Equation Ledger

This ledger preserves mathematical formulas, research-state rules, and discarded-but-important ideas from the NFC pivot context. Status labels are conservative: implemented, experimental, scaffolded, theory only, archived, deferred, falsified, or needs evidence.

equation_id	equation	category	source/context	current project status	where included in final dossier	notes
EQ-001	Attention(Q,K,V)=softmax(QK^T/sqrt(d))V	operator learning	Transformer baseline	implemented baseline	this ledger, master dossier	Baseline math, not NFC novelty.
EQ-002	h_{t+1}=A(x_t)h_t+B(x_t)x_t	operator learning	Mamba/selective SSM schematic	deferred baseline	this ledger, master dossier	Mamba is not wired as a NeuroTwin backbone.
EQ-003	F_s(x,t,omega) in R^d	latent field definition	NFC primitive	experimental	constitution, master dossier	Core NFC primitive.
EQ-004	Y_m=O_m(F_s,A_s,U,epsilon_m)	observation operators	NFC observation model	experimental	constitution, master dossier	Replaces direct-fusion framing.
EQ-005	`p(Y_{1:M}	U,A_s)=int p(F	U,A_s) prod_m p(Y_m	F,A_s) dF`	generative probabilistic model	NFC generative statement
EQ-006	F(t+Delta)=Phi_{theta,Delta}(F(t),U_[t-H,t],A_s)	controlled dynamics	NFC dynamics	experimental	constitution, master dossier	Implemented only as simplified neural modules.
EQ-007	dF_t=f_theta(F_t)dt+g_theta(F_t)dU_t	controlled dynamics	Neural CDE-style update	theory only	master dossier	Preserved as future dynamics option.
EQ-008	tau partial_t F(x,t)=-F(x,t)+int_Omega K_theta(x,x',t)sigma(F(x',t))dx'+B_theta U(t)+eta(x,t)	neural-field PDE / integro-differential dynamics	NFC field equation	theory only	constitution, master dossier	Guides model hierarchy.
EQ-009	Z_{t+1}=Z_t+Delta t[-DZ_t+K_t sigma(Z_t)+BU_t]+xi_t	controlled dynamics	Discretized field dynamics	experimental	constitution, master dossier	Current implementation is a simplified form.
EQ-010	K_t approx U_t V_t^T	low-rank pair kernel	Pair-kernel approximation	implemented experimental	constitution, master dossier	Implemented as low-rank pair update.
EQ-011	M_t=softmax((U_t V_t^T)/sqrt(r)+S)	low-rank pair kernel	Pair influence	implemented experimental	constitution, master dossier	Structural prior S is optional.
EQ-012	Z'_t=Z_t+M_t Z_t W	low-rank pair kernel	Node update	implemented experimental	constitution, master dossier	Pair-Operator survives as ablation idea.
EQ-013	NPU_ij(t)=E[abs(K_ij(t)-Khat_ij(t))]	uncertainty / calibration	Neural pair uncertainty	scaffolded	ledger, master dossier	Pair uncertainty exists only in limited form.
EQ-014	nabla_w F(i,j)=sqrt(w_ij)(F_j-F_i)	graph calculus	Weighted graph derivative	theory only	constitution, master dossier	Future regularizer.
EQ-015	L=D-W	graph calculus	Graph Laplacian	theory only	constitution, master dossier	Future anatomy/graph constraint.
EQ-016	R_graph(F)=sum_(i,j in E) w_ij norm(F_i-F_j)^2 = Tr(F^T L F)	graph/temporal/spectral regularizers	Graph smoothness	theory only	constitution, master dossier	Not an implemented loss.
EQ-017	(H_HRF a)(t)=int_0^infty h(tau)a(t-tau)dtau	fMRI HRF observation	HRF convolution	experimental	constitution, master dossier	Implemented as simplified convolution.
EQ-018	Y_fMRI(p,t)=R_p H_HRF g_theta(F(.,t))+epsilon	fMRI HRF observation	fMRI operator	experimental	constitution, master dossier	Current operator is a simplified readout.
EQ-019	tilde U_t=sum_{ell=0}^L alpha_ell U_{t-ell}	observation operators	Stimulus lag adapter	experimental	master dossier	Implemented as causal averaging adapter.
EQ-020	Y_EEG(t)=L_s J_theta(F_t)+epsilon_EEG	EEG/MEG observation	EEG operator	experimental	constitution, master dossier	Current operator is simplified pooling/readout.
EQ-021	Y_MEG(t)=M_s J_theta(F_t)+epsilon_MEG	EEG/MEG observation	MEG operator	theory only	constitution, master dossier	No MEG support claim.
EQ-022	Y_{n,t} ~ Poisson(Delta t softplus(w_n^T F(x_n,t)))	spike/calcium observation	Spike observation	theory only	constitution, master dossier	Not implemented.
EQ-023	c_{n,t}=(k_Ca*r_n)(t)+epsilon	spike/calcium observation	Calcium observation	theory only	constitution, master dossier	Not implemented.
EQ-024	`p(a_t	F_t,U_t)=softmax(C pool(F_t)+D U_t)`	behavior observation	Behavior observation	scaffolded	constitution, master dossier
EQ-025	`q_phi(F_{0:T}	Y_obs,U,A_s)`	variational inference / ELBO	Variational posterior	theory only	ledger, master dossier
EQ-026	`L_ELBO=E_q[sum_{m,t} log p_theta(Y_{m,t}	F_t,A_s)] - beta KL(q_phi(F	Y,U,A_s)		p_theta(F	U,A_s))`
EQ-027	F'=AF	gauge symmetry / identifiability	Gauge ambiguity	theory only	constitution, master dossier	Explains latent non-identifiability.
EQ-028	O'_m=O_m A^{-1}	gauge symmetry / identifiability	Gauge ambiguity	theory only	constitution, master dossier	Must not overclaim latent semantics.
EQ-029	O'_m(F')=O_m(F)	gauge symmetry / identifiability	Gauge invariance	theory only	constitution, master dossier	Observation equivalence.
EQ-030	R_time=sum_t norm(F_{t+1}-F_t)^2	graph/temporal/spectral regularizers	Temporal smoothness	theory only	ledger, master dossier	Not implemented as loss.
EQ-031	R_band=sum_omega gamma_omega norm(Fhat(omega))^2	graph/temporal/spectral regularizers	Spectral regularizer	theory only	ledger, master dossier	Not implemented.
EQ-032	A_s=I+U_s V_s^T	operator learning	Low-rank subject adapter	scaffolded	ledger, master dossier	Subject adaptation exists elsewhere, not NFC proof.
EQ-033	dF/dt=f_theta(F,U,t)	stability / spectral radius	Continuous stability setup	theory only	ledger, master dossier	Future stability audit.
EQ-034	J_t=partial f_theta / partial F	stability / spectral radius	Jacobian	theory only	ledger, master dossier	Not computed.
EQ-035	Re(lambda_i(J_t))<0	stability / spectral radius	Local stability	theory only	ledger, master dossier	Not enforced.
EQ-036	rho(A)<1	stability / spectral radius	Discrete stability	theory only	ledger, master dossier	Not enforced.
EQ-037	R_stab=max(0,rho(A)-1)^2	stability / spectral radius	Stability penalty	theory only	ledger, master dossier	Future loss term.
EQ-038	L_NLL=sum_{m,t,i} (y-mu)^2/(2 sigma^2)+0.5 log sigma^2	uncertainty / calibration	Uncertainty NLL	theory only	constitution, master dossier	Current uncertainty calibration is not full NLL.
EQ-039	P[Y in C_alpha(X)] approx 1-alpha	uncertainty / calibration	Calibration condition	theory only	constitution, master dossier	Requires real calibration artifacts.
EQ-040	L_InfoNCE=-log exp(sim(z_a,z_b)/tau)/sum_{b'} exp(sim(z_a,z_b')/tau)	information theory / contrastive learning	Contrastive alignment	theory only	ledger, master dossier	Future representation audit.
EQ-041	g(F_{t+1})=K g(F_t)	Koopman / residual dynamics	Koopman form	theory only	ledger, master dossier	Future dynamics baseline.
EQ-042	F_{t+1}=K F_t+R_theta(F_t,U_t)	Koopman / residual dynamics	Koopman residual update	theory only	ledger, master dossier	Not implemented.
EQ-043	L=L_pred+lambda_R norm(R_theta)^2+lambda_K max(0,rho(K)-1)^2	Koopman / residual dynamics	Koopman residual loss	theory only	ledger, master dossier	Not implemented.
EQ-044	C=(1/T)XX^T	Riemannian EEG covariance	EEG covariance	theory only	ledger, master dossier	Future EEG baseline.
EQ-045	Chat=LL^T+epsilon I	Riemannian EEG covariance	SPD covariance head	theory only	ledger, master dossier	Not implemented.
EQ-046	d_AIRM(C1,C2)=norm(log(C1^{-1/2}C2C1^{-1/2}))_F	Riemannian EEG covariance	AIRM distance	theory only	ledger, master dossier	Not implemented.
EQ-047	T_m={k Delta_m:k in Z}	sampling lattice / Fourier / coprime windows	Sampling grid	theory only	ledger, master dossier	Future sampling stress test.
EQ-048	t in union_m T_m	sampling lattice / Fourier / coprime windows	Union of sampling grids	theory only	ledger, master dossier	Future multimodal timing model.
EQ-049	phi(t)=[sin(2 pi k t/T), cos(2 pi k t/T)]_{k=1}^K	sampling lattice / Fourier / coprime windows	Fourier features	theory only	ledger, master dossier	Not implemented.
EQ-050	L v_k=lambda_k v_k	sampling lattice / Fourier / coprime windows	Graph spectral modes	theory only	ledger, master dossier	Not implemented.
EQ-051	gcd(w,s)=1	sampling lattice / Fourier / coprime windows	Coprime window stress test	theory only	ledger, master dossier	Future split/window audit.
EQ-052	log(phi_1(t;theta)/phi_0) approx J delta mu_a(t;theta)	fNIRS optical/Rytov observation	Rytov approximation	theory only	fNIRS notes, master dossier	fNIRS is docs only.
EQ-053	phi_1(t;theta) approx phi_0 exp(J delta mu_a(t;theta))	fNIRS optical/Rytov observation	Predicted intensity	theory only	fNIRS notes, master dossier	No fNIRS implementation claim.
EQ-054	Delta OD(t;theta)=log phi_base_bar - log phi_1(t;theta)	fNIRS optical/Rytov observation	Optical density	theory only	fNIRS notes, master dossier	Docs only.
EQ-055	Y_fNIRS(t)=Delta OD(phi_0 exp(J delta mu_a(t)))+epsilon	fNIRS optical/Rytov observation	fNIRS observation	theory only	fNIRS notes, master dossier	Docs only.
EQ-056	F_hemo(t)=(h_HRF*g(F_neural))(t)	fNIRS optical/Rytov observation	Neural to hemodynamic field	theory only	fNIRS notes, master dossier	Informs future generator realism.
EQ-057	Y_fMRI(t)=R F_hemo(t)+epsilon	fMRI HRF observation	Synthetic fMRI	experimental	ledger, master dossier	Current synthetic generator approximates this.
EQ-058	Y_EEG(t)=L F_neural(t)+A_motion(t)+epsilon	EEG/MEG observation	Synthetic EEG	experimental	ledger, master dossier	Motion artifacts are future realism.
EQ-059	Q:R^d -> {0,1}^{bd}	TurboQuant/TurboVec math	TurboQuant map	deferred	TurboQuant notes, master dossier	Optional infrastructure only.
EQ-060	E norm(x-xhat)_2^2	TurboQuant/TurboVec math	MSE objective	deferred	TurboQuant notes, master dossier	Not implemented.
EQ-061	E[(<x,q>-<xhat,q>)^2]	TurboQuant/TurboVec math	Inner-product objective	deferred	TurboQuant notes, master dossier	Not implemented.
EQ-062	r=norm(v)_2, u=v/norm(v)_2	TurboQuant/TurboVec math	Vector normalization	deferred	TurboQuant notes, master dossier	Optional future adapter.
EQ-063	z=Ru	TurboQuant/TurboVec math	Random rotation	deferred	TurboQuant notes, master dossier	Optional future adapter.
EQ-064	norm(z)_2=norm(u)_2=1	TurboQuant/TurboVec math	Norm preservation	deferred	TurboQuant notes, master dossier	Optional future adapter.
EQ-065	z_i approx N(0,1/d)	TurboQuant/TurboVec math	Coordinate approximation	deferred	TurboQuant notes, master dossier	Risky at low dimension.
EQ-066	q_i=c_argmin_k abs(z_i-c_k)	TurboQuant/TurboVec math	Scalar quantization	deferred	TurboQuant notes, master dossier	Optional future adapter.
EQ-067	x=xhat_MSE+r_residual	TurboQuant/TurboVec math	Residual split	deferred	TurboQuant notes, master dossier	Optional future adapter.
EQ-068	xhat_prod=xhat_MSE+QJL(r_residual)	TurboQuant/TurboVec math	QJL residual correction	deferred	TurboQuant notes, master dossier	Optional future adapter.
EQ-069	score(q,x)=<q,x>	retrieval-kNN baseline	Inner-product score	deferred	TurboQuant notes, master dossier	Use exact numpy first.
EQ-070	I_stim={Q(e_stim,i)}_{i=1}^N	semantic duplicate audit	Stimulus index	deferred	TurboQuant notes, master dossier	Future semantic audit.
EQ-071	z_t=pool(F_t)	latent field memory	Latent summary	deferred	TurboQuant notes, master dossier	Future memory layer.
EQ-072	z_{t,k}=pool_{i in network k}(F_{t,i})	latent field memory	Network summary	deferred	TurboQuant notes, master dossier	Future audit feature.
EQ-073	N_k(q)={i_1,...,i_k}	retrieval-kNN baseline	Retrieval neighborhood	deferred	TurboQuant notes, master dossier	Optional baseline.
EQ-074	yhat_test=sum_{i in N_k(q)} w_i y_i	retrieval-kNN baseline	Retrieval-kNN prediction	deferred	TurboQuant notes, master dossier	Must not use test targets.
EQ-075	w_i=exp(tau <q,z_i>)/sum_{j in N_k(q)} exp(tau <q,z_j>)	retrieval-kNN baseline	Retrieval weights	deferred	TurboQuant notes, master dossier	Train-only labels only.
EQ-076	abs(<q,z_i>-<q,zhat_i>) <= norm(q)_2 norm(z_i-zhat_i)_2	TurboQuant/TurboVec math	Score error bound	deferred	TurboQuant notes, master dossier	Quantization risk.
EQ-077	d_min(x_test,D_train)=min_{x_i in D_train} norm(q(x_test)-q(x_i))	semantic duplicate audit	Nearest train distance	deferred	TurboQuant notes, master dossier	Future leakage audit.
EQ-078	L=sum_m lambda_m L_m+lambda_latent L_latent+lambda_graph Tr(F^T L F)+lambda_stab R_stab+lambda_unc L_cal	falsification gates	Full NFC loss	theory only	master dossier	Not implemented as one loss.
EQ-079	L_fMRI=norm(M(Yhat_fMRI-Y_fMRI))^2	fMRI HRF observation	fMRI reconstruction loss	experimental	master dossier	MSE implemented, mask semantics vary.
EQ-080	L_EEG=norm(M(Yhat_EEG-Y_EEG))^2+gamma norm(PSD(Yhat)-PSD(Y))^2	EEG/MEG observation	EEG loss	theory only	master dossier	PSD term not implemented.
EQ-081	L_spike=-sum_{n,t} log Poisson(y_{n,t};lambda_{n,t})	spike/calcium observation	Spike loss	theory only	master dossier	Not implemented.
EQ-082	L_{a->b}=norm(O_b(I_theta(y_a))-y_b)	falsification gates	Cross-modal consistency	theory only	master dossier	Field-mediated route under test.
EQ-083	T_{a->b}=O_b circ I_a	falsification gates	Commutative translation factorization	theory only	master dossier	Research framing.
EQ-084	y_a -> y_b	falsification gates	Direct translation baseline	implemented baseline	master dossier	Baseline lane.
EQ-085	y_a -> F -> y_b	falsification gates	Field-mediated translation	experimental	master dossier	NFC lane.

Status Rules

• Pair-Operator is archived as the main primitive and retained as an ablation.

• NFC is the experimental architecture path.

• fNIRS is theory/evidence only and not implemented support.

• TurboVec/TurboQuant is optional retrieval/audit infrastructure, not the model contribution.

• MOABB Track A is a reproducibility evidence path.

• Algonauts/CNeuroMod fMRI is the future Track B model battlefield.

• MDD or clinical claims are not allowed.

• A100 experiment rules require strict 1x synthetic diagnostic before Algonauts or 6x DDP.