Paper M8 — From Tolerance to Invariance in Cohesive Continua

Abstract

Cohesion Dynamics introduces a finite tolerance window $W$ constraining admissible mismatch within cohesive phases. Prior work has shown how interference, decoherence, and emergent structure arise while mismatch remains within tolerance. However, an essential transition has remained implicit: how continuously accumulated mismatch becomes discrete, conserved structure. In this paper, we formalise this transition using only existing substrate mechanics—relaxation, closure, and partition. We show that certain structural asymmetries are invariant under all admissible closures and therefore persist as identity-defining features of a cohesive phase. We identify three independent mechanisms by which mismatch becomes closure-stable: symmetry alignment failure, structural overload, and closure synchrony loss. This clarification explains the emergence of conserved quantities, superselection sectors, abrupt decoherence, and no-hair boundaries, and renders the W-narrowing programme well-posed without introducing new axioms or physical primitives.

1. Scope and Dependencies

1.1 Assumed Results

This paper assumes without re-derivation:

From Paper A (Substrate Mechanics):

Discrete substrate with finite alphabet $\Sigma$ and locations $V$
Local constraint system $\mathcal{C}$ defining admissibility
Mismatch measure $M(v;X)$ for configurations $X$
Commit semantics and tolerance $W$ as structural primitives
Relaxation as locally admissible mismatch-reducing rewrites
Closure as stability under further admissible relaxation
Partition when mismatch exceeds tolerance $W$

From Papers M1–M7 (Mechanism Papers):

Constructive viability and mismatch (M1)
Constraint dynamics and emergent constructors (M2)
Modes as emergent structures (M3)
Phase, path, and coherence structure (M4)
Constructor emergence (M5)
Tolerance window constraint programme (M6)
Layered definitions and explanatory weight (M7)

From B-series (as context for quantum structure):

Quantum state representation and recovery results (B1–B2)
Recognition that conserved quantities appear in quantum recovery

From G-series (as context for boundary behaviour):

No-hair behaviour at maximal divergence boundaries (G1)

1.2 What This Paper Does NOT Do

M8 does not:

Introduce new axioms or substrate primitives
Modify existing axioms or foundational commitments
Make numerical claims about specific physical quantities
Derive effective physical laws (that is B-series work)
Perform empirical simulations (that is E-series work)
Assume Hilbert space, metrics, or spacetime formalisms

1.3 Relationship to Other Series

A-series (Substrate): M8 clarifies how closure mechanics already established in Paper A give rise to invariant structure
M-series (prior): M8 builds on tolerance (M1, M4, M6) and uses definitional framework from M7
B-series: M8 explains why discrete conserved quantities emerge, unblocking quantum recovery
G-series: M8 explains no-hair boundaries without additional geometric assumptions
E-series: M8 renders W-narrowing empirically meaningful by distinguishing tolerance phenomena from closure-stable structure

2. Introduction

Cohesion Dynamics (CD) seeks to explain the emergence of physical structure from minimal informational substrate mechanics. Central to this framework is the notion of tolerance: a finite window $W$ bounding admissible mismatch under relaxation and closure. Previous M-series papers have established how coherence, interference, and emergent dynamics arise within this tolerance regime. Yet physical reality is characterised not only by tolerance-limited variation but also by discrete, conserved structure—quantities that do not smear, average, or decohere gradually.

This paper addresses the missing link between these regimes. We show that conserved structure does not arise from new postulates or additional dynamics, but from properties already implicit in closure. Certain mismatches cannot be eliminated or redistributed by any admissible closure; such mismatches are therefore closure-stable and persist across the evolution of a cohesive phase. Making this transition explicit resolves longstanding ambiguities surrounding tolerance, conservation, and decoherence, and provides a principled foundation for narrowing the admissible range of $W$ .

3. Argument Outline: Why This Mechanism Matters

3.1 The Problem: Tolerance vs Invariance

Cohesion Dynamics establishes that admissible mismatch is bounded by tolerance $W$ . This successfully explains:

Coherence within tolerance regimes (M4)
Decoherence when tolerance is violated (M4)
Constructor persistence (M5)
Emergence of phase structure (M4)

However, a crucial question remains unanswered:

How do discrete, sharply conserved quantities emerge from a tolerance-limited continuum?

Physical reality exhibits:

Conserved charges that are exactly preserved, not approximately
Abrupt decoherence boundaries, not gradual transitions
Superselection sectors that cannot be merged
No-hair theorems at maximal divergence boundaries

These features appear incompatible with tolerance-based dynamics, which permit continuous variation within $W$ . If tolerance governs all mismatch, why do some quantities become completely invariant?

3.2 The Resolution: Closure-Stable Mismatch

The resolution lies in recognising a crucial distinction:

Tolerance constrains admissibility of closure, not the properties of closure-stable configurations.

While mismatch may accumulate continuously during relaxation, closure determines which features persist. When certain mismatches cannot be eliminated by any admissible closure, they become closure-stable invariants that:

Are not subject to tolerance
Persist across all future evolution
Function as ledger constraints on the cohesive phase

This transition from tolerance-governed mismatch to closure-stable invariance is not a new mechanism—it is inherent in closure itself.

3.3 Why This Unblocks the Programme

Making the tolerance-to-invariance transition explicit:

Explains conserved structure without new axioms
Unifies symmetry breaking, structural limits, and boundary formation under a single mechanism
Renders W-narrowing well-posed by distinguishing tolerance phenomena from closure-stable artefacts
Unblocks B-series quantum recovery by explaining why discrete structure emerges
Unblocks G-series derivations by explaining no-hair boundaries without geometric assumptions

4. Tolerance as a Constraint on Closure

We begin by clarifying the role of tolerance within the substrate mechanics established in Paper A.

Relaxation explores locally admissible rewrites that reduce or redistribute mismatch. Each relaxation step must preserve admissibility under local constraints $\mathcal{C}$ .

Closure selects configurations that are stable under further admissible relaxation. A configuration is closed when no further admissible rewrites that reduce mismatch are available.

Tolerance $W$ constrains which configurations may participate in a shared closure. If mismatch between configurations exceeds $W$ , they cannot close together—partition occurs.

This leads to a crucial distinction:

Tolerance governs admissibility of closure, not the behaviour of features that are invariant under all admissible closures.

While mismatch may accumulate continuously during relaxation (subject to tolerance), closure determines what features persist. Some features may be invariant under all admissible closures—these are not subject to tolerance limits.

This clarification resolves the apparent tension between tolerance-limited dynamics and sharply conserved quantities.

Structure vs Tolerance Clarification
Structural emergence (e.g. bonding, fusion, collapse) is driven by constraint binding and closure dependency.
The tolerance parameter $W$ governs reconciliation among competing candidate closures prior to categorical structure formation, but does not determine the structure itself.
Once constraint binding removes independent closure, categorical structure is established and $W$ no longer governs that outcome.

This prevents misreading $W$ as a structural “merge control” and aligns theory with simulator behavior. Tolerance regulates pre-structural exploration and admissibility; structure arises from constraint binding and closure mechanics.

5. Structural Asymmetry and Encapsulability

We now formalise the key distinction that underlies the tolerance-to-invariance transition.

5.1 Structural Asymmetry

Definition (Structural Asymmetry): A structural asymmetry is any mismatch pattern that distinguishes configurations within a cohesion domain.

Structural asymmetries include:

Differences in closure-cycle phase alignment
Differences in provenance histories
Differences in internal constraint satisfaction patterns
Differences in compositional structure

5.2 Encapsulability

Definition (Encapsulable Asymmetry): A structural asymmetry is encapsulable if there exists at least one admissible closure in which it can be:

Eliminated (reduced to zero mismatch)
Redistributed (merged with other mismatch)
Rendered irrelevant to future evolution (isolated from active constraints)

Encapsulable asymmetry remains subject to tolerance $W$ and may cancel, average, or merge across divergent paths.

Definition (Non-Encapsulable Asymmetry): A structural asymmetry is non-encapsulable if no admissible closure exists that can eliminate, redistribute, or isolate it.

Non-encapsulable asymmetry is necessarily closure-stable—it persists across all future closures of the same cohesive phase.

5.3 The Transition

The transition from tolerance-governed dynamics to invariant structure occurs precisely when asymmetry becomes non-encapsulable.

This transition is not gradual—it is abrupt and binary. Either an admissible closure exists (asymmetry is encapsulable), or it does not (asymmetry is closure-stable).

6. Closure-Stable Mismatch and Ledger Formation

We now formalise the central mechanism of this paper.

6.1 Closure-Stable Mismatch

Definition (Closure-Stable Mismatch): A mismatch is closure-stable if and only if it is invariant under all admissible closures.

Closure-stable mismatch has the following properties:

Not eliminable by any sequence of admissible relaxation steps
Not redistributable through closure with other configurations
Not subject to tolerance (it is a property of closure itself, not admissibility)
Globally accountable (must be tracked across the entire cohesive phase)

6.2 Ledger Constraints

Closure-stable mismatch functions as a ledger constraint: it cannot be altered by relaxation and constrains all future admissible evolution.

In practice, closure-stable mismatches manifest as:

Discrete conserved quantities (charge, angular momentum)
Topological invariants (winding numbers, linking numbers)
Symmetry sector labels (superselection rules)
Boundary conditions (no-hair constraints)

Importantly, this transition does not require a new dynamical process. It is a property of closure itself. When closure cannot eliminate a mismatch without violating admissibility, that mismatch necessarily persists and becomes a ledger constraint.

6.3 Relation to Tolerance

The key insight is this:

Tolerance $W$ bounds mismatch that can be accommodated within shared closure
Closure-stable mismatch is invariant under closure and therefore not bounded by $W$

This resolves the apparent paradox: tolerance governs what can merge, but does not bound what cannot be eliminated.

7. Conceptual Aside: Closure-Stable Invariants (Illustrative)

Note: The following analogy is provided solely to build intuition for the mechanism described in Sections 5–6. It plays no formal role in the theory and is explicitly non-normative.

Consider the configuration space of a Rubik’s cube. The cube admits only local admissible moves (face rotations), yet its global state space decomposes into disconnected components distinguished by parity and orientation invariants.

These invariants are not accumulated gradually. Rather, they are revealed when no sequence of admissible moves can reconcile two configurations. States may appear arbitrarily close while remaining irreconcilable—the invariant is discrete and abrupt.

This illustrates a key feature of Cohesion Dynamics: global invariants need not be postulated or dynamically enforced. They arise when closure cannot eliminate distinguishing structure under admissible operations. Invariants are therefore discrete, abrupt, and non-tolerant—not because tolerance has failed continuously, but because admissible reconciliation has ceased to exist.

8. Three Mechanisms of Closure Stability

We identify three independent mechanisms by which mismatch becomes closure-stable. These mechanisms are not mutually exclusive and may operate simultaneously.

8.1 Symmetry Alignment Failure

Mechanism: Divergent provenance paths may fail to admit a common representation under which they are equivalent.

When symmetry alignment fails:

No admissible closure can merge the paths
Any structural differences distinguishing them become closure-stable
The configuration space partitions into disconnected sectors

Physical manifestations:

Superselection sectors in quantum mechanics
Charge conservation sectors
Baryon number conservation
Abrupt decoherence (loss of admissible joint closure)

Key insight: Symmetry alignment failure is not a violation of symmetry—it is the inability to establish a common symmetry frame under admissible closure.

8.2 Structural Overload

Mechanism: Even with intact symmetry alignment, structural mismatch may exceed what can be eliminated by any admissible relaxation.

When structural overload occurs:

Closure exists but only with residual mismatch
This residual is invariant across closures
The residual becomes a ledger constraint

Physical manifestations:

Angular momentum conservation
Energy conservation (in appropriate limits)
Topological invariants
Winding numbers

Key insight: Structural overload does not require symmetry breaking—it arises from the combinatorial limits of admissible closure.

8.3 Closure Synchrony Loss

Mechanism: When closure ordering cannot be jointly maintained across regions, shared closure becomes impossible.

When closure synchrony is lost:

Partition occurs (as analysed in Paper A)
Only closure-stable quantities persist at the boundary
Internal structure is erased except for closure-stable features

Physical manifestations:

Horizon formation (black holes, cosmological horizons)
No-hair theorems (only mass, charge, angular momentum persist)
Maximal divergence boundaries
Boundary conditions in gravitational collapse

Key insight: Closure synchrony loss explains why maximal divergence boundaries retain only a minimal set of closure-stable quantities—all other structure is encapsulable and therefore lost.

9. Consequences

The mechanisms identified in this paper explain several core physical features that have previously been treated as independent:

9.1 Discrete Conserved Quantities

Conserved quantities arise when mismatch is closure-stable, not by postulate or symmetry enforcement.

This explains:

Why conservation is exact, not approximate
Why conserved quantities are discrete
Why they persist across interactions
Why they constrain all future evolution

9.2 Abrupt Decoherence

Loss of admissible closure leads to sudden, not gradual, separation of histories.

This explains:

Why quantum decoherence is abrupt (superselection)
Why measurement outcomes are discrete
Why histories cannot re-merge after decoherence
Why decoherence boundaries are sharp

9.3 Superselection Sectors

Symmetry-aligned closure partitions configuration space into non-mergeable domains.

This explains:

Why certain quantum superpositions are forbidden
Why charge sectors cannot be superposed
Why fermion/boson distinctions are absolute
Why certain observables are always classical

9.4 No-Hair Boundaries

Maximal internal divergence erases all but closure-stable features.

This explains:

Why black holes retain only mass, charge, angular momentum
Why horizon formation eliminates internal degrees of freedom
Why boundary conditions simplify at maximal divergence
Why no-hair theorems hold without additional assumptions

These results follow directly from substrate mechanics and do not rely on Hilbert space, Noether symmetries, or spacetime primitives.

10. Implications for the W-Narrowing Programme

The admissible range of tolerance $W$ is constrained by the requirement that closure-stable structure emerges neither too early nor too late.

10.1 Constraints on W

If tolerance is too small:

Closure-stable mismatch appears prematurely
Coherence is suppressed
Interference cannot develop
Quantum-like structure fails to emerge

If tolerance is too large:

Closure-stable structure fails to form
Stable identity cannot develop
Discrete conserved quantities do not emerge
Decoherence boundaries are too permissive

10.2 Rendering W-Narrowing Well-Posed

By making the tolerance-to-invariance transition explicit, this paper enables:

Clear distinction between tolerance-driven phenomena and closure-stable artefacts
Operational criteria for when closure-stability is expected to occur
Empirical programmes (E-series) that can measure the onset of closure-stability
Simulation methods that can distinguish tolerance effects from invariant structure
Principled narrowing of $W$ without reference to specific physical laws

This makes the W-narrowing programme (initiated in M6) epistemically meaningful and empirically tractable.

11. Relation to Physical Law

The mechanisms identified in this paper are not proposed as replacements for known physical laws, nor as derivations of specific interactions such as electromagnetism or the Standard Model. Rather, we hypothesise that tolerance-limited closure and the emergence of closure-stable invariants constitute deeper structural constraints under which any effective physical laws must operate.

In this respect, the role of these mechanisms is analogous to that of entropy or causal order: they constrain the space of admissible law-like behaviour without specifying the detailed dynamics themselves. The relevance of this hypothesis is supported by the independent convergence of closure-stable mismatch with multiple features of observed physics—including discrete conserved quantities, abrupt decoherence, superselection sectors, and no-hair boundary behaviour—despite being derived without assuming those phenomena.

This positioning is intentionally conservative. The present work does not claim that Cohesion Dynamics uniquely determines the laws of our universe, only that it identifies structural conditions that any physically realisable universe must satisfy. Downstream recovery of quantum or classical laws therefore remains contingent on additional structure, to be explored in subsequent work.

12. Relation to Prior Work

M1–M4 established tolerance, phase alignment, and coherence within admissibility regimes.

M5–M7 explored persistence, constructor emergence, and definitional frameworks.

M8 occupies the hinge between these results and downstream physics recovery. It clarifies how tolerance-limited dynamics give rise to invariant structure, resolving the apparent tension between continuous mismatch and discrete conservation.

B-series and G-series results assume this transition implicitly—M8 makes it explicit and demonstrates that it follows from substrate mechanics without new axioms.

13. Explicitly Out of Scope

M8 does not:

Calculate numerical values for conserved quantities (E-series)
Derive specific Noether symmetries (B-series)
Recover Hamiltonian dynamics (B-series)
Perform simulations to demonstrate closure-stability (E-series)
Claim that closure-stability uniquely determines quantum mechanics
Introduce new substrate primitives or axioms
Modify the tolerance window $W$ or propose a numerical value

These remain future work for the appropriate series.

14. Summary

We have shown that conserved, identity-defining structure arises when mismatch becomes invariant under all admissible closures. This transition is implicit in the substrate mechanics of Cohesion Dynamics and requires no new axioms.

By clarifying the limits of tolerance and the role of closure-stable mismatch, we unify symmetry alignment failure, structural overload, and closure synchrony loss under a single mechanism. This result:

Strengthens the foundations of Cohesion Dynamics
Explains the emergence of discrete conserved quantities
Renders the W-narrowing programme well-posed
Unblocks B-series quantum recovery
Explains G-series no-hair boundaries
Provides operational criteria for future E-series empirical work

The tolerance-to-invariance transition is not a new physical principle. It is a structural consequence of closure mechanics that was implicit in Paper A and is now made explicit.

15. Conclusion and Next Steps

M8 completes the formal mechanism layer needed to understand how discrete, conserved structure emerges from tolerance-limited substrate dynamics. With this clarification in place:

B-series work can proceed with confidence that discrete quantum structure has a substrate-mechanical explanation.

G-series work can explain no-hair boundaries without geometric assumptions.

E-series work (particularly E-W subseries) can distinguish tolerance phenomena from closure-stable structure, enabling principled narrowing of $W$ .

Future M-series work may explore:

Conditions under which closure-stability is computationally decidable
Relationship between closure-stable mismatch and topological invariants
Whether closure-stability admits a canonical decomposition

The programme can now proceed to recover effective physics with a complete understanding of how tolerance and invariance relate within substrate mechanics.