Jcost_unit0
plain-language theorem explainer
The lemma establishes that the recognition cost J vanishes at its fixed point of unity. Acoustics and action-principle derivations cite it to set the zero baseline for matched signals or constant paths. The proof is a one-line simplification that unfolds the definition of Jcost.
Claim. $J(1) = 0$ where $J(x) = (x + x^{-1})/2 - 1$ for real $x > 0$.
background
Jcost is defined by J(x) = (x + 1/x)/2 - 1, matching the T5 uniqueness form cosh(log x) - 1 that satisfies the Recognition Composition Law. This lemma sits in the Cost module and supplies the zero value at the self-similar fixed point. It relies on the core definition and the continuum bridge identification of the Laplacian action as (1/2) sum w_ij (ε_i - ε_j)^2.
proof idea
One-line wrapper that applies simp to unfold the definition of Jcost and evaluate at 1.
why it matters
This anchors the cost minimum in the Recognition framework and feeds directly into pitchCost_at_unison, hearingLossPenalty_zero, srCost_zero_at_threshold, actionJ_const_one, Jcost_taylor_quadratic, and pleasure_max_at_one. It realizes J-uniqueness at x=1, enabling the phi-ladder, eight-tick octave, and downstream quadratic limits.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.
papers checked against this theorem (showing 23 of 23)
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Generative models stay exactly on particle phase space manifold
"the 'pure noise' forward process endpoint corresponds to the uniform distribution on phase space"
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Known operators shrink risk bounds and scale down sample needs
"The estimator decomposes the total risk into a sum over learned layers; every known operator contributes zero to this sum... Replacing learned layer m by a known operator removes the term Am E∥em∥22 from (4)."
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Density triggers conductor-insulator crossover in ultracold plasmas
"F(r,ε) = A_F exp{−ε/(kBT_vir)} / sqrt(ε − U_eff(r))"
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Mutual information beats entropy in benchmark selection
"f1(S) = H(X_S) = ½ log det(2πe Σ_SS) ... maximizing log det(Σ_AA), since additive constants do not affect the argmax."
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Segmentation masking improves sign language AI accuracy
"L = (1/Ω) Σ |I(p) − Î(p)|² ... mean squared error loss based only on masked patches"
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Two quantum Wasserstein distances coincide on qubits with one cost operator
"DDPT(ϱ,ϱ)² = Σ Iϱ(Hn), where Iϱ(H) = Tr(H²ϱ) − Tr(H√ϱ H√ϱ) is the Wigner–Yanase skew information."
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Morphing wings cut control cost 65% in obstacle avoidance
"we adopt a physics-based cost model that directly computes actuation effort from aerodynamic loading using the virtual work principle"
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MARL controller lifts 5G cell throughput with small fairness cost
"a PRB agent learns user-level resource shares ... and a power agent distributes the base-station power budget ... a fairness-aware reward based on smoothed throughput and Jain's fairness index"
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VLM judge reproduces human video rankings with perfect match
"PHAS(m) = (1/|P|) Σ_p Σ_d w_d s_{m,p,d} / Σ_d w_d × λ(m,p), where w_d are calibrated by non-negative ridge logistic regression on human preference annotations."
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Flow-matching super-resolves ground astro images without fakes
"W(k) = H*(k) / (|H(k)|² + λ_SNR⁻¹) ... Wiener-regularized approximate adjoint"
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Run-and-tumble motion keeps chemotaxis robust to noise
"Weber's law / log-sensing regime where v_drift ∼ g independent of c"
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Supervision preferences and confrontation control rumor spread
"c_i(t) = ξ/(1+e^{-d_{i→r}(t)}) ... ĉ_i(t) = min{c_i(t),1}"
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DINOv3 segments remote sensing images without RS fine-tuning
"CAT-Seg [8] introduced cost aggregation for OVSS by framing similarity scores between CLIP text and image embeddings as cost maps. ... The cost aggregation refines the raw cost maps into per-class probability maps."
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Fidelity stays exactly 1 when coupling tuned to -1 in four-qubit chain
"F = |cos(φ/2)| ... C_l1 = sin²(φ/2) ... freezing at α = −1"
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Reward centering now works for episodic RL
"Φ(s) := b/(1−γ) ... F(s,a,s') = γ·b/(1−γ) − b/(1−γ) = −b"
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Hybrids make image restoration up to 3.4x faster on edge CPUs
"We minimize the loss L_distill = |O_S − O_T|_2^2 to make MambaIR blocks serve as lightweight surrogates."
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Encoder models beat larger decoders on noisy medical report QA
"ϕ(u, v) = 1 − d_lev(u, v)/max(|u|,|v|)"
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BerLU smooths activations with Bernstein polynomials
"BerLU(x) = αx for x<-ε; (1-α)/(4ε) x² + (1+α)/2 x + (1-α)ε/4 for -ε≤x≤ε; x for x>ε"
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One measurement stabilizes bistable qubit frequency
"the optimal probing time τ_opt ≈ 1/(2 Δ_TLS) ... maximizes sensitivity to the TLS-induced frequency shift"
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Unanswerable math prompts deviate in LLM geometry before output
"we extract last-layer hidden states, apply mean pooling over all input tokens, and subtract the global mean vector... All distances are cosine distances (1−cos θ)."
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Mirror descent gives O(1/n) rate for convex optimal control
"Take h(u)=½|u|². Then D_h(v|u)=½|v−u|². Hence the mirror step becomes... Euclidean projected gradient descent is the special case of mirror descent associated with the quadratic mirror map h(u)=½|u|²."
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Recurrent RL hidden states match optimal control co-states
"Assuming a standard continuous action space and a quadratic control penalty c(u) = u⊤Ru ... yields a closed-form optimal control law u⋆ = -½ R⁻¹ G_θ(y) h⋆."
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Exact entropy and temperature for any number of equispaced levels
"Ω_p(E,N) = [x^q] ((1−x^p)/(1−x))^N ... the contour-integral form and asymptotic evaluation follow standard methods of coefficient extraction and saddle-point analysis"