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mathematics
tensor-tomography
inverse-problems
positive-sensing
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| {"id": "kernel-spectrum", "title": "Marginal kernel and spectrum", "statement": "The coordinate-line sum operator has rank N-D and kernel equal to the tensor product of coordinatewise zero-sum subspaces.", "assumptions": "Finite product alphabet, q_i >= 2; unnormalized coordinate-line sum operator.", "formula": "N=product(q_i); D=product(q_i-1); rank(A)=N-D", "source_file": "research/manuscript/theory.tex", "source_label": "thm:kernel", "verification_module": "test_separable_completion", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "Normalization affects the nonzero Gram spectrum."} | |
| {"id": "all-temperatures", "title": "Complete linewise information", "statement": "All unmasked temperatures determine exactly each coordinate line score multiset; trades can remain invisible.", "assumptions": "Fixed finite real score field; exact coordinate-line observations over all positive temperatures.", "formula": "All-temperature equality iff linewise multiset equality", "source_file": "research/manuscript/theory.tex", "source_label": "thm:multisets", "verification_module": "test_trades", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "No coordinate ordering within a line is recovered from its multiset alone."} | |
| {"id": "smallest-edit", "title": "Sharp smallest invisible change", "statement": "Distinct all-temperature-indistinguishable fields differ at at least 2^n sites; equality has a rectangular two-value checkerboard exchange.", "assumptions": "Product of n finite alphabets of size at least two; exact all-temperature data.", "formula": "minimum Hamming distance = 2^n", "source_file": "research/manuscript/theory.tex", "source_label": "thm:edit", "verification_module": "test_trades", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "Product-code and trade antecedents are attributed in the manuscript."} | |
| {"id": "multiplicity", "title": "Sharp score-multiplicity rigidity", "statement": "A score value occurring fewer than 2^(n-1) times has its positions fixed by exact linewise multiset data.", "assumptions": "Exact repeated real scores on an n-coordinate finite product; complete linewise multisets.", "formula": "multiplicity < 2^(n-1)", "source_file": "research/manuscript/theory.tex", "source_label": "thm:multiplicity", "verification_module": "test_trades", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "Threshold is sharp; stability under approximate score repetition is not asserted."} | |
| {"id": "positive-channels", "title": "Optimal positive-channel completion", "statement": "Strictly positive two-level masks provide universal recovery with the optimal total channel count and exactly N selected scalar responses.", "assumptions": "Calibrated linear exponential-domain sensing of a fixed positive field; arbitrary mask addressing allowed.", "formula": "m_star=ceil(N/(N-D)); total selected scalar responses=N", "source_file": "research/manuscript/completion.tex", "source_label": "thm:channels", "verification_module": "test_coded_channels", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "Counts include a baseline when used; masks may require O(N) addressing/storage."} | |
| {"id": "separable-channels", "title": "Separable-channel nullity", "statement": "The exact joint nullity of separable masked coordinate-line observations is the product of the coordinate factor-span codimensions.", "assumptions": "Strictly positive separable calibrated masks; r_i is the rank of all coordinate-i factor vectors.", "formula": "joint_nullity=product(q_i-r_i); generic channel threshold=min(q_i)", "source_file": "research/manuscript/completion.tex", "source_label": "thm:separable", "verification_module": "test_separable_completion", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "Generic threshold requires full expected coordinate factor ranks."} | |
| {"id": "scalar-completion", "title": "Optimal scalar completion", "statement": "Exactly D additional scalar linear probes after baseline are necessary and sufficient; positive probes can be constructed.", "assumptions": "Full baseline coordinate-line data already available; arbitrary linear probes of a positive field.", "formula": "additional scalar probes=D=product(q_i-1)", "source_file": "research/manuscript/completion.tex", "source_label": "thm:scalar", "verification_module": "test_separable_completion", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "Additional scalar count differs from total channel and selected-response budgets."} | |
| {"id": "fourier-completion", "title": "Bounded Fourier completion", "statement": "Real tensor Fourier contrasts shifted by a positive constant provide bounded positive completion masks.", "assumptions": "Finite product alphabet; complete baseline; correctly normalized real orthogonal Fourier contrast basis.", "formula": "D positive completion probes", "source_file": "research/manuscript/completion.tex", "source_label": "prop:fourier", "verification_module": "test_separable_completion", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "Noise certificates depend on contrast normalization and shared baseline mass error."} | |
| {"id": "one-row-stability", "title": "Exact nonnegative one-row optimum", "statement": "A parity-class indicator attains the optimal smallest Gram eigenvalue for one nonnegative unit-energy extra row.", "assumptions": "Binary n-cube; full unscaled edge-sum baseline; squared Euclidean row energy one.", "formula": "mu_n=(2*n+1-sqrt(4*n*n+1))/2", "source_file": "research/manuscript/stability.tex", "source_label": "thm:positiveopt", "verification_module": "test_positive_design", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "Attaining rows have zero entries; strictly positive rows can approach but do not attain the optimum."} | |
| {"id": "rank-energy", "title": "Positive rank-energy bound", "statement": "Total probe energy weighted by nonnegative coordinatewise tensor rank bounds the augmented smallest Gram eigenvalue.", "assumptions": "Binary n-cube; H0 is the full unscaled edge-sum baseline Gram matrix; nonnegative probe rows b_j; R_j is a nonnegative coordinatewise CP rank bound.", "formula": "lambda_min(H0+sum_j b_j*b_j^T) <= 2^(-n)*sum_j R_j*norm(b_j)^2", "source_file": "research/manuscript/stability.tex", "source_label": "thm:rankenergy", "verification_module": "test_positive_design", "status": "Manuscript proof supplied; finite checks PASS; external review pending", "limitations": "The inequality does not identify runtime or practical acquisition cost with row energy."} | |