Holographic Harmonic Model — Axioms & Metatheorems

Why start with axioms?

Axioms give us testable starting points. In HHM the foundation is not space, time, or matter, but Ψ(x,t)—the structured field that collapses into what we observe. Axioms specify how Ψ is allowed to behave so we can measure, compare, and falsify claims.

AX000 — Epistemic Primacy of Ψ

Statement: All observation begins with Ψ(x,t)—the modal field of the moment.

Why it’s needed

Rather than assume a fixed background (space/time), we start from structured patterns that are actually measured.

How it’s tested

Analyses proceed directly on field data (EEG segments, glyph sequences, harmonic maps) prior to external clocks or coordinates.

What it changes

Reality is read as structure-first; time/space emerge from relations among Ψ-states.

AX001 — Hilbert-Based Projection

Statement: Ψ must be measured via projection into a basis (harmonic, symbolic, spatial) in a Hilbert-like space.

Why it’s needed

Ψ is too rich to observe “raw.” Basis projections make it computable and comparable.

How it’s tested

• EEG: Fourier/wavelet bands
• Symbols: vectorized glyph streams
• Cosmology: spherical harmonics a_ℓm

What it changes

Truth is structure-through-projection; multiple valid views can agree on the same underlying Ψ.

AX002 — Rhythmic Recurrence

Statement: Ψ(x,t) recurs. Repetition underlies measurement, memory, and meaning.

How it’s tested

• OP003 Echo (autocorrelation / RQA)
• OP002 Rec (similarity across steps)
• OP014 CollapseOrbit (cycle length)

Implication

“New vs old” shifts to “recurs vs not”—identity grows with recurrence strength.

AX003 — Modal Collapse Stability

Statement: Collapses tend to stabilize into consistent modal forms (loops/orbits).

How it’s tested

• OP014 CollapseOrbit for orbit length
• OP004 CollapseInfinity for convergence

Implication

Predictable attractors enable memory and long-form patterning across domains.

AX004 — Scale-Invariant Dynamics

Statement: The same modal dynamics appear across scales and domains.

How it’s tested

Cross-domain comparisons using OP002/OP003/OP014 show matched structure despite differing units/frequencies/symbols.

Implication

Insights transfer between neuroscience, cosmology, bioacoustics, and symbols.

AX005 — Information Conservation in Ψ

Statement: Information is transformed, not destroyed, across collapses.

How it’s tested

• OP002 Rec and OP015 TraceSignature show persistence across transformations
• OP003 Echo / OP014 CollapseOrbit reveal reactivation after drift

Implication

Identity and meaning can reappear after change; “loss” often measures as structural redistribution.

📥 Download Axiom Dataset

JSON structure of the axioms and links to validations:

Metatheorems of HHM

Metatheorems are cross-domain consequences of the axioms. Each has measurable criteria using HHM operators and can be extended or falsified with new data.

• Derived from one or more axioms
• Stated with operator-level tests and thresholds
• Validated across multiple domains/scales where possible

MT0001 — Axial Saturation

Definition: Under iterative application, Ψ tends toward a stable modal amplitude (attractor).

Tests

• OP004 CollapseInfinity convergence
• Entropy plateau via OP006 UnifiedEntropy

Domains

EEG (meditation loops), CMB bands, compressed streams, decohering quantum states, gene oscillators.

Depends on

AX000, AX003, AX004

MT0002 — Temporal Recurrence

Definition: Ψ exhibits measurable recurrence across time.

Tests

• OP003 Echo (lag correlation / RQA)
• OP002 Rec between Ψ_t and Ψ_t+n
• OP014 CollapseOrbit (cycle length)

Depends on

AX000, AX002

MT0003 — Modal Identity Preservation

Definition: Identity persists under transformation when structural metrics remain within bounds.

Typical bounds

• Rec ≥ 0.85 (OP002)
• |ΔEntropy| ≤ 0.15 (OP006)
• TraceSignature overlap ≥ 90% (OP015)

Depends on

AX001, AX005

MT0004 — Field Similarity

Definition: Structurally similar Ψ-states can be recognized across domains.

Tests

• OP002 Rec, OP003 Echo
• Optional: OP018 ModalCoherence if defined in your bundle

Depends on

AX002, AX005

MT0005 — Modal Transition

Definition: Transitions follow measurable pathways in modal space.

Tests

• OP011 ModalGradient (rate/direction)
• OP008 CollapseDelay, OP012 RhythmBoundary, OP019 DriftPath (if present)

Depends on

AX002, AX004

Metatheorem T014 — Robust Modal Identity

Statement: If two Ψ-states cross agreed thresholds on structure metrics, they are expressions of the same identity, independent of domain/scale/substrate.

Operator thresholds (typical)

• OP002 Rec ≥ 0.85
• OP003 Echo ≥ 0.90
• OP006 |ΔEntropy| ≤ 0.15
• OP014 CollapseOrbit within ±1 step

Why it matters

Identity is structural. When structure recurs, identity persists—even across media, systems, or eras.

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