Notation by work
PRISM-1
All notation · Glossary · Source coverage
120 symbol spellings in 179 mathematical passages across 4 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 120 entries are available below.
=Definition passage · 38 passages
Search aliases: =
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
aDefinition passage · 2 passages
Search aliases: a
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
iDefinition passage · 2 passages
Search aliases: i
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
\mathbf nDefinition passage · 2 passages
Search aliases: n mathbf
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
\mathsf PDefinition passage · 1 passage
Search aliases: P mathsf
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
pDefinition passage · 2 passages
Search aliases: p
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
\piDefinition passage · 2 passages
Search aliases: π pi
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
\sigmaDefinition passage · 2 passages
Search aliases: σ sigma
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
\tauDefinition passage · 4 passages
Search aliases: τ tau
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
\ellDefinition passage · 2 passages
Search aliases: ℓ ell
Definition passage: Packet descriptor · Komorebi_PRISM_1_Final_Updated.tex, line 501
[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.
Exact mathematical passage
\mathsf P=(i,p,a,\sigma,\mathbf n,\ell,\pi,\tau),
*Use in source · 13 passages
Search aliases: *
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 535
Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.
Exact mathematical passage
A=D+UV^*,
+Use in source · 17 passages
Search aliases: +
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
AUse in source · 14 passages
Search aliases: A
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 313
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
AB=(AS)(S^{-1}B).
A_eUse in source · 1 passage
Search aliases: A_e
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 701
[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.
Exact mathematical passage
A=\sum_{e\in E}A_e, \qquad \widehat A=\sum_{e\in E}(A_e+E_e).
\AcalUse in source · 3 passages
Search aliases: Acal A mathcal
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 518
Atlas descriptor. An atlas descriptor records an analysis graph, a synthesis graph, normalization data, and packet selectors. A unitary transform uses its adjoint for synthesis. A redundant exact frame records the canonical dual or another declared synthesis map satisfying S A=I.
Exact mathematical passage
\Scal\Acal=I.
\mathrm{amortization}Use in source · 1 passage
Search aliases: amortization mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
\mathrm{architecture}Use in source · 1 passage
Search aliases: architecture mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
BUse in source · 6 passages
Search aliases: B
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 313
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
AB=(AS)(S^{-1}B).
bUse in source · 17 passages
Search aliases: b
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 538
Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.
Exact mathematical passage
b
b^2Use in source · 1 passage
Search aliases: b^2
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 553
The block-diagonal part stores (n/b)b^2=nb scalars. The two rank factors store 2nr. The total is n(b+2r). A full array stores n^2 scalars.
Exact mathematical passage
(n/b)b^2=nb
cUse in source · 8 passages
Search aliases: c
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 338
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
c(o)
c_eUse in source · 2 passages
Search aliases: c_e
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 725
Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.
Exact mathematical passage
\varepsilon_e =\frac{\tau c_e}{s_e\sum_f c_f},
c_fUse in source · 1 passage
Search aliases: c_f
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 725
Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.
Exact mathematical passage
\varepsilon_e =\frac{\tau c_e}{s_e\sum_f c_f},
\operatorname{child}Use in source · 1 passage
Search aliases: child operatorname
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
DUse in source · 12 passages
Search aliases: D
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 317
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
A=L D_A S^{-1}, \qquad B=S D_B R.
D^{-1}Use in source · 1 passage
Search aliases: D^-1
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 688
Woodbury solve. The chip uses packet solves with D_α and one small dense correction: (D+UV^*)^-1 =D^-1-D^-1U(I+V^*D^-1U)^-1V^*D^-1. The solver engines handle the rank-sized middle system.
Exact mathematical passage
(D+UV^*)^{-1} =D^{-1}-D^{-1}U(I+V^*D^{-1}U)^{-1}V^*D^{-1}.
D_AUse in source · 3 passages
Search aliases: D_A
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 317
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
A=L D_A S^{-1}, \qquad B=S D_B R.
D_BUse in source · 3 passages
Search aliases: D_B
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 317
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
A=L D_A S^{-1}, \qquad B=S D_B R.
D_\alphaUse in source · 1 passage
Search aliases: D_α alpha
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 687
Woodbury solve. The chip uses packet solves with D_α and one small dense correction: (D+UV^*)^-1 =D^-1-D^-1U(I+V^*D^-1U)^-1V^*D^-1. The solver engines handle the rank-sized middle system.
Exact mathematical passage
D_\alpha
EUse in source · 7 passages
Search aliases: E
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 607
Exact analysis and synthesis. Let U be unitary and let E_α be coordinate selectors. The pulled-back projectors are P_α=U^*E_α U. The hardware analysis applies U and selects the coordinates. Synthesis scatters the selected coordinates and applies U^*.
Exact mathematical passage
E_\alpha
eUse in source · 10 passages
Search aliases: e
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 338
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
e(o)
E_eUse in source · 3 passages
Search aliases: E_e
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 701
[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.
Exact mathematical passage
A=\sum_{e\in E}A_e, \qquad \widehat A=\sum_{e\in E}(A_e+E_e).
E_\alphaUse in source · 4 passages
Search aliases: E_α alpha
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 607
Exact analysis and synthesis. Let U be unitary and let E_α be coordinate selectors. The pulled-back projectors are P_α=U^*E_α U. The hardware analysis applies U and selects the coordinates. Synthesis scatters the selected coordinates and applies U^*.
Exact mathematical passage
E_\alpha
FUse in source · 4 passages
Search aliases: F
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 330
Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array represented by the chain.
Exact mathematical passage
A=F_LF_{L-1}\cdots F_1
fUse in source · 1 passage
Search aliases: f
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 725
Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.
Exact mathematical passage
\varepsilon_e =\frac{\tau c_e}{s_e\sum_f c_f},
F_1Use in source · 1 passage
Search aliases: F_1
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 330
Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array represented by the chain.
Exact mathematical passage
A=F_LF_{L-1}\cdots F_1
F_LUse in source · 1 passage
Search aliases: F_L
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 330
Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array represented by the chain.
Exact mathematical passage
A=F_LF_{L-1}\cdots F_1
F_{L-1}Use in source · 1 passage
Search aliases: F_L-1
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 330
Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array represented by the chain.
Exact mathematical passage
A=F_LF_{L-1}\cdots F_1
\mathrm{graph}Use in source · 1 passage
Search aliases: graph mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 1030
PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.
Exact mathematical passage
T= \max\left( \frac{F}{\eta_cP}, \frac{B}{\eta_bW} \right) +T_{\mathrm{graph}},
HUse in source · 4 passages
Search aliases: H
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 579
Local refinement. Suppose a parent packet task v is replaced by an acyclic child module H=(W,F). Predecessors of v connect to child entries. Child exits connect to successors of v.
Exact mathematical passage
H=(W,F)
IUse in source · 5 passages
Search aliases: I
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
JUse in source · 1 passage
Search aliases: J
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
\mathrm{kernel}Use in source · 1 passage
Search aliases: kernel mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
LUse in source · 6 passages
Search aliases: L
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 317
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
A=L D_A S^{-1}, \qquad B=S D_B R.
\logUse in source · 2 passages
Search aliases: log
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 333
Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array represented by the chain.
Exact mathematical passage
L=O(\log n)
\maxUse in source · 1 passage
Search aliases: max
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 1030
PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.
Exact mathematical passage
T= \max\left( \frac{F}{\eta_cP}, \frac{B}{\eta_bW} \right) +T_{\mathrm{graph}},
\minUse in source · 1 passage
Search aliases: min
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
\min_{o\in\mathcal O_v}Use in source · 1 passage
Search aliases: min_o∈ O_v min in mathcal
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
nUse in source · 29 passages
Search aliases: n
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 333
Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array represented by the chain.
Exact mathematical passage
O(n)
n^2Use in source · 5 passages
Search aliases: n^2
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 542
Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.
Exact mathematical passage
n^2
\normUse in source · 1 passage
Search aliases: norm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 707
[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.
Exact mathematical passage
\boxed{ \norm{(\widehat A-A)x} \le \sum_{e\in E}\norm{E_e}\norm{P_{\alpha(e)}x}.}
OUse in source · 4 passages
Search aliases: O
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 333
Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array represented by the chain.
Exact mathematical passage
O(n)
\mathcal OUse in source · 1 passage
Search aliases: O mathcal
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
oUse in source · 4 passages
Search aliases: o
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 338
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
o
\mathcal O_vUse in source · 1 passage
Search aliases: O_v mathcal
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
PUse in source · 8 passages
Search aliases: P
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
P_\alphaUse in source · 4 passages
Search aliases: P_α alpha
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
P_{\alpha(e)}Use in source · 2 passages
Search aliases: P_α(e) alpha
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 707
[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.
Exact mathematical passage
\boxed{ \norm{(\widehat A-A)x} \le \sum_{e\in E}\norm{E_e}\norm{P_{\alpha(e)}x}.}
P_\betaUse in source · 3 passages
Search aliases: P_β beta
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\precUse in source · 5 passages
Search aliases: prec
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 582
[Topological splice] Let prec be a topological order of the original graph and let prec_H be a topological order of the child module. Replacing v in prec by prec_H gives a topological order of the refined graph. Every unaffected pair of context tasks keeps its relative order.
Exact mathematical passage
\prec
\prec_HUse in source · 3 passages
Search aliases: prec_H prec
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 582
[Topological splice] Let prec be a topological order of the original graph and let prec_H be a topological order of the child module. Replacing v in prec by prec_H gives a topological order of the refined graph. Every unaffected pair of context tasks keeps its relative order.
Exact mathematical passage
\prec_H
qUse in source · 12 passages
Search aliases: q
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 646
BDLR dataflow. For A=D+UV^*, the apply is AX=DX+U(V^*X). The diagonal blocks remain distributed. The r× q contraction is reduced and broadcast.
Exact mathematical passage
r\times q
RUse in source · 3 passages
Search aliases: R
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 317
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
A=L D_A S^{-1}, \qquad B=S D_B R.
\mathfrak RUse in source · 1 passage
Search aliases: R mathfrak
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 248
No dense preservation requirement. The architecture does not begin with a dense matrix and try to compress it after the fact. It begins with a typed operator. That operator may be represented as R={ · explicit array, block, block sparse, low rank, BDLR, · Kronecker, sparse chain, transform plan, packet graph}. A dense representation is useful when no better form exists or when a downstream user explicitly asks for every entry. It is not a state that the chip must preserve.
Exact mathematical passage
\boxed{ \begin{aligned} \mathfrak R=\{&\text{explicit array},\ \text{block},\ \text{block sparse},\ \text{low rank},\ \text{BDLR},\\ &\text{Kronecker},\ \text{sparse chain},\ \text{transform plan},\ \text{packet graph}\}. \end{aligned} }
rUse in source · 16 passages
Search aliases: r
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 538
Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.
Exact mathematical passage
r
\mathrm{representation}Use in source · 1 passage
Search aliases: representation mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
SUse in source · 5 passages
Search aliases: S
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 312
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
S
sUse in source · 2 passages
Search aliases: s
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 725
Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.
Exact mathematical passage
\varepsilon_e =\frac{\tau c_e}{s_e\sum_f c_f},
S^{-1}Use in source · 2 passages
Search aliases: S^-1
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 313
Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simultaneous diagonalization, symmetry sectors, and tensor factorizations are instances of the same rule.
Exact mathematical passage
AB=(AS)(S^{-1}B).
S_{\mathrm{amortization}}Use in source · 1 passage
Search aliases: S_amortization mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
S_{\mathrm{architecture}}Use in source · 1 passage
Search aliases: S_architecture mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
s_eUse in source · 2 passages
Search aliases: s_e
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 725
Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.
Exact mathematical passage
\varepsilon_e =\frac{\tau c_e}{s_e\sum_f c_f},
S_{\mathrm{kernel}}Use in source · 1 passage
Search aliases: S_kernel mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
S_{\mathrm{representation}}Use in source · 1 passage
Search aliases: S_representation mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
S_{\mathrm{total}}Use in source · 1 passage
Search aliases: S_total mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
\ScalUse in source · 3 passages
Search aliases: Scal S mathcal
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 518
Atlas descriptor. An atlas descriptor records an analysis graph, a synthesis graph, normalization data, and packet selectors. A unitary transform uses its adjoint for synthesis. A redundant exact frame records the canonical dual or another declared synthesis map satisfying S A=I.
Exact mathematical passage
\Scal\Acal=I.
TUse in source · 5 passages
Search aliases: T
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 210
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
T:X\to X
T_{\mathrm{graph}}Use in source · 1 passage
Search aliases: T_graph mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 1030
PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.
Exact mathematical passage
T= \max\left( \frac{F}{\eta_cP}, \frac{B}{\eta_bW} \right) +T_{\mathrm{graph}},
T_{\beta\alpha}Use in source · 2 passages
Search aliases: T_βα beta alpha
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 211
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
T_{\beta\alpha}=P_\beta T P_\alpha.
\mathrm{total}Use in source · 1 passage
Search aliases: total mathrm
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 975
Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_architecture S_kernel S_amortization. This is a bookkeeping identity only when every factor comes from matched rows with the same output contract and timin…
Exact mathematical passage
S_{\mathrm{total}} = S_{\mathrm{representation}} S_{\mathrm{architecture}} S_{\mathrm{kernel}} S_{\mathrm{amortization}}.
UUse in source · 14 passages
Search aliases: U
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 535
Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.
Exact mathematical passage
A=D+UV^*,
uUse in source · 1 passage
Search aliases: u
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
U^*Use in source · 3 passages
Search aliases: U^*
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 608
Exact analysis and synthesis. Let U be unitary and let E_α be coordinate selectors. The pulled-back projectors are P_α=U^*E_α U. The hardware analysis applies U and selects the coordinates. Synthesis scatters the selected coordinates and applies U^*.
Exact mathematical passage
P_\alpha=U^*E_\alpha U.
VUse in source · 11 passages
Search aliases: V
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 535
Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.
Exact mathematical passage
A=D+UV^*,
vUse in source · 8 passages
Search aliases: v
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
V^*Use in source · 10 passages
Search aliases: V^*
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 535
Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.
Exact mathematical passage
A=D+UV^*,
WUse in source · 3 passages
Search aliases: W
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 579
Local refinement. Suppose a parent packet task v is replaced by an acyclic child module H=(W,F). Predecessors of v connect to child entries. Child exits connect to successors of v.
Exact mathematical passage
H=(W,F)
\widehat AUse in source · 3 passages
Search aliases: widehat A widehat
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 701
[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.
Exact mathematical passage
A=\sum_{e\in E}A_e, \qquad \widehat A=\sum_{e\in E}(A_e+E_e).
XUse in source · 8 passages
Search aliases: X
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
xUse in source · 3 passages
Search aliases: x
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 623
For a complete selector family, ∑_α E_α=I. Hence S A x =U^*(∑_α E_α)Ux =U^*Ux =x. The incomplete case replaces the identity by the declared selector sum.
Exact mathematical passage
\Scal\Acal x =U^*\left(\sum_\alpha E_\alpha\right)Ux =U^*Ux =x.
X_\alphaUse in source · 1 passage
Search aliases: X_α alpha
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
yUse in source · 3 passages
Search aliases: y
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 956
[Certified commit] Suppose a command result is provisional until a certificate c(y) is evaluated. If the commit controller releases y only when c(y) satisfies the declared acceptance rule, then every visible result has passed that rule. A failed result cannot satisfy a downstream completion token.
Exact mathematical passage
c(y)
\timesUse in source · 21 passages
Search aliases: × times
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 132
The software basis already exists. On the recorded RTX 3070 Ti Laptop GPU, an equal-block packet operator was measured at 0.046032 ms and ran 14.8919× faster than its matched device-resident dense GPU row. A BDLR packet path with CUDA Graph replay was measured at 0.079872 ms and ran 10.9355× faster than the matched dense GPU row. Graph replay improved the same packet representation by about 1.218×. A compressed BDLR product took 0.1582 ms, while materializing the output took 0.5073 ms. These mea…
Exact mathematical passage
14.8919\times
\alphaUse in source · 12 passages
Search aliases: α alpha
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\betaUse in source · 4 passages
Search aliases: β beta
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\deltaUse in source · 1 passage
Search aliases: δ delta
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\delta_{\alpha\beta}Use in source · 1 passage
Search aliases: δ_αβ delta alpha beta
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\varepsilonUse in source · 1 passage
Search aliases: ε varepsilon
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 725
Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.
Exact mathematical passage
\varepsilon_e =\frac{\tau c_e}{s_e\sum_f c_f},
\varepsilon_eUse in source · 1 passage
Search aliases: ε_e varepsilon
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 725
Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.
Exact mathematical passage
\varepsilon_e =\frac{\tau c_e}{s_e\sum_f c_f},
\etaUse in source · 3 passages
Search aliases: η eta
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 1030
PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.
Exact mathematical passage
T= \max\left( \frac{F}{\eta_cP}, \frac{B}{\eta_bW} \right) +T_{\mathrm{graph}},
\eta_bUse in source · 2 passages
Search aliases: η_b eta
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 1030
PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.
Exact mathematical passage
T= \max\left( \frac{F}{\eta_cP}, \frac{B}{\eta_bW} \right) +T_{\mathrm{graph}},
\eta_cUse in source · 2 passages
Search aliases: η_c eta
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 1030
PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.
Exact mathematical passage
T= \max\left( \frac{F}{\eta_cP}, \frac{B}{\eta_bW} \right) +T_{\mathrm{graph}},
\lambdaUse in source · 1 passage
Search aliases: λ lambda
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
\muUse in source · 6 passages
Search aliases: μ mu
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 1055
Fused multi-atlas analysis · 14.73 mus · 8.96 mus
Exact mathematical passage
\mu
\toUse in source · 2 passages
Search aliases: → to
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 210
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
T:X\to X
\inUse in source · 4 passages
Search aliases: ∈ in
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\sumUse in source · 9 passages
Search aliases: ∑ sum
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\sum_eUse in source · 1 passage
Search aliases: ∑_e sum
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 717
Exact packet reconstruction gives (widehat A-A)x=∑_eE_eP_α(e)x. The triangle inequality gives the bound.
Exact mathematical passage
(\widehat A-A)x=\sum_eE_eP_{\alpha(e)}x.
\sum_{e\in E}Use in source · 2 passages
Search aliases: ∑_e∈ E sum in
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 701
[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.
Exact mathematical passage
A=\sum_{e\in E}A_e, \qquad \widehat A=\sum_{e\in E}(A_e+E_e).
\sum_fUse in source · 1 passage
Search aliases: ∑_f sum
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 725
Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.
Exact mathematical passage
\varepsilon_e =\frac{\tau c_e}{s_e\sum_f c_f},
\sum_{u\in\operatorname{child}(o)}Use in source · 1 passage
Search aliases: ∑_u∈child(o) sum in operatorname
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 339
For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.
Exact mathematical passage
J(v)=\min_{o\in\mathcal O_v} \left[ c(o)+\lambda e(o)^2+ \sum_{u\in\operatorname{child}(o)}J(u) \right].
\sum_\alphaUse in source · 2 passages
Search aliases: ∑_α sum alpha
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 622
For a complete selector family, ∑_α E_α=I. Hence S A x =U^*(∑_α E_α)Ux =U^*Ux =x. The incomplete case replaces the identity by the declared selector sum.
Exact mathematical passage
\sum_\alpha E_\alpha=I
\sum_{\alpha,\beta}Use in source · 1 passage
Search aliases: ∑_α,β sum alpha beta
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 215
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
\boxed{T=\sum_{\alpha,\beta}T_{\beta\alpha}.}
\sum_{\alpha\in I}Use in source · 1 passage
Search aliases: ∑_α∈ I sum alpha in
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\approxUse in source · 1 passage
Search aliases: ≈ approx
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 433
Modeled arithmetic peaks. With 256 execution groups and one 16×16 matrix array per group, the model yields FP16 or BF16 · ≈ 196.6 TFLOP/s, FP32 · ≈ 98.3 TFLOP/s, FP64 · ≈ 24.6 TFLOP/s. These are model peaks. The design is not organized around a claim of superior generic dense throughput. Its purpose is to execute fewer operations and move fewer bytes when the selected representation proves that the explicit array is unnecessary.
Exact mathematical passage
\boxed{ \begin{aligned} \text{FP16 or BF16} &\approx 196.6\text{ TFLOP/s},\\ \text{FP32} &\approx 98.3\text{ TFLOP/s},\\ \text{FP64} &\approx 24.6\text{ TFLOP/s}. \end{aligned}}
\leUse in source · 1 passage
Search aliases: ≤ le
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 707
[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.
Exact mathematical passage
\boxed{ \norm{(\widehat A-A)x} \le \sum_{e\in E}\norm{E_e}\norm{P_{\alpha(e)}x}.}
\bigoplusUse in source · 1 passage
Search aliases: ⊕ bigoplus
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\bigoplus_{\alpha\in I}Use in source · 1 passage
Search aliases: ⊕_α∈ I bigoplus alpha in
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 203
Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities are part of the problem. PRISM-1 carries them into the execution model instead of asking software to rediscover them after every kernel boundary.
Exact mathematical passage
X=\bigoplus_{\alpha\in I}X_\alpha, \qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\cdotsUse in source · 1 passage
Search aliases: ⋯ cdots
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 330
Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array represented by the chain.
Exact mathematical passage
A=F_LF_{L-1}\cdots F_1
\longrightarrowUse in source · 2 passages
Search aliases: ⟶ longrightarrow
Use in source · Komorebi_PRISM_1_Final_Updated.tex, line 266
CUDA-like execution hierarchy. CUDA made heterogeneous arithmetic programmable through a hierarchy of lanes, warps, blocks, grids, and memories. PRISM uses an analogous execution hierarchy while retaining the mathematical structure that CUDA kernels normally receive only through software conventions: lane ⟶ packet warp ⟶ execution group ⟶ Packet Streaming Multiprocessor ⟶ packet graph. A packet warp may invoke a matrix group, vector lanes, a transform engine, a small solver, exact arithmetic, or…
Exact mathematical passage
\boxed{ \text{lane} \longrightarrow \text{packet warp} \longrightarrow \text{execution group} \longrightarrow \text{Packet Streaming Multiprocessor} \longrightarrow \text{packet graph}. }
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