Observer Viability: Formation, Preservation, and Detection

provisional

Overview

An observer is a localized, persistent, detectable coherence pattern. “Existing as an observer” has three distinct structural components, and this derivation establishes them as the three independent viability conditions of the framework:

• Formation. The observer must come into being — substrate fluctuations must tunnel into the observer’s code space with enough probability for crystallization to occur somewhere in cosmological history. This is Mass Hierarchy’s WKB mechanism, operating at formation time.

• Preservation. Once formed, the observer’s integer-invariant profile must survive substrate noise over its coherence lifetime. This is the QEC preservation condition $d_{\mathrm{achieved}} \geq d_{\mathrm{req}}$ derived from Substrate Noise and Profile-Dependent Coupling Modulation on the three-axis code of Observer as an Error-Correcting Code, operating throughout the observer’s lifetime.

• Detection. The observer must have a well-defined operational edge — a finite radius at which external observers’ resolution matches the observer’s pattern signal. This is the edge equation of Observer Edges and Mutual Opacity, operating in inter-observer relation.

The viability theorem. An observer is viable iff all three conditions hold. The three are pairwise-distinct in functional form — no two are reducible to a single mechanism via simple-functional identification. The SM spectrum is the simultaneous-satisfaction set of all three.

Structural distinctness. The three mechanisms have characteristic functional dependences:

• Formation: $P_{\mathrm{form}} \sim e^{-c/g^2}$, exponential in inverse coupling squared.
• Preservation: the QEC exponent $(d_{\mathrm{req}}/2)\log(p_{th}/p_{\mathrm{phys}}^{\mathrm{eff}})$ simplifies algebraically to $c_{\mathrm{logic}} \log(T_{\mathrm{coh}}/\tau_P)$ — coupling-independent after cancellation.
• Detection: $r_{\mathrm{edge}} \sim r_C = \hbar/(mc)$, Compton-scaled, mass-linear.

Pairwise scaling-mismatch arguments confirm that no two of these functional forms can be identified — each is a distinct mechanism. The three together constitute the complete axis set for observer existence under the framework’s current commitments.

The structural preservation margin. At a WKB-selected mass $m_k = m_P e^{-c_k/g_k^2}$ with profile $\mathcal{I}_A$:

$\frac{d_{\mathrm{achieved}}(m_k)}{d_{\mathrm{req}}(\mathcal{I}_A)} \;\sim\; \frac{e^{c_k/g_k^2}}{c_{\mathrm{logic}}\, \log(T_{\mathrm{coh}}/\tau_P)}$

— an exponential in WKB action over a logarithmic preservation demand. For SM couplings ($c/g^2 \in [20, 60]$) and cosmological lifetime ($\log(T_{\mathrm{coh}}/\tau_P) \approx 138$): ratio $\sim 10^{9}$–$10^{25}$. The large observed preservation margins are a structural consequence of the mechanism split at SM couplings.

Confinement as (D)-axis failure. Free quarks form (they crystallize at the QCD bootstrap level, satisfying F) and would preserve (their QEC code is viable within confining hadrons, satisfying P). They fail detection: by Profile-Dependent Edges and Confinement Theorem 7.1, isolated color-charged profiles have no finite edge — the color channel’s linear-confining signal has no Yukawa-like radial decay. Color-neutral composites (hadrons) satisfy all three conditions.

Scope. Each of (F), (P), (D) is a theorem or structural-correspondence argument in its own right (inheriting the rigor of its source derivation). Joint sufficiency — the claim that no fourth independent axis is needed — is a framework-completeness assertion rather than a theorem. A fourth axis could in principle be identified in future work.

Statement

Observer Viability Theorem (Tripartite). A candidate observer profile $\mathcal{I}_A$ with mass $m_A$ is viable — corresponds to a stable, detectable, persistent observer in the framework’s sense — iff all three conditions hold:

(F) Formation. $m_A$ is a WKB crystallization scale at some bootstrap level: tunneling probability is sufficient for crystallization over cosmological history,

$P_{\mathrm{form}}(m_A; \mathcal{I}_A) \cdot \frac{V_4^{\mathrm{cosmo}}}{V_4^{\mathrm{Planck}}} \geq 1$

with $P_{\mathrm{form}} \sim e^{-c/g^2}$ by WKB (Mass Hierarchy Theorem 3.1).

(P) Preservation. The three-axis QEC code achieves sufficient distance to preserve its integer invariants under substrate noise over its coherence lifetime,

$d_{\mathrm{achieved}}(m_A) \geq d_{\mathrm{req}}(\mathcal{I}_A; T_{\mathrm{coh}})$

on spatial, algebraic, and temporal axes (Substrate Noise and Profile-Dependent Coupling Modulation Proposition 2.5).

(D) Detection. The edge equation of Observer Edges and Mutual Opacity has a finite-$r$ solution for the profile’s pattern signal; equivalently, $A(r; \mathcal{I}_A)$ has Yukawa-like decay at large $r$ (not pathological linear-confining divergence),

$\exists\, r_{\mathrm{edge}} \in (0, \infty): \quad A(r_{\mathrm{edge}}; \mathcal{I}_A) = m_B c^2 \text{ for any reference detector } B$

(Observer Pattern Signal; Observer Edges and Mutual Opacity; Profile-Dependent Edges and Confinement Theorem 7.1).

Theorem (Pairwise distinctness). No two of (F), (P), (D) are reducible to each other via simple-functional identification:

• Formation ≠ Preservation: WKB action $S/\hbar = c/g^2$ is $1/g^2$-dependent; QEC exponent $(d_{\mathrm{req}}/2)\log(p_{th}/p_{\mathrm{phys}}^{\mathrm{eff}})$ simplifies algebraically to $c_{\mathrm{logic}} \log(T_{\mathrm{coh}}/\tau_P)$, coupling-independent. No linear or monomial identification matches.
• Preservation ≠ Detection: preservation quantities cancel coupling dependence; detection edge $r_{\mathrm{edge}} \approx W(1) r_C$ is Compton-scaled and mass-dependent via $r_C$.
• Formation ≠ Detection: formation is exponential in $1/g^2$, detection Compton-linear in $1/m$.

Theorem (Structural preservation margin). At the WKB-selected mass $m_k = m_P e^{-c_k/g_k^2}$:

$\frac{d_{\mathrm{achieved}}(m_k)}{d_{\mathrm{req}}(\mathcal{I}_A)} \;\sim\; \frac{e^{c_k/g_k^2}}{c_{\mathrm{logic}}\, \log(T_{\mathrm{coh}}/\tau_P)}$

— an exponential in WKB action over a logarithmic preservation demand. For SM couplings and cosmological lifetime: ratio $\sim 10^{9}$–$10^{25}$. The observed large preservation margins are derived, not arbitrary.

Corollary (Standard Model spectrum). The observed SM spectrum is the set of profiles satisfying (F) ∧ (P) ∧ (D):

• SM fundamental fermions, gauge mediators, and their composites satisfy all three.
• Free quarks fail (D) only: they form, preserve within hadrons, but have no well-defined detection edge as isolated particles — hence confinement.
• Color-neutral hadrons satisfy all three with Yukawa-screened exterior color signal at $\Lambda_{\mathrm{QCD}}$.

Corollary (Confinement as (D)-axis failure). Under the triplet, confinement is a specific viability-axis failure — free quarks fail detection while satisfying formation and preservation — rather than a mixed formation/preservation issue.

Derivation

Step 1: The three mechanisms

Formation — WKB tunneling. The mass spectrum of viable observers is set by crystallization at each bootstrap level: $\Lambda_k \sim \Lambda_{k-1} e^{-c/g_k^2}$ (Mass Hierarchy Theorem 3.1). The exponential suppression in inverse coupling squared is the framework’s natural mechanism for the observed mass hierarchy — logarithmically natural ratios of modest couplings produce exponentially separated mass scales.

Definition 1.1 (Formation). The formation probability $P_{\mathrm{form}}(m_A; \mathcal{I}_A)$ is the probability per unit cosmological 4-volume that substrate fluctuations crystallize into a stable observer with profile $\mathcal{I}_A$ at mass scale $m_A$. Under Mass Hierarchy structural postulate S1 (tunneling–crystallization correspondence), this is the WKB tunneling probability through a coherence barrier: $P_{\mathrm{form}}(m_A; \mathcal{I}_A) \sim e^{-S(\mathcal{I}_A)/\hbar}$ with $S(\mathcal{I}_A)/\hbar = c/g^2$.

Preservation — QEC code viability. Once formed, an observer’s profile is a three-axis integer-invariant code (Observer as an Error-Correcting Code) subject to substrate bit-flip rates $p_{\mathrm{phys}}^{\mathrm{eff}}(\mathcal{I}_A) = p_{\mathrm{phys}}^{\mathrm{geom}} + \sum g_i^2 \eta_i$ (Substrate Noise and Profile-Dependent Coupling Modulation Corollary 4.3).

Definition 1.2 (Preservation). The preservation condition is the QEC viability inequality

$d_{\mathrm{achieved}}(m_A) \;\geq\; d_{\mathrm{req}}(\mathcal{I}_A; T_{\mathrm{coh}}) \;=\; \frac{2 c_{\mathrm{axis}}(\mathcal{I}_A)}{\log(p_{th}/p_{\mathrm{phys}}^{\mathrm{eff}})} \cdot \log(T_{\mathrm{coh}}/\tau_P)$

on each axis, with $d_{\mathrm{achieved}} \sim m_P/m_A$ (polynomial in inverse mass) and $d_{\mathrm{req}}$ logarithmic in lifetime.

Detection — pattern-edge viability. The observer’s pattern signal must cross external observers’ resolution threshold at some finite radius (Observer Edges and Mutual Opacity Step 2).

Definition 1.3 (Detection). The detection condition is that the edge equation

$A(r_{\mathrm{edge}}; \mathcal{I}_A) \;=\; m_B c^2$

has a finite-$r$ solution, where $A(r; \mathcal{I}_A)$ is the pattern signal (Observer Pattern Signal). Equivalently, the profile’s signal must be Yukawa-like (or Coulomb-like for massless components) at large $r$, not linearly divergent.

Step 2: Pairwise distinctness — scaling arguments

Proposition 2.1 (Formation ≠ Preservation). WKB action: $S/\hbar = c/g^2$, exponential in $1/g^2$. QEC candidate: $(d_{\mathrm{req}}/2)\log(p_{th}/p_{\mathrm{phys}}^{\mathrm{eff}})$, which by direct substitution equals

$\frac{d_{\mathrm{req}}(\mathcal{I}_A)}{2} \cdot \log\!\left(\frac{p_{th}}{p_{\mathrm{phys}}^{\mathrm{eff}}}\right) \;=\; c_{\mathrm{logic}}(\mathcal{I}_A) \cdot \log\!\left(\frac{T_{\mathrm{coh}}}{\tau_P}\right)$

— coupling-independent after algebraic cancellation (the $d_{\mathrm{req}}$ definition and $\log$ factor are inverses-up-to-constant by construction). The two have incompatible functional dependence on coupling. No linear or monomial identification $S/\hbar = F(d_{\mathrm{req}}, \log(p_{th}/p_{\mathrm{phys}}^{\mathrm{eff}}))$ matches.

Proposition 2.2 (Preservation ≠ Detection). Preservation quantities cancel coupling dependence (Proposition 2.1); detection edge $r_{\mathrm{edge}} \approx W(1) r_C = W(1) \hbar/(mc)$ is Compton-scaled and mass-dependent via $r_C$, not coupling-canceled. Different functional forms; no identification possible.

Proposition 2.3 (Formation ≠ Detection). Formation: $P_{\mathrm{form}} \sim e^{-c/g^2}$, exponential in coupling. Detection: $r_{\mathrm{edge}} \sim r_C$, Compton-linear in $1/m$. Different scalings; no functional identification.

Step 3: Necessity of each condition

Proposition 3.1 (Formation is necessary). A profile that fails (F) never crystallizes from substrate in cosmological history. Even if it would preserve and have a well-defined edge, no instance exists.

Proposition 3.2 (Preservation is necessary). A profile that fails (P) — $d_{\mathrm{achieved}} < d_{\mathrm{req}}$ on some axis — decoheres under substrate noise within its intended coherence lifetime. Even if it forms and has a well-defined edge, formed instances do not persist as stable observers.

Proposition 3.3 (Detection is necessary). A profile that fails (D) — no finite-$r$ edge solution — lacks a well-defined spatial boundary as a pattern. By Observer Edges and Mutual Opacity, inter-observer relations require that each observer have a finite edge beyond which it is mutually opaque. A pathological-signal profile has no such edge; it cannot participate in the observer network.

Proof. By the framework’s observer-centrism (Observer Holographic Equivalence Corollary 4.6), every observer is defined by its relations with other observers. A pattern with no finite edge cannot be in well-defined relation with peers at any distance. The profile fails to constitute a framework observer. $\square$

Corollary 3.4 (Tripartite necessity). All three conditions are necessary for observer viability; no single condition implies the others.

Step 4: Joint sufficiency

Proposition 4.1 (Joint sufficiency). Under the framework’s current commitments — Axioms 1–3, bootstrap hierarchy, three-axis QEC code, pattern-edge structure — (F) ∧ (P) ∧ (D) are jointly sufficient for observer viability: any profile satisfying all three is realized as a stable, detectable observer, and the observed SM spectrum consists of such profiles.

Structural argument. A profile satisfying (F) has at least one instance formed. A profile satisfying (P) preserves its integer-invariant profile. A profile satisfying (D) has a well-defined operational edge. These three together exhaust the framework’s current conditions on observer existence.

Remark 4.2. Joint sufficiency is a framework-completeness claim. A fourth axis could in principle be identified — for instance, a bootstrap-composition viability condition distinct from (F) — but none is currently known to be independent of the three.

Step 5: Structural preservation margin

Proposition 5.1 (Margin as exponential-over-logarithmic). At the WKB-selected mass $m_k = m_P e^{-c_k/g_k^2}$ and profile $\mathcal{I}_A$:

$\frac{d_{\mathrm{achieved}}(m_k)}{d_{\mathrm{req}}(\mathcal{I}_A)} \;=\; \frac{m_P/m_k}{c_{\mathrm{logic}} \log(T_{\mathrm{coh}}/\tau_P)/\log(p_{th}/p_{\mathrm{phys}}^{\mathrm{eff}})} \;\sim\; \frac{e^{c_k/g_k^2}}{c_{\mathrm{logic}} \log(T_{\mathrm{coh}}/\tau_P)}$

— the ratio of WKB-exponential formation suppression to QEC-logarithmic preservation demand.

Proof. Substitute $m_k = m_P e^{-c_k/g_k^2}$ into $d_{\mathrm{achieved}} \sim m_P/m_k$ and the $d_{\mathrm{req}}$ formula from Substrate Noise and Profile-Dependent Coupling Modulation Proposition 2.5. $\square$

Remark 5.2 (Magnitude for SM couplings).

• Electroweak-scale WKB ($c/g^2 \sim 40$): margin $\sim e^{40}/(10^2) \sim 10^{15}$.
• Lighter-fermion ratios ($c/g^2 \sim 60$): margin $\sim e^{60}/(10^2) \sim 10^{24}$.
• Planck-near scales ($c/g^2 \sim 10$): margin $\sim e^{10}/(10^2) \sim 10^2$ — still $> 1$.

The $10^{15}$–$10^{25}$ range is the structural prediction; observed SM preservation margins fall within it.

Remark 5.3 (The margin is derived, not arbitrary). The enormous preservation margins observed when applying the QEC machinery to the SM spectrum reflect the ratio of formation-exponential to preservation-logarithmic at SM couplings. The margin value is derived from the functional forms of the two mechanisms, not a degree of freedom adjustable by choice.

Step 6: SM spectrum as simultaneous-satisfaction set

Proposition 6.1 (SM satisfies all three conditions). Each observed SM particle passes (F), (P), and (D):

• SM fundamental fermions (electrons, muons, taus, neutrinos, up-type and down-type quarks): (F) by WKB at appropriate bootstrap level; (P) with $10^{15}$–$10^{25}$ margins (Remark 5.2); (D) at Compton scale via Yukawa signal.
• SM gauge mediators (photon, $W$, $Z$, gluon-within-hadron): (F) at gauge-sector crystallization; (P) via gauge-mediator massless exemption (Substrate Noise and Profile-Dependent Coupling Modulation Corollary 6.1); (D) via Coulomb tail (photon) or hadron-confined Yukawa (gluon).

Proposition 6.2 (Free quarks fail (D) only). Free quarks satisfy (F) — they form at QCD bootstrap level. They would satisfy (P) — their QEC code preservation is viable within confining hadrons. They fail (D) alone: isolated color-charged profiles have pathological non-Yukawa signal (Profile-Dependent Edges and Confinement Proposition 5.1). The triplet renders confinement a (D)-axis failure.

Proposition 6.3 (Color-neutral hadrons satisfy all three). Hadrons satisfy (F) by binding at QCD scale, (P) with preservation margin, (D) with Yukawa-screened exterior color signal at the hadron scale (Profile-Dependent Edges and Confinement Proposition 6.3). Hadrons are the viable observer profiles at QCD bootstrap level.

Step 7: Detection and the earlier-open joint-fixed-point thread

Earlier framing of this derivation (when it was a pair theorem) flagged a “joint fixed-point consistency” thread as a future-work direction: whether formation and preservation might jointly enter a single fixed-point condition selecting observer masses. With detection as the third leg, the joint-mechanism picture takes a different form than that thread anticipated.

Remark 7.1 (Resolution of the joint-fixed-point thread). Under the triplet, the three mechanisms are not unified into a single fixed-point equation; they are three independent viability axes. The “joint-fixed-point” conception — where one quantity self-consistently determines mass — is replaced by the simultaneous-satisfaction picture — where three independent conditions must each hold. This is structurally stronger than a single fixed-point would have been: three distinct constraints are jointly tighter than any single constraint.

The three mechanisms do interact at the phenomenological level:

• Detection reflects masses selected by formation: the detection edge is at Compton scale of the WKB-selected mass.
• Preservation and detection both depend on profile structure — profiles that fail detection (free quarks) typically also fail preservation at the free-quark level, but the mechanisms are logically distinct.
• None of the three determines masses independently of the others; formation is the mass-selection mechanism, while detection and preservation are constraints the formation-selected masses must also satisfy.

Consequences

C1. Observer viability is tripartite. (F) ∧ (P) ∧ (D) is the complete axis set for observer existence under the framework’s current commitments.

C2. The three mechanisms are structurally distinct. Formation: exponential in coupling (WKB). Preservation: logarithmic in lifetime, coupling-cancelled (QEC). Detection: Compton-linear in mass (pattern edges). Pairwise-incompatible scalings.

C3. Confinement is a (D)-axis failure. Free quarks fail detection (no finite edge for isolated color); hadrons satisfy (D). Confinement is a specific viability-axis failure.

C4. Massless gauge mediators are limits of all three conditions. Photon and gluon satisfy (F), (P), (D) in their respective massless / confined-composite limits. No fine-tuning; structural.

C5. SM preservation margins $10^{15}$–$10^{25}$ are derived. Large SM preservation margins fall out of the triplet: formation-exponential dominates preservation-logarithmic at SM couplings, while detection-Compton completes the viability picture.

C6. Mass selection is formation’s domain. Detection reflects masses selected by formation (via $r_C = \hbar/(mc)$); it does not independently select masses. The mass-selection mechanism is WKB (Mass Hierarchy).

C7. Bootstrap composition is not automatically a fourth axis. Higher-level composite observers satisfy their own (F) ∧ (P) ∧ (D) at the composite level. Whether bootstrap composition requires an additional viability condition (cross-level consistency) is a question for future work.

C8. The observed SM spectrum is the intersection. SM = (F)-viable ∩ (P)-viable ∩ (D)-viable. Each SM particle passes all three; each non-observed profile fails at least one.

Rigor Assessment

Rigorous (direct consolidation of source theorems):

• Proposition 2.1 (Formation ≠ Preservation scaling mismatch): rigorous algebraic cancellation + scaling argument.
• Propositions 3.1, 3.2, 3.3 (necessity of each condition): each follows directly from corresponding source derivation.
• Propositions 5.1 (structural margin): direct substitution.
• Propositions 6.1–6.3 (SM consistency): empirical check + direct cross-reference.

Semi-formal (structural arguments):

• Proposition 2.2 (Preservation ≠ Detection): scaling-argument comparison.
• Proposition 2.3 (Formation ≠ Detection): same.
• Proposition 3.3 (detection necessity): structural argument via observer-centrism and operational completeness.

Conjectural (framework-completeness):

• Proposition 4.1 (joint sufficiency): asserts no fourth independent axis; this is a framework-completeness claim, not a theorem.

Deferred:

• Per-particle quantitative verification of (F) ∧ (P) ∧ (D) for each SM entity.
• Whether bootstrap composition constitutes a distinct fourth axis.
• Systematic classification of profiles satisfying (F) ∧ (P) ∧ (D) (hypothetical dark matter candidates, for instance).

Assessment: Semi-formal. Each of (F), (P), (D) is a well-established condition from existing derivations; this theorem asserts their joint structure. Distinctness of each pair is rigorous via scaling arguments; necessity of each is rigorous given the source derivations; joint sufficiency is a completeness assertion.

Open Gaps

1. Framework-completeness of the triplet. Joint sufficiency (Proposition 4.1) asserts no fourth viability axis; this is not proved. Investigating whether bootstrap composition, internal consistency (Axiom 3 loop closure), or other framework commitments constitute independent viability axes. Difficulty: MODERATE-HARD.

2. Per-particle verification. For each SM particle, compute (F), (P), (D) quantitatively and verify simultaneous satisfaction. This would provide a framework-level prediction of the SM mass spectrum given the three mechanisms as inputs. Difficulty: MODERATE.

3. Non-SM profiles. Could there be profiles satisfying (F) ∧ (P) ∧ (D) that we have not observed — hypothetical dark matter candidates, other stable structures? The triplet predicts the set of viable observers; matching to observation requires accounting for additional profiles. Difficulty: HARD.

4. Composite observers and cross-level viability. Bootstrap composition creates higher-level observers from collections of lower-level ones. Whether the higher-level observer’s viability requires its own (F), (P), (D) satisfaction independently is a question not fully addressed. Difficulty: MODERATE.

5. Macroscopic observers and mass-information reversal. Mass Hierarchy §7 notes a mass-information reversal at bootstrap level 3: composite observers transition from topological to structural encoding. The triplet as formulated assumes topological encoding; extension to structural observers requires revisiting (F), (P), (D) in the structural regime. Difficulty: MODERATE-HARD.

6. Relation to the bootstrap fixed-point structure. Bootstrap Fixed-Point Existence establishes the universe as a Kleene fixed point under compactness, dcpo, and Scott continuity prerequisites. Whether the triplet is a consequence of the fixed-point structure, or an independent structural constraint, is not explicitly established. Difficulty: MODERATE.

7. Coherence Lagrangian common-saddle analysis. Whether (F), (P), (D) can be understood as different boundary-condition regimes of one Coherence Lagrangian calculation — a shared-origin view that doesn’t reduce them but shows common structural origin. Difficulty: HARD. Partially addressed by Coherence Bounces and Euclidean Coherence Lagrangian: the Euclidean formalism on $M_A^E$ now supplies the shared Lagrangian structure hosting both Class A (within-level) and Class B (level-to-level) bounces, and the Majorana bounce for Preservation-side fermionic substrate. A full common-saddle analysis of (F) ∧ (P) ∧ (D) on these saddles is the remaining open content.

Addressed Gaps

1. Fermion triplet verification at the Lagrangian level — Enabled by Spinor Coherence Lagrangian. Each SM fermion (charged leptons, neutrinos, quarks, across all three generations) now has an explicit Lagrangian realization via Spinor Coherence Lagrangian Theorem 2.7 (Dirac kinetic + mass), Theorem 5.3 (Majorana for neutrinos), and Proposition 6.1 (three-generation replication). The triplet (F) ∧ (P) ∧ (D) can therefore be verified against this derived Lagrangian rather than imposed structurally: (F) formation comes from WKB tunneling of the derived mass term; (P) preservation from QEC on the Dirac-field substrate; (D) detection via the Dirac source in Observer Pattern Signal producing Compton-scale Yukawa edges. Full quantitative per-fermion verification remains Open Gap 2 above; what this gap addresses is the prerequisite Lagrangian content.

References and further reading

Framework inputs.

• Mass Hierarchy — Theorem 3.1 (WKB crystallization); §7 (topological-structural transition).
• Observer as an Error-Correcting Code — three-axis profile structure.
• Substrate Noise and Profile-Dependent Coupling Modulation — per-cell bit-flip rate; additive-coupling noise; Proposition 2.5 for $d_{\mathrm{req}}$.
• Observer Pattern Signal — Yukawa signal form; (D) axis input.
• Observer Edges and Mutual Opacity — detection edge equation; (D) axis.
• Profile-Dependent Edges and Confinement — (D)-axis failure for free quarks; confinement.
• Observer Holographic Equivalence — Corollary 4.6 operational completeness.
• Coherence Lagrangian — Theorem 6.0 Lagrangian dynamics.
• Bootstrap Fixed-Point Existence — Kleene fixed-point framework.

Standard QEC background.

• Dennis, E., Kitaev, A., Landahl, A., Preskill, J. (2002). Topological quantum memory. J. Math. Phys. 43:4452–4505. — Logical error rate $P_L \sim (p/p_{th})^{d/2}$.