Paper M6 — Constraining the Tolerance Window W

Abstract

This paper establishes what it means to measure, constrain, or recover the tolerance window $W$ within Cohesion Dynamics (CD), and provides a systematic programme of candidate recovery strategies ranked by feasibility, resource requirements, and expected tightness of constraints.

The tolerance parameter $W$ plays a structurally central role throughout the CD programme: it governs continuum cohesion, bounds constructor persistence, constrains mergeability of histories, enforces quantum-classical boundaries, and selects spectral discreteness. Despite this centrality, $W$ has not yet been empirically constrained or recovered from observable phenomena.

Building on substrate mechanics (Paper A), phase/compatibility/tolerance definitions (M4), and quantum recovery results (B1–B5), we define operational criteria for $W$ recovery, enumerate observable “handles” where $W$ enters the formalism, and develop a prioritised roadmap of recovery strategies. These strategies range from solo-feasible simulation sweeps and theoretical inequality bounds to resource-intensive astrophysical constraints and laboratory experiments.

The paper does not claim a numerical value for $W$, does not assume $W = \hbar$, and does not introduce new axioms or substrate primitives. Instead, it provides a meta-exploratory method paper that unblocks empirical work by clarifying what recovery paths exist, which are tractable, and which are most vulnerable to degeneracy or confounders.

Epistemic role: This paper is meta-exploratory in epistemic role — it introduces no new mechanisms and derives no new physical structure, but systematises how an existing primitive becomes empirically constrained.

This paper enables future E-series work by identifying specific empirical programmes, defines success criteria for $W$ recovery, and establishes falsification conditions. It concludes by spawning 3 E-paper briefs (E-W1, E-W2, E-W3) for the highest-priority recovery strategies.

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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: configurations diverge and reconverge through admissible resolution
• Tolerance $W$ as a structural primitive

From Paper M4 (Phase, Path, and Coherence Structure):

• Phase $\phi$ as closure-cycle alignment
• Tolerance vector $W = (W_{\text{shape}}, W_{\text{clock}}, W_{\text{spin}})$
• Compatibility condition: $|\Delta \phi_{ij}| \le W_{\text{spin}}$
• Coherence as finite tolerance-bounded regime
• Decoherence as tolerance violation with provenance partition
• Closure as joint satisfaction of internal and external constraints

From Paper B1 (Quantum State Representation):

• Linear amplitude spaces emerge from substrate mergeability
• Amplitudes track uncommitted alternatives
• Phase structure emerges from substrate provenance
• Coherence depends on tolerance-bounded compatibility

From Paper B3 (Spectral Discreteness):

• Discrete spectra emerge from closure stability
• Continuous configurations generically fail closure and violate tolerance
• Quantisation is a selection effect imposed by tolerance-limited admissibility
• Stable modes correspond to configurations satisfying closure within $W$

From Paper B5 (Measurement and Born Rule):

• Outcome probabilities emerge as branch-relative epistemic weights
• Quantum probabilities forced by compatibility accounting under finite tolerance $W$
• Measurement is commit: substrate resolution under tolerance constraints
• Branch weighting stability depends on tolerance preservation

Axiom References:

• AX-TOL (Finite tolerance window $W$) — central to all recovery strategies
• AX-PAR (Partition on tolerance violation) — governs failure modes
• AX-COH (Cohesion) — defines continuum boundaries via $W$

Capability Assumptions:

This paper references quantum-capable substrates (Derived Capability Class: DCC-QM, defined in R-DCC) and constructor-capable substrates (Core Capability Class: CCC, defined in R-CCC). These classifications inform which recovery strategies are viable and which observable handles exist.

1.2 What This Paper Does NOT Assume

This paper does not import:

• A numerical value for $W$ (recovery is the objective, not an assumption)
• The assumption that $W = \hbar$ or any other specific constant
• Effective physics formalisms as foundational (metrics, Hilbert spaces)
• Empirical claims about $W$ from unverified sources
• Laboratory measurement results (those are future E-series work)
• Astrophysical data analysis results (those are future E-series work)

1.3 Explicitly Out of Scope

This paper does not address:

• Numerical recovery of $W$ — that is E-series work
• Simulation implementation details — those are delegated to E-papers
• Specific experimental apparatus design — future applied work
• Comparison with alternative theories — not a recovery strategy
• Extensions to variable $W$ or regime-dependent $W$ — assumes global constant $W$ for now
• Implications for cosmology — deferred to G-series and future work

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2. Argument Outline: Why $W$ Must Be Recoverable (or Constrainable)

This section provides a high-level reasoning chain before detailed analysis.

Step 1: $W$ is structurally necessary

Tolerance $W$ is not a free parameter or aesthetic choice. It is a structural primitive required for:

• Continuum cohesion (AX-COH)
• Constructor persistence (M1, M2)
• Quantum coherence (M4, B1)
• Spectral discreteness (B3)
• Outcome weighting stability (B5)

Without finite $W$, continua collapse, constructors fail, and quantum mechanics does not emerge.

Step 2: $W$ enters observable phenomena

Because $W$ governs structural transitions (continuum boundaries, mode stability, decoherence thresholds, classical emergence), it must leave observable signatures in:

• Interference survival / decoherence timescales
• Spectral discreteness / stability under closure
• Branch-relative weighting stability
• Classical emergence thresholds
• (Future) Geometry / gravity emergence thresholds

These are not hidden variables—they are representational necessities derived in B-series papers.

Step 3: Recovery is a consistency requirement

If CD is correct and $W$ is structurally necessary, then:

• Observable quantum phenomena must reflect $W$-dependent structure
• Empirical constraints on $W$ must be extractable from known physics
• Failure to recover $W$ would constitute a falsification of CD

Therefore, $W$ recovery is not optional—it is a programme coherence requirement.

Step 4: Recovery strategies exist

We can identify multiple pathways to constrain or recover $W$:

• Theory-led: Derive bounds from known quantum coherence regimes
• Simulation-led: Sweep $W$ and map phase transitions in CD substrate
• Observational: Extract constraints from astrophysical or cosmological data
• Experimental: Design laboratory tests targeting $W$-dependent signatures

Step 5: Prioritisation enables tractable progress

Different strategies have vastly different resource requirements, degeneracy vulnerabilities, and expected constraint tightness. By ranking strategies systematically, we can:

• Focus effort on solo-feasible, high-yield methods first
• Defer resource-intensive or degenerate methods
• Spawn targeted E-papers for top-priority strategies

Conclusion of argument outline:

$W$ recovery is structurally necessary, observationally tractable, and programme-unblocking. This paper provides the systematic roadmap required to make recovery an empirical programme rather than an open question.

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3. Operational Definition of $W$

3.1 $W$ as Tolerance Threshold for Mergeability and Partition

The tolerance window $W$ is defined operationally in M4 and Paper A as the admissibility budget for mutual cohesion.

Formally:

Two informational states remain mutually cohesive if and only if their mismatch (configuration distance, phase difference, or structural incompatibility) remains within $W$.

Operationally:

$$|\Delta \phi_{ij}| \le W_{\text{spin}} \implies \text{states $i$ and $j$ remain mergeable}$$ $$|\Delta \phi_{ij}| > W_{\text{spin}} \implies \text{partition (decoherence)}$$

In M4, tolerance is decomposed into components:

$$W = (W_{\text{shape}}, W_{\text{clock}}, W_{\text{spin}})$$

For quantum recovery (B-series), the spin-like tolerance $W_{\text{spin}}$ governs phase compatibility and is the primary target for recovery.

3.2 Global Constant vs Regime-Dependent Effective $W$

This paper assumes $W$ is a global constant—the same structural primitive applies across all layers (continuum cohesion, constructor persistence, quantum coherence).

However, different effective tolerances may emerge in different regimes:

• Low-energy quantum systems may probe $W_{\text{spin}}$ directly
• High-energy or classical systems may exhibit coarse-grained effective tolerances due to closure averaging or decoherence

For recovery purposes, we seek:

1. Global $W$ — the fundamental structural constant
2. Regime-specific effective $W$ — emergent thresholds in particular systems

Distinguishing these is critical for avoiding false constraints.

Important clarification: This distinction between global and effective $W$ is a known theoretical pressure point. Effective tolerances are emergent coarse-grainings over a single primitive $W$, not evidence for multiple independent primitives. Allowing unconstrained regime-dependent $W$ would collapse B-series derivations, which assume a single global tolerance parameter. This distinction is therefore constrained by programme coherence, not a free modeling choice.

3.3 Bounds vs Point Estimate

Recovery strategies may yield:

• Upper bounds: "$W \le X$" from interference survival or coherence preservation
• Lower bounds: "$W \ge Y$" from spectral discreteness or mode stability
• Scaling laws: "$W \propto Z^{\alpha}$" from dimensional analysis or symmetry arguments
• Point estimates: "$W \approx C$" from best-fit numerical recovery (most vulnerable to degeneracy)

This paper prioritises bounds and scaling laws over point estimates, as they are more robust to confounders and degeneracy.

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4. Observable “Handles” on $W$

This section enumerates the programme places where $W$ enters the formalism and produces observable consequences.

4.1 Interference Survival and Decoherence

From M4 and B1:

Coherence persists when phase differences remain within $W_{\text{spin}}$. Decoherence occurs when tolerance is violated.

Observable signature:

• Interference fringe visibility as a function of path separation
• Decoherence timescales in quantum systems
• Coherence length in matter-wave interferometry

$W$-dependent prediction:

$$\text{Coherence persists} \iff |\Delta \phi| \le W_{\text{spin}}$$

Recovery strategy: Compare observed decoherence thresholds with CD predictions to bound $W$.

4.2 Spectral Discreteness and Stability Under Closure

From B3:

Discrete spectra emerge because only configurations satisfying closure within tolerance $W$ can persist. Continuous configurations generically fail closure.

Observable signature:

• Discrete energy levels in bound systems (atoms, molecules, quantum dots)
• Stability of spectral lines under perturbation
• Quantum number structure (integer indexing)

$W$-dependent prediction:

$$\text{Mode stability} \implies \text{closure satisfied within } W$$

Recovery strategy: Analyse spectral discreteness in known systems to infer $W$ bounds from closure requirements.

4.3 Branch-Relative Weighting Stability

From B5:

Outcome probabilities are stable under evolution only if tolerance is preserved. Branch weights $w_k \propto |\alpha_k|^2$ depend on compatibility accounting within $W$.

Observable signature:

• Born rule validity in quantum measurements
• Probability conservation under unitary evolution
• Outcome distribution stability in repeated measurements

$W$-dependent prediction:

$$\text{Born rule holds} \implies W \text{ preserves compatibility across branches}$$

Recovery strategy: Test Born rule violations or deviations to constrain $W$ indirectly.

4.4 Classical Emergence Thresholds

From B4 and B5:

Classical behaviour emerges when systems exceed quantum coherence regimes. The quantum-classical boundary is $W$-dependent.

Observable signature:

• Macroscopic superposition suppression
• Pointer state selection in decoherence
• Classical limit of quantum systems

$W$-dependent prediction:

$$\text{Classical regime} \implies \text{effective tolerance exceeded}$$

Recovery strategy: Map classical emergence thresholds to infer $W$ from quantum-to-classical transitions.

4.5 (Future) Geometry and Gravity Emergence Thresholds

From G-series (future work):

If geometry and gravity emerge from cohesion gradients (G-series programme), then $W$ will govern:

• Spatial discreteness scales
• Gravitational coupling strength
• Curvature limits or singularity structure

Observable signature (speculative):

• Quantum gravity phenomenology
• Spacetime discreteness constraints
• Black hole thermodynamics

$W$-dependent prediction (future):

$$\text{Geometry emerges} \implies W \text{ sets discreteness scale}$$

Recovery strategy (deferred): G-series must complete before this handle is usable.

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5. Candidate Recovery Strategies

Each strategy is described using a standard template:

• Mechanism: What aspect of CD it leverages
• Data source: Simulation / public datasets / experiments
• What it yields: Upper bound / lower bound / scaling law / estimate
• Primary confounders: Degeneracy, assumptions, noise
• Minimal viable implementation: Solo feasible?
• Effort rating: S (small), M (medium), L (large), XL (extra-large)

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Strategy 1: Simulation Sweep and Phase-Transition Mapping

Mechanism:
Sweep $W$ in CD substrate simulations and identify phase transitions for interference persistence, spectral discreteness, and decoherence thresholds.

Data source:
Simulation (internal CD implementation)

What it yields:

• Scaling laws: $W \propto f(\text{system size}, \text{coupling})$
• Critical thresholds for coherence/decoherence transitions
• Bounds from interference survival criteria

Primary confounders:

• Finite-size effects in simulation
• Discretisation artifacts
• Computational precision limits

Minimal viable implementation:
Yes—requires CD simulation framework (assumed to exist or be buildable)

Effort rating: M
(Medium: requires simulation infrastructure but no external data or lab)

Methodological guardrail: Simulation results are used to identify phase transitions and scaling behaviour, not to “fit” $W$ to known constants. Any numerical coincidence with $\hbar$ or other physical constants would require independent theoretical explanation and cannot be taken as evidence of recovery without corroborating theoretical derivation. This prevents numerological curve-fitting.

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Strategy 2: Theoretical Inequality Bounds from Known Quantum Coherence Regimes

Mechanism:
Use known quantum coherence persistence (e.g., atomic interferometry, superconducting qubits) to derive upper bounds on $W$ via the condition $|\Delta \phi| \le W_{\text{spin}}$.

Data source:
Literature review of quantum coherence experiments

What it yields:

• Upper bounds: "$W \le X$" from interference survival
• Order-of-magnitude estimates

Primary confounders:

• Effective vs global $W$ distinction
• Environmental decoherence masking intrinsic tolerance limits
• Regime-dependent coarse-graining

Minimal viable implementation:
Yes—requires literature analysis and theoretical derivation only

Effort rating: S
(Small: desk research and analytical work)

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Strategy 3: Internal Consistency Constraints

Mechanism:
Show that $W$ must lie in a specific window for quantum-capable substrates (DCC-QM, R-DCC) to exist at all. Derive bounds from self-consistency requirements.

Data source:
Theoretical analysis (no empirical data required)

What it yields:

• Lower bounds: "$W \ge Y$" from mode stability requirements
• Upper bounds: "$W \le Z$" from coherence preservation
• Existence proof for viable $W$ window

Primary confounders:

• Circular reasoning if not carefully isolated from derived results
• Assumptions about substrate structure

Minimal viable implementation:
Yes—pure theoretical analysis

Effort rating: S
(Small: analytical derivation)

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Strategy 4: Spectral Discreteness Analysis in Bound Systems

Mechanism:
Analyse known discrete spectra (hydrogen, harmonic oscillator, quantum dots) and derive constraints on $W$ from closure stability requirements (B3).

Data source:
Literature: atomic spectra, precision spectroscopy

What it yields:

• Bounds from observed spectral line stability
• Constraints on $W$ from quantum number structure

Primary confounders:

• Effective Hamiltonians may obscure substrate-level $W$ dependence
• Standard quantum mechanics already assumes discrete spectra, risk of circular logic

Minimal viable implementation:
Yes—literature analysis and theoretical mapping

Effort rating: M
(Medium: requires careful mapping from effective QM to substrate structure)

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Strategy 5: Astrophysical Core Stability Constraints (Dark Matter Cores)

Mechanism:
If dark matter halos exhibit cohesive cores with stability thresholds, these may reflect $W$-bounded persistence. Derive constraints (not point estimates) from observed core stability.

Data source:
Observational astronomy (public datasets: galaxy rotation curves, halo profiles)

What it yields:

• Bounds on $W$ from stability scales
• Order-of-magnitude constraints (not precise estimates)

Primary confounders:

• Degeneracy with baryonic matter effects
• Model-dependent halo profiles
• Uncertain dark matter physics

Minimal viable implementation:
Possible solo but heavy (requires astrophysical data analysis)

Effort rating: L
(Large: data-intensive, degeneracy-prone)

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Strategy 6: CMB and Large-Scale Structure Qualitative Constraints

Mechanism:
If $W$ governs early-universe cohesion dynamics, it may leave signatures in CMB power spectra or large-scale structure formation.

Data source:
Observational cosmology (public datasets: Planck CMB, SDSS)

What it yields:

• Bounds from structure formation timescales
• Constraints from coherence preservation in early universe

Primary confounders:

• Extremely degenerate with standard cosmological parameters
• Difficult to isolate $W$-dependent signatures
• Requires full cosmological CD model (not yet developed)

Minimal viable implementation:
No—requires collaborators and full cosmological framework

Effort rating: XL
(Extra-large: major research programme, high degeneracy)

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Strategy 7: Mesoscopic Interference and Decoherence Experiments

Mechanism:
Design or analyse experiments targeting the quantum-classical boundary (e.g., matter-wave interferometry with large molecules, optomechanical systems).

Data source:
Laboratory experiments (requires experimental collaborators)

What it yields:

• Direct bounds on $W$ from interference survival thresholds
• Tight constraints from controlled experiments

Primary confounders:

• Environmental decoherence dominates over intrinsic $W$ effects
• Difficult to isolate substrate-level tolerance from effective decoherence

Minimal viable implementation:
No—requires experimental collaborators and lab access

Effort rating: XL
(Extra-large: requires lab, collaborators, experimental design)

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Strategy 8: Precision Spectroscopy and Clock Comparisons

Mechanism:
Use ultra-high-precision atomic clocks and spectroscopy to test $W$-dependent phase accumulation or stability limits.

Data source:
Laboratory experiments (requires experimental collaborators)

What it yields:

• Bounds from spectral line stability under perturbation
• Constraints from phase coherence preservation

Primary confounders:

• Standard QED effects may dominate
• Unclear how to isolate $W$ signature from other precision effects

Minimal viable implementation:
No—requires experimental collaborators and precision apparatus

Effort rating: XL
(Extra-large: specialised equipment, collaborators required)

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Strategy 9: Controlled Quantum Measurement Chain Experiments

Mechanism:
Design experiments to test Born rule validity and outcome weighting stability under varying system sizes or coupling strengths. Map deviations (if any) to $W$ bounds.

Data source:
Laboratory experiments (requires experimental collaborators)

What it yields:

• Bounds from Born rule tests
• Constraints from branch-weighting stability

Primary confounders:

• Born rule extremely well-tested; deviations unlikely
• Difficult to distinguish $W$ effects from statistical noise or systematic errors

Minimal viable implementation:
No—requires lab and collaborators

Effort rating: XL
(Extra-large: major experimental programme)

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Strategy 10: Dimensional Analysis and Symmetry Arguments

Mechanism:
Use dimensional analysis to relate $W$ to known physical constants (e.g., $\hbar$, $c$, fundamental masses) and constrain possible forms of $W$.

Data source:
Theoretical analysis (no empirical data required)

What it yields:

• Scaling laws: $W \propto \hbar^{\alpha} c^{\beta} m^{\gamma}$
• Constraints on functional form of $W$

Primary confounders:

• Risk of assuming $W = \hbar$ without justification
• Dimensional analysis cannot fix numerical coefficients

Minimal viable implementation:
Yes—pure theoretical work

Effort rating: S
(Small: analytical derivation)

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Strategy 11: Constructor Persistence Bounds

Mechanism:
Use M1–M2 results on constructor persistence and derive bounds on $W$ from observed stability of atoms, molecules, or other persistent structures.

Data source:
Literature: atomic stability, molecular bond strengths

What it yields:

• Lower bounds: "$W \ge Y$" from constructor persistence requirements
• Constraints from reusability and structural stability

Primary confounders:

• Effective vs global $W$ distinction
• Standard chemistry already assumes stable atoms; risk of circularity

Minimal viable implementation:
Yes—literature analysis and theoretical mapping

Effort rating: M
(Medium: requires mapping from effective chemistry to substrate structure)

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Strategy 12: Quantum Correlation Bounds (Bell Tests, Entanglement)

Mechanism:
Use quantum correlation experiments (Bell tests, entanglement witnesses) to constrain $W$ via compatibility requirements for non-factorisable states (B2).

Data source:
Literature: quantum correlation experiments

What it yields:

• Bounds from entanglement stability
• Constraints from non-local correlation preservation

Primary confounders:

• Standard QM already predicts maximal Bell violations; unclear how $W$ modifies this
• Difficult to distinguish $W$ effects from QM predictions

Minimal viable implementation:
Possible solo but medium effort

Effort rating: M
(Medium: requires careful theoretical mapping)

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6. Prioritised Plan

The following table ranks the 12 candidate strategies by feasibility, effort, and expected yield.

Rank	Strategy	Effort	Solo Feasible?	Yield Type	Degeneracy Risk	Priority Group
1	Internal Consistency Constraints	S	Yes	Bounds (lower/upper)	Low	Solo feasible now
2	Dimensional Analysis & Symmetry	S	Yes	Scaling laws	Medium	Solo feasible now
3	Theoretical Inequality Bounds (Quantum Coherence)	S	Yes	Upper bounds	Medium	Solo feasible now
4	Simulation Sweep & Phase-Transition Mapping	M	Yes	Bounds, scaling laws	Low	Solo feasible now
5	Spectral Discreteness Analysis	M	Yes	Bounds	Medium	Solo feasible but heavy
6	Constructor Persistence Bounds	M	Yes	Lower bounds	Medium	Solo feasible but heavy
7	Quantum Correlation Bounds (Bell Tests)	M	Yes	Bounds	Medium	Solo feasible but heavy
8	Astrophysical Core Stability Constraints	L	Possible	Bounds	High	Solo feasible but heavy
9	CMB / Large-Scale Structure Constraints	XL	No	Bounds	Very high	Needs collaborators
10	Mesoscopic Interference Experiments	XL	No	Tight bounds	Medium	Needs lab
11	Precision Spectroscopy / Clocks	XL	No	Tight bounds	Medium	Needs lab
12	Controlled Quantum Measurement Chains	XL	No	Bounds	Medium	Needs lab

Grouping:

Group A: Solo Feasible Now (Highest Priority)

• Strategy 1: Simulation Sweep
• Strategy 2: Theoretical Inequality Bounds
• Strategy 3: Internal Consistency Constraints
• Strategy 10: Dimensional Analysis

Note on Strategy 3 (Internal Consistency Constraints): This strategy is logically prior to simulation and empirical strategies. It establishes whether a viable $W$ window exists at all within the theoretical framework. If Strategy 3 fails (i.e., no consistent $W$ interval exists), downstream empirical strategies become moot, and CD faces falsification. This is why Strategy 3 ranks first despite similar effort rating—it is falsification-capable and degeneracy-resistant in a way other strategies are not.

Group B: Solo Feasible But Heavy

• Strategy 4: Spectral Discreteness Analysis
• Strategy 11: Constructor Persistence Bounds
• Strategy 12: Quantum Correlation Bounds
• Strategy 5: Astrophysical Core Stability (borderline)

Group C: Needs Collaborators / Lab

• Strategy 6: CMB / Large-Scale Structure
• Strategy 7: Mesoscopic Interference Experiments
• Strategy 8: Precision Spectroscopy / Clocks
• Strategy 9: Controlled Quantum Measurement Chains

Recommended immediate focus: Strategies 1, 2, 3 (Group A, top priority)

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7. E-Paper Spawn Points

For the top 3 ranked strategies, we define E-paper briefs that can spawn dedicated empirical investigations.

E-Paper Spawn 1: Internal Consistency Constraints on $W$

Working Title: E-W1 — Internal Consistency Bounds on Tolerance $W$

Objective:
Derive rigorous upper and lower bounds on $W$ from self-consistency requirements: quantum-capable substrates (DCC-QM) must support coherence, mode stability, and branch weighting, which impose constraints on viable $W$ values.

Success Criteria:

• Proof that $W$ must lie in a non-empty interval $[W_{\min}, W_{\max}]$
• Lower bound $W_{\min}$ from mode stability requirements (B3)
• Upper bound $W_{\max}$ from coherence preservation (M4, B1)
• Demonstration that $W$ outside this interval breaks quantum recovery

Required Artifacts:

• Formal derivation of bounds
• Consistency proofs
• Falsification criteria (what would count as “no viable $W$”)

Dependencies:

• Normative: A!>depends, M4!>depends, B1!>depends, B3!>depends
• Non-normative: R-DCC?>informs

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E-Paper Spawn 2: Simulation Sweep and Phase-Transition Mapping

Working Title: E-W2 — Simulation-Based Phase-Transition Mapping for $W$ Recovery

Objective:
Implement CD substrate simulations sweeping $W$ across orders of magnitude, and map phase transitions for interference survival, spectral discreteness, and decoherence thresholds. Derive scaling laws and bounds.

Success Criteria:

• Identification of critical $W$ thresholds for coherence/decoherence transitions
• Scaling law $W \propto f(\text{system size}, \text{coupling})$
• Demonstration that observed quantum phenomena constrain $W$ to a specific regime

Required Artifacts:

• CD simulation implementation (or extension of existing framework)
• Phase diagrams: $W$ vs system observables
• Numerical bounds on $W$ with error analysis

Dependencies:

• Normative: A!>depends, M4!>depends, B1!>depends, B3!>depends
• Non-normative: R-DCC?>informs
• Tooling: CD simulation framework (to be developed or adapted)

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E-Paper Spawn 3: Theoretical Inequality Bounds from Quantum Coherence Regimes

Working Title: E-W3 — Upper Bounds on $W$ from Known Quantum Coherence Persistence

Objective:
Survey literature on quantum coherence experiments (atomic interferometry, superconducting qubits, matter-wave interference) and derive upper bounds on $W$ using the condition $|\Delta \phi| \le W_{\text{spin}}$.

Success Criteria:

• Collection of coherence persistence data from literature
• Derivation of upper bounds: "$W \le X$" for various quantum systems
• Analysis of effective vs global $W$ distinction
• Identification of tightest experimental constraints

Required Artifacts:

• Literature survey of coherence experiments
• Theoretical mapping from observed coherence to $W$ bounds
• Error analysis accounting for effective tolerance vs global $W$

Dependencies:

• Normative: A!>depends, M4!>depends, B1!>depends
• Non-normative: R-DCC?>informs
• Data: Published quantum coherence experiments (literature review)

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8. Failure Modes and Falsification

8.1 What Would Count as ”$W$ Not Recoverable in Principle”?

The following failure modes exhaust the ways in which $W$ recovery could fail while the rest of the programme remains intact. If $W$ recovery fails across all strategies, one of the following must be true:

Failure Mode 1: $W$ is not a global constant

If $W$ varies arbitrarily across regimes or systems, no single recovery value exists. This would require:

• Re-scoping M4 to allow regime-dependent $W$
• Major revision of A-series substrate definition
• Likely collapse of B-series quantum recovery (which assumes global $W$)

Falsification implication:
CD as currently formulated is falsified or requires major restructuring.

Failure Mode 2: $W$ is unobservably fine-tuned

If $W$ is so fine-tuned that all observable phenomena are insensitive to its value (e.g., $W$ only matters at Planck scale or below detectability), then:

• CD predictions become empirically vacuous
• No empirical programme can constrain $W$
• CD becomes unfalsifiable

Falsification implication:
CD loses empirical content; theoretical interest only.

Failure Mode 3: Observable phenomena are degenerate with respect to $W$

If all $W$-dependent predictions are degenerate with standard QM or other parameters, then:

• No unique $W$ recovery is possible
• CD and standard QM make identical predictions
• CD collapses to instrumentalist reinterpretation of QM

Falsification implication:
CD does not provide new physics; only metaphysical reinterpretation.

Failure Mode 4: Internal consistency bounds are empty

If Strategy 3 (internal consistency) shows no viable $W$ interval exists (i.e., $W_{\min} > W_{\max}$), then:

• Quantum-capable substrates (DCC-QM) are impossible within CD
• B-series quantum recovery is invalid
• CD is falsified

Falsification implication:
CD is internally inconsistent; programme fails.

8.2 How to Respond to Each Failure Mode

Response to Failure Mode 1 (variable $W$):

• Investigate whether effective $W$ emergence is consistent with global primitive $W$
• Develop regime-specific effective theories within CD framework
• If no consistent story exists, CD requires major revision

Response to Failure Mode 2 (fine-tuning):

• Re-examine whether CD makes any empirical predictions
• If not, reconsider programme viability
• Possibly accept CD as theoretical/conceptual framework only

Response to Failure Mode 3 (degeneracy):

• Identify novel predictions that break degeneracy with standard QM
• Focus on regime where CD and QM diverge (if any)
• If no divergence exists, accept CD as ontological reinterpretation

Response to Failure Mode 4 (empty consistency bounds):

• Re-examine A-series, M4, B-series derivations for errors
• If derivations are sound, CD is falsified
• Abandon or radically revise programme

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9. Programme Role

9.1 What This Paper Enables

Enables E-series empirical work:
By clarifying recovery strategies and spawning E-paper briefs, M6 unblocks systematic empirical investigation of $W$.

Enables programme coherence:
By defining what recovery means and what failure modes exist, M6 provides falsification criteria for CD.

Enables resource prioritisation:
By ranking strategies, M6 allows solo researchers to focus on tractable methods and defer resource-intensive methods.

Enables future G-series work:
If $W$ governs geometry emergence (G-series programme), M6 establishes the empirical foundation for constraining geometric scales.

9.2 What This Paper Constrains

Constrains premature numerical claims:
By showing $W$ recovery is non-trivial and degenerate, M6 prevents unjustified claims like "$W = \hbar$" without evidence.

Constrains unfalsifiable CD formulations:
By defining failure modes, M6 ensures CD remains empirically accountable.

Constrains scope creep:
By deferring lab-based and cosmological strategies to collaborator-dependent work, M6 keeps solo-feasible programme focused.

9.3 What This Paper Rules Out

Rules out $W$ as a free parameter:
M6 establishes that $W$ must be recoverable or CD fails. It cannot be arbitrary.

Rules out purely theoretical CD:
M6 shows that empirical constraints on $W$ are necessary for programme viability.

Rules out ignoring degeneracy:
M6 highlights that many naive recovery strategies are degenerate; careful analysis is required.

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10. Implications for Other Series

10.1 Implications for B-Series (Derived Physics)

• B-series quantum recovery assumes finite global $W$; M6 shows how to test this assumption
• If $W$ recovery succeeds, B-series predictions gain empirical support
• If $W$ recovery fails, B-series must be revised or abandoned

10.2 Implications for E-Series (Empirical Narrowing)

• M6 spawns 2–3 E-papers (E-W1, E-W2, E-W3) for immediate empirical work
• E-series simulations and literature analysis can proceed with clear success criteria
• Future E-papers may target additional strategies from M6 prioritised plan

10.3 Implications for G-Series (Gravity and Geometry)

• If $W$ governs geometric discreteness scales, G-series derivations must be consistent with $W$ bounds from M6
• Cross-series coherence requirement: Any bounds on $W$ recovered through M6-enabled E-series work must be respected by G-series derivations. This ensures programme-level consistency without prematurely entangling the two series.
• M6 establishes empirical foundation for future G-series $W$-dependent predictions
• G-series may add new observable handles (Strategy 4.5) once developed

10.4 Implications for R-Series (Reference and Classification)

• R-W (Tolerance as a Structural Primitive) should be updated to reference M6 recovery programme
• Future R-series classifications may incorporate $W$ bounds once recovered

10.5 Implications for M-Series (Formal Mechanisms)

• M6 completes the M-series meta-exploratory work by bridging formal mechanisms (M1–M5) to empirical hooks
• Future M-papers may reference M6 when discussing $W$-dependent predictions

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11. Explicitly Out of Scope

This paper does not:

• Claim a numerical value for $W$ — that is E-series work
• Perform simulations — delegated to E-W2
• Perform literature data analysis — delegated to E-W3
• Design experiments — deferred to collaborator-dependent strategies
• Extend to cosmology — G-series prerequisite
• Introduce new axioms — uses existing A-series and M4 results
• Assume $W = \hbar$ — avoids premature identification
• Address regime-dependent $W$ — assumes global constant for now
• Resolve degeneracy issues — highlights them as challenges for E-series

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12. Conclusion

This paper has established a systematic programme for recovering or constraining the tolerance window $W$ within Cohesion Dynamics. By defining operational criteria for $W$ recovery, enumerating observable handles, and prioritising 12 candidate recovery strategies, M6 transforms $W$ recovery from an open question into a tractable empirical programme.

The key results are:

1. $W$ recovery is a programme coherence requirement, not optional
2. Multiple recovery pathways exist, ranging from solo-feasible theoretical bounds to collaborator-dependent experiments
3. Prioritisation enables tractable progress, focusing on Strategies 1–3 (internal consistency, simulation sweeps, theoretical bounds)
4. E-paper spawn points provide actionable next steps for empirical work (E-W1, E-W2, E-W3)
5. Failure modes are well-defined, ensuring CD remains falsifiable

M6 does not claim to have recovered $W$—it provides the roadmap for doing so. The next steps are:

• Immediate: Pursue Strategy 3 (internal consistency bounds) — E-W1
• Near-term: Develop or extend CD simulation framework and pursue Strategy 1 (simulation sweeps) — E-W2
• Medium-term: Conduct literature review and pursue Strategy 2 (theoretical inequality bounds) — E-W3

If these strategies succeed in narrowing $W$ to a specific regime, CD gains empirical support. If they fail, CD faces falsification or major revision. Either outcome advances the programme.

The tolerance window $W$ is no longer a hidden parameter—it is now an empirical target with a clear recovery programme. This is what M6 enables.