CORE Preprint
Spectral Brand Theory
SBT (2026a)
8-dimensional decomposition with observer cohorts, coherence types, and conviction dynamics
Research Atlas
A structured map of the Spectral Brand Theory and Organizational Schema Theory research programs. Seven layers, from mathematical foundations to empirical validation. Find your entry point, follow the reading path, identify where contribution is possible.
How to Use This Map
Each paper occupies a block in a layered architecture with a primary layer (where its core contribution lives) and optional secondary layers (where it provides supporting context). Papers marked CORE carry the essential ideas for that layer — read them to get 80% of the layer's content. Supplementary papers deepen, extend, or formalize.
A potential collaborator uses this page to identify their expertise — geometry, empirical methods, org theory, practice — and immediately sees which papers matter and where their contribution could fit.
What Is Your Entry Point?
Select your background. The layers below reorder to prioritize what matters most to you. All layers remain visible.
L0 What is SBT?
Foundations
The foundational architecture — eight dimensions, observer cohorts, coherence types, and conviction dynamics. Start here for orientation regardless of background.
Core — read first
CORE Under Review
Under review
Spec-gap is universal: biology, brands, organizations, code
Supplementary
supp Preprint
Why Eight?
Minimal completeness proof + empirical robustness test (22 LLMs, drop-one/drop-pair/10D)
supp Preprint
Alibi
Domain-agnostic observation-to-knowledge architecture from financial NLP
supp Preprint
Literature Survey
What existed before SBT; why it was not enough; six open problems
supp
primary: L3 Under Review
Under review
Bridge from Aaker's 4 perspectives to SBT's 8 dimensions
L1 What is the mathematical structure?
Geometry
The formal machinery — Fisher-Rao metric, projection bounds, concentration of measure, sphere packing, and multi-observer triangulation.
Core — read first
CORE Preprint
Brand Space Geometry: Formal Metric
R1 (2026d)
Fisher-Rao metric, warped product, 36 independent components; the mathematical foundation
Supplementary
supp Preprint
Metamerism
JL lemma: >152% distortion projecting 8D to 1D; 31-39% brand pairs are metameric
supp Preprint
Cohort Boundaries
57% of simplex volume is boundary at delta=0.10; discrete segmentation is geometrically lossy
supp Preprint
Sphere Packing
E8 kissing number (240) bounds nearest competitors; dimensional correlation collapses capacity
supp Preprint
Triangulation
v1.2.1 · Multi-observer disagreement is signal; Perception DOP predicts estimation error (R^2=.926)
supp
primary: L0 Preprint
Why Eight?
Minimal completeness proof + empirical robustness test (22 LLMs, drop-one/drop-pair/10D)
supp
primary: L0 Preprint
Literature Survey
What existed before SBT; why it was not enough; six open problems
supp
primary: L4 Preprint
Spec Impossibility
48D org space; exhaustive specification is geometrically impossible
supp
primary: L2 Preprint
Spectral Dynamics
v1.1 · Phase space resolves Bonnet ambiguity; trajectory clustering detects competitive convergence
supp
primary: L5 Preprint
Rate-Distortion
J-shaped R(D); intermediate formats beat high-rate formats; 17 architectures converge
L2 How does perception change over time?
Dynamics
Non-ergodic tracking bias, Fokker-Planck diffusion, coherence-resilience under crisis, velocity and acceleration in phase space, and threshold inequality for separability recovery after coherence shocks.
Core — read first
CORE Preprint
Non-Ergodic Brand Perception
R9 (2026o)
Cross-sectional tracking violates ergodicity; three structural sources of bias
CORE Preprint
Restoring Perceptual Separability After Coherence Shocks: A μ > λ Threshold Inequality in Brand Perception
R22 (2026ad)
μ > λ threshold inequality (Kato-Rellich + Diaconis-Stroock + Froyland-Padberg). Monte Carlo: gap 1.10 vs .02 (52x); IRF 1.4 vs 13.1 mo. 52 refs, 5.8% self-cite. Target: Marketing Science.
CORE Preprint
Dove Case Study: Dimensional Activation
R10 (2026p)
20-year longitudinal decomposition across 4 cohorts; the worked example
Supplementary
supp Preprint
Diffusion
Fokker-Planck SDEs; absorbing boundaries; non-ergodicity proved formally
supp Preprint
Coherence-Resilience
Coherence type (not score) predicts crisis survival via drift geometry
supp Preprint
Spectral Dynamics
v1.1 · Phase space resolves Bonnet ambiguity; trajectory clustering detects competitive convergence
L3 What does SBT predict for real problems?
Applications
Practitioner tools — the Spectral Audit, the Dove longitudinal case study, resource allocation, spectral immunity, and AI-native identity.
Core — read first
CORE Under Review
Under review
Six-step diagnostic: run it on any brand today for ~$0.80
CORE Preprint
Dove Case Study: Dimensional Activation
R10 (2026p)
20-year longitudinal decomposition across 4 cohorts; the worked example
CORE Preprint
Non-Ergodic Brand Perception
R9 (2026o)
Cross-sectional tracking violates ergodicity; three structural sources of bias
CORE Preprint
Dimensional Collapse in AI-Mediated Brand Perception: LLMs as Metameric Observers
R15 (2026v)
v3.1 · 21,350 core calls (31,275 total with supplementary), 24 models, 9 cultural traditions; dimensional collapse is universal (cosine .977) and temporally stable (H13); primacy is GPT-specific (F1); Run 15b isolated a JSON-format primacy effect (η² = .217) that motivated the dimension_order parameter (canonical / latin_square / random) shipped in sbt-framework v2.3.1
Supplementary
supp Preprint
Resource Allocation
Optimal investment proportional to cohort weights; alignment gap bounds loss
supp Superseded
Portfolio Theory
SUPERSEDED by R21 (2026ac) — merged with R20 into Spectral Immunity. Original theory: multi-brand interference (LVMH constructive vs Unilever destructive).
supp Preprint
AI-Native Identity
v1.8.1 · Behavioral signatures replace logos for AI observers; Brand Function verification
supp Under Review
Under review
Bridge from Aaker's 4 perspectives to SBT's 8 dimensions
supp
primary: L2 Preprint
Coherence-Resilience
Coherence type (not score) predicts crisis survival via drift geometry
supp Working Paper
OrgSchema Audit
OST equivalent of the Spectral Audit; Spectra Coffee worked example; 2 propositions
L4 Where else does this structure appear?
Cross-Domain
The rendering problem is universal — organizations, biology, code, canon. OST, specification impossibility, and coordinate-free positioning.
Core — read first
CORE Under Review
Under review
Spec-gap is universal: biology, brands, organizations, code
Supplementary
supp Under Review
Under review
6-level cascade; acceptance testing as the missing construct
supp Preprint
Spec Impossibility
48D org space; exhaustive specification is geometrically impossible
supp Under Review
Under review
Org positions project process space onto personnel dimension
supp Under Review
Under review
Version-controlled IP specification; Shakespeare fork demo
supp Under Review
Under review
Temporal stability: value > process > org form
L5 Does it hold against data?
Empirical
21,350 API calls across 24 LLMs and 9 cultural traditions confirm dimensional collapse. Rate-distortion curve maps 17 encoder architectures.
Core — read first
CORE Preprint
Dimensional Collapse in AI-Mediated Brand Perception: LLMs as Metameric Observers
R15 (2026v)
v3.1 · 21,350 core calls (31,275 total with supplementary), 24 models, 9 cultural traditions; dimensional collapse is universal (cosine .977) and temporally stable (H13); primacy is GPT-specific (F1); Run 15b isolated a JSON-format primacy effect (η² = .217) that motivated the dimension_order parameter (canonical / latin_square / random) shipped in sbt-framework v2.3.1
Supplementary
supp Preprint
Rate-Distortion
J-shaped R(D); intermediate formats beat high-rate formats; 17 architectures converge
supp
primary: L1 Preprint
Metamerism
JL lemma: >152% distortion projecting 8D to 1D; 31-39% brand pairs are metameric
supp
primary: L3 Preprint
AI-Native Identity
v1.8.1 · Behavioral signatures replace logos for AI observers; Brand Function verification
supp
primary: L1 Preprint
Triangulation
v1.2.1 · Multi-observer disagreement is signal; Perception DOP predicts estimation error (R^2=.926)
supp Superseded
Portfolio Interference (R20)
SUPERSEDED by R21 (2026ac) — merged with R8 into Spectral Immunity (DOI 10.5281/zenodo.19765401). Original empirical: 9,925 obs, 40 brands, 13 models, 7 traditions; 0/20 FDR-significant.
supp Working Paper
PRISM Instrument
v1.0.2 · Five-layer scaffold; PRISM-B items, 1-5 ordinal, DCI scoring; 4 propositions
L6 How should research itself be structured?
Meta-Science
Verification crisis as specification crisis. Paper as YAML spec. Git-native publishing protocol. The epistemological pipeline that made SBT verifiable.
Core — read first
CORE Under Review
Under review
Verification crisis = specification crisis; Paper Spec (YAML) with 5 layers
Supplementary
supp Preprint
Research Repo
v2.0 · Git-native protocol; papers are renders, repos are the artifact
supp
primary: L0 Preprint
Alibi
Domain-agnostic observation-to-knowledge architecture from financial NLP
supp
primary: L4 Under Review
Under review
Version-controlled IP specification; Shakespeare fork demo
Reading Paths by Audience
Ten structured paths through the program. Each starts where your expertise intersects, then expands outward.
Brand practitioner L3 → L0 → L3
You run brand strategy, communications, or marketing. Start with the diagnostic, then understand why it works.
Brand equity / Aaker tradition L3 → L0
You work in or cite the Aaker brand equity framework and want to see the relationship.
- Engage Aaker L3
- SBT L0
- Why Eight? L0
Empirical researcher L5 → L0 → L5
You evaluate this as a contribution to quantitative marketing science.
- PRISM-B L5
- SBT L0
- Rate-Distortion L5
AI / ML researcher L5 → L3 → L1
You study how language models represent concepts. The empirical and geometry layers are most relevant.
- PRISM-B L5
- AI-Native Identity L3
- Rate-Distortion L5
- Metamerism L1
Mathematician / Geometer L1 straight through
You want to verify the formal machinery before engaging with applications.
- Brand Geometry L1
- Metamerism L1
- Cohort Boundaries L1
- Sphere Packing L1
Information theorist L5 → L1 → L5
You think in bits, rate-distortion, and channel capacity.
- Rate-Distortion L5
- Metamerism L1
- PRISM-B L5
Psychometrician / Survey designer L1 → L5 → L1
You design measurement instruments. The triangulation and PRISM papers are most directly relevant.
- Triangulation L1
- PRISM-B L5
- Brand Geometry L1
Strategy / management scholar L2 → L3
You study competitive dynamics and long-run brand trajectories.
- Non-Ergodic L2
- Dove Case L3
- Coherence-Resilience L2
Organizational theorist L4
You study how structure relates to performance. The rendering problem is your entry point.
- Rendering Problem L4
- OST L4
- Projection L4
Philosopher of science L6 → L0
You care about how knowledge is structured and verified. Start with the meta-science layer.
- Paper Spec L6
- Research Repo L6
- Alibi L0
Open Problems
Identified gaps in the current program. A collaborator who fills any of these makes a direct contribution to the architecture.
| # | Gap | Layers | Priority | Notes |
|---|---|---|---|---|
| 1 | Human-subject empirical validation of cohort divergence | L5 | HIGH | All current evidence is LLM-mediated. PRISM-B instrument sensitivity confirmed (Run 15); primacy effects bounded as model-specific (F1, GPT-only) and domain-specific (F2, brand-only). Run 15b additionally isolated a JSON-format primacy effect (η² = .217) at the elicitation layer; resolved in sbt-framework v2.3.1 via the dimension_order parameter (Latin-square ordering averages out positional bias). Human conjoint/MaxDiff study with elicited dimensional weights is the next milestone. |
| 2 | Long-horizon longitudinal tracking | L2 L5 | MEDIUM | H13 closed short-horizon stability across 4 model pairs (cosines > .97). Open: tracking the same cohort across 6+ months and through multiple disruption events to test whether the μ > λ inequality predicts in-vivo trajectories. |
| 3 | Real-world agentic deployment | L3 L4 | MEDIUM | Exps A/D/Q1 demonstrated compounding and showed constraint framing reduces variance 62% (Q1). Open: field deployment in live agent workflows with revealed-purchase outcomes, beyond simulated agentic commerce. |
| 4 | Collapse onset and early-warning indicators | L2 L5 | HIGH | R22 closed the recovery side (μ > λ at scale δ restores separability). The symmetric onset problem is open: which observable signals precede the spectral collapse, how early can the gap-decay rate be estimated, and what is the practitioner-facing lead time before separability is lost. |
| 5 | Cohort discovery from raw observation | L1 L5 | MEDIUM | Most papers stipulate cohorts (priors, demographics, weight vectors). Open: unsupervised identification of latent cohorts from observation streams without pre-specified weights, and conditions under which discovered cohorts coincide with the alibi-style invariant structure. |
| 6 | Formal cross-domain operator identification | L4 | MEDIUM | R22 + OST companion cite independent convergence in capital-markets and DeFi composability work showing the same threshold-inequality and projection-operator structure. Open: a formal identification paper showing that brand-perception, organizational verification, and DeFi composability share the same operator-theoretic structure under a specified mapping. |
| 7 | Causal identification beyond observational designs | L5 | MEDIUM | All current empirical work is observational and LLM-mediated. Open: quasi-experimental designs (regression-discontinuity around brand events, instrumental variables, natural experiments) that identify the perception-shift effect of a specified disruption rather than its correlation. |
| 8 | Multi-shock and cascade dynamics | L2 L3 | LOW | R22 models a single coherence shock. Open: interaction effects of sequenced shocks, simultaneous shocks across cohorts, and contagion across linked brands in a portfolio. R21 portfolio immunity result suggests the cascade structure differs for AI vs. human observers. |
| 9 | Specification-to-measurement empirical bridge | L0 L3 | MEDIUM | SBT v3.2.0 §5.2.1 formalized the DO/WHAT bridge between organizational specification (OST) and observable dimensions (SBT). Open: an empirical study mapping a documented organizational specification onto its measured perception cloud and quantifying specification-perception coupling strength. |
| 10 | Capstone synthesis | L6 | PREMATURE | Unified theory review across L0–L5. Premature until human-subject validation (#1) and a longitudinal field result (#2) are in hand. |