Research-Stack/6-Documentation/docs/semantics/BASIN_STABILITY_CERTIFICATE.md

Basin-Stability Certificate

Date: 2026-05-05 Status: SEED PRIMITIVE Truth Seal: [ SSS-ENE-TRUTH-2026-05-05 ]

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0. The Seed

A mass number should not only encode semantic weight; it should carry a basin-stability certificate.

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.

Stated as a working form:

mass number = semantic ratio + admissible basin behavior + residual-risk witness

The first term is what we already had. The second and third terms are what this document adds.

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1. Why This Is Needed

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:

1. converge to a lawful attractor,
2. converge to a strange attractor (a non-root fixed point that is locally attracting but globally wrong),
3. or fail to converge at all.

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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2. The Template

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:

Color	Meaning
Red	Critical orbit converges to a lawful root
Green	Critical orbit converges to a lawful root (other branch)
Black	Critical orbit fails to converge — regime is unstable

That iteration map is the certificate. A point in K-space with no black is admissible. A point with black is not.

Translated to this stack:

operator family       →  semantic transform family
conjugacy reduction   →  canonical conjugate operator
fixed points          →  lawful attractors
strange fixed points  →  strange semantic attractors
critical-point probes →  critical orbit witnesses
parameter-space map   →  K-space / mass-number-space map
stable / unstable     →  admissible / non-admissible compression regimes

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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3. What a Certificate Must Contain

A basin-stability certificate attached to a mass number must declare:

1. Operator family. The transform class the mass number is meant to govern.
2. Canonical reduction. The conjugate operator after coordinate normalization (the analogue of the Möbius reduction to S(z)).
3. Lawful attractors. The fixed points that correspond to lawful outcomes.
4. Strange attractors. Fixed points that are locally attracting but not lawful — the regime hazards.
5. Critical probes. A finite set of distinguished orbits whose long-term behavior witnesses the regime.
6. Admissibility verdict. Pass / fail / boundary, with the witness recorded.
7. Residual-risk witness. What survives uncertified — the analogue of black regions in parameter space, carried forward as honest residual rather than discarded.

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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4. Connection to Existing Primitives

Existing primitive	Relation to certificate
lawful_loss (invariant survival)	Local check at the bridge — certificate adds the global orbit check
CompressionPattern	Detects mismatch — certificate detects regime instability
HutterContext	Context restoration at marginal cost — certificate determines whether iteration in that context stays bounded
AdaptiveBlock	Stateful update rules — certificate is the precondition for trusting that adaptation will not drift to a strange attractor
CodingCost	Final accounting — residual-risk witness must be paid here, not hidden

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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5. Failure Modes the Certificate Catches

5.1 Strange-Attractor Capture

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.

5.2 Boundary Drift

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.

5.3 Imaginary-Axis Failure

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.

5.4 Residual Hiding

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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6. What This Does Not Claim

• It does not claim the Chebyshev rational map is the right operator for any specific transform in this stack.
• It does not claim that orbit certification is sufficient for lawfulness — it is necessary, downstream of invariant survival.
• 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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7. Next Actions

1. Pick one operator family already in the stack (candidate: an AdaptiveBlock update rule) and identify its canonical conjugate.
2. Enumerate its fixed points and mark which are lawful vs strange.
3. Define a finite set of critical probes for it.
4. Run the parameter-space iteration. Produce the first basin map in this stack.
5. Use the result to issue or refuse a basin-stability certificate for that operator's mass number.

This is the smallest end-to-end build that turns the seed into a working primitive.

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8. Reference

• Linares, D. and Cadenas, C. Dynamics of the Modified Chebyshev's Method to Multiple Roots. arXiv:2601.10751v1, 2026-01-19.
• See also: INCOMPATIBLE_MANIFOLDS_AND_LAWFUL_LOSS.md, CARTOGRAPHY_OF_COMPRESSION_FAILURE.md.

Status: SEED PRIMITIVE | AWAITING FIRST BASIN MAP