# 空間域證明包圍論 — Series Route Map v1.0

**Date:** 2026-08-14  
**Status:** Phase I complete after Paper 08

## P01 — Survivor Soundness

$$
\mathcal C\subseteq\Omega_t,
\qquad
\Omega_{t+1}=\Omega_t\cap H_t.
$$

Theorem cut 不得誤刪真反例。

## P02 — Representation Faithfulness

建立 RouteCert、fiber saturation、singular-fiber semantics、Route Atlas。

Representation 不得壓掉 proof-relevant counterexample distinctions。

## P03 — Global Coverage / Closure

建立 typed gaps 與 Global Closure Certificate：

$$
\boxed{\mathsf{GCC}.}
$$

Local correctness 不等於 global completeness。

## P04 — Trace Compilation / Incremental Replay

建立 proof DAG、closure basis、support index、support-aware rollback、dirty replay、amortization。

## P05 — Discovery–Verification Dynamics

區分：

$$
D^{\rm resolve}
\neq
D^{\rm frontier}.
$$

建立 Weak / Strong / Productive DVI 與 frontier-hardening falsifiability。

## P06 — Exceptional-Core Geometry

建立：

$$
\Omega_\infty=\bigcap_t\Omega_t,
$$

measure / emptiness separation、closure-separating diagnostics、relative theorem-language core、source-preserving measure semantics。

## P07 — Enclosure Routing

建立 nonredundant closure value、Pareto routing、submodular / synergy geometry distinction、bridge-aware routing、closure fairness。

## P08 — Runtime / Benchmark / Proof-Space Observatory

整合 canonical state：

$$
\boxed{
\mathbb S_t
=
\langle
\Omega_t,
RouteCert_t,
\mathbf G_t,
GCC_t,
\mathcal H_t,
Compiled_t,
DVI_t,
SurvProf_t,
RouteDecision_t,
Env_t
\rangle.
}
$$

建立：

- append-only event ledger；
- deterministic reducer；
- certificate-gated commit；
- checkpoint / tail replay；
- support-aware rollback；
- non-authoritative observatory；
- state / event / benchmark JSON schemas；
- eight benchmark tracks；
- finite reference runtime；
- Hard-Zeta adapter specification。

## Phase-I closure stack

$$
\boxed{
\begin{aligned}
&\text{Sound Problem / Survivor Semantics}\\
&\to\text{Faithful Representation}\\
&\to\text{Global Coverage / Closure}\\
&\to\text{Compiled Proof History}\\
&\to\text{Measured Cost Dynamics}\\
&\to\text{Exceptional-Core Diagnostics}\\
&\to\text{Cost-Aware Routing}\\
&\to\text{Executable Proof-Space Runtime}.
\end{aligned}
}
$$

## Phase II is intentionally not auto-expanded

Paper 08 closes the first foundational series. Follow-up work should be opened only as concrete tracks, not by automatically extending the abstract sequence.

Candidate tracks:

1. **Formalization track:** Lean / Coq formalization of selected SDPE runtime theorems.
2. **Benchmark track:** longitudinal DVI + routing benchmark on theorem families.
3. **Case-study track:** Hard-Zeta, SAT/cube-and-conquer, finite graph classification.
4. **Observatory track:** UI / database / visualization without proof authority.
5. **Closure-basis track:** minimizing final replay certificates.

The foundational eight-paper Phase I is considered complete at v1.0.