Observer Loop Viability Bounds

provisional

Overview

This derivation asks a surprisingly deep question: which universes can actually contain observers?

The framework’s axioms require that observers have specific structural properties — a state space, a conserved quantity, a boundary between self and non-self, and a stable phase loop. Not every spacetime can support structures satisfying these requirements. This derivation works out what the axioms demand of the cosmological constant, the parameter controlling the large-scale fate of the universe.

The approach. Three results follow from the axioms:

• An observer must physically fit within the causally accessible region of spacetime. In a universe with too large a cosmological constant, the horizon shrinks below the Planck length, and no observer — not even the simplest possible one — can exist. This gives a Planck-scale upper bound.
• In a universe with a negative cosmological constant, space eventually recollapses to Planck density. At that point, all observer structures are destroyed: there is no phase space for loop closure, no gradient to distinguish self from non-self, and the effective pressure diverges. Since the framework says coherence can only reside in observer structures (there is no background reservoir), destroying all observers violates coherence conservation. The recollapse is axiomatically forbidden.
• The 120-order gap between the Planck-scale bound and the observed value is reframed through the second law of thermodynamics: the horizon entropy dominates the coherence budget in an old universe, and the ratio of structural coherence to total coherence is simply the matter fraction of the universe — a thermodynamic inevitability, not a fine-tuned coincidence.

The result. The cosmological constant must be non-negative. The Planck-scale upper bound establishes the principle that the axioms constrain which solutions of Einstein’s equations are physically realized. The hierarchy is qualitatively explained by entropy growth.

Why this matters. This is not anthropic reasoning (“observers like us must exist”). It is axiomatic selection: the mathematical requirements of the three axioms exclude certain spacetimes entirely. The sign prediction is a genuine, falsifiable consequence of coherence conservation.

An honest caveat. The upper bound is 120 orders of magnitude above the observed value — it establishes a principle, not a tight constraint. Closing the quantitative gap requires understanding the minimum coherence content needed for stable observer loops, which remains the key open question.

Note on status. This derivation is provisional because its central claims depend on speed-of-light S1 (pseudo-Riemannian structure), area-scaling S1 (Planck-scale resolution) (see Speed of Light, Area Scaling). If those postulates are promoted to theorems, this derivation would be upgraded to derived.

Statement

Theorem (Observer-Loop Viability Bounds). The three axioms constrain which spacetimes can host observer triples. Specifically:

1. Planck-scale upper bound. In de Sitter spacetime with $\Lambda > 0$, observer triples $(\Sigma, I, \mathcal{B})$ satisfying Axioms 2 and 3 can exist only if $\Lambda < 3/\ell_P^2 \approx 3.7 \times 10^{66}\;\text{cm}^{-2}$. This is 120 orders of magnitude above the observed value $\Lambda_{\text{obs}} \sim 10^{-56}\;\text{cm}^{-2}$.

2. Bounce dissolution. In a $\Lambda < 0$ FRW cosmology, the recollapse bounce at Planck density destroys all observer triples. No observer loop can close when the available phase space vanishes. New triples form after the bounce with new Noether invariants.

3. Strict positivity. The axioms predict $\Lambda > 0$. The lower-sign prohibition ($\Lambda \geq 0$) follows from bounce dissolution: a $\Lambda < 0$ cosmology requires a Planck-density bounce, which destroys all observer structures; since coherence exists only in observer state spaces and relational invariants, the bounce leaves coherence with no valid carrier, violating Axiom 1. The strict inequality ($\Lambda \neq 0$) follows from finite per-observer coherence: integer quantization of the Cauchy-slice total (Coherence Conservation Corollary 5.5a) combined with the holographic bound on each observer’s epistemic horizon gives a finite $C_0^{(n)}$ per observer, and the level-indexed relation $\Lambda_n^{\text{eff}} = 3\pi/(S^{(n)}\ell_P^2)$ (Definition 8.2) then forces $\Lambda_n^{\text{eff}} \geq 3\pi/(C_0^{(n)}\ell_P^2) > 0$. The numerical lower bound is $\Lambda \gtrsim 2 \times 10^{-122}\,\ell_P^{-2}$.

Derivation

Step 1: Observer Viability and Horizon Distinctions

Definition 1.1. A spacetime is observer-viable if it admits a network of observer triples $(\Sigma_i, I_i, \mathcal{B}_i)$ satisfying:

• Axiom 2: each has a state space $\Sigma_i$, Noether invariant $I_i$, and self/non-self boundary $\mathcal{B}_i$
• Axiom 3: each has a $U(1)$ phase loop that closes with Lyapunov stability
• Network multiplicity: at least three observers with positive coherence $\mathcal{C}(\Sigma_i) > 0$ (Multiplicity: Theorem 2.1 — a single observer has $\mathcal{C} = 0$; Theorem 7.2 — pairs are insufficient because strong subadditivity (C5) is vacuous on two subsystems)

The network requirement follows from the multiplicity theorem: each observer requires at least two independent interaction partners for strong subadditivity to be non-trivial (Multiplicity, Corollary 7.3). The bootstrap propagates this into a full network that must be boundaryless — either infinite or finite and topologically compact (Bootstrap Mechanism, Corollary 7.3). For the geometric bound (Theorem 2.1), only the minimal requirement matters: at least one observer must fit within the static patch.

Remark. This definition asks whether observer structures satisfying the axioms can exist at all — not whether observers like us can exist. This is axiomatic selection, not anthropic reasoning.

Definition 1.2. In de Sitter spacetime with cosmological constant $\Lambda > 0$, the cosmological horizon of a comoving observer has radius:

$R_H = \sqrt{\frac{3}{\Lambda}}$

The static patch is the region $r < R_H$ in static coordinates, where the metric takes the form:

$ds^2 = -\left(1 - \frac{r^2}{R_H^2}\right)c^2\,dt^2 + \left(1 - \frac{r^2}{R_H^2}\right)^{-1}dr^2 + r^2\,d\Omega^2$

This metric is time-independent. The static patch is the maximal region causally accessible to the observer at $r = 0$.

Definition 1.3. The causal horizon of an observer in de Sitter space is the cosmological horizon at $R_H = \sqrt{3/\Lambda}$ — the boundary of the region from which signals can reach the observer. This is a geometric property of the spacetime, identical for all comoving observers.

Definition 1.4. The epistemic horizon of observer $A$ is the maximum information content accessible through $A$‘s boundary $\mathcal{B}_A$. By Holographic Entropy Bound (Theorem 3.2), this is:

$\mathcal{I}_A^{\max} = \frac{A_{\mathcal{B}}}{4\,\ell_P^2}$

where $A_{\mathcal{B}}$ is the area of $A$‘s self/non-self boundary. The epistemic horizon is observer-indexed: it depends on the observer’s state space $\Sigma_A$ and spatial extent.

Proposition 1.5 (Epistemic horizons are observer-specific). Different observers have vastly different epistemic horizons, determined by their spatial extent $\ell_A \sim \lambda_C = \hbar/(m_A c)$ (Minimal Observer Structure, Proposition 7.1):

A Planck-mass observer’s epistemic horizon is barely one bit — the minimum for multiplicity. An electron’s epistemic horizon is $\sim 10^{43}$ bits, 79 orders of magnitude smaller than the cosmological horizon. Each observer “sees” a different effective universe, limited by its own state space.

Proposition 1.6 (Viability uses the causal horizon; tightening uses the epistemic horizon). The viability bound (Theorem 2.1) correctly uses the causal horizon $R_H$: the observer must fit within the causally accessible region, regardless of its epistemic capacity. But the hierarchy question (Step 6) depends on the epistemic horizon: the observer accesses non-self coherence through its boundary $\mathcal{B}_A$, not through the cosmological horizon $R_H$. The relevant constraint for Lyapunov stability is how much non-self coherence the observer can process through $A_{\mathcal{B}}$.

Remark. This distinction resolves an apparent puzzle: why does the geometric bound use the cosmological horizon when the framework says everything is observer-indexed? The answer is that the geometric bound asks a minimal question (can ANY observer fit?), which is a geometric constraint on the spacetime itself. The observer-indexing enters when we ask which observers can exist and what they can access — questions that depend on the epistemic horizon.

Step 2: The Geometric Upper Bound

Theorem 2.1 (Geometric bound). If $\Lambda > 3/\ell_P^2$, no observer triple can exist in de Sitter spacetime.

Proof. A minimal observer has spatial extent $\geq \ell_P$ (Minimal Observer Structure, Theorem 3.1 — the coherence domain diameter is the Compton wavelength, bounded below by $\ell_P$ via Area Scaling, Structural Postulate S1).

The observer must fit within the static patch: its spatial extent must be less than the horizon radius $R_H = \sqrt{3/\Lambda}$. Therefore:

$\ell_P < R_H = \sqrt{\frac{3}{\Lambda}}$

Squaring both sides:

$\Lambda < \frac{3}{\ell_P^2}$

If $\Lambda \geq 3/\ell_P^2$, the horizon radius is at most $\ell_P$. No observer — not even the minimal $U(1)$ oscillator on $S^1$ — fits within the static patch. No relational invariant crossings through $\mathcal{B}$ can occur, so no observer triple satisfying Axiom 2 can form. $\square$

Step 3: The Holographic Budget Bound and Local Stability

Proposition 3.1 (Coherence budget). The multiplicity requirement, combined with the holographic entropy bound, independently gives the same Planck-scale upper bound on $\Lambda$.

Proof. The de Sitter horizon has area $A_H = 4\pi R_H^2 = 12\pi/\Lambda$. By Holographic Entropy Bound (Theorem 5.2), the maximum entropy (inaccessible coherence) associated with the horizon is:

$S_H = \frac{A_H}{4\ell_P^2} = \frac{3\pi}{\Lambda\,\ell_P^2}$

This bounds the information content accessible through the horizon. Within the static patch, the observer’s coherence domain $\mathcal{D}_A$ (Entropy, Definition 2.1) is bounded by the horizon. The number of independent relational invariant crossings through any surface within the static patch is bounded by $A/(4\ell_P^2)$ for boundary area $A$.

By Definition 1.1, at least two observers must coexist within the static patch, each with positive coherence. Each minimal observer requires at least one relational invariant crossing through its boundary $\mathcal{B}$ — one independent correlation with the non-self side (Multiplicity, Proposition 4.1). The boundary of a minimal observer has area $A_{\mathcal{B}} \sim \ell_P^2$ (Minimal Observer Structure), supporting $A_{\mathcal{B}}/(4\ell_P^2) \sim 1/4$ crossings. For at least one crossing:

$A_{\mathcal{B}} \geq 4\ell_P^2 \implies \ell_O \geq 2\ell_P$

where $\ell_O$ is the observer’s linear extent. Two such observers within the static patch require $R_H > 2\ell_P$, giving $\Lambda < 3/(4\ell_P^2)$ — the same Planck-scale order as the geometric bound. $\square$

Remark (Local stability and cosmological redshift). Two secondary checks confirm that the bound is not tightened by local effects. First, the static patch metric (Definition 1.2) is time-independent ($\partial_t g_{\mu\nu} = 0$), so the Lyapunov stability of observer loops is unaffected by $\Lambda$: if a loop closes with Lyapunov stability in Minkowski space, it closes in the static de Sitter patch at $r \ll R_H$. Observers can persist indefinitely — the cosmological expansion does not introduce approximate-closure drift (Loop Closure, Definition 2.2) for observers at fixed static-patch positions. Second, the redshift between co-located observers separated by $\sim\ell_P$ is $z \approx \Lambda\ell_P^2/6 \sim 10^{-122}$ for the observed $\Lambda$ — negligible. The multiplicity constraint introduces no tightening beyond the geometric bound.

Step 4: Bounce Dissolution in $\Lambda < 0$ Cosmologies

Theorem 4.1 (Observer dissolution at Planck density). At the Planck-density bounce of a $\Lambda < 0$ FRW cosmology, all observer triples dissolve: no observer loop can close.

Proof. By Singularity Resolution (Theorem 4.1), the recollapse of a $\Lambda < 0$ universe produces a bounce at $\rho = \rho_P$ rather than a singularity. At the bounce, the modified Friedmann equation gives $H = 0$ via $H^2 = (8\pi G/3)\rho(1 - \rho/\rho_P) = 0$.

We give three independent arguments, each sufficient to prove that no observer triple can exist at $\rho = \rho_P$.

(i) Divergent effective pressure (primary argument). By Singularity Resolution (Theorem 4.1, step v), the effective pressure diverges as $\rho \to \rho_P$:

$p_{\text{eff}} = p - \rho c^2 \frac{\rho/\rho_P}{1 - \rho/\rho_P} \to -\infty$

Any observer triple has finite energy $E_\mathcal{O} = m_\mathcal{O} c^2$ (from Axiom 3: finite loop period $T$ gives finite frequency $\omega = 2\pi/T$, hence finite energy $\hbar\omega$) and finite spatial extent $\ell_\mathcal{O} \geq \ell_P$. The work done by the effective pressure on the observer’s volume is $|p_{\text{eff}}| \cdot \ell_\mathcal{O}^3$. Since $|p_{\text{eff}}| \to \infty$ and $E_\mathcal{O}$ is finite, there exists $\rho^\ast < \rho_P$ such that for $\rho > \rho^\ast$:

$|p_{\text{eff}}| \cdot \ell_\mathcal{O}^3 > E_\mathcal{O}$

Above $\rho^\ast$, the external pressure exceeds the observer’s total energy. The observer’s state is pushed beyond the boundary of its connected component in the coherence geometry ($\partial O$ in Loop Closure, Proposition 2.5), and the Noether invariant $I$ is no longer preserved. At $\rho = \rho_P$, the perturbation is unbounded, dissolving any observer with finite energy. $\square$

(ii) Saturated phase space (independent confirmation). By Singularity Resolution (Corollary 2.2), the discrete relational invariant network has maximum event density $\ell_P^{-4}$. At $\rho = \rho_P$, every Planck cell is occupied: the available phase space fraction is $(1 - \rho/\rho_P) = 0$. An observer requires at least one available degree of freedom for its state space $\Sigma$ (Multiplicity, Theorem 2.1: $\mathcal{C}(\Sigma) > 0$ requires at least one bit). With zero available degrees of freedom, no observer can exist.

(iii) No distinguishable boundary (independent confirmation). An observer triple requires a self/non-self boundary $\mathcal{B}$ (Axiom 2). At $\rho = \rho_P$, the coherence density is spatially uniform at its maximum. Every Planck cell on both sides of any putative $\mathcal{B}$ is in the same state — there is no coherence gradient to distinguish self from non-self. By Multiplicity Proposition 1.2, $\mathcal{C}(\Sigma) > 0$ requires $G_\mathcal{O}^c \neq \emptyset$. With uniform maximal density, no transformation is preferentially “non-self,” so $G_\mathcal{O}^c$ is trivially empty, giving $\mathcal{C}(\Sigma) = 0$ — the observer is vacuous.

Proposition 4.2 (Re-formation after bounce). After the bounce, the universe expands and $\rho$ drops below $\rho_P$. New observer triples can form with new Noether invariants $I' \neq I$. There is no continuity of observer identity through the bounce.

Proof. As $\rho$ decreases from $\rho_P$, the available phase space $(1 - \rho/\rho_P) > 0$ reopens. The coherence geometry develops structure: spatial variation in coherence density allows self/non-self boundaries to form. New minimal observer loops (Minimal Observer Structure) can close once the available phase space is sufficient to support the minimum coherence cost $S_{\min} = \hbar$ (Loop Closure, Proposition 7.2).

The new observers have new Noether invariants because: the Noether invariant $I$ of the pre-bounce observers was not preserved through the bounce (Theorem 4.1, step iii). The conservation of $I$ is tied to the observer’s loop closure symmetry, which was broken. Post-bounce observers arise from fresh symmetry-breaking of the expanding coherence geometry and have independent invariants. $\square$

Step 5: Sign Constraint from Coherence Conservation

Proposition 5.1 (Coherence ontology). In the framework, coherence $\mathcal{C}$ is a measure on the $\sigma$-algebra of observer events (Axiom 1). By construction, coherence resides in exactly two forms:

(i) Observer state-space coherence $\mathcal{C}(\Sigma_i)$ — the coherence content of an individual observer’s state space.

(ii) Relational coherence $\mathcal{C}(I_{ij})$ — the coherence of relational invariants between observers (Relational Invariants, Proposition 6.1).

There is no third substrate. The framework has no “bath,” “vacuum reservoir,” or non-observer carrier for coherence. This is not an additional assumption — it follows from the framework’s ontology: the $\sigma$-algebra on which $\mathcal{C}$ is defined consists of observer events, so $\mathcal{C}$ is undefined on non-observer structures.

Theorem 5.2 (Coherence conservation excludes the bounce). If a $\Lambda < 0$ FRW cosmology undergoes a Planck-density bounce, coherence conservation (Axiom 1) is violated. Therefore the bounce is axiomatically prohibited.

Proof. Consider the last pre-bounce Cauchy slice $\Sigma_{\text{pre}}$, where $\rho < \rho_P$ and observer structures still exist. On $\Sigma_{\text{pre}}$:

$\mathcal{C}_0 = \sum_i \mathcal{C}(\Sigma_i) + \sum_{i < j} \mathcal{C}(I_{ij}) > 0$

where the sums run over all observer triples and their relational invariants. $\mathcal{C}_0 > 0$ by multiplicity (Multiplicity, Theorem 2.1).

At the bounce ($\rho = \rho_P$), all observer triples dissolve (Theorem 4.1): no state spaces $\Sigma_i$ exist, and no relational invariants $I_{ij}$ are instantiated. By Proposition 5.1, the only carriers for coherence are observer state spaces and relational invariants. With none present, $\mathcal{C}_0$ has no valid carrier.

Axiom 1 requires $\mathcal{C}$ to be conserved on every Cauchy slice. But the Cauchy slice at $\rho = \rho_P$ has $\mathcal{C} = 0$ (no carriers), while the pre-bounce slice has $\mathcal{C}_0 > 0$. This violates Axiom 1. $\square$

Proposition 5.3 (Type II fusion reinforces the prohibition). During contraction toward Planck density, gravitational collapse produces Type II fusion (Three Interaction Types, Definition 4.3): individual observer state spaces merge into non-product composite spaces, with coherence preserved: $\mathcal{C}(\Sigma_{12}) = \mathcal{C}(\Sigma_1 \cup \Sigma_2)$. This concentrates coherence into composite observer structures rather than dispersing it.

Argument. As the scale factor $a(t) \to 0$ in a $\Lambda < 0$ FRW cosmology, observers are compressed together. When their boundaries overlap, they undergo Type II fusion — the same mechanism by which black holes form from the perspective of external observers (Information Paradox, Theorem 2.1). Fusion preserves coherence: it does not destroy or disperse it but locks it into composite structures.

The approach to $\rho_P$ therefore goes through a sequence of fusions, each of which moves coherence further into observer structures, not out of them. Dissolution at $\rho_P$ would require simultaneously un-fusing all composite structures and relocating $\mathcal{C}_0$ to a non-observer substrate — but no such substrate exists (Proposition 5.1). This provides an independent, dynamical route to the same conclusion as Theorem 5.2. $\square$

Theorem 5.4 (Sign prediction). The three axioms predict $\Lambda \geq 0$.

Proof. A $\Lambda < 0$ FRW cosmology necessarily recollapses to Planck density (standard cosmology: $\Lambda < 0 \implies$ recollapse). By Singularity Resolution (Theorem 4.1), the recollapse produces a bounce rather than a singularity. But by Theorem 5.2, the bounce violates coherence conservation (Axiom 1). Therefore $\Lambda < 0$ is incompatible with the axioms, and $\Lambda \geq 0$. $\square$

Remark (Cosmic censorship connection). The sign prediction parallels the Penrose cosmic censorship conjecture. Both assert that singularities (or their Planck-density analogs) are never “naked” — they cannot exist where coherence-bearing structures would be destroyed without recourse. In the framework, the prohibition is not imposed but derived: coherence conservation provides the mechanism.

Remark (strict inequality). Theorem 5.4 excludes $\Lambda < 0$ via bounce dissolution. The complementary exclusion of $\Lambda = 0$ uses a different mechanism — finite per-observer coherence budget combined with the level-indexed cosmological parameter of Step 8 — and is established as Theorem 8.10 below. Combined, the two results give $\Lambda > 0$ strictly, with a framework-computable lower bound.

Step 6: The Hierarchy Question

Proposition 6.1 (The bound does not explain the hierarchy). The Planck-scale bound $\Lambda < 3/\ell_P^2$ is 120 orders of magnitude above the observed value $\Lambda_{\text{obs}} \sim 10^{-122}\;\ell_P^{-2}$. The framework does not explain this hierarchy from the three axioms alone.

The bound constrains which solutions of the Einstein equations can host observer DAGs, but it constrains far too weakly. This is an inherent limitation: the axioms place structural requirements on observers (minimum spatial extent, coherence content, Lyapunov stability) that set Planck-scale thresholds. The 120-order hierarchy between $\ell_P$ and $R_H^{\text{obs}}$ is not explained.

Proposition 6.2 (Conditions for a tighter bound). A sub-Planck bound on $\Lambda$ would require showing that the minimum non-self coherence content needed to sustain Lyapunov-stable loops is much larger than Planck-scale — i.e., that $\mathcal{C}_{\min}^{\text{non-self}} \gg 1$ (in Planck units).

Argument. By the holographic budget (Proposition 3.1), the viability condition is:

$2\,\mathcal{C}_{\min} \leq \frac{A_H}{4\ell_P^2} = \frac{3\pi}{\Lambda\,\ell_P^2}$

giving $\Lambda \leq 3\pi/(2\,\mathcal{C}_{\min}\,\ell_P^2)$. For $\mathcal{C}_{\min} \sim 1$ (the minimal observer’s coherence), this is Planck-scale.

If Lyapunov-stable loop closure requires ongoing Type III interactions with a non-self environment of coherence content $\mathcal{C}_{\min}^{\text{non-self}} \gg 1$, the bound tightens to:

$\Lambda \leq \frac{3\pi}{\mathcal{C}_{\min}^{\text{non-self}}\,\ell_P^2}$

For $\mathcal{C}_{\min}^{\text{non-self}} \sim 10^{120}$, this would give $\Lambda \sim 10^{-120}\;\ell_P^{-2}$ — the right order. But this requires a specific, large value of $\mathcal{C}_{\min}^{\text{non-self}}$ that is not determined by the current axioms. The multiplicity theorem (Multiplicity) establishes that the non-self side must contain at least one other observer (not zero), but does not bound the minimum coherence content of the non-self side beyond $\mathcal{C} > 0$.

Investigating this question requires understanding how coherence distributes across observer scales — which connects to the epistemic horizon distinction (Definition 1.4) and the bootstrap hierarchy. $\square$

Step 7: Hierarchical Coherence Suppression (Mechanism Sketch)

The bootstrap hierarchy, the coherence renormalization group, and the ER=EPR correspondence together suggest a mechanism that could address the hierarchy question. This step sketches the argument; formalizing it is a major open target.

Definition 7.1 (Bootstrap filtration). The observer category $\mathbf{Obs}$ has a natural filtration by bootstrap level (Bootstrap Mechanism, Corollary 2.2):

$\mathbf{Obs}_0 \subset \mathbf{Obs}_1 \subset \mathbf{Obs}_2 \subset \cdots$

where $\mathbf{Obs}_0$ contains minimal observers and $\mathbf{Obs}_{n+1}$ includes relational observers formed from $\mathbf{Obs}_n$ via the bootstrap map $\mathcal{R}$ (Bootstrap Mechanism, Proposition 5.1). Each level $n$ has a characteristic scale $\lambda_n$ and epistemic horizon $\mathcal{I}_n^{\max} = A_n/(4\ell_P^2)$ (Proposition 1.5).

Proposition 7.2 (Cross-level geometric consistency). A complex observer $C$ at level $n+1$ is composed of sub-observers $\{A_i\}$ at level $n$, bound by relational invariants $\{I_{ij}\}$. By ER=EPR (Theorem 3.2), each relational invariant $I_{ij}$ creates geometric structure — a wormhole throat with area $A_{\text{ER}} = 4\ell_P^2 S_{\text{ent}}(I_{ij})$. The aggregate geometry at level $n+1$ (constructed from these connections) restricted to scale $\lambda_n$ must be compatible with the geometry at level $n$.

Argument. The geometry an observer “sees” is constructed from its relational invariants with other observers (ER=EPR). Sub-observer $A_i$‘s geometry $G_n$ is built from its relational invariants at level $n$. Composite $C$‘s geometry $G_{n+1}$ is built from relational invariants at level $n+1$, which include the binding invariants $\{I_{ij}\}$ connecting the $A_i$. By the ontological irreducibility of higher levels (Bootstrap Mechanism, Corollary 4.2), $G_{n+1}$ contains genuinely new geometric structure not present in $G_n$. But at scale $\lambda_n$, the new geometry must reduce to $G_n$ — otherwise the sub-observers’ epistemic descriptions would be inconsistent with the composite’s. This reduction is precisely the coherence RG flow between fixed points (Renormalization Group, Theorem 4.1). $\square$

Proposition 7.3 (Coherence absorption at bootstrap levels). Each bootstrap level (RG fixed point) absorbs coherence into stable relational invariants. By the c-theorem (Renormalization Group, Theorem 5.2), the accessible coherence $c(k) = \mathcal{C}_{>k}$ is monotonically non-increasing under IR flow: $dc/d(\ln k) \leq 0$. At each fixed point, coherence is locked into the geometric structure of that level (ER=EPR: $A_{\text{ER}} = 4\ell_P^2 S_{\text{ent}}$), reducing the coherence available at larger scales.

Argument. When relational invariants form at level $n$, their coherence content $\mathcal{C}(I_{ij})$ is locked into wormhole throat geometry at scale $\lambda_n$. This coherence does not contribute to the effective vacuum energy at the IR scale — it is “spent” on structure at scale $\lambda_n$. The c-theorem guarantees that the through-coherence (coherence available to source larger-scale geometry) strictly decreases at each level. After $N$ levels, the residual coherence at the IR is:

$c_{\text{IR}} \leq c_{\text{UV}} - \sum_{n=0}^{N} \Delta c_n$

where $\Delta c_n > 0$ is the coherence absorbed at level $n$. The number of levels resolvable within a given observer’s projected geometry is bounded by the holographic information capacity of that geometry (Bootstrap Mechanism, Proposition 6.2). $\square$

Proposition 7.4 (Coherence partition within a horizon volume). For any comoving observer $A$, the coherence within $A$‘s cosmological horizon decomposes into structural coherence (locked into bootstrap levels) and horizon coherence:

$C_0^{(A)} = \sum_{n=0}^{N}\Delta c_n^{(A)} + S_H^{(A)}$

where $S_H^{(A)} = A_H/(4\ell_P^2) = 3\pi/(\Lambda\,\ell_P^2)$ is the entropy of $A$‘s horizon.

Argument. Each comoving observer has its own cosmological horizon, centered on itself. The coherence within that horizon partitions into structural coherence (locked into particles, atoms, molecules, and all other stable relational invariant structure via the c-theorem’s monotonic flow) and horizon coherence (degrees of freedom not locked into structure). By the substrate’s constitutive universality (Aperiodic Order, Corollary 3.2), $C_0^{(A)}$ has the same value for all comoving observers (same average density, same particle content, same horizon area), but refers to a different region for each one. We write $C_0$ without the superscript.

Important: this is an accounting identity, not a constraint. The equation defines how $C_0$ partitions. It is automatically satisfied for any $\Lambda$ and any matter fraction $\Omega_m$, because $C_0$, $\sum \Delta c_n$, and $S_H$ are all defined in terms of the same spacetime. Solving for $\Lambda$ gives:

$\Lambda = \frac{3\pi}{\left(C_0 - \sum_{n}\Delta c_n\right)\ell_P^2}$

but this does not determine $\Lambda$ — it merely re-expresses the partition. A genuine constraint on $\Lambda$ requires an independent computation of $\sum \Delta c_n$ from the bootstrap structure, without using $\Lambda$ as input. This is the role of the geometry functor (Gap 5) and Conjectures 8.9a–8.9b (topological vs epoch-conditional bridge).

The observed universe has $S_H \sim 10^{122}$, meaning the horizon coherence vastly exceeds the structural coherence (matter content). This means $\sum \Delta c_n \ll C_0$ — the bootstrap hierarchy absorbs a tiny fraction of the total coherence. The hierarchy question becomes: why does the bootstrap absorb so little? $\square$

Remark ($C_0$ is per-horizon, not global). The quantity $C_0$ is the coherence within a specific observer’s cosmological horizon — not the total coherence of the universe. Axiom 1 conserves $\mathcal{C}_{\text{global}}$ on full Cauchy slices, which may extend far beyond any observer’s horizon (and might be infinite if the observer network is spatially infinite). No observer can access $\mathcal{C}_{\text{global}}$. What enters the partition equation is $C_0$ — the per-horizon-volume budget. Its universality among comoving observers follows from the substrate’s constitutive universality — the uniform patch frequencies guaranteed by the aperiodic order of the observer network (Aperiodic Order, Corollary 3.2). This is a structural consequence of the axioms, not an external assumption.

Proposition 7.5 (The hierarchy is the second law). The self-consistency equation (Proposition 7.4) is the entropy decomposition (Entropy, Definition 3.1) applied at the cosmological scale. The hierarchy $\sum \Delta c_n \ll C_0$ is a consequence of the second law of thermodynamics.

Proof. For any bounded observer $A$ inside the cosmological horizon, the entropy decomposition gives:

$C_0 = \mathcal{C}_A(\tau) + S_A(\tau)$

where $\mathcal{C}_A$ is the accessible coherence and $S_A$ is the inaccessible coherence (entropy) relative to $A$ (Entropy, Definition 3.1).

The inaccessible coherence $S_A$ is dominated by the cosmological horizon: $S_A \approx S_H = 3\pi/(\Lambda\,\ell_P^2)$. This follows because every bounded observer’s coherence domain $\mathcal{D}_A$ is a proper subset of the full relational invariant graph (Entropy, Proposition 2.4), and the horizon counts the degrees of freedom beyond $A$‘s causal reach.

The accessible coherence $\mathcal{C}_A$ is the coherence locked into relational invariant structure that $A$ can resolve — the structural coherence $\sum \Delta c_n$ from the bootstrap hierarchy. Therefore:

$C_0 = \underbrace{\sum \Delta c_n}_{\mathcal{C}_A} + \underbrace{S_H}_{S_A}$

The self-consistency equation and the entropy decomposition are the same equation. The hierarchy question “why is $\sum \Delta c_n \ll C_0$?” is identical to “why is entropy large?” — and the second law answers this: $S_A$ increases monotonically (Entropy, Theorem 4.1). In a universe whose interaction graph has been growing for cosmic time $t \gg t_P$, the inaccessible fraction has grown large.

This decomposition is approximately observer-independent for observers whose epistemic horizon satisfies $\mathcal{I}_n^{\max} \gg 1$. Any two such observers within the same cosmological horizon see $S_A \approx S_B \approx S_H$, because the horizon entropy dominates — observer-specific corrections (the epistemic horizon differences from Proposition 1.5) are negligible relative to $S_H \sim 10^{122}$.

Caveat (validity domain). This approximation breaks down for observers at low bootstrap levels. For a Planck-mass observer at level 0, the epistemic horizon is $\mathcal{I}_0^{\max} \sim 1$ bit (Proposition 1.5). Its “correction” is not small relative to $S_H$ — the observer’s entire accessible coherence is the correction. Nearly all coherence is inaccessible from level 0’s perspective. Step 8 makes this level-dependence precise: the entropy decomposition is genuinely level-indexed, and the hierarchy question looks fundamentally different from different levels of the bootstrap.

For observers at the electron scale or above ($\mathcal{I}_n^{\max} \gtrsim 10^{43}$), the approximation is excellent and the cosmological density fractions are the entropy fractions:

$\Omega_\Lambda = \frac{S_H}{C_0}, \qquad \Omega_m = 1 - \frac{S_H}{C_0}$

The second law drives $\Omega_\Lambda \to 1$ as the universe approaches de Sitter equilibrium. The observed $\Omega_\Lambda \approx 0.7$ is a snapshot of this dynamical process at the current cosmic epoch — not a fine-tuned parameter. $\square$

Remark (What the second law does and does not explain). The identification $\Omega_\Lambda = S_H/C_0$ dissolves the 120-order hierarchy as a fine-tuning problem. The ratio $\Lambda_{\text{obs}}/\Lambda_P \sim 10^{-122}$ reflects the fact that the universe is old enough for entropy to dominate the coherence budget — a thermodynamic inevitability, not a coincidence.

What remains open:

• The crystallization fraction: The partition equation (Proposition 7.4) is an accounting identity — it describes how $C_0$ splits but cannot determine $\Lambda$ on its own. A genuine constraint requires computing $\sum \Delta c_n$ independently from the bootstrap structure and particle content, without $\Lambda$ as input. This is the role of the geometry functor (Gap 5). Note that $C_0$ itself is not an independent “initial condition” — it is the per-horizon coherence budget, determined by $\Lambda$ and the expansion history. The genuine unknown is the crystallization fraction $\Omega_m = \sum \Delta c_n / C_0$.
• The coincidence problem: why is $\Omega_m \sim \Omega_\Lambda$ at the current epoch (both order unity), rather than $\Omega_\Lambda$ being extremely close to 1? The second law says $\Omega_\Lambda$ grows, but does not select the current epoch. This is observer-selection territory — complex observers exist during the matter-dominated and transition eras, not in the de Sitter vacuum — but formalizing this within the framework’s non-anthropic axiom-selection approach (Definition 1.1) is open.

The absorbed fraction $\sum \Delta c_n / C_0 = \Omega_m \approx 0.3$ is a cosmological observable, not a Planck-scale ratio. The bootstrap ceiling (Bootstrap Mechanism, Proposition 6.2) bounds the number of resolvable levels by the holographic information capacity of the observer’s projected geometry, but the actual number of stable levels is determined by the division algebra chain (Bootstrap → Division Algebras) and the mass hierarchy tunneling factors (Mass Hierarchy).

Remark (The geometry functor and level-indexed $\Lambda$). The entropy identification (Proposition 7.5) explains the hierarchy qualitatively: the second law ensures $S_H \gg \sum \Delta c_n$ in an old universe. A quantitative prediction of $\Lambda$ would additionally require a geometry functor $G: \mathbf{Obs} \to \mathbf{Geom}$ mapping each observer’s epistemic horizon to an effective geometry, compatible with the bootstrap map $\mathcal{R}$ (functoriality across levels). Crucially, Step 8 shows that $G$ must produce level-indexed effective cosmological parameters $\Lambda_n^{\text{eff}}$ (Definition 8.2): each bootstrap level projects its own effective geometry with its own effective $\Lambda$. The constraint is not that all levels agree on a single $\Lambda$, but that the sequence $\{\Lambda_n^{\text{eff}}\}$ satisfies cross-level consistency (Proposition 8.3). This would constrain $C_0$ through the relationship between the bootstrap structure, the division algebra chain, and the spacetime geometry — potentially reducing the free parameter count from one ($C_0$) to zero. The existing gap in Bootstrap Mechanism about promoting $\mathcal{R}$ to a full functor on morphisms is the key prerequisite.

Step 8: Level-Dependent Coherence Partition

The framework says geometry is emergent: each observer projects an effective geometry from its accessible relational invariants via ER=EPR (Theorem 3.2). The epistemic horizon distinction (Proposition 1.5) shows that different bootstrap levels project vastly different effective geometries. Step 7 used the self-consistency equation with $S_H$ treated as level-independent. This step makes the level-dependence explicit and identifies a potential category error in the standard 120-order hierarchy comparison.

Definition 8.1 (Level-$n$ entropy partition). For an observer at bootstrap level $n$ (Definition 7.1), the entropy decomposition (Entropy, Definition 3.1) takes the form:

$C_0 = \mathcal{C}_{\text{acc}}^{(n)} + S^{(n)}$

where $\mathcal{C}_{\text{acc}}^{(n)}$ is the coherence accessible through the level-$n$ epistemic horizon $\mathcal{I}_n^{\max} = A_n/(4\ell_P^2)$, and $S^{(n)}$ is the inaccessible coherence from level $n$‘s perspective. The inaccessible portion $S^{(n)}$ includes:

(i) The cosmological horizon entropy $S_H = 3\pi/(\Lambda\,\ell_P^2)$ — coherence beyond causal reach for any observer.

(ii) Coherence locked into bootstrap levels $> n$ that level-$n$ cannot resolve — relational invariants between composite observers at scales $> \lambda_n$ that are invisible to level $n$.

Therefore $S^{(n)} \geq S_H$ for all $n$, with equality only at the highest realized level $N$ where all sub-cosmological structure is resolved. For the minimal observer at level 0: $S^{(0)} \approx C_0 - O(1)$.

Definition 8.2 (Effective cosmological parameter). The geometry projected at level $n$ is constructed from relational invariants accessible to level-$n$ observers via ER=EPR. This projected geometry has an effective horizon determined by what level $n$ can access. Define $\Lambda_n^{\text{eff}}$ as the cosmological parameter of this effective geometry:

$\Lambda_n^{\text{eff}} = \frac{3\pi}{S^{(n)}\,\ell_P^2}$

At the highest level ($n = N$): $S^{(N)} = S_H$, so $\Lambda_N^{\text{eff}} = \Lambda$ — the effective parameter coincides with the geometric constant in the Einstein equations. At level 0: $S^{(0)} \approx C_0$, so $\Lambda_0^{\text{eff}} \approx 3\pi/(C_0\,\ell_P^2)$.

Proposition 8.3 (Cross-level consistency). The effective cosmological parameters at different levels are not independent — they are constrained by the cross-level geometric consistency requirement (Proposition 7.2).

Argument. By Proposition 7.2, the geometry $G_{n+1}$ at level $n+1$ restricted to scale $\lambda_n$ must reduce to the geometry $G_n$ at level $n$. The effective horizon at level $n+1$ is determined by $S^{(n+1)} = S^{(n)} - \Delta S_n$, where $\Delta S_n > 0$ is the coherence that becomes accessible when moving from level $n$ to level $n+1$ (the coherence locked into relational invariants at scale $\lambda_{n+1}$ that level $n$ could not resolve). Therefore:

$\Lambda_{n+1}^{\text{eff}} = \frac{3\pi}{(S^{(n)} - \Delta S_n)\,\ell_P^2} > \Lambda_n^{\text{eff}}$

The sequence $\{\Lambda_n^{\text{eff}}\}$ is strictly increasing with level — higher-level observers project geometries with larger effective $\Lambda$ (smaller effective horizons in the inaccessible sector, because they have resolved more of the coherence into accessible structure). The sequence is bounded above by $\Lambda = \Lambda_N^{\text{eff}}$ and below by $\Lambda_0^{\text{eff}} = 3\pi/(C_0\,\ell_P^2)$. The increments $\Delta S_n$ are determined by the bootstrap structure: the division algebra chain, the mass hierarchy tunneling factors, and the coherence absorbed at each RG fixed point (Proposition 7.3). $\square$

Proposition 8.4 (Reframing the hierarchy). The standard statement of the cosmological constant hierarchy problem compares the Planck-scale bound $\Lambda < 3/\ell_P^2$ (Theorem 2.1) with the observed value $\Lambda_{\text{obs}} \sim 10^{-122}\;\ell_P^{-2}$. Within the framework, this comparison mixes two different levels of geometric projection.

Argument. The Planck bound (Theorem 2.1) asks whether a minimal observer at level 0 can fit within the causal horizon. It constrains the causal structure of the spacetime, which is ontic and level-independent — the bound is $\Lambda < 3/\ell_P^2$ regardless of which level is doing the observing. This is correct.

However, the “observed value” $\Lambda_{\text{obs}}$ is measured by human-scale observers at a high bootstrap level $N$. What they measure is $\Lambda_N^{\text{eff}}$ — the effective cosmological parameter of the geometry projected at their level. The ratio:

$\frac{\Lambda_{\text{obs}}}{\Lambda_P} = \frac{\Lambda_N^{\text{eff}}}{3/\ell_P^2} \sim 10^{-122}$

is a cross-level comparison. It compares a bound set by level-0 viability with a parameter measured at level $N$. Within the framework, the natural comparison at each level is between $\Lambda_n^{\text{eff}}$ and the viability bound for level-$n$ observers.

This does not dissolve the hierarchy — the ontic cosmological constant $\Lambda$ that appears in the Einstein equations is still $10^{-122}$ in Planck units, and the framework cannot derive this value. But it reframes the question: the hierarchy is not “why is $\Lambda$ so small?” but “what determines the sequence $\{\Lambda_n^{\text{eff}}\}$ and its endpoint $\Lambda_N^{\text{eff}} = \Lambda$?” The answer lies in the bootstrap structure — how much coherence each level absorbs — which is determined by the division algebra chain, the mass hierarchy, and the geometry functor (Gap 5).

A numerical consistency check (using the empirical matter fraction $\Omega_m \approx 0.3$) sharpens this reframing — see Proposition 8.7. $\square$

Proposition 8.5 (Breakdown of the level-independence approximation). Proposition 7.5 identifies the self-consistency equation with the entropy decomposition and claims approximate observer-independence. This approximation has a specific validity domain:

For observers at or above the electron scale, $S^{(n)}$ differs from $S_H$ by at most $10^{43}$ — negligible compared to $S_H \sim 10^{122}$. The approximation is excellent. For level-0 observers, $S^{(0)}/S_H \approx C_0/S_H \gg 1$ — the approximation fails completely. $\square$

Proposition 8.6 (The hierarchy from level 0’s perspective). A minimal observer at level 0 “sees” an almost entirely de Sitter universe. Its effective cosmological fraction is:

$\Omega_\Lambda^{(0)} = \frac{S^{(0)}}{C_0} \approx 1 - \frac{O(1)}{C_0} \approx 1$

From level 0’s perspective, there is no hierarchy problem in the partition — nearly all coherence is inaccessible. The universe it projects is almost entirely horizon, with vanishing structural content.

Proposition 8.7 (Numerical consistency check). Using the empirical matter fraction $\Omega_m \approx 0.3$ as input, the level-indexed quantities can be evaluated numerically. The result sharpens the reframing: the cross-level hierarchy is tiny, and the 120-order gap lives entirely in the absolute scale of $C_0$.

Calculation. From $\Lambda_{\text{obs}} \approx 2.9 \times 10^{-122}\;\ell_P^{-2}$ and $\Omega_\Lambda \approx 0.7$:

$S_H = \frac{3\pi}{\Lambda_{\text{obs}}\,\ell_P^2} \approx 3.25 \times 10^{122}, \qquad C_0 = \frac{S_H}{\Omega_\Lambda} \approx 4.6 \times 10^{122}$

$\sum \Delta c_n = \Omega_m \times C_0 \approx 1.4 \times 10^{122}$

The level-indexed effective parameters (Definition 8.2):

$\Lambda_0^{\text{eff}} = \frac{3\pi}{C_0\,\ell_P^2} \approx 2.0 \times 10^{-122}\;\ell_P^{-2}, \qquad \Lambda_N^{\text{eff}} = \Lambda_{\text{obs}} \approx 2.9 \times 10^{-122}\;\ell_P^{-2}$

$\frac{\Lambda_N^{\text{eff}}}{\Lambda_0^{\text{eff}}} = \frac{C_0}{C_0 - \sum \Delta c_n} = \frac{1}{\Omega_\Lambda} \approx 1.43$

The cross-level ratio is 1.43 — not $10^{122}$. The entire 120-order hierarchy lives in the absolute scale of $C_0 \sim 10^{122}$, not in the difference between levels. Level 0 and level $N$ agree on $\Lambda$ to within a factor of $1/\Omega_\Lambda$.

This means the hierarchy question is precisely: what determines the crystallization fraction $\Omega_m$? The partition equation (Proposition 7.4) is an accounting identity that holds for any $\Lambda$. The level-indexing does not dissolve the question — it clarifies that $C_0$ is not a free “initial condition” (it is determined by $\Lambda$ and the expansion history) and that the genuine unknown is the crystallization fraction. $\square$

Proposition 8.8 (The hierarchy requires an independent constraint). The partition equation $C_0 = \sum \Delta c_n + S_H$ cannot determine $\Lambda$ on its own — it is an accounting identity. The cosmological constant is determined only if $\sum \Delta c_n$ can be computed independently from the bootstrap structure.

Argument. The partition equation defines $C_0 \equiv \sum \Delta c_n + S_H$. Expressing each term via $\Lambda$ and $\Omega_m$:

$\frac{S_H}{1 - \Omega_m} = \frac{\Omega_m \cdot S_H}{1 - \Omega_m} + S_H$

This simplifies to $1 = 1$ — a tautology. The equation is automatically satisfied for any $\Lambda$ and any $\Omega_m$. It describes the structure of the partition but cannot select which partition is realized.

A genuine constraint requires computing $\sum \Delta c_n$ from the bootstrap structure without $\Lambda$ as input. The bootstrap structure is finite and determined:

• 4 algebra levels (R, C, H, O) from Bootstrap → Division Algebras (Theorem 7.1 — Hurwitz obstruction)
• 3 generations from Three Generations (Theorem 3.1 — dim so(3) = 3)
• Coupling ratios $\alpha_1 : \alpha_2 : \alpha_3 = 4 : 2 : 1$ from Coupling Constants (Proposition 2.1)
• Mass hierarchy pattern $\Lambda_k \sim \Lambda_{k-1} \cdot e^{-c_k/g_k^2}$ from Mass Hierarchy (Theorem 3.1)

If the geometry functor (Gap 5) can compute $\sum \Delta c_n = F(\text{SM structure})$ as a function of the known particle content alone — independently of $\Lambda$ — then the partition equation becomes:

$C_0(\Lambda) = F(\text{SM structure}) + S_H(\Lambda)$

This is one equation in one unknown ($\Lambda$), with $S_H(\Lambda) = 3\pi/(\Lambda\ell_P^2)$ and $C_0(\Lambda) = F + S_H$. It has a solution for any positive $F$: $\Lambda = 3\pi/(C_0 - F)\ell_P^{-2}$. But determining $F$ — the total coherence crystallized by the SM — is the open problem. $\square$

The bridge structure between saturated endpoints

Endpoint saturation is unconditional. Both endpoints of the bootstrap hierarchy are gravitationally saturated as structural facts of the framework, not as conditions to impose.

At the bottom of the hierarchy, the minimal observer has spatial extent $\sim \ell_P$, Schwarzschild radius $R_S(m_P) = 2Gm_P/c^2 \sim \ell_P$, and Compton wavelength $\lambda_C(m_P) = \hbar/(m_P c) \sim \ell_P$. All three scales coincide by construction (Minimal Observer Structure, Proposition 7.1). The holographic bound is saturated automatically: the observer’s boundary area $A_0 \sim \ell_P^2$ encodes exactly $\sim 1$ bit, and the observer is simultaneously the smallest possible and already at its own gravitational collapse limit.

At the top, each observer’s cosmological horizon has radius $R_H = \sqrt{3/\Lambda}$. The mass enclosed within that horizon is $M_H = c^2 R_H/(2G)$ (in the de Sitter limit). Its Schwarzschild radius is $R_S(M_H) = 2GM_H/c^2 = R_H$. The horizon sits at its own gravitational collapse scale. The holographic bound is again saturated: $S_H = A_H/(4\ell_P^2)$ encodes the maximum information for a region of this size. (Each comoving observer has its own horizon, centered on itself. By constitutive universality the budget $C_0$ has the same value for all, but it is a per-observer quantity — not a global total. See Proposition 7.4, Remark.)

Because both endpoints are always saturated, “double saturation” is not a candidate boundary condition to impose — it is a structural fact of the framework. What remains is to ask whether the bridge between the universally-saturated endpoints is uniquely determined by SM structure.

Between the two saturated endpoints, the bootstrap hierarchy builds a finite bridge: 4 algebra levels (Bootstrap → Division Algebras, Theorem 7.1), 3 generations (Three Generations, Theorem 3.1), coupling ratios $\alpha_1:\alpha_2:\alpha_3 = 4:2:1$ (Coupling Constants, Proposition 2.1), and exponential mass separations (Mass Hierarchy, Theorem 3.1). Each step absorbs coherence $\Delta c_n$ determined by the particle content and couplings at that level. The c-theorem (Renormalization Group, Theorem 5.2) ensures the flow is monotonic.

The total coherence is:

$C_0 = \underbrace{S_{\min}}_{\substack{\text{Planck}\\\text{saturation}}} + \underbrace{\sum_{n=1}^{N} \Delta c_n}_{\substack{\text{bootstrap}\\\text{bridge}}} + \underbrace{S_H}_{\substack{\text{cosmological}\\\text{saturation}}}$

Mass Hierarchy (Step 7) shows that the bridge decomposes into two regimes with different information scaling:

$\sum_{n=1}^{N} \Delta c_n = \underbrace{\sum_{n \leq 3} \Delta c_n^{\text{topo}}}_{\text{topological encoding}} + \underbrace{\sum_{n > 3} \Delta c_n^{\text{struct}}}_{\text{structural encoding}}$

Within the division algebra chain ($n \leq 3$), information is stored topologically (gauge charges, winding numbers, generation indices) and the mass-information relationship is inverse: heavier particles have smaller boundaries and less epistemic capacity. Beyond the octonionic level ($n > 3$), the algebras are exhausted and information is stored structurally (relational invariant networks), where the mass-information relationship reverses: more complex composites have larger boundaries and more epistemic capacity. The topological contribution is bounded by the finite particle content that the division algebras determine. The structural contribution grows with composite complexity. The transition occurs at the confinement scale $\Lambda_{\text{QCD}} \sim 0.3$ GeV.

The two regimes differ sharply in their dependence on SM structure alone. The topological regime is bounded by the finite SM particle content; the structural regime grows with composite complexity within a per-observer horizon, which depends on cosmic time and the horizon volume itself. This motivates splitting the literal “bridge is SM-determined” claim into two conjectures with different status.

Conjecture 8.9a (Topological bridge is SM-determined). The topological contribution to the bootstrap bridge

$\sum_{n \leq 3}\Delta c_n^{\mathrm{topo}} \;=\; F_{\mathrm{topo}}(\text{SM structure})$

is uniquely determined by the Standard Model particle content — 4 division algebra levels, 3 generations, coupling ratios $4:2:1$, and the topological invariants (gauge charges, winding numbers, generation indices) carried by these structures. The resulting bound $\sum_{n \leq 3}\Delta c_n^{\mathrm{topo}} = O(10^2)$ bits in Planck units is SM-computable and $\Lambda$-independent.

Status. Framework content for this conjecture is in place — the division algebra chain, generation count, and coupling ratios are all framework-derived. What remains is explicit enumeration of topological information content at each level, a finite bookkeeping task given the SM particle content. The resulting $O(10^2)$ bits is negligible compared to $S_H \sim 10^{122}$ — the topological regime cannot account for the observed structural coherence.

Conjecture 8.9b (Structural bridge is epoch-conditional). The structural contribution to the bootstrap bridge,

$\sum_{n > 3}\Delta c_n^{\mathrm{struct}},$

is not SM-determined alone. By Mass Hierarchy Step 7 Remark, the structural regime scales with composite density $\times$ horizon volume, both of which are cosmic-epoch-dependent. By Proposition 7.5’s caveat, $\Omega_m \to 0$ as $t \to \infty$ in the de Sitter approach, so $\sum_{n > 3}\Delta c_n^{\mathrm{struct}}$ is a decaying observable of the current cosmic epoch, not a framework-fundamental quantity. Converting the strict lower bound $\Lambda \geq 3\pi/(C_0\ell_P^2)$ (Theorem 8.10 below) into a framework-determined value for $\Lambda$ requires both:

(i) a geometry functor $G: \mathbf{Obs} \to \mathbf{Geom}$ that computes coherence absorption at each bootstrap level from the SM and the saturation boundary conditions (Gap 5); and

(ii) a cosmic-epoch selection principle beyond the minimal-observer requirement of Definition 1.1 — some framework-intrinsic specification of which cosmic time corresponds to a measured $\Omega_m$.

Status. Optional under existing framework structure — neither forced nor excluded. A literal “SM-only determines $\Lambda$” claim is excluded by the epoch-dependence of the dominant structural regime; a reformulated “SM + cosmic epoch determines $\Omega_m$” claim is consistent but requires both (i) and (ii), neither of which is in place.

Remark (The literal “double saturation uniquely determines $\Lambda$” claim is excluded). An earlier formulation of this conjecture asserted that simultaneous saturation of the two gravitationally-saturated endpoints, combined with an SM-determined bridge, uniquely determined $\Lambda$. Conjecture 8.9b shows this is excluded: the dominant term of the bridge — the structural contribution $\sum_{n > 3}\Delta c_n^{\mathrm{struct}}$ dominating the observed $\sum \Delta c_n \sim 10^{122}$ bits — is epoch-dependent, and no SM-only function $F(\text{SM structure})$ can capture a decaying cosmic-evolution observable. A restricted “topological-only” version (Conjecture 8.9a combined with $\sum_{n > 3}\Delta c_n^{\mathrm{struct}} = 0$) is internally consistent but observationally excluded: it would force $\Omega_m \sim 10^{-120}$ and $\Omega_\Lambda \to 1$, contradicting the measured $\Omega_\Lambda \approx 0.7$.

Remark (The $\Lambda_{\mathrm{obs}}/\Lambda_0^{\mathrm{eff}} \approx 1.43$ factor is cosmic-evolution). The deviation factor $1/\Omega_\Lambda \approx 1.43$ between the observed $\Lambda_{\mathrm{obs}}$ and the framework-derived lower bound $\Lambda_0^{\mathrm{eff}}$ of Corollary 8.11 is a cosmic-evolution quantity, not a framework-fundamental ratio. By Proposition 7.5’s caveat, $\Omega_\Lambda \to 1$ as the universe approaches de Sitter equilibrium, so the 1.43 factor tracks the current non-equilibrium state. Existing framework structure bounds $\Omega_\Lambda \in [0.5, 1)$ (via the holographic upper bound $\sum \Delta c_n \leq S_H$), consistent with the observed $0.7$, but does not pin the specific current-epoch value — that selection belongs to Conjecture 8.9b.

Observational check. With the empirical matter fraction $\Omega_m \approx 0.3$, the holographic bound on the structural coherence gives $\sum \Delta c_n \leq S_H$ (the information within the horizon cannot exceed the horizon entropy), which constrains $\Omega_\Lambda \geq 0.5$. The observed $\Omega_\Lambda \approx 0.7$ satisfies this framework bound. The structural coherence $\sum \Delta c_n \approx 0.43\,S_H$ is well within the holographic limit — the bridge uses less than half the available “room” between the two saturated endpoints at the current epoch.

Theorem 8.10 (Strict positivity of $\Lambda$). $\Lambda > 0$, with lower bound $\Lambda \geq \frac{3\pi}{C_0^{(N)}\,\ell_P^2}$ where $C_0^{(N)}$ is the per-observer Cauchy-slice total at the highest bootstrap level $N$.

Proof. By Coherence Conservation Corollary 5.5a, the Cauchy-slice total is an integer multiple of $\hbar\omega_0$: $C_0^{(n)} = N^{(n)}\,\hbar\omega_0$ with $N^{(n)} \in \mathbb{Z}_{\geq 0}$. By Proposition 7.4 of the present derivation, $C_0$ is a per-observer quantity: each observer’s budget is determined by its own epistemic horizon with boundary area $A_\mathcal{B}$, bounded above by the holographic capacity $A_\mathcal{B}/(4\ell_P^2)$. Per-observer finiteness therefore applies universally — no observer has infinite accessible coherence, regardless of the global spatial extent of the interaction graph.

Definition 8.1 decomposes $C_0 = \mathcal{C}_{\text{acc}}^{(n)} + S^{(n)}$ with both components non-negative, so $S^{(n)} \leq C_0^{(n)}$. Definition 8.2 gives $\Lambda_n^{\text{eff}} = 3\pi/(S^{(n)}\ell_P^2)$. Combining: $\Lambda_n^{\text{eff}} \;\geq\; \frac{3\pi}{C_0^{(n)}\,\ell_P^2} \;>\; 0$ at every bootstrap level. At the highest level $N$, $S^{(N)} = S_H$ and $\Lambda_N^{\text{eff}} = \Lambda$ — the cosmological constant in the Einstein field equations. Combined with Theorem 5.4 ($\Lambda \geq 0$), strict positivity $\Lambda > 0$ follows. $\square$

Corollary 8.11 (Numerical lower bound). Using the per-observer coherence budget $C_0 \approx 4.6 \times 10^{122}\,\hbar\omega_0$ of Proposition 8.7, the framework-derived lower bound is $\Lambda \;\gtrsim\; 2 \times 10^{-122}\,\ell_P^{-2}$ — coinciding with the level-0 projection $\Lambda_0^{\text{eff}}$ of Proposition 8.7. The observed value $\Lambda_{\text{obs}} \approx 2.9 \times 10^{-122}\,\ell_P^{-2}$ lies at factor $\sim 1/\Omega_\Lambda \approx 1.43$ above this bound, consistent with measurement at bootstrap level $N$ where $S^{(N)} < C_0$.

Remark (connection to the cosmological arrow of time). The structural arrow of Time as Phase Ordering (Theorem 6.1) is monotone non-decreasing relational invariant depth along any directed path. At cosmological scale, Theorem 8.10 forces $\Lambda > 0$, which via Einstein’s equations gives eternal expansion — the Gibbons-Hawking bath of the resulting de Sitter phase supplies persistent event generation at constant rate per Hubble volume. Matter-dominated $\Lambda = 0$ cosmologies, by contrast, give a total event count per comoving volume that converges as $t \to \infty$ (integrated two-body interaction rate $\propto t^{-2}$), so the structural arrow saturates in finite depth. Strict positivity of $\Lambda$ therefore identifies the global cosmological arrow (eternal expansion) with the structural arrow (unbounded $d(v)$) at the level of the framework’s coherence accounting.

Remark (What Step 8 does and does not show). Step 8 defines the level-indexed quantities ($S^{(n)}$, $\Lambda_n^{\text{eff}}$), shows where the level-independence approximation breaks down (Proposition 8.5), localizes the hierarchy to the crystallization fraction rather than the absolute scale of $C_0$ (Proposition 8.7), shows that the partition equation is an identity rather than a constraint (Proposition 8.8), establishes strict positivity of $\Lambda$ with a computable lower bound (Theorem 8.10, Corollary 8.11), and splits the bridge-determination question into a topological part (Conjecture 8.9a, SM-determined in principle) and a structural part (Conjecture 8.9b, epoch-conditional). The open question at this derivation’s level is whether the topological part can be explicitly enumerated from the SM and whether the structural part can be pinned by a framework-intrinsic cosmic-epoch selection principle combined with the geometry functor (Gap 5).

Consistency Check

The observed universe has $\Lambda_{\text{obs}} \sim 10^{-122}\;\ell_P^{-2}$, giving $R_H \sim 10^{61}\;\ell_P$. The viability bound $\Lambda < 3/\ell_P^2$ is easily satisfied.

• Geometric bound: $R_H \sim 10^{61}\,\ell_P \gg \ell_P$. $\checkmark$
• Holographic budget: $S_H = 3\pi/(\Lambda_{\text{obs}}\,\ell_P^2) \sim 10^{122}$ bits — vastly exceeds the coherence requirement for two minimal observers. $\checkmark$
• Cosmological redshift: For observers separated by $\ell_P$, the redshift is $z \sim 10^{-122}$ — negligible. $\checkmark$
• Strict positivity: $\Lambda_{\text{obs}} \approx 2.9 \times 10^{-122}\,\ell_P^{-2} > 0$, consistent with Theorem 5.4 ($\Lambda \geq 0$) and Theorem 8.10 ($\Lambda > 0$). The observed value lies at factor $\sim 1.43$ above the framework’s lower bound $\Lambda \geq 3\pi/(C_0\ell_P^2) \approx 2 \times 10^{-122}\,\ell_P^{-2}$ (Corollary 8.11). $\checkmark$
• Static patch lifetime: The static patch metric is time-independent, and observers can persist indefinitely. $\checkmark$

The framework is consistent with the observed cosmological constant, predicts strict positivity with a computable lower bound, and identifies the remaining task as determining the absolute value of $\Lambda$ from the bootstrap structure — via the topological bridge enumeration (Conjecture 8.9a), a cosmic-epoch selection principle (Conjecture 8.9b), and the geometry functor program (Gap 5 below).

Comparison with Anthropic Reasoning

The anthropic bound is $\sim 10^{120}$ times tighter because it requires complex observers (galaxies, carbon chemistry), not minimal ones. This derivation’s rigorous bound (Theorem 2.1) is correspondingly weaker. However, Step 8 provides a structurally different path to the hierarchy that does not rely on a landscape or typicality arguments: the finite bootstrap structure (4 algebra levels, 3 generations, known coupling ratios) converts $\Lambda$ from an unconstrained free parameter to a self-consistency condition on a computable system with one remaining unknown ($C_0$). The numerical check (Proposition 8.7) confirms that the cross-level ratio is only $\sim 1.4$ — the hierarchy is entirely about the absolute scale of $C_0$, which is determined by the self-consistency loop. Whether this loop has a unique fixed point is the open question (Gap 5).