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140 lines
7.2 KiB
Markdown
140 lines
7.2 KiB
Markdown
# Basin-Stability Certificate
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**Date:** 2026-05-05
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**Status:** SEED PRIMITIVE
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**Truth Seal:** `[ SSS-ENE-TRUTH-2026-05-05 ]`
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---
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## 0. The Seed
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> **A mass number should not only encode semantic weight; it should carry a basin-stability certificate.**
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Mass number, in this stack, has been treated as a semantic ratio — a compact scalar that summarizes how much lawful weight a structure carries. This document promotes it: a usable mass number must also certify that the operator family acting on the structure is regime-stable, witnessed by where its critical orbits land.
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Stated as a working form:
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```text
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mass number = semantic ratio + admissible basin behavior + residual-risk witness
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```
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The first term is what we already had. The second and third terms are what this document adds.
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---
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## 1. Why This Is Needed
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The lawfulness filter currently accepts a structure when its invariants survive projection (see `INCOMPATIBLE_MANIFOLDS_AND_LAWFUL_LOSS.md`). That is necessary but not sufficient. Invariant survival is a local property at the bridge point. It does not tell us whether the operator we are about to apply will:
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1. converge to a lawful attractor,
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2. converge to a *strange* attractor (a non-root fixed point that is locally attracting but globally wrong),
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3. or fail to converge at all.
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A mass number that does not encode this distinction cannot tell a valid compression regime from one that has merely passed a local check and is about to drift.
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---
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## 2. The Template
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The template comes from complex dynamics of root-finding methods. Linares and Cadenas (2026, arXiv:2601.10751v1) study the Modified Chebyshev iteration on two-root polynomials `(z-a)^m (z-b)^n` and reduce the entire family to a single rational map `S(z)` parameterized by the multiplicity ratio `K = m/n`. They classify regimes not by inspecting fixed-point eigenvalues alone, but by iterating the **critical points** over the parameter plane and recording where they land:
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| Color | Meaning |
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|---|---|
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| Red | Critical orbit converges to a lawful root |
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| Green | Critical orbit converges to a lawful root (other branch) |
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| Black | Critical orbit fails to converge — regime is unstable |
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That iteration map is the certificate. A point in `K`-space with no black is admissible. A point with black is not.
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Translated to this stack:
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```text
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operator family → semantic transform family
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conjugacy reduction → canonical conjugate operator
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fixed points → lawful attractors
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strange fixed points → strange semantic attractors
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critical-point probes → critical orbit witnesses
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parameter-space map → K-space / mass-number-space map
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stable / unstable → admissible / non-admissible compression regimes
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```
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The structural move that matters is not the specific iteration. It is the discipline of certifying a regime by orbit behavior of distinguished probes, not by local linearization.
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---
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## 3. What a Certificate Must Contain
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A basin-stability certificate attached to a mass number must declare:
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1. **Operator family.** The transform class the mass number is meant to govern.
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2. **Canonical reduction.** The conjugate operator after coordinate normalization (the analogue of the Möbius reduction to `S(z)`).
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3. **Lawful attractors.** The fixed points that correspond to lawful outcomes.
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4. **Strange attractors.** Fixed points that are locally attracting but not lawful — the regime hazards.
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5. **Critical probes.** A finite set of distinguished orbits whose long-term behavior witnesses the regime.
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6. **Admissibility verdict.** Pass / fail / boundary, with the witness recorded.
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7. **Residual-risk witness.** What survives uncertified — the analogue of black regions in parameter space, carried forward as honest residual rather than discarded.
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A mass number missing any of these fields is a *naked* mass number. It may still be useful as a coarse weight, but it is not yet a lawful regime descriptor.
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---
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## 4. Connection to Existing Primitives
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| Existing primitive | Relation to certificate |
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|---|---|
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| `lawful_loss` (invariant survival) | Local check at the bridge — certificate adds the global orbit check |
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| `CompressionPattern` | Detects mismatch — certificate detects regime instability |
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| `HutterContext` | Context restoration at marginal cost — certificate determines whether iteration in that context stays bounded |
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| `AdaptiveBlock` | Stateful update rules — certificate is the precondition for trusting that adaptation will not drift to a strange attractor |
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| `CodingCost` | Final accounting — residual-risk witness must be paid here, not hidden |
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The certificate slots in *between* the lawfulness check and the cost accounting. It is the layer that asks: *this transform passed local lawfulness — will it still be lawful under iteration?*
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---
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## 5. Failure Modes the Certificate Catches
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### 5.1 Strange-Attractor Capture
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A semantic transform that locally lowers cost but iterates into a non-lawful fixed point. The local gradient looks correct; the long-term orbit is wrong. Without the certificate, this looks like compression progress. With the certificate, it is flagged as a strange attractor and rejected.
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### 5.2 Boundary Drift
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A regime that is admissible at one mass number and inadmissible at a nearby mass number, with no obvious local signal at the boundary. The certificate forces the boundary to be marked, so adapter selection cannot silently cross it.
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### 5.3 Imaginary-Axis Failure
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The Chebyshev study found that purely imaginary `K` values produce dominantly black parameter regions. The analogue here: when the mass number is forced into a formally valid but semantically transverse direction, regime stability collapses. The certificate refuses to issue admissibility on that axis.
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### 5.4 Residual Hiding
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Without the residual-risk witness, unconverged orbits get folded silently into "the rest." With the witness, they remain typed and costed — preserving the discipline of `INCOMPATIBLE_MANIFOLDS_AND_LAWFUL_LOSS.md` that loss must be lawful, not erased.
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---
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## 6. What This Does Not Claim
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- It does not claim the Chebyshev rational map is the right operator for any specific transform in this stack.
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- It does not claim that orbit certification is sufficient for lawfulness — it is necessary, downstream of invariant survival.
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- It does not claim that every mass number must be re-issued. Existing uses remain valid as semantic weights; the certificate is what they need to *also* govern operator selection.
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---
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## 7. Next Actions
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1. Pick one operator family already in the stack (candidate: an `AdaptiveBlock` update rule) and identify its canonical conjugate.
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2. Enumerate its fixed points and mark which are lawful vs strange.
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3. Define a finite set of critical probes for it.
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4. Run the parameter-space iteration. Produce the first basin map in this stack.
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5. Use the result to issue or refuse a basin-stability certificate for that operator's mass number.
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This is the smallest end-to-end build that turns the seed into a working primitive.
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---
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## 8. Reference
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- Linares, D. and Cadenas, C. *Dynamics of the Modified Chebyshev's Method to Multiple Roots.* arXiv:2601.10751v1, 2026-01-19.
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- See also: `INCOMPATIBLE_MANIFOLDS_AND_LAWFUL_LOSS.md`, `CARTOGRAPHY_OF_COMPRESSION_FAILURE.md`.
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**Status: SEED PRIMITIVE | AWAITING FIRST BASIN MAP**
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