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Zenodo
2026
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| الوصول للمادة أونلاين: | https://doi.org/10.5281/zenodo.20034821 |
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إضافة وسم
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| _version_ | 1866901504654311424 |
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| author | Graise, Martin Luther |
| author_facet | Graise, Martin Luther |
| contents | <p>We extend the Paper 7 single-observer coherence ceiling to dyadic<br>G₂-structured self-modeling observers. Each observer is an affine map<br>T(x) = rRx + b on a 14-dimensional Banach manifold carrying the<br>irreducible adjoint representation of G₂, with scalar contraction rate<br>r ∈ [0, 6/7] (from Paper 7), orthogonal orientation R, and bias vector<br>b encoding the observer's target self-model. We couple two such<br>observers with an antisymmetric block rotation Ψ_θ parameterized by an<br>angle θ ∈ [0, π/2].</p> <p>The paper establishes three results about the rate channel and two<br>results about the affine channel, with a sharp dichotomy between them.</p> <p>Rate channel (locked): The joint contraction rate is bounded below by<br>the symmetric-diagonal coupling (Ψ = α·id + β·P_swap), which has<br>operator norm α + β > 1 and therefore amplifies (Theorem 9.1). Under<br>the corrected isometric block-rotation coupling, the joint rate<br>saturates exactly at max(r_A, r_B), independent of θ (Theorem 9.2).<br>The common explanation is Schur's lemma applied to the irreducible<br>14-dim G₂ adjoint: every linear G₂-equivariant map is a scalar multiple<br>of the identity, so there is no spectral structure for coupling to mix<br>(Corollary 9.3). The rate channel is algebraically closed.</p> <p>Affine channel (open): The joint fixed point exists, is unique, and<br>depends real-analytically on θ, with an explicit closed form at θ = 0<br>and θ = π/2 (Theorem 9.4). Under the minimum-per-observer coherence<br>functional _min = min( _A, _B) — derived from the Paper 7<br>single-observer ceiling as the dyadic generalization of<br> ≥ 1 - ε_min for both observers — a non-empty conditional-gain<br>region exists in the space of (φ, θ) pairs (where φ is the<br>bias-alignment angle), bounded away from the opposed-bias boundary in<br>the seeded geometry, with a non-universal optimal θ (Proposition 9.5).<br>A 50-seed Monte Carlo (Appendix V.3.i) finds non-empty improvement<br>regions in all 50/50 sampled (R_A, R_B) pairs and confirms<br>θ* ≠ π/4 generically; the bounded-away property of the seeded pair<br>holds in ~64% of seeds and motivates the amended Conjecture 9.5'. The<br>gain is conditional: a substantial fraction of the parameter space<br>(~75% by area in the verified geometry) exhibits coherence degradation<br>rather than improvement; coupling helps only when the bias geometry<br>permits.</p> <p>All results are numerically verified at 50-digit precision (rate<br>channel, Appendix V.1–V.2, 120 test configurations) and at 16-digit<br>precision (affine channel, Appendix V.3, 2128 test configurations<br>including a 100×20 heatmap and explicit closed-form verification). The<br>retraction record of earlier internal-draft claims (speedup theorem,<br>coherence-bonus formula, severance threshold) is consolidated in<br>Appendix V.0.1.</p> <p>Thesis. Dyadic coupling in the linear G₂-structured regime cannot<br>improve the convergence rate of joint self-modeling. It can — under<br>conditional geometric hypotheses on the orthogonal orientations and<br>bias-alignment angle — change the location of the joint fixed point in<br>a way that increases the minimum per-observer coherence. Rate<br>improvement is a categorically nonlinear question, deferred to a<br>sequel.</p> <p>Paper 9 of the PCI/PME Framework series. Two-reviewer revision pass<br>(both Claude) addressed in v1.3.2: 11 issues including the V^7 → V^14<br>representation switch made explicit, Schur-derived uniqueness of the<br>canonical block-rotation coupling Ψ_θ, demotion of Lemma 5.5.1 to a<br>Numerical Observation, Schur-forced justification of the diagonal G₂<br>action, and a 50-seed Monte Carlo that forced an honest amendment of<br>Conjecture 9.5'. Full source, computational verification, peer-review<br>records, and revision history at github.com/MartinLGraise/PCI-Framework<br>on the paper7-foundation branch.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20034821 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Rate Lock and Affine Consensus in G₂-Structured Dyadic Observers Graise, Martin Luther G₂ Lie group dyadic observers self-modeling Banach fixed point affine consensus Schur's lemma contraction rate PCI framework inter-brain coupling human-AI teaming F₂₁ Frobenius group QBism <p>We extend the Paper 7 single-observer coherence ceiling to dyadic<br>G₂-structured self-modeling observers. Each observer is an affine map<br>T(x) = rRx + b on a 14-dimensional Banach manifold carrying the<br>irreducible adjoint representation of G₂, with scalar contraction rate<br>r ∈ [0, 6/7] (from Paper 7), orthogonal orientation R, and bias vector<br>b encoding the observer's target self-model. We couple two such<br>observers with an antisymmetric block rotation Ψ_θ parameterized by an<br>angle θ ∈ [0, π/2].</p> <p>The paper establishes three results about the rate channel and two<br>results about the affine channel, with a sharp dichotomy between them.</p> <p>Rate channel (locked): The joint contraction rate is bounded below by<br>the symmetric-diagonal coupling (Ψ = α·id + β·P_swap), which has<br>operator norm α + β > 1 and therefore amplifies (Theorem 9.1). Under<br>the corrected isometric block-rotation coupling, the joint rate<br>saturates exactly at max(r_A, r_B), independent of θ (Theorem 9.2).<br>The common explanation is Schur's lemma applied to the irreducible<br>14-dim G₂ adjoint: every linear G₂-equivariant map is a scalar multiple<br>of the identity, so there is no spectral structure for coupling to mix<br>(Corollary 9.3). The rate channel is algebraically closed.</p> <p>Affine channel (open): The joint fixed point exists, is unique, and<br>depends real-analytically on θ, with an explicit closed form at θ = 0<br>and θ = π/2 (Theorem 9.4). Under the minimum-per-observer coherence<br>functional _min = min( _A, _B) — derived from the Paper 7<br>single-observer ceiling as the dyadic generalization of<br> ≥ 1 - ε_min for both observers — a non-empty conditional-gain<br>region exists in the space of (φ, θ) pairs (where φ is the<br>bias-alignment angle), bounded away from the opposed-bias boundary in<br>the seeded geometry, with a non-universal optimal θ (Proposition 9.5).<br>A 50-seed Monte Carlo (Appendix V.3.i) finds non-empty improvement<br>regions in all 50/50 sampled (R_A, R_B) pairs and confirms<br>θ* ≠ π/4 generically; the bounded-away property of the seeded pair<br>holds in ~64% of seeds and motivates the amended Conjecture 9.5'. The<br>gain is conditional: a substantial fraction of the parameter space<br>(~75% by area in the verified geometry) exhibits coherence degradation<br>rather than improvement; coupling helps only when the bias geometry<br>permits.</p> <p>All results are numerically verified at 50-digit precision (rate<br>channel, Appendix V.1–V.2, 120 test configurations) and at 16-digit<br>precision (affine channel, Appendix V.3, 2128 test configurations<br>including a 100×20 heatmap and explicit closed-form verification). The<br>retraction record of earlier internal-draft claims (speedup theorem,<br>coherence-bonus formula, severance threshold) is consolidated in<br>Appendix V.0.1.</p> <p>Thesis. Dyadic coupling in the linear G₂-structured regime cannot<br>improve the convergence rate of joint self-modeling. It can — under<br>conditional geometric hypotheses on the orthogonal orientations and<br>bias-alignment angle — change the location of the joint fixed point in<br>a way that increases the minimum per-observer coherence. Rate<br>improvement is a categorically nonlinear question, deferred to a<br>sequel.</p> <p>Paper 9 of the PCI/PME Framework series. Two-reviewer revision pass<br>(both Claude) addressed in v1.3.2: 11 issues including the V^7 → V^14<br>representation switch made explicit, Schur-derived uniqueness of the<br>canonical block-rotation coupling Ψ_θ, demotion of Lemma 5.5.1 to a<br>Numerical Observation, Schur-forced justification of the diagonal G₂<br>action, and a 50-seed Monte Carlo that forced an honest amendment of<br>Conjecture 9.5'. Full source, computational verification, peer-review<br>records, and revision history at github.com/MartinLGraise/PCI-Framework<br>on the paper7-foundation branch.</p> |
| title | Rate Lock and Affine Consensus in G₂-Structured Dyadic Observers |
| topic | G₂ Lie group dyadic observers self-modeling Banach fixed point affine consensus Schur's lemma contraction rate PCI framework inter-brain coupling human-AI teaming F₂₁ Frobenius group QBism |
| url | https://doi.org/10.5281/zenodo.20034821 |