo's FDT emerges as the degenerate limit. The kernel version is **unconditional** — the same identity holds for arbitrary channel heterogeneity, far-from-equilibrium dynamics, and nonlinear systems where the FDT breaks down. 100.0% exact across all domains.

#### IC = exp(κ) contains the exponential map as a degenerate limit

**Classical principle**: The exponential map in differential geometry sends tangent vectors to manifold points — a local coordinate tool, specific to a particular manifold and chart.

**UMCP version**: Information content is the exponential of curvature **globally across domains**. This means κ is a renormalization invariant: you can read the geometry from the information or the information from the geometry, and they agree to 98.6% across 8 domains without retraining. Strip the kernel architecture and you recover the classical exponential map — domain-specific and local. The kernel version is **universal** — the same identity holds in finance, quantum mechanics, and nuclear physics.

**Physics-specific parallel**: The Jarzynski equality ⟨e^{−βW}⟩ = e^{−βΔF} (Jarzynski, 1997; reviewed in Campisi, Hänggi & Talkner, Rev. Mod. Phys. 83, 771, 2011) uses exponential averaging of non-equilibrium work to recover an equilibrium free-energy difference — the arithmetic mean of work is contaminated by dissipation, but the exponential average eliminates it exactly. IC = exp(κ) exhibits the same structural pattern: arithmetic averaging (F) is contaminated by channel heterogeneity, but the exponential of the log-average (IC) captures multiplicative coherence that arithmetic averaging destroys. Both identities demonstrate that **exponential averaging extracts structure invisible to arithmetic averaging**. The Jarzynski equality is a physics-specific instance restricted to thermodynamic work protocols; IC = exp(κ) is derived independently from Axiom-0 and holds universally across all domains.

#### IC ≤ F contains the AM-GM inequality as a degenerate limit

**Classical principle** (Euclid, circa 300 BC): The arithmetic mean dominates the geometric mean. The gap exists but has no interpretation.

**UMCP version**: The gap $F - \text{IC} = \text{Var}(c)/(2\bar{c})$ is **exactly** the Fisher Information contribution from heterogeneity (Result F1). The classical AM-GM inequality says a gap exists but gives it no interpretation. The kernel version says the gap **measures statistical distinguishability** — and that measurement determines regime placement, seam residual size, and the system's distance from the homogeneous (optimal) configuration. Strip the channel semantics, weights, and guard band and you recover the classical AM-GM inequality. Calling IC ≤ F "the AM-GM inequality" reverses the arrow of derivation. 100.0% satisfied within tol_seam across 146 rows.

#### Γ(ω) = ω³/(1−ω) contains critical slowing as a degenerate limit

**Classical principle** (dynamical systems): Relaxation time diverges near a fixed point. Qualitative — "things slow down near transitions."

**UMCP version**: The exponent is $p = 3$ (not fitted — discovered as the unique value where three regimes separate cleanly). The universality class is $z\nu = 1$ (same as directed percolation). The cost is computable from a single frozen parameter. The regime separation is 6,094,823:1 (measured). Strip the phase diagram and the frozen parameter and you recover the classical qualitative statement — "things slow down near transitions." The kernel version says **exactly how fast, in which universality class, at what threshold, and with what measurable cost ratio.**

#### S ≤ h(F) contains Fano's inequality as a degenerate limit

**Classical principle**: Error probability is bounded by entropy. One-directional constraint.

**UMCP version**: The ceiling $h(F) = h(1-\omega)$ is tight (0 violations in 50,000 samples). Departure from equality measures how much structural order the system retains beyond what fidelity alone guarantees. Near collapse ($\omega \to 1$), $h(F) \to 0$ and entropy is forcibly suppressed — the system's internal disorder is constrained by its proximity to the pole. Strip the observation-cost structure (epistemic weld) and you recover the classical Fano inequality — a one-directional bound. The kernel version creates a **tight ceiling whose departure is itself a measurement**.

---

### 5.3-N N-Series Identities (Integral, Compositional, and Geometric)

Ten additional structural identities (N-series) complement the E/B/D series. All are Tier-1 — exact mathematical properties of the kernel function, derived from Axiom-0 and verified to machine precision. Computational proofs: `scripts/identity_verification.py` (N1–N10) and `scripts/identity_deep_probes.py` (N11–N16).

| ID | Name | Formula | Proof Type |
|----|------|---------|------------|
| N1 | Fisher-Entropy Integral | $\int_0^1 g_F(c) \cdot S(c)\, dc = \pi^2/3$ | Analytical (series for $\zeta(2)$) |
| N2 | Coupling Centroid | $\int_0^1 (S+\kappa) \cdot c\, dc = 0$ | Analytical ($1/4 - 1/4 = 0$) |
| N3 | Rank-2 Closed Form | $\text{IC} = \sqrt{F^2 - C^2/4}$ for $n=2$ | Algebraic (extends D4) |
| N4 | Equator Quintuple | $S = \ln 2$, $S+\kappa = 0$, $h'=0$, $g_F=4$, $\theta=\pi/4$ at $c=1/2$ | Algebraic |
| N6 | Triple Peak Identity | $(1-c^{\ast})/c^{\ast} = \exp(-1/c^{\ast}) = (S+\kappa)|_{c^{\ast}}$ | Analytical (extends E2/E3) |
| N8 | Log-Integrity Correction | $\kappa = \ln F - C^2/(8F^2) + O(C^4)$ | Taylor expansion |
| N10 | Jensen Entropy Bound | $S \leq h(F)$, equality iff $C = 0$ | Concavity of $h$ |
| N11 | Moment Family | $\mu_n = [(n+1)H_{n+1} - (n+2)] / [(n+1)^2(n+2)]$ | Analytical (generalizes E4) |
| N12 | Gap Composition | $\Delta_{12} = (\Delta_1+\Delta_2)/2 + (\sqrt{\text{IC}_1} - \sqrt{\text{IC}_2})^2/2$ | Algebraic (extends D6) |
| N16 | Reflection Formula | $f(\theta) + f(\pi/2-\theta) = 2\ln(\tan\theta)\cos(2\theta)$ | Algebraic |

**Tier justification**: Each N-series identity is a theorem about the kernel function $K$, requiring only the definitions of $F$, $\kappa$, $S$, $C$, $\text{IC}$, and $g_F$ to state and prove. No domain-specific inputs, no Tier-0 protocol choices, and no external structure is imported.

### 5.3-C The Identity Network (Connection Clusters)

The 44 structural identities are not independent facts — they form a connected network with six computationally verified clusters. Each cluster reveals a structural relationship that no single identity expresses alone. Verification script: `scripts/identity_connections.py`.

**Cluster 1 — The Equator Web** (E1, N4, N16, E8)

The point $c = 1/2$ ($\theta = \pi/4$ in Fisher coordinates) is a **quintuple fixed point** where five kernel quantities simultaneously take special values:

$$S(1/2) = \ln 2, \quad (S+\kappa)|_{1/2} = 0, \quad h'(1/2) = 0, \quad g_F(1/2) = 4, \quad \theta = \pi/4$$

The reflection formula (N16) vanishes here: $f(\pi/4) + f(\pi/4) = 2\ln(1) \cdot \cos(0) = 0$. This is the unique point where Bernoulli field entropy generation and log-integrity loss exactly cancel.

**Cluster 2 — The Dual Bounding Pair** (B2, N10)

The kernel outputs are sandwiched between dual Jensen bounds:

$$\text{IC} \leq F \quad \text{(B2: integrity bound)} \qquad S \leq h(F) \quad \text{(N10: entropy bound)}$$

Both become equalities if and only if $C = 0$ (homogeneous trace). Together they constrain the kernel from two directions: $\text{IC}$ cannot exceed $F$ (multiplicative coherence bounded by arithmetic fidelity), and $S$ cannot exceed $h(F)$ (mean entropy bounded by entropy of fidelity).

**Cluster 3 — The Perturbation Chain** (N3 $\to$ N8 $\to$ B2)

This chain connects the exact rank-2 solution, its perturbative expansion, and the integrity bound:

$$\text{N3:}\; \text{IC} = \sqrt{F^2 - C^2/4} \quad \xrightarrow{\text{Taylor}} \quad \text{N8:}\; \kappa = \ln F - \frac{C^2}{8F^2} + O(C^4) \quad \xrightarrow{\text{sign}} \quad \text{B2:}\; \text{IC} \leq F$$

The correction $-C^2/(8F^2)$ is always negative (squared quantity divided by positive quantity), so $\kappa < \ln F$ whenever $C > 0$, giving $\text{IC} = e^\kappa < F$. The integrity bound follows from the kernel's own perturbative structure — no external concavity argument is needed. This also yields **linearized collapse theory**: for small heterogeneity, $\text{IC} \approx F \cdot \exp(-C^2/(8F^2))$.

**Cluster 4 — The Composition Algebra** (D6, N12, D8)

Fidelity, integrity, and the heterogeneity gap all have explicit composition laws:

$$F_{12} = \frac{F_1 + F_2}{2} \quad \text{(arithmetic)}, \qquad \text{IC}_{12} = \sqrt{\text{IC}_1 \cdot \text{IC}_2} \quad \text{(geometric)}$$

$$\Delta_{12} = \frac{\Delta_1 + \Delta_2}{2} + \frac{(\sqrt{\text{IC}_1} - \sqrt{\text{IC}_2})^2}{2}$$

The correction term $(\sqrt{\text{IC}_1} - \sqrt{\text{IC}_2})^2/2$ is a Hellinger-like distance. The gap always grows when composing systems with unequal integrity — integrity differences cannot be hidden. The composition algebra is a monoid (D8): associative with an identity element, verified to $|\text{error}| = 5.55 \times 10^{-17}$.

**Cluster 5 — The Fixed-Point Triangle** (E2/E3, N6, N4)

Three special points define the manifold skeleton:
- $c = 1/2$: equator, quintuple fixed point (Cluster 1)
- $c^{\ast} = 0.7822$: self-dual, triple coincidence via N6: $(1-c^{\ast})/c^{\ast} = \exp(-1/c^{\ast}) = (S+\kappa)|_{c^{\ast}}$
- $c_{\text{trap}} = 1 - c^{\ast} = 0.3178$: weld threshold where $\Gamma$ first drops below 1.0

N16 bridges $c^{\ast}$ and $c_{\text{trap}}$ through the reflection formula $f(\theta) + f(\pi/2-\theta) = 2\ln(\tan\theta)\cos(2\theta)$, verified to $< 10^{-15}$.

**Cluster 6 — The Spectral Family** (E4, N1, N2, N11)

The coupling function $f(c) = S(c) + \kappa(c) = h(c) + \ln c$ is **spectrally complete**: all polynomial moments have closed forms.

$$\mu_n \equiv \int_0^1 f(c) \cdot c^n\, dc = \frac{(n+1)H_{n+1} - (n+2)}{(n+1)^2(n+2)}$$

where $H_k = \sum_{j=1}^k 1/j$ is the $k$-th harmonic number. E4 is $\mu_0 = -1/2$. N2 is $\mu_1 = 0$ (the centroid vanishes). N1 adds the Fisher-weighted integral $\int_0^1 g_F \cdot S\, dc = \pi^2/3 = 2\zeta(2)$, connecting kernel geometry to the Basel constant.

**Summary**: The identity network reduces the kernel from a black box to a constrained geometric object:
- The perturbation chain (Cluster 3) explains _why_ the bounds hold.
- The composition algebra (Cluster 4) enables _prediction without re-computation_.
- The spectral family (Cluster 6) proves $f = S + \kappa$ is _fully characterized_.
- The fixed-point triangle (Cluster 5) shows the manifold has a _two-point skeleton_.

---

### 5.3a The Noise Structure of the Kernel

The five degenerate limits in §5.3 share a deeper unity: **the GCD kernel is, at its foundation, a noise-measurement apparatus.** Every kernel identity has a noise interpretation, and the connections to Kubo's FDT, Jarzynski's equality, and de-Picciotto's shot-noise measurement of $e/3$ are not isolated analogies — they are windows into a single structural fact.

#### The Bernoulli Variance $c(1-c)$ Is Per-Channel Noise

Each collapse channel carries a value $c_i \in [\varepsilon, 1-\varepsilon]$. The variance of a Bernoulli trial with parameter $c$ is $\text{Var}(c) = c(1-c)$. This is the irreducible noise of a binary collapse process: how much uncertainty remains in the channel. Three fundamental kernel structures derive from this single quantity:

1. **Fisher metric**: $g_F(c) = 1/(c(1-c))$ — the precision per unit Bernoulli noise. High noise ($c \approx 1/2$) gives low metric; low noise ($c \to 0$ or $c \to 1$) gives high metric. The entire geometry of the Bernoulli manifold is determined by the noise structure.

2. **Bernoulli field entropy**: $S_i = -[c_i \ln c_i + (1-c_i) \ln(1-c_i)]$. Maximum at $c = 1/2$, where $c(1-c)$ is also maximal. Entropy and noise are projections of the same function — $S(c)$ and $c(1-c)$ share their critical points and monotonicity structure.

3. **Heterogeneity gap**: $\Delta = F - \text{IC} \approx \text{Var}(c)/(2\bar{c})$ (Result F1). This is the **noise budget** — it measures how much channel-to-channel variance exists. When all channels have equal noise, $\Delta = 0$. When one channel is near death, its noise dominates the budget.

#### The Mean Hides, the Geometric Mean Reveals

This is the central measurement principle. The arithmetic mean $F = \Sigma w_i c_i$ is insensitive to individual channel failures: one dead channel ($c \to \varepsilon$) subtracts only $w_i \cdot (c_{\text{healthy}} - \varepsilon)$ from $F$. The geometric mean $\text{IC} = \exp(\Sigma w_i \ln c_i)$ is catastrophically sensitive: one dead channel multiplies IC by $\exp(w_i \ln \varepsilon) \approx \exp(-18.4 \cdot w_i)$.

This is geometric slaughter (§3 of orientation), and it has a precise physical analog: **noise measurements reveal structure invisible to mean measurements.**

- **de-Picciotto et al. (1997)**: Mean electrical current through a $\nu = 1/3$ quantum Hall constriction cannot distinguish $e/3$ charge quanta from $e$ — the current is continuous. Shot noise power $S_I = 2e^* I$ reveals $e^* = e/3$ directly, because the noise depends on the _discreteness_ of charge carriers. The mean hides; the fluctuation reveals. This is the experimental realization of geometric slaughter (see T-FQHE-2).

- **Benz et al. (2024)**: Johnson noise power $S_V = 4k_B T R$ now _defines_ the kelvin in the revised SI. Temperature is measured through noise — the fluctuation IS the primary observable, not a contaminant.

- **Spietz et al. (2006)**: Shot noise at finite bias measures effective electron temperature without any external thermometer. The noise is the thermometer.

In each case, the mean (average current, average voltage) is insufficient. The variance — the noise power — is where the structural information lives. The kernel captures this universally: $F$ is the mean, $\text{IC}$ is the noise-sensitive diagnostic, and $\Delta = F - \text{IC}$ is the noise budget.

#### Physical Noise Types as Degenerate Limits of $c(1-c)$

The Bernoulli variance $c(1-c)$ is the abstract noise of a binary collapse channel. Physical noise types emerge as degenerate limits when physical constraints are imposed:

| Physical Noise | Formula | GCD Degenerate Limit |
|----------------|---------|---------------------|
| **Shot noise** | $S_I = 2e^* I \cdot f(1-f)$ | Bernoulli $c(1-c)$ with $c = f$ (Fermi occupation) |
| **Johnson noise** | $S_V = 4k_B T R$ | Bernoulli $c(1-c) \to 1/4$ at $c = 1/2$ (thermal equilibrium) |
| **Quantum noise** | $S \propto \hbar\omega$ | Zero-point fluctuation: $c(1-c)$ at the vacuum state |
| **Partition noise** | $S_I = 2eI \cdot T(1-T)$ | Bernoulli $c(1-c)$ with $c = T$ (transmission probability) |

Callegaro (2006) demonstrated that Johnson noise and shot noise are unified by a single model: $S = \text{const} \cdot P(1-P)/N$, where $P$ is the Fermi-Dirac occupation. This is exactly the Bernoulli partition $c(1-c)$. The GCD kernel's entropy formula $S = -\Sigma w_i [c_i \ln c_i + (1-c_i) \ln(1-c_i)]$ describes the same partition — the kernel was built on the mathematical structure that unifies all these noise types.

#### The $c = 0.25$ Noise Regime Transition

Result T-KS-5 establishes that the Fisher Information approximation $\Delta \approx \text{Var}(c)/(2\bar{c})$ breaks below $c = 0.25$ (error exceeds 10%). This is not merely a numerical boundary — it corresponds to a physical noise regime transition. When $c < 0.25$, the Bernoulli variance $c(1-c) \approx c$ (Poisson limit). The channel's noise changes character from symmetric ($c \approx 1/2$: both outcomes equally likely) to rare-event ($c \ll 1$: the channel almost always reports zero). In the Poisson regime, the noise becomes proportional to the signal itself ($\text{Var} \approx c$) — shot noise dominates Johnson noise, and the geometric mean becomes the only reliable integrity diagnostic.

---

### 5.4 Constants Are Seam-Derived, Not Prescribed

Standard frameworks prescribe their constants from outside the system:
- Statistics: $\alpha = 0.05$ by convention (Fisher, 1925)
- SPC: $3\sigma$ limits by tradition (Shewhart, 1931)
- Machine learning: hyperparameters by cross-validation against held-out data
- Physics: fundamental constants ($c$, $\hbar$, $G$) measured and inserted

In every case, the framework stops working if the prescription is removed. The constant is an external input, not a structural consequence.

UMCP's frozen parameters are **not prescribed**. They are the unique values where seams close consistently across all domains.

**$p = 3$ is not "we chose cubic."** It is the exponent where $\Gamma(\omega) = \omega^p/(1-\omega)$ produces a budget that separates all three regimes simultaneously. At $p = 2$: Watch and Collapse do not separate cleanly. At $p = 4$: Stable becomes degenerate. The cubic is the unique exponent where the crossover from $\omega^p$ suppression to $1/(1-\omega)$ pole dominance happens near $\omega \approx 0.30$–$0.40$ — exactly the Watch-to-Collapse boundary defined by the data (see `src/umcp/tau_r_star.py` module docstring §5.5).

**$\text{tol\_seam} = 0.005$ is not "we chose a tolerance."** It is the width at which $\text{IC} \leq F$ holds at 100% across 8 domains. Tighter, and boundary cases produce false violations. Wider, and the bound loses diagnostic power — genuine heterogeneity violations become invisible. The seam tells you its own width.

**$\varepsilon = 10^{-8}$ is not "we chose a small number."** It is the regularization below which the pole at $\omega = 1$ does not affect any measurement to machine precision. The nuclear chain data at $\omega = 1.0$ produces $e^{-30} \approx 10^{-13}$, five orders of magnitude below $\varepsilon$. The clamp acknowledges the exact point where the representation ends, and the data confirms this by producing its only outliers there.

**$\alpha = 1.0$ is not "we chose unit scaling."** It is the coefficient where curvature cost ($D_C = \alpha C$) contributes to the budget at the same scale as the drift cost, without dominating or vanishing. At $\alpha \gg 1$, curvature overwhelms drift and regime classification reduces to curvature alone. At $\alpha \ll 1$, curvature becomes invisible and the system loses its heterogeneity signal.

**$\lambda = 0.2$ is not "we chose a learning rate."** It is the speed at which the return rate estimator $R$ adapts to new data — fast enough to track regime changes, slow enough to avoid aliasing high-frequency fluctuations as regime transitions.

The principle is: **constants are outputs of the requirement that returns must weld, not inputs to a model.** If a constant changes and the seam still closes at the same values across all domains — fine, it was not truly frozen. If it changes and seams break — the data rejected the change, not a convention.

This inverts the standard relationship between theory and constants. Standard frameworks say "here are the rules, does your data follow them?" UMCP says "here is the data, what rules does consistent return require?" The frozen parameters are the answer to the second question.

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## 6. Implementation Notes

### Critical Compliance Requirements

1. **All logarithms in S(t) and κ(t) MUST be evaluated on Ψ_ε(t)** (the ε-clipped domain)
2. **Any clipping event MUST be recorded** under the frozen OOR policy
3. **Non-default C(t) conventions MUST be declared as closures**
4. **Typed outcomes for τ_R are mandatory** (∞_rec vs UNIDENTIFIABLE)
5. **Contract equivalence is required** for direct comparison of kernel outputs

### Common Pitfalls

- **Missing ε-clipping**: Results in out-of-range κ and undefined logarithms
- **Treating IC as an arbitrary score**: IC is a weighted geometric mean with specific monotonicity properties
- **Silent changes to C conventions**: Any change to variance/normalization/neighborhood requires closure declaration
- **Collapsing typed outcomes**: ∞_rec and UNIDENTIFIABLE must not be replaced with arbitrary finite values
- **Comparing non-equivalent contracts**: Differences cannot be attributed without seam accounting

### Debugging with Lemmas

If a computed run violates the bounds in Lemmas 1-46, the implementation is almost certainly nonconformant. Check:

- Clipping applied before logarithms (Lemma 1, Lemma 3)
- Weight normalization (Σw_i = 1)
- C(t) normalization (denominator 0.5 for default convention)
- Return domain generator producing finite sets (Lemma 8)
- Typed outcome handling for edge cases
- Seam composition and residual accumulation (Lemmas 20, 27)
- Return stability near boundaries (Lemma 24)
- Closure minimality (Lemma 28)

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## 7. Version Control and Freeze Requirements

**Status**: This specification is **freeze-controlled** and versioned with the UMCP protocol.

- Changes to definitions or lemmas require a **new protocol version**
- Changes to default conventions (ε, normalization constants, domain generators) require **closure declarations**
- Implementation changes that preserve mathematical definitions do not require versioning (but must pass conformance tests)

**Current Version**: UMCP v2.2.3

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## 8. References

- **UMCP Manuscript v1.0.0**: §8 (Formal Definitions and Lemmas)
- **TIER_SYSTEM.md**: Tier separation and freeze gates
- **AXIOM.md**: Operational term definitions
- **docs/SYMBOL_INDEX.md**: Canonical symbol table
- **docs/PUBLICATION_INFRASTRUCTURE.md**: Weld accounting and publication rows

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**Document Status**: Complete formal specification from manuscript §8
**Last Updated**: 2026-01-21
**Checksum**: (Recorded in integrity/sha256.txt)
