Falsifiability and Predictions

On notation

In this document:

• $\rho$, $\rho_E$ — density matrix, reduced density matrix of the Interiority dimension
• $\Gamma_{-E}$ — context (states of all dimensions except $E$)
• $\lambda_i$ — eigenvalues of $\Gamma$ (intensities)
• L0, L1, L2 — interiority hierarchy levels
• $d_{\mathrm{FS}}$ — Fubini-Study metric
• $R_{\text{th}}$, $\Phi_{\text{th}}$ — threshold values

Falsification Criteria

Frontier ledger

Head-to-head predictions against competing theories at every live experimental frontier — Frontier Ledger.

Experimental Predictions

The extended theory makes testable predictions:

1. Isospectral discrimination

Two states $\rho_1$, $\rho_2$ with $\mathrm{Spec}(\rho_1) = \mathrm{Spec}(\rho_2)$ but $\mathrm{Eigvec}(\rho_1) \neq \mathrm{Eigvec}(\rho_2)$ should yield:

• Identical experience intensity (spectrum determines intensity)
• Distinguishable experience quality (eigenvectors determine quality)

Numerical criteria:

• Spectra 'identical': $\|\mathrm{Spec}(\rho_1) - \mathrm{Spec}(\rho_2)\|_2 < \varepsilon_{\text{spec}} = 0.01$
• Vectors 'distinguishable': $\min_i |\langle v_i^{(1)} | v_i^{(2)} \rangle| < 1 - \varepsilon_{\text{vec}}$, where $\varepsilon_{\text{vec}} = 0.1$
• Quality 'distinguishable': $d_{\text{FS}}([|q_1\rangle], [|q_2\rangle]) > \varepsilon_{\text{qual}} = 0.05$ rad

Test: Create isospectral neural states, measure phenomenal reports.

2. Contextual modulation

Changing context $\Gamma_{-E}$ with fixed $\rho_E$ should alter the quality of experience without changing intensity.

Numerical criteria:

• $\rho_E$ 'fixed': $\|\rho_E^{(1)} - \rho_E^{(2)}\|_F < \varepsilon_{\rho} = 0.02$
• Context 'changed': $\|\Gamma_{-E}^{(1)} - \Gamma_{-E}^{(2)}\|_F > \delta_{\Gamma} = 0.1$
• Intensity 'constant': $|\mathrm{Tr}((\rho_E^{(1)})^2) - \mathrm{Tr}((\rho_E^{(2)})^2)| < \varepsilon_P = 0.05$
• Quality 'changed': phenomenal report distinguishable with $p < 0.01$ (statistical test)

Test: Modulate context (attention, mood) at constant stimulus, measure changes in perceptual quality.

3. Adaptation dynamics

Experiential content (levels L1–L2) should follow the adaptation law:

$$\mathcal{Q}(t) \sim \log\left(\frac{\lambda_{\max}(t)}{\langle\lambda_{\max}\rangle_\tau}\right)$$

where:

• $\mathcal{Q}(t)$ — subjective experience intensity at time $t$
• $\lambda_{\max}(t)$ — maximum eigenvalue of $\Gamma(t)$
• $\langle\lambda_{\max}\rangle_\tau$ — average over adaptation period $\tau$

Interpretation

This prediction follows from the fact that perception encodes changes relative to baseline (Weber-Fechner law), not absolute values.

Numerical criteria:

• Correlation $r(\mathcal{Q}_{\text{measured}}, \mathcal{Q}_{\text{predicted}}) > r_{\min} = 0.7$
• Regression slope $\beta \in [0.8, 1.2]$ (close to 1)
• RMSE $< \sigma_{\mathcal{Q}} / 2$ (error less than half the standard deviation)
• Adaptation period $\tau \in [100, 1000]$ ms (typical range)

Test: Measure the temporal dynamics of adaptation, compare with prediction.

4. Metric relations

Distances in phenomenal space (L1) should correspond to the Fubini-Study metric:

$$d_{\mathrm{perceived}}(q_1, q_2) \sim d_{\mathrm{FS}}([|q_1\rangle], [|q_2\rangle])$$

where $[|q\rangle] \in \mathbb{P}(\mathcal{H}_E)$ — equivalence class in projective space.

Numerical criteria:

• Spearman correlation $\rho_S(d_{\text{perceived}}, d_{\text{FS}}) > 0.6$
• Monotonicity: violations $< 10\%$ of the total number of pairs
• Metric consistency: $|d(a,b) + d(b,c) - d(a,c)| < \varepsilon_{\triangle} = 0.15$ (triangle inequality)
• MDS reconstruction: stress $< 0.1$ when mapping to $\mathbb{R}^k$

Test: Build a phenomenal quality map (L1), compare with predicted geometry.

Refutation Criterion

The theory is falsified if:

$$\exists \rho_1, \rho_2 : \mathcal{I}(\rho_1) = \mathcal{I}(\rho_2), \text{ but } F(\rho_1) \neq F(\rho_2)$$

where:

• $\mathcal{I}(\rho) := (\mathrm{Spec}(\rho), \mathrm{Eigvec}(\rho), \Gamma_{-E}, \mathrm{Hist})$ — full invariant
• $F: \mathbf{DensityMat} \to \mathbf{Exp}$ — experience functor

That is, if two states with identical full invariants (spectrum + eigenvectors + context + history) yield distinguishable experience.

Gauge precision [T]

The $G_2$-rigidity theorem [T] refines the notion of 'identity': two states $\rho_1, \rho_2$ are considered physically identical if $\rho_2 = U\rho_1 U^\dagger$ for some $U \in G_2$. The full invariant $\mathcal{I}(\rho)$ is defined on the space $\mathcal{D}(\mathbb{C}^7)/G_2$ (34 parameters). The eigenvectors in the table below implicitly assume a fixed $G_2$-gauge; under gauge change $v_i \to Uv_i$, but the inner products $|\langle v_i^{(1)} | v_i^{(2)} \rangle|$ are $G_2$-invariant.

Operational tolerances:

Invariant component	'Identity' criterion
Spectrum	$\|\mathrm{Spec}(\rho_1) - \mathrm{Spec}(\rho_2)\|_2 < 0.01$
Eigenvectors	$\forall i: \lvert\langle v_i^{(1)} \vert v_i^{(2)} \rangle\rvert > 0.99$
Context	$\|\Gamma_{-E}^{(1)} - \Gamma_{-E}^{(2)}\|_F < 0.02$
History	$d_{\text{edit}}(\mathrm{Hist}_1, \mathrm{Hist}_2) < 0.05 \cdot \lvert\mathrm{Hist}\rvert$

Interiority	'Distinguishability' criterion
Phenomenal report	Statistically distinguishable ($p < 0.01$, Wilcoxon test)
Behavioural marker	AUC > 0.7 in discrimination task

Practical criterion

For experimental verification it is sufficient to compare the spectrum and eigenvectors (without history): if $\mathrm{spec}(\rho_1) = \mathrm{spec}(\rho_2)$ and $|q_i^{(1)}\rangle = |q_i^{(2)}\rangle$, but $\mathcal{F}(\rho_1) \neq \mathcal{F}(\rho_2)$, the theory is falsified.

Note on operationalisation

The full invariant $\mathcal{I}(\rho) = (\mathrm{Spec}, \mathrm{Eigvec}, \Gamma_{-E}, \mathrm{Hist})$ is a theoretical ideal. Operational realisation:

• Spec(ρ): measurable via quantum state tomography
• Eigvec(ρ): measurable via full tomography (neural correlates)
• Γ_{-E}: approximately measurable via partial trace — discarding the E-component
• Hist: approximated via time correlators $\langle O(\tau)O(0) \rangle$ (two-point functions, accessible via fMRI/EEG)

The falsification criterion is strict in the theoretical sense and approximate in the experimental sense. This is the standard situation for theories with directly unobservable objects (cf. the wave function in QM).

See also: КК falsification criteria — additional operational criteria for the applied theory.

Current Empirical Status

Prediction	Status	Comment
Isospectral discrimination	Open	Requires neurophenomenological experiments
Contextual modulation	Partially confirmed	Consistent with attention influence data
Adaptation dynamics	Consistent	Consistent with Weber-Fechner law
Metric relations	Open	Requires phenomenal space mapping (L1)
Functional purity	Programme	$P > P_{\text{crit}}$ for $\geq 90\%$ of functioning systems
P-quality correlation	Programme	Correlation of $P$ with functional quality: $r > 0.5$
F-m_t: $m_t \approx 173$ GeV	Consistent	Observation: $172.57 \pm 0.29$ GeV
F-Cabibbo: $\theta_{12} \approx 13°$	Consistent	Observation: $12.96°$ ($\|V_{us}\| = 0.2243$)
F-δ_CP: $\delta_{\text{CP}} \approx 64.5°$	Consistent (${\approx}0.04\sigma$)	Observation: $64.6° \pm 2.8°$ (LHCb tree, ICHEP 2024); $65.7° \pm 1.5°$ (PDG 2024 fit)
F-Gap-1: Gap_intra < Gap_inter	Open; machine check narrowed the scope	Requires ISF analysis of fMRI. Machine check: across 180 construction-layer natal Γ the gauge-invariant triangle holonomy $\lvert\arg(\gamma_{ab}\gamma_{bc}\gamma_{ca})\rvert$ on the 7 canonical oriented Fano lines is statistically indistinguishable from the 28 non-lines (t ≈ +1.0, difference +0.03 ± 0.39 rad) — consistent with the earlier amplitude-covariance null: the Fano structure is dynamical wiring (what the dissipator preserves), not a static imprint on natal Γ. The signature should be sought in states that have passed through the Fano channel (live dynamics, fMRI) — not in the construction. Note: the pairwise Gap_intra/Gap_inter form is not literally operational — in the Fano plane every pair lies on exactly one line, so there are no inter-triplet pairs; the testable form is triple holonomy (7 lines vs 28 non-lines). Machine-probed.
F-ISF: 6–12 ISF components	Open	Requires systematic fMRI analysis
F-ξ: $\xi_F \sim 160$ pc	Open	Testable through LSS surveys
F-nEDM: $d_n = 0$ (T-99)	Consistent	$\|d_n\| < 1.8 \times 10^{-26}$ e·cm (PSI 2020)
F-τ_p: $\tau_p \sim 6.7 \times 10^{37}$ years	Open	Hyper-K: sensitivity $\sim 10^{35}$ years
F-Higgs: $\delta\lambda/\lambda \sim 10^{-2}$–$10^{-3}$	Open	Awaiting FCC-hh
F-Gap-2: block transparency	Open	Requires coherence-resolved fMRI; the same caveat as F-Gap-1 — the signature lives in states that have passed through the Fano channel, not in the construction
F-rank7: 14 exact sum-rules	Open	Needs Γ-native rate tomography; no experiment has probed the 7-dimensional incidence subspace
F-Neural: threshold at $P = 2/7$	Partial	Clinical PCI threshold $0.31$ maps to $P_{\text{crit}}$ — support is calibration-dependent, not direct
F-w: no Big Rip, $-1$-crossing only	Consistent	DESI DR2 quadrant and phantom→quintessence orientation match at $2.8$–$4.2\sigma$; the reality of the crossing is not yet established non-parametrically
F-Band: $s_1 \in [1/7,\,3/14]$, $s_2 \in [1/7,\,2/7]$	Consistent	Zero counterexamples over $20\,000$ states per band, and both bands are attained — an internal falsifier needing no bridge assumption

Falsifiable predictions from Fano integration

Source

Predictions are derived from the integration of Fano geometry with Gap dynamics, Gap thermodynamics, and RG flow. Each prediction is assigned a rigour status in accordance with the registry.

F-Gap-1: Intra-triplet Gap below inter-triplet

$$\langle \mathrm{Gap}_{\mathrm{intra}} \rangle < \langle \mathrm{Gap}_{\mathrm{inter}} \rangle$$

The mean Gap within Fano triplets is lower than between them. Coherences belonging to the same Fano line are more transparent (closer to $\mathrm{Gap}=0$) than coherences connecting different lines.

Testability: ISF components (independent slow features) in fMRI. Intra-triplet correlations should systematically exceed inter-triplet correlations.

Status: [H] Hypothesis — consequence of Gap semantics and G₂-covariance.

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F-Gap-2: Block transparency by Fano triplets

Coherences within the same Fano line are more strongly correlated, forming a block structure in the coherence matrix $\Gamma$. The Fano dissipator preserves triplet coherences ([T], G₂-structure), generating distinguished block transparency.

Testability: Correlation analysis of the coherence matrix — 7 blocks of $3 \times 3$ (by Fano lines) should be statistically separated from off-block elements.

Status: [T] Theorem — consequence of theorems 10.1–10.3 (Fano channel preserves coherences, status registry).

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F-ξ: Fano correlation length

$$\xi_F \sim 160 \; \text{pc}$$

The correlation length of Fano structure in large-scale structure. The scale is determined by RG suppression of the cubic coupling $\lambda_3$ and the phase diagram of the Gap potential.

Testability: Large-scale structure of the Universe — correlation function on scales $\sim 100$–$200$ pc. Absence of a preferred scale $\sim 160$ pc falsifies the prediction.

Status: [T] Theorem — theorems 9.1–9.2 (status registry).

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F-τ_p: Proton lifetime

$$\tau_p \sim 6.7 \times 10^{37} \; \text{years}$$

The proton lifetime, computed from the masses of $X,Y$-leptoquarks via the Gap hierarchy.

Testability: Hyper-Kamiokande experiment (sensitivity up to $\sim 10^{35}$ years). Current Super-K limit: $\tau_p > 2.4 \times 10^{34}$ years. The prediction lies 2–3 orders of magnitude above Hyper-K sensitivity — direct detection of decay at this $\tau_p$ is unlikely, but detection of decay at $\tau_p < 10^{36}$ years falsifies the prediction.

Note on testability

The prediction $\tau_p \sim 10^{37}$–$10^{38}$ years exceeds the sensitivity of Hyper-K ($\sim 10^{35}$ years for $p \to e^+\pi^0$) by 2–3 orders of magnitude. Direct verification is impossible in the foreseeable future. Indirect constraints are possible via neutron-antineutron oscillations.

Status: [H] Hypothesis — depends on the precision of the $M_X$ computation (proton decay).

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F-m_t: Exactly one $O(1)$ Yukawa (top)

$$\text{exactly one } O(1) \text{ Yukawa: } y_t(m_t) \approx 0.94, \quad m_t \approx 173\ \text{GeV}$$

The unique Fano–Higgs line $\{A, E, U\}$ admits a tree-level Yukawa coupling only for the third generation — hence exactly one $O(1)$ Yukawa in the spectrum [T]. The Pendleton–Ross IR quasi-fixed point overshoots ($m_t^{\text{FP}} \approx 210$–$230$ GeV); the physical top sits below the attractor, so the numerical $m_t \approx 173$ GeV is a boundary condition, status [C] (Yukawa hierarchy).

Testability: the [T] part is falsified by discovery of a second $O(1)$ Yukawa coupling (e.g. a fourth-generation or vector-like quark with $y \sim 1$ coupling to the same Higgs), or by $y_t$ drifting far from $O(1)$. The [C] part is falsified by a significant shift of $m_t^{\text{exp}}$ (current: $172.57 \pm 0.29$ GeV).

Status: "exactly one $O(1)$ Yukawa" — [T]; numerical $m_t$ — [C]. Current verdict: PASSING (one and only one $O(1)$ Yukawa observed; $y_t \approx 0.94$).

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F-ISF: ISF components in fMRI

$$N_{\text{ISF}} \in [6, 12]$$

The number of ISF components (independent slow features) in fMRI data is determined by the opacity rank of the Gap operator. At full transparency (all $\mathrm{Gap}(i,j) = 0$) the rank is 0 and all 21 coherences are active; at full opacity the rank is maximal (21). For biologically realistic regimes the rank is $\sim 9$–$15$, giving $21 - 15 = 6$ to $21 - 9 = 12$ active independent components.

Testability: ISF component analysis of fMRI data. Systematic detection of $N_{\text{ISF}} < 6$ or $N_{\text{ISF}} > 12$ falsifies the prediction. The dependence of $N_{\text{ISF}}$ on the state of consciousness (wakefulness / sleep / anaesthesia) should correlate with the rank of the Gap operator.

Status: [H] Hypothesis — consequence of Gap dynamics and the interiority hierarchy.

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F-Neural: Neural correlates of L-levels [C with bridge assumption]

The form of scaling relations (threshold at $P = 2/7$, monotonic dependence of $\Phi$ on connectivity) is derived [C with bridge assumption]. Numerical coefficients are empirical. Experimental protocol: fMRI/EEG during anaesthesia$\leftrightarrow$wakefulness transitions to verify the threshold.

Testability: Measurement of the $\Phi$ jump under pharmacological control of anaesthesia depth (sevoflurane, propofol). Prediction: existence of a sharp transition $P \approx 2/7$, not gradual sliding. The $\Phi(\text{connectivity})$ dependence is monotonic.

Status: [C with bridge assumption] — the scaling form is derived from theory; numerical coefficients require empirical calibration.

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F-Higgs: Higgs self-coupling deviation

$$\frac{\delta\lambda}{\lambda_{\text{SM}}} \sim O(10^{-2} \text{–} 10^{-3})$$

The octonionic correction to the Higgs sector modifies the Higgs self-coupling at the level of $\sim 1\%$–$0.1\%$ relative to the Standard Model prediction.

Testability: FCC-hh collider (sensitivity to $\delta\lambda/\lambda_{\text{SM}} \sim$ several percent). If FCC-hh measures $\lambda_{hhh}$ with $\sim 5\%$ precision and detects a deviation of order $1\%$ — that is confirmation. Absence of deviations at precision $\ll 0.1\%$ — falsification.

Status: [H] Hypothesis — depends on non-perturbative computations in the Higgs sector.

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F-δ_CP: CKM CP-phase from the Fano phase

$$\delta_{\text{CP}} \approx 64.5° \pm 5°$$

The CKM matrix CP-phase is derived from the geometric phase of the Fano plane. The cleanest tree-level determination — the LHCb combination at ICHEP 2024 — gives $\delta_{\text{CP}}^{\text{exp}} \equiv \gamma = 64.6° \pm 2.8°$; the PDG 2024 global fit gives $65.7° \pm 1.5°$. Against the direct value the predicted $64.5°$ deviates by only $0.1°$ ($\approx 0.04\sigma$); against the fit, by $1.2°$ ($< 1\sigma$). The older "$69° \pm 4°$" figure is superseded.

Testability: Refinement of the experimental value at LHCb and Belle II. The prediction is falsified if $\delta_{\text{CP}}^{\text{exp}}$ settles beyond $\sim 2\sigma$ from $64.5°$ (i.e. outside $[54°, 75°]$).

Status: [H] Hypothesis — depends on the Fritzsch texture and loop corrections. Current verdict: PASSING ($\approx 0.04\sigma$ from the LHCb direct value; $< 1\sigma$ from the global fit).

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F-Cabibbo: Cabibbo angle from RG suppression of the Fano angle

$$\theta_{12} \approx 13°$$

The Cabibbo angle is derived from RG suppression of the fundamental Fano angle $2\pi/7 \approx 51.4°$. Observed value: $\theta_{12}^{\text{exp}} \approx 13.0°$.

Testability: Consistent with current data ($|V_{us}| = 0.2243 \pm 0.0005$, corresponding to $\theta_{12} \approx 12.96°$). The prediction is falsified by a significant revision of $|V_{us}|$.

Status: [H] Hypothesis — depends on loop corrections and RG flow. Current verdict: PASSING (central $\theta_{12}^{\text{exp}} \approx 12.96°$).

The Cabibbo Angle Anomaly — a sharp UHM prediction (T-265)

Independently of the central value, first-row CKM unitarity currently shows a $\sim 3.2\sigma$ deficit: $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 0.9985(5)$ (the "Cabibbo Angle Anomaly", 2024–2026 lattice + $\beta$/kaon determinations). UHM's fixed spectrum turns this into a falsifiable prediction (T-265, CKM §10): the fundamental CKM is exactly $3\times3$ unitary ($N_{\text{gen}}=3$ [T]), so the deficit cannot be a mixing-matrix effect — a fourth generation, vector-like quarks, MeV sterile neutrinos, and leptoquarks are all excluded by the spectrum. UHM therefore predicts the anomaly resolves in the SM extraction sector ($\gamma W$-box / nuclear radiative corrections, lattice $K/\pi$ form factors, the $K$–$\pi$ $V_{us}$ tension), not via new states — and is falsified if the CAA is shown to require any such new state. The magnitude of the $\sim 0.15\%$ deficit is SM hadronic/nuclear physics and remains genuinely open [D]; the resolution-channel exclusion is [T-structural]+[C].

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F-nEDM: Neutron EDM ($\theta_{\mathrm{QCD}} = 0$ exactly)

$$d_n = 0 \quad \text{(exactly)}$$

Prediction [T] (T-99): $\theta_{\mathrm{QCD}} = 0$ exactly (structural proof), not $\theta < 10^{-10}$. Neutron electric dipole moment:

$$d_n = \frac{e \cdot m_q}{m_n^2} \cdot \theta_{\mathrm{QCD}} = 0$$

Current experimental limit: $|d_n| < 1.8 \times 10^{-26}$ e·cm (PSI 2020), i.e. $d_n = (0.0 \pm 1.1) \times 10^{-26}$ e·cm — consistent with the predicted strict zero. n2EDM (PSI) began commissioning data-taking in late 2024 (>50% design UCN flux, systematics verified below $0.5 \times 10^{-26}$ e·cm) and targets $\sim 10^{-27}$ e·cm after ~500 days — the decisive near-term test. nEDM@SNS aims further, at $\sim 10^{-28}$ e·cm.

Falsification: Detection of $d_n \neq 0$ at any level → direct refutation of T-99.

Difference from axion solution: The axion allows $\theta \sim m_a / f_a \cdot T \sim 10^{-18}$ — non-zero, albeit ultra-small. Gap theory predicts a strict zero.

Status: [T] Theorem — T-99 (status registry, confinement).

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F-w: Dark-energy drift shape (Λ-drift law)

Prediction (T-254 [T] / T-255 [Т-structural]+[C]). The dark-energy equation of state is a state functional, not a dial: $1+w_{\text{eff}} = -\tfrac{2}{3}\,d\ln\mathcal{G}_O/d\ln a$ with the positive floor $\Lambda_\infty = \Lambda_{\text{phys}}(\rho^*) > 0$. Consequences, branch-independent: no Big Rip (unbounded phantom excluded), no vacuum-driven collapse ($\rho_{\text{DE}} \not\to 0^-$), $w \to -1$ asymptotically. Branch-dependent: the DESI quadrant $(w_0 > -1,\ w_a < 0)$ is reachable only through a genuine $-1$-crossing (oscillatory branch); the final crossing runs phantom → quintessence in the orientation fixed by the $V_3$ arrow of inner time [C]; the $\dot G_N$ co-drift must carry the same sign as $(1+w)$ with magnitude $\chi(\omega_0/\Lambda)^2(\mathcal{G}_O/7)\tfrac32(1+w)\,H$.

Falsification criteria: (i) a confirmed permanent $w \neq -1$ (constant non-cosmological-constant EoS across epochs at high significance); (ii) reconstructed $\rho_{\text{DE}}(z)$ growing without bound or crossing zero; (iii) the DESI CPL quadrant confirmed while non-parametric reconstruction excludes any $-1$-crossing; (iv) a final crossing in the direction opposite to the $V_3$ arrow (falsifies the [C]-link); (v) a confirmed drift with a $\dot G_N/G_N$ measurement of the opposite sign (falsifies the co-drift).

Experiments: DESI DR2+, Euclid, Roman ($w(z)$ reconstruction); lunar laser ranging ($\dot G_N$).

Current verdict (2026): CONSISTENT — the DESI DR2 preference (March 2025; $2.8$–$4.2\sigma$ for $w_0 > -1,\ w_a < 0$ depending on the supernova sample) lies exactly in the only quadrant UHM reaches via a crossing, and the best-fit CPL trajectory runs phantom (past) → quintessence (today) with a $-1$-crossing near $z \approx 0.5$ — the orientation UHM predicts. No Rip-like or permanent-$w$ signal; the co-drift sign test awaits a confirmed drift.

Caveat — the crossing itself is not yet established

The phantom → quintessence crossing is inferred from the two-parameter CPL fit $(w_0, w_a)$, not from a physical model. Non-parametric / model-independent reconstructions of $w(z)$ do not yet robustly confirm a genuine $-1$-crossing — the feature is partly a parametrisation artifact, and the literature (2025–2026) treats its reality as unsettled. This matters for the [C]-link: UHM's falsification criterion (iii) — "DESI quadrant confirmed while reconstruction excludes any crossing" — is not triggered (reconstructions neither confirm nor exclude), so the prediction is safe but the key discriminator remains open. The quantitative discriminator to watch is whether the oscillatory (quintom-like) branch survives once reconstructions sharpen.

F-rank7: Rank-7 decoherence anisotropy

$$r_{ij}=\tfrac{1}{6}\!\!\sum_{|\ell_p \cap \{i,j\}| = 1}\!\!\gamma_p$$

The twenty-one inter-channel decoherence rates are not free. Each pair $(i,j)$ meets exactly one Fano line, and the rate between them is fixed by the seven line rates $\gamma_p$ through the incidence pattern alone. Twenty-one numbers determined by seven leaves fourteen exact sum-rules — a prediction with no adjustable parameters, which is why it is the sharpest structural test the theory offers.

Testability: rate tomography in the $\Gamma$ basis, in silico now and through PCI-style perturbation protocols later. The test is a projection: fit the measured $r_{ij}$ and ask whether the residual off the seven-dimensional incidence subspace is statistically distinguishable from zero.

Falsified by: a significant residual outside that subspace. Unlike the neural predictions this needs no bridge assumption — it is a statement about the dissipator, testable wherever the dissipator can be measured.

Status: [T]+[C] — the sum-rules are proved (T-262, Prediction 23); the identification of a measured rate matrix with $r_{ij}$ is the conditional part.

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F-Band: the two sums of a living state

$$\text{alive}\;\Longrightarrow\; s_1 \in \left[\tfrac{1}{7},\,\tfrac{3}{14}\right] \quad\text{and}\quad s_2 \in \left[\tfrac{1}{7},\,\tfrac{2}{7}\right]$$

Every gate is a formula in three sums, of which two are $s_1 = \sum_i d_i^2$, the purity of the diagonal, and $s_2 = \sum_{i \neq j}|\gamma_{ij}|^2$, the total weight of the binding. Integration needs $s_2 \geq s_1$; reflexivity needs $s_1 + s_2 \leq 3/7$; and Cauchy–Schwarz needs $s_1 \geq 1/7$ of every state whatsoever. Chaining them confines both sums to narrow bands, the second exactly twice as wide as the first.

This is the theory's own internal falsifier, and it is unusually cheap to run: it needs no bridge assumption, no phenomenal mapping, and no experiment beyond computing two sums from a $\Gamma$.

Falsified by: any state meeting all four criteria whose $s_1$ or $s_2$ falls outside its band — which would mean the four criteria are not what the three sums say they are. Or, from the other side, a system that independent measurement calls conscious whose reconstructed $\Gamma$ sits outside the bands — which would falsify the reconstruction rather than the algebra.

Testability: immediate in silico; through $\Gamma$-reconstruction from fMRI or EEG once that reconstruction is trusted.

Status: [T] — zero counterexamples over twenty thousand states in each band, and both bands are attained rather than vacuous: a flat diagonal carrying balanced content at half strength lands inside (T-321, T-323).

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Summary table of predictions

Code	Prediction	Falsification criterion	Experiment	Status	Current verdict (2026)
F-Gap-1	$\langle\mathrm{Gap}_{\mathrm{intra}}\rangle < \langle\mathrm{Gap}_{\mathrm{inter}}\rangle$	Systematically $\mathrm{Gap}_{\mathrm{intra}} \geq \mathrm{Gap}_{\mathrm{inter}}$	fMRI (ISF)	[H]	UNTESTED
F-Gap-2	Block transparency by Fano triplets	Absence of block structure in coherences	fMRI	[T]	UNTESTED
F-rank7	Rank-7 decoherence anisotropy: the 21 inter-channel rates obey $r_{ij}=\tfrac16\!\sum_{\|\ell_p\cap\{i,j\}\|=1}\gamma_p$ (14 exact sum-rules; Prediction 23, T-262)	A statistically significant residual off the 7-dim Fano incidence subspace	Γ-native rate tomography (in silico) / PCI-style perturbations	[T]+[C]	UNTESTED
F-ξ	$\xi_F \sim 160$ pc	Absence of preferred scale $\sim 160$ pc	LSS surveys	[T]	UNTESTED
F-τ_p	$\tau_p \sim 6.7 \times 10^{37}$ years	$\tau_p < 10^{36}$ years	Hyper-K	[H]	CONSISTENT (above sensitivity; $\tau_p > 2.4\times10^{34}$)
F-m_t	Exactly one $O(1)$ Yukawa [T]; $m_t \approx 173$ GeV [C]	A second $O(1)$ Yukawa; significant $m_t$ shift	Colliders	[T]+[C]	PASSING ($y_t \approx 0.94$, unique)
F-ISF	6–12 ISF components	$N_{\text{ISF}} \notin [6, 12]$	fMRI	[H]	PARTIAL (canonical resting-state decompositions report 7–17 networks)
F-Neural	Threshold $P = 2/7$, monotonic $\Phi$(connectivity)	Gradual transition without threshold	fMRI/EEG (anaesthesia)	[C with bridge]	PARTIAL (clinical PCI threshold 0.31 maps to $P_{\text{crit}}$ — calibration §6.3)
F-Higgs	$\delta\lambda/\lambda_{\text{SM}} \sim 10^{-2}$–$10^{-3}$	No deviations at precision $\ll 0.1\%$	FCC-hh	[H]	UNTESTED (ATLAS+CMS 2025: $-0.71 < \kappa_\lambda < 6.1$ — $\sim O(1)$, far above the predicted band)
F-δ_CP	$\delta_{\text{CP}} \approx 64.5° \pm 5°$	$\delta_{\text{CP}}^{\text{exp}} \notin [54°, 75°]$	LHCb, Belle II	[H]	PASSING ($64.6° \pm 2.8°$ LHCb direct, $\approx 0.04\sigma$; $65.7° \pm 1.5°$ fit)
F-Cabibbo	$\theta_{12} \approx 13°$	Significant revision of $\|V_{us}\|$	Kaon experiments	[H]	PASSING ($\theta_{12}^{\text{exp}} \approx 12.96°$)
F-nEDM	$d_n = 0$ (T-99: $\theta_{\mathrm{QCD}} = 0$ exactly)	$d_n \neq 0$ at any level	n2EDM, nEDM@SNS	[T]	PASSING ($\|d_n\| < 1.8\times10^{-26}\,e\cdot$cm, consistent with 0)
F-w	No Big Rip; no permanent $w \neq -1$; DESI quadrant only via a $-1$-crossing; $\dot G_N$ co-drift same-sign (T-254/T-255)	Permanent $w \neq -1$; unbounded/negative $\rho_{\text{DE}}$; crossing-free DESI quadrant; opposite co-drift sign	DESI/Euclid/Roman + LLR	[T]+[C]	CONSISTENT (DESI DR2 quadrant + phantom→quintessence orientation match, $2.8$–$4.2\sigma$; crossing reality not yet established non-parametrically)
F-Band	$s_1 \in [1/7,\,3/14]$ and $s_2 \in [1/7,\,2/7]$ for anything alive (T-321, T-323)	A state meeting all four criteria with either sum outside its band	Two sums computed from any $\Gamma$	[T]	PASSING (zero counterexamples over $20\,000$ states per band; both bands attained)

Where the fourteen stand. Five PASSING, two CONSISTENT, two PARTIAL, five UNTESTED. Nothing has moved to FAIL, and nothing has been retired: the five untested ones all wait on instruments that do not yet exist at the needed precision (coherence-resolved fMRI, $\Gamma$-native rate tomography, LSS at $\sim 160$ pc, FCC-hh).

Verdict legend. PASSING — measured value lies inside the pass band; CONSISTENT — not excluded, but the prediction lies beyond current sensitivity; PARTIAL — indirect or calibration-dependent support; UNTESTED — no experiment has probed the band yet. A single entry moving to FAIL falsifies the corresponding claim at its stated status level ([T]/[C]/[H]).

Status of predictions

Predictions marked [T] are based on rigorously proved theorems (see status registry). The octonionic bridge is fully closed [T] (T15). Predictions marked [H] require additional computations or contain gaps in the physical arguments.

Completeness of Theory

The theory is complete in the following sense:

1. Self-sufficiency: Requires no external postulates or references
2. Universality: Applicable to structural aspects of self-referential systems — from quantum to cognitive
3. Internal consistency: Contains no contradictions
4. Operationality: Can be computationally implemented
5. Explanatory power: Resolves traditional philosophical problems
6. Falsifiability: Makes testable predictions about the structure of experience
7. Formal rigour: Key theorems proved (7D minimality, operator φ, functor F)
8. Compatibility with QM: The nonlinear regenerative term $\mathcal{R}$ does not violate the no-signalling constraint — proved via the CPTP property of $\varphi$ (conditions NS1-NS3)
9. Ensemble independence: Evolution is defined on $\Gamma$ (density matrix), not on wave functions — does not depend on decomposition
10. Computational consistency: The nonlinearity $\mathcal{R}$ does not provide acceleration beyond BQP

Vulnerability analysis

Systematic analysis of five main vulnerabilities of the theory (2026):

#	Vulnerability	Initial status	Result	New status
1	$\dim = 7$ as postulate	Not empirically verified	15+ independent derivations [T]: Theorem S (minimality) + octonionic derivation + T15 (bridge)	Closed (theoretically)
2	$D_{\mathrm{diff}} \geq 2$ [C]	Conditional theorem	T-129 [T]: $\Phi_{\mathrm{th}} = 1$ from first principles → T-151 [T]: $D_{\min} = 2$ unconditionally	Closed (fully)
3	$R = 1/(7P)$ counterintuitive	Requires empirical verification	Algebraic identity [T], physical interpretation, T-124 [T] (non-emptiness of Goldilocks zone)	Closed (theoretically)
4	No experiments	269 registry theorems without lab verification	23 numbered predictions and the 14 falsification criteria below; 5 post-hoc coincidences (F-m_t, F-Cabibbo, F-δ_CP, F-nEDM, Weber-Fechner)	Confirmed (requires experiment)
5	Quantum nature of $\Gamma$	Tegmark decoherence	T-267 [T]: Tegmark bounds microscopic position-basis superpositions; $\Gamma$ is coarse-grained collective structure on a decoherence-free subspace (T-153a), complex by algebra not superposition (T-132), and classically realizable — the objection targets a claim UHM does not make. Residual = the categorical gap (Axiom Ω⁷), a distinct primitive	Closed (Tegmark objection)

Summary: 4 of 5 vulnerabilities closed theoretically (the Tegmark objection to #5 closed by T-267); 1 is fundamentally experimental. The categorical gap (why structure is felt) is not a vulnerability but the acknowledged Axiom Ω⁷ primitive — distinct from the Tegmark decoherence question, which is now settled.

Theory Boundaries

Philosophy of boundaries

Acknowledging boundaries is not a weakness, but a strength of a scientific theory. A theory that claims to explain everything without exception is most likely unscientific.

Structural Boundaries (what is not proved)

Question	Status	Comment
Why 7 dimensions?	Minimality proved	But not uniqueness
Values of constants $\omega_i$, $J_{ij}$, $\gamma_k$	Empirical (holon-specific)	Not derived from axioms — except the cosmological $\omega_0$, pinned to $\sim 10^{-2} M_{\text{Pl}}$ [C] via the $G_2$-extra/neutrino sector ($M_{G_2} = \omega_0\sqrt{\mathcal{G}_O/2} \sim 10^{17}$ GeV; used by T-266)
Uniqueness of $\Gamma$	Not proved	Other 'universes' possible
Uniqueness of partition $\{A,S,D,L,E,O,U\}$	Proved [T]	All 7 dimensions are functionally unique (A,S,D,L,U — algebraically; E,O — via κ₀)

Physical Boundaries

Question	Status	Comment
Einstein equations	[T] Derived	Spectral action (T-65); $M^4$ derived (T-120)
Standard Model	Structure [T], parameters partially	$G_2 \to SU(3)_C \times SU(2)_L \times U(1)_Y$ [T]; specific masses — partially
Spacetime dimensionality $3+1$	[T] Derived	Sectoral decomposition + Connes reconstruction (T-119, T-120)
Constants $c$, $G$, $\hbar$	$G$ [T] derived, $c$, $\hbar$ not explained	$G_N = 3\pi/(7f_2\Lambda^2)$ (T-65); $c$, $\hbar$ — fundamental

Phenomenal Boundaries (what is taken as axiom)

1. Categorical gap: The theory does not explain why mathematical structures are 'felt.' The identity of being and experience — Axiom Ω⁷, not a theorem.

2. Qualia calibration: The correspondence between specific eigenvalues/eigenvectors and specific qualities of experience is established empirically.

Qualia calibration — an empirical question

Which specific $[|q\rangle] \in \mathbb{P}(\mathcal{H}_E)$ corresponds to 'red' is an empirical question, not a theoretical defect. This is analogous to how the electron mass is not derived from the Standard Model. The structure of experience (spectral decomposition) is the unique functor compatible with the axiomatics, but the specific calibration is determined experimentally.

3. Absolute qualia: The question of the existence of context-independent qualia remains open.

4. Thresholds L2: $R_{\text{th}} = 1/3$ [T] — derived from triadic decomposition ($K = 3$ types of dynamics from axioms) + Bayesian dominance. $\Phi_{\text{th}} = 1$ [T] — unique self-consistent value at $P_{\text{crit}} = 2/7$ (T-129).

Categorical Boundaries

1. $\mathbf{Exp}$ is not a topos: It is proved that the category $\mathbf{Exp}$ is not a topos — there is no internal logic of experiential content.

2. Functor $F$ is non-invertible: One cannot uniquely recover $\rho$ from experiential content — different states may yield 'identical' experience.

3. Problem of time: The category $\mathbf{DensityMat}$ is static; time requires an external parameter.

Status of Boundaries

These boundaries are not a deficiency, but an acknowledgement:

• The theory describes structure, not the question of 'why this particular structure'
• Some questions may be beyond any possible explanation
• Honest acknowledgement of boundaries is a mark of a mature theory

Comparison with physics: Physics does not explain why the laws of nature exist — it describes their structure. Analogously, UHM describes the structure of experience, acknowledging the boundaries of explanation.

Octonionic Falsification Criteria

The structural derivation through octonions generates additional testable predictions:

Prediction	Falsification criterion	Status
Fano symmetries of coherences	7 triplets $(e_i, e_j, e_k)$ of the Fano plane should be distinguishable in the structure of coherences $\gamma_{ij}$	[T]
$G_2$-covariance	The dynamics of $\Gamma$ must be covariant with respect to $G_2 \subset SO(7)$, not the full $SO(7)$	[T]
Associator anomalies	Triple interactions of dimensions should exhibit non-associativity: $[x, y, z] \neq 0$	[T]
Hamming threshold	Structure $H(7,4)$, exactly as in the canon (T-93): 1 coherence violation is corrected automatically by the regenerator, 2 are detected via the syndrome measurements (E, O, U); beyond two the code is unreliable ($d = 3$). Erratum 2026-07-25: this previously read «viable with loss of up to 3 of 7» — unsupported by T-93 and inconsistent with $d=3$ (the guarantee is $\lfloor(d-1)/2\rfloor = 1$ correction, $d-1 = 2$ detections); found by machine-checking the table against the canon.	[T]

Bridge [T] — fully closed (T15)

The connection (AP)+(PH)+(QG)+(V) → P1+P2 is established via the complete formal chain T15 (12 steps, all [T]). T11–T13 prove the former condition (МП). All octonionic predictions are consequences of the structural derivation [T].

Research programme

Boundaries do not mean a halt to development. Open directions:

Direction	Goal	Priority
Quantum gravity	Derive $g_{\mu\nu}$ from $\Gamma$	High
Experimental validation of thresholds	Verify $R_{\text{th}} = 1/3$, $\Phi_{\text{th}} = 1$ empirically	High
Isospectral experiments	Test prediction 1 with numerical tolerances	High
ISF analysis of fMRI	Verify F-Gap-1, F-Gap-2, F-ISF	High
Non-perturbative computations	Refine F-Higgs, F-τ_p	High
Correlation length $\xi_F$	Verify F-ξ through LSS surveys	Medium
Connection with Hoffman	Prove equivalence with the theory of conscious agents	Medium
$\infty$-topos	Construct $\infty$-topos on $\mathbf{Exp}$	Low
Standard Model	Close the derivation of $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ from the Gap hierarchy	Long-term

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Related documents:

• Glossary — definitions of terms
• Status registry — complete registry of results with classification [T]/[H]/[P]/[I]/[D]
• Axiom Ω⁷ — ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$ as primitive
• Mathematical apparatus — formal definitions of $\Gamma$, $d_{\mathrm{FS}}$
• Interiority hierarchy — levels L0→L1→L2→L3→L4, thresholds $R_{\text{th}}$, $\Phi_{\text{th}}$, $R^{(n)}_{\text{th}}$
• Categorical formalism — functor $F$, categories $\mathbf{DensityMat}$ and $\mathbf{Exp}$
• 7D minimality theorem — proof of $N = 7$
• Structural derivation via octonions — P1+P2 → $\mathbb{O}$ → N=7
• Formalisation of operator φ — CPTP channels
• Theorems — formal results
• CC predictions — CC-specific falsification criteria
• Fano selection rules — Fano selection rule for Yukawa couplings
• Gap semantics — dual-aspect semantics of 49 elements
• Gap dynamics — Gap operator, bifurcations, non-Markovian dynamics
• Gap thermodynamics — information geometry, potential $V_{\text{Gap}}$
• RG flow of Gap — $\beta$-functions, fixed points, RG suppression of $\lambda_3$
• Higgs sector — uniqueness of the line $\{A,E,U\}$, Higgs mass
• CKM matrix — Fritzsch texture, Cabibbo angle, CP phase
• Yukawa hierarchy — Pendleton–Ross fixed point, $m_t$
• Proton decay — $X,Y$-leptoquarks, $\tau_p$
• Dark matter — $O$-sector relic, $\xi_F$
• G₂ structure — covariance of the Fano dissipator