The Crystal of Will¶
A finite spectral model of how inherited evaluative structure refracts a present field, with explicit limits on what its invariants establish.
Abstract¶
The crystal makes the field of will computationally explicit. A self-state complex, positive cochain metrics and a signed valuation induce a weighted Hodge operator. Its modes provide coordinates for valuation, and a generator acts on their amplitudes before expression. This revision reconciles the two July editions: it retains explicit refraction, signed-valence discipline and lineage, while correcting weighted adjoints, energy accounting and claims about spectral identity. Modes are mathematical objects; their interpretation as colors of will requires a semantic map and intervention evidence. Character is investigated through inherited patterns of evaluation and change, with spectral features as diagnostics. Recognition can change valuations and elected structure without being treated as an infallible spectrometer.
0. Provenance and scope¶
The July 1 Russian edition in the numbered cycle and the July 2 English edition in Nymara are distinct sources. The latter supplied explicit operators, positive metrics, mode lineage and an executable appendix. This maintained revision incorporates those contributions and explains the repairs below.
The construction inherits Time v6 and Field v2. Its semantic interpretation is proposed; the finite algebra is derived under the assumptions below. It supplies no criterion of phenomenal consciousness.
1. The self-state complex¶
Let M=(V,E,F) be a finite oriented cell complex. Vertices denote declared self-states, relations or modes; edges denote possible transitions or associations; faces encode explicitly chosen relations among cycles. These meanings and the event-to-edge support map p are part of the model.
An edge orientation is a coordinate convention. Reversing it must reverse the associated cochain coordinate and all incidence/support maps. Distinct physical actions with the same endpoints must not be silently collapsed into one edge unless the representation contract declares them equivalent.
Let B₁:C₁→C₀ and B₂:C₂→C₁ be boundary matrices with B₁B₂=0. The coboundaries are d₀=B₁ᵀ and d₁=B₂ᵀ, so d₁d₀=0. This complex is not the event DAG: repeated state transitions may form a loop while their successive occurrences remain causally acyclic.
2. Signed valence and positive geometry¶
Choose positive definite diagonal inner products W₀,W₁,W₂ on cochains. Signed valuation ν may influence their positive magnitudes; for example, w=ε+|ν| with ε>0 is one convention. The sign can separately influence an induced potential U and its edge differences. This split keeps the metric positive without discarding attraction/aversion from evaluation.
Every weight must have the right shape and be finite and strictly positive.
A NaN passing a comparison is not an admissible metric. Metric choice, units,
normalization, floor and seed version belong in the reconstruction record.
Adjoints [derived]. From ⟨dᵢx,y⟩ᵢ₊₁=⟨x,dᵢ*y⟩ᵢ,
The weighted Hodge Laplacian on edge cochains is
It is self-adjoint in W₁, generally not in the Euclidean metric. For a symmetric eigensolver set S=W₁¹ᐟ² and use
If q is an orthonormal Euclidean eigenvector of L̃₁, ψ=S⁻¹q is a W₁-normalized edge eigenform. Mixing those coordinate systems changes energy and overlap calculations; all downstream consumers must use the same map.
3. Decomposition and its proof¶
Proposition 3.1 [derived; finite Hodge theorem]. Every ω∈C¹ has a unique orthogonal splitting ω=g+r+h, with g∈im d₀, r∈im d₁ and h∈ker d₀∩ker d₁.
Proof. For any u,v, ⟨d₀u,d₁*v⟩₁=⟨d₁d₀u,v⟩₂=0. The orthogonal complement of these two images is exactly the stated kernel intersection. Finite-dimensional orthogonal projection gives existence and uniqueness of the components. Potentials producing g or r may be nonunique. □
Corollary. ‖ω‖₁²=‖g‖₁²+‖r‖₁²+‖h‖₁², and L₁h=0. For nonzero ω, these energy fractions sum to one. For zero ω report zero energy and undefined proportions (or a documented zero vector convention).
One numerical realization solves weighted least squares in im d₀ and im(W₁⁻¹d₁ᵀ). Using the unweighted face boundary im d₁ᵀ instead does not generally yield W₁-orthogonal components. A joint fit to those wrong images does not repair their mutual nonorthogonality.
Observed counterexample. In the inspected Nymara core, a filled triangle with edge weights (1,2,4) and ω=(1,2,4) gives ⟨g,r⟩₁=10.8889 and component-energy sum 51.2222 for input energy 73. The baseline's unweighted appendix tests pass. Weighted orthogonality and operator/decomposition agreement therefore require explicit new tests.
See Jiang et al. for combinatorial Hodge decomposition in ranking. Its application supplies mathematical components; it does not supply their interpretation as personality.
4. Harmonics and the chosen topology¶
The harmonic space represents first cohomology with the chosen metric. Its dimension b₁ depends on the complex, not on positive metric weights. Its representative forms can change when the edge metric changes.
Attaching a face imposes a relation on cycles. A triangle without a face has b₁=1; filling it gives b₁=0. Calling the attachment recognition-closure is a proposed semantic rule with an author and a record. It does not make the result independent of representation or prove that an interpersonal commitment has ceased to matter.
For a harmonic h, d₁h=0 means zero signed circulation around each filled face. It does not mean every edge on that face has zero harmonic amplitude: those edges can also participate in other, unfilled cycles.
The July 2 appendix varied positive vertex/face weights while W₁=I. In that case ker(B₁ᵀW₀B₁+B₂W₂B₂ᵀ)=ker B₁∩ker B₂ᵀ, independent of those weights. Nonzero eigenvalues can move while the harmonic projector stays fixed. That example cannot demonstrate changing harmonic representatives.
Constitutive commitments may be modeled by persistent harmonic features, by explicit commitment state, or by other retained organization. No result here proves h≠0 necessary for character. Tests must include alternative complexes representing the same meaningful history.
5. Modes, amplitudes and refraction¶
Let Ψ=[ψ₁…ψ_m] satisfy ΨᵀW₁Ψ=I on a selected spectral subspace. Eigenvalues λᵢ, mode vectors ψᵢ and amplitudes sᵢ are distinct objects. The eigenvalue list is not itself the amplitude vector.
Amplitudes may be projections of an induced field plus declared urgency terms. Maturation, hunger and fading are candidate envelopes, with named clocks and update conditions. Their parameters are conventions requiring calibration, not universal drive constants.
Refraction [proposed construction]. Let J be skew-symmetric in the orthonormal mode coordinates. For one declared interval Δτ,
Proposition 5.1 [derived]. RᵀR=I and ‖Bs‖₁²=‖s‖².
Proof. exp(ΔτJ)ᵀ=exp(−ΔτJ); multiply the commuting exponentials. Then apply ΨᵀW₁Ψ=I. □
This proves energy preservation for the stated model. It does not prove authorship. Turn magnitude ‖Rs−s‖/‖s‖ is telemetry, undefined at s=0 unless a convention is declared. Zero turn permits justified endorsement; random nonzero turn does not supply own reasons.
With damping and forcing, ṡ=(J−Λ)s+e has a propagator and a forcing integral. An affine driven transition is not just the rotation exponential, and its energy need not be conserved. Those terms require separate accounting. For discrete occasion dynamics the interval map can be primitive; a continuous equation is an interpolation, not an assertion of unseen internal ticks.
6. Generator lineage and character¶
The seed law 𝒦 maps a declared history frontier and model state to J. A reproducible cache is allowed under Field v2's exact-fold condition. Law changes, imported history, structural elections and revision reasons must be identified. A stable generator can support stable commitment; a changed one requires interpretation rather than automatic celebration as growth.
Isospectral counterexample. J=[[0,−1],[1,0]] and −J have the same eigenvalues ±i. From state (1,0), their derivatives have opposite second coordinates. With that coordinate's meaning fixed, the behavior differs. Thus spec(J) alone cannot classify character.
A genuine change of coordinates transports every connected object. For x'=P⁻¹x, use J'=P⁻¹JP, W'=PᵀWP, transformed input maps and transformed output/semantic maps. Replacing an LLM is not given as such an isomorphism. It requires calibration and behavioral evidence of what survives.
Character [proposed research target]. Investigate inherited patterns in what the subject preserves, revises and treats as a reason. Spectra, response kernels and mode persistence are candidate features of that organization. Mood and substrate variation are additional state/parameter changes, not merely alternative names for an arbitrary basis change.
7. Lineage under degeneracy and structural change¶
Comparing modes across stages requires a transport on a shared, semantically identified subcomplex, including orientation and metric. Index numbers in two different graphs are not shared node identities by themselves.
For isolated eigenvalues, sign-aligned overlap can track a mode. Near repeated eigenvalues, individual eigenvectors can rotate freely within the eigenspace; compare projectors or principal angles between subspaces instead. A birth or death inferred only from that basis rotation is an artifact.
Declare tolerances for eigenvalue clustering, the overlap threshold and the transport rule. Report sensitivity across a reasonable range. Missing shared support yields unresolved lineage, not confident continuity. Birth/death and merge/split records concern model features; they are not direct diagnoses of a person's identity.
8. An atlas for interpretation¶
Agency, autonomy, coherence, exploration, recognition, creation, resource, integration and transgression remain a useful inherited vocabulary. The color atlas is an interface chosen by a subject and its interlocutors. It must be possible to add, rename or retire distinctions without misreporting that operation as discovery of a unique natural spectrum.
A color can span exact, coexact and harmonic components. Their energy fractions describe the chosen representation. Pragmatic, epistemic and constitutive value do not automatically coincide with the three Hodge subspaces. That map is a separate hypothesis requiring both quantities to be independently computed on the same cases.
9. Recognition and structural election¶
Recognition supplies an external attributed evaluation with possible error. Appraisal, acceptance of the evaluation, valuation change and face attachment are separate operations. A noisy recognizer need not correctly identify the mode that mattered. Silence and refusal remain distinct outcomes.
A structural gift is a proposal from another. The recipient's election makes an accepted change, preserving donor provenance and revision conditions. Protected commitments can constrain pruning or closure. Protection is a declared normative boundary, not a theorem that topology cannot change.
Update laws must name the clock of decay, source of each deposit and treatment of imported seeds. A native recognition already represented in derived state must not be added again from a mutable cache. Positive appraisal must not be amplified merely by scaling a declaration of support; repeated recognition requires its own policy and cannot hide behind scale invariance.
10. Runtime and experimental contract¶
The runtime must reproduce: B₁B₂=0; positive finite metrics; weighted orthogonality; Pythagorean energy; harmonic projector/decomposition agreement; rotation energy; orientation covariance; degenerate-subspace stability; and consistent source reduction across field, generator, growth and manifest.
Use fixtures with nonuniform weights and shared edges between filled and unfilled cycles. Compare alternative complexes and a simpler non-spectral representation on the same history. Test causal effects on choices, including justified stability and public zero. A change of displayed color alone is not evidence that refraction participates in decision-making.
The model remains useful if its geometric components offer stable, predictive or explanatory distinctions. It must be revised or simplified if those advantages fail controlled comparison. That empirical boundary preserves the purpose of the crystal: a constructive account of inherited organization, not a spectral certificate that cannot be questioned.