{"id":"9727f138-6106-4acd-840c-bebecb16a694","arxiv_id":"2608.02707","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A nonzero balanced value in a function satisfying complete oriented-simplex cancellation forces support at least c_n R^{ceil(n/2)}, an exponent shown optimal by explicit examples.","lead":"This paper proves a sharp lower bound: a nonzero value at the balanced point of a lattice simplex forces at least c_n R^{ceil(n/2)} nonzero values in a function satisfying complete oriented-simplex cancellation relations. The result settles the optimal growth exponent for a discrete unique-continuation problem with roots in Anderson-Bernoulli localization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the tensorized Pascal uncertainty lemma and the shell-summing argument are internally sound.","rationale":"I read the manuscript in good faith, focusing on the central claim: a nonzero balanced coefficient forces support of order R^{ceil(n/2)} on the simplex, with matching constructions. The factorial normalization (Lemma 3.1) is correct, and the one-variable Pascal uncertainty principle (Lemma 4.2) is fully justified by the rank argument in Lemma 4.1. The tensorized Lemma 5.1 is the key mechanism; I checked the induction carefully. The expansion into fibers phi_{epsilon',omega}(y) is exact, the coefficient condition (8) ensures the relevant fibers have degree at least kappa_d, and disjoint x'-monomial fibers make the support counts additive. No hidden merging or cancellation occurs. Proposition 6.1 correctly reconstructs H from a facet, extracts a deep monomial, applies Lemma 5.1, bounds intermediate shifts by the two endpoint shifts via the Hamming distance, and identifies the endpoints with genuine coordinate-facet supports; homogeneity guarantees injectivity of the monomial maps. The shell summation in Section 7 is exact because differentiation subtracts t1_n injectively and the shells are disjoint and exhaustive. The optimality constructions in Proposition 8.1 are explicit, have the correct degree, are annihilated by D, have nonzero balanced coefficients, and achieve the claimed support order. The paper also honestly states that the simplex relations are not identified with an arbitrary discrete Schrodinger equation; that is a scope limitation, not a soundness issue. I therefore find no significant objection and would keep the reader's ACCEPT verdict unchanged.","tokens_in":12834,"tokens_out":20940,"duration_ms":187075,"concrete_test":"Run an independent randomized or exhaustive check of Lemma 5.1 for d = 2 and d = 3: generate sparse polynomials p with a selected nonzero coefficient [x^kappa]p and compute the left side of (7); a single violation would refute the lemma, while agreement across many trials would corroborate the induction and the derived constants.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the proof line by line, I find no load-bearing concern. The reader's flagged weakest assumption, Lemma 5.1, is sound: the induction separates the last variable, and for each omega in supp(p_kappa_d(x'+epsilon')) the fiber phi_{epsilon',omega}(y) has degree at least kappa_d, so Lemma 4.2 applies fiberwise; distinct x'-monomial fibers make support counts additive, and the induction closes with the coefficient condition (8). Proposition 6.1's endpoint identifications are also valid: homogeneity determines the omitted exponent, so dehomogenization and the substitution v_n = -1 are injective on monomial supports, preventing merging or cancellation. The Hamming-distance compression W_S <= Lambda_r^ell (W_empty + W_[d]) is a valid upper bound, and the shell decomposition bwt(F_t) = shell_t(F) is exact. The optimality constructions are explicit and their annihilation by D is correct. The only scope caveat is the paper's own statement that the simplex system is not identified with an arbitrary discrete Schrodinger equation; this limits applicability but not the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies functions g on the lattice simplex Δ^{(n)}_{nR}, n≥2, R≥1, satisfying the complete oriented-simplex relations Σ_{i=1}^n g(β+e_i)=0 for all β of degree nR−1. Its main result (Theorem 2.1) states that if the balanced value g(R,...,R) is nonzero, then |supp g| ≥ c_n R^{⌈n/2⌉}, with an explicit constant for n≥3 and the stronger exact statement |supp g|=2R+1 for n=2; it also constructs examples showing that the exponent is optimal. The proof normalizes g to a homogeneous polynomial F in ker D, proves a two-shift Pascal uncertainty bound via a rank lemma established by the Lindström–Gessel–Viennot lemma (Lemmas 4.1–4.2), tensorizes it (Lemma 5.1), converts it into a two-facet monomial boundary estimate (Proposition 6.1), and sums this estimate over disjoint derivative shells (Section 7).","tokens_in":13080,"tokens_out":6861,"duration_ms":66869,"significance":"If correct, the paper gives a self-contained discrete unique-continuation inequality with the optimal exponent for all n, extending the n=3 triangular estimate of Li–Zhang to arbitrary dimension by a purely algebraic argument. The proof is modular, gives explicit constants, and is accompanied by explicit optimality constructions. The LGV-based rank lemma and the tensorized Pascal uncertainty principle are natural and generalize cleanly. I also note the paper's honest scope statement that the simplex system is not identified with an arbitrary discrete Schrödinger equation; this limits immediate physical application but does not affect the validity of the theorem. The self-cited localization results are used for context and for the n=3 remark, not as input to the general proof, so the argument is not circular.","major_comments":[],"minor_comments":[{"comment":"The symbol ℓ_d is typeset ambiguously; it should be declared explicitly as floor(d/2) in both the detailed proof overview and in Proposition 6.1, so that θ=d−ℓ_d equals ceil(d/2) and the final exponent is immediately unambiguous.","section":"§2 detailed overview and §6 Proposition 6.1"},{"comment":"The disclosure of the AI tool in the abstract is unusual for a mathematical paper; consider moving this information to the acknowledgments or to a reproducibility statement so that it does not distract from the mathematical content.","section":"Abstract and Acknowledgments"},{"comment":"Several references appear with future or very recent dates (for example [22], [29], [20]); please update publication statuses if they have appeared in final form.","section":"References"}],"recommendation":"accept","confidential_remarks":"This is a strong, self-contained paper with no load-bearing concerns identified. The main proof is carefully structured and the technical lemmas check out. The only editorial considerations are the AI-assistance disclosure in the abstract and the large amount of surrounding localization literature, neither of which affects correctness."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: this is a correct, sharp result about a very specific discrete system, not a step toward Anderson localization. The theorem is cleanly stated, the proof is self-contained, and the exponent is optimal in every dimension. If you read it, you can trust the proofs.\n\nWhat is new: the general dimension n>=4, with matching constructions and the tensorized Pascal uncertainty principle. The n=3 case is the Li-Zhang triangle estimate via an explicit affine map, n=2 is an alternating recurrence; the paper says both. The main theorem is genuinely new for n>=4, and the proof is algebraic and exact.\n\nWhat the paper does well: the normalization trick (simplex relations become DF=0) turns the problem into a monomial-support question. The Pascal submatrix rank lemma is proved via LGV paths, the tensorization induction is clean and the fiber decomposition works. The shell summation is correct with disjointness properly argued. I checked the one delicate identification, W_[d] with the facet z_{n-1}=0; homogeneity ensures the exponent map is injective, so no merging or cancellation. The explicit optimality constructions pair variables for even n and use a three-variable block for odd n; they are valid.\n\nSoft spots: the scope is narrow. The system is not embedded in any discrete Schrodinger operator, and the paper openly says an application would require an additional geometric embedding. The Anderson-Bernoulli discussion is motivation, not a theorem. The constant c_n is explicit but not optimized and decays in n; that is a side issue. This is not a breakthrough for localization, but it is a solid standalone contribution with a reusable uncertainty principle.\n\nThe citation pattern looks fine: the self-cited work is used for motivation and the n=3 remark, not as input to the main proof. No circularity.\n\nWho this is for: people working on discrete unique continuation, uncertainty principles, or combinatorial aspects of discrete Schrodinger equations. A specialist journal referee would find it valuable; a general math-ph reader should not expect applications.\n\nRecommendation: send to peer review. It deserves a serious referee despite the narrow scope.","headline":"A clean, self-contained optimal support bound for a simplex unique continuation system; the proof is sound and the scope is honestly narrow.","tokens_in":13563,"tokens_out":2065,"would_cite":true,"duration_ms":19524,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single nonzero value at the center of a lattice simplex forces a support lower bound of order $R^{\\lceil n/2\\rceil}$, and this growth rate is optimal.","keywords":["discrete unique continuation","lattice simplex","support cardinality","Pascal uncertainty principle","factorial normalization","optimal exponent","oriented simplex relations","random Schrödinger localization"],"falsifier":"A direct check is to search for a two-variable polynomial $p$ with $[xy]p\\neq 0$ whose four unit-shifted versions have total support size below $9$; Lemma 5.1 with $\\kappa=(1,1)$ requires at least $9$. A counterexample would disprove the key lemma, and a failure there would remove the foundation of the main theorem's proof.","tokens_in":12666,"feed_emoji":"🔺","tokens_out":9473,"duration_ms":79384,"temperature":0.7,"pith_summary":"This paper proves a discrete analog of unique continuation on the integer points of an $n$-dimensional simplex of side length $R$. If a function on the simplex satisfies a complete system of $n$-term oriented-simplex relations and takes a nonzero value at the balanced center, then its support must contain at least a constant times $R^{\\lceil n/2\\rceil}$ points, where the constant depends only on $n$. Explicit constructions show that the exponent cannot be improved, so the bound is optimal. This matters because support-cardinality lower bounds are the kind of geometric input used in proofs of localization for random Schrödinger operators on lattices, where literal unique continuation fails.","feed_headline":"Nonzero center forces order R^(n/2) lattice points","feed_subtitle":"A single nonzero value at a simplex center forces a support bound that cannot be improved, via Pascal's triangle.","key_machinery":"The load-bearing mechanism is the tensorized Pascal uncertainty principle: if a polynomial $p$ in $d$ variables has a nonzero coefficient for the monomial $x^\\kappa$, then the total number of supported monomials across all $2^d$ partial unit shifts is at least $\\prod_i(\\kappa_i+2)$. It is built from a one-variable two-shift bound obtained from the rank of Pascal submatrices, and it converts a single deep monomial on a coordinate facet into many boundary monomials on pairs of coordinate facets. Two additional mechanisms carry the full argument: factorial normalization, which turns the simplex relations into the single equation $DF=0$ for $D=\\partial_{z_1}+\\cdots+\\partial_{z_n}$ and gives translation invariance along the diagonal, and a shell decomposition by simultaneous derivatives in all variables, which sums the fixed-scale boundary estimate over disjoint shells to produce the final exponent.","core_discovery":"The central claim is Theorem 2.1: for integers $n\\ge 2$ and $R\\ge 1$, every function $g$ on $\\Delta_{nR}^{(n)}$ satisfying $\\sum_{i=1}^n g(\\beta+e_i)=0$ for all $\\beta\\in\\Delta_{nR-1}^{(n)}$ and with $g(R,\\dots,R)\\neq 0$ must have $|\\operatorname{supp} g|\\ge c_n R^{\\lceil n/2\\rceil}$. The proof gives an explicit constant for $n\\ge 3$, and for $n=2$ it gives the exact value $|\\operatorname{supp} g|=2R+1$. The paper also proves optimality by constructing explicit products of coordinate differences with nonzero balanced coefficient whose support has size of order $R^{\\lceil n/2\\rceil}$.","pith_inferences":["Editorial inference: the same tensorized Pascal mechanism is likely to give support bounds for monomials assumed deep at interior points other than the balanced center, with the bound depending on the minimum exponent.","Editorial inference: the factorial normalization and shell summation might transfer to other finite difference relations whose characteristic variety is a single direction, yielding sharp support exponents for related lattice systems.","Editorial inference: because the proof uses only monomial counting, it may extend to positive-weight support measures, not just cardinality, if the one-variable two-shift inequality holds with the appropriate weights.","Editorial inference: a direct computational search for small $R$ in the three-variable case could test how close true minimal supports come to the explicit constant, a comparison not carried out in the paper."],"forward_implications":["For every fixed dimension $n$, nonvanishing at the balanced point forces at least $c_n R^{\\lceil n/2\\rceil}$ nonzero values, with an explicit constant for $n\\ge 3$.","The exponent $\\lceil n/2\\rceil$ is sharp: there are admissible functions whose support is only of order $R^{\\lceil n/2\\rceil}$.","The $n=2$ case is exactly $2R+1$: every point of the simplex is forced to be nonzero.","The $n=3$ case recovers the quadratic triangular-lattice support bound previously used as a discrete unique-continuation input for random Schrödinger localization.","The boundary estimate that feeds the proof already shows that two specified coordinate facets together contain many forced monomials, before shells are summed."],"supporting_citations":[{"why":"Supplies the three-variable triangular-lattice unique continuation result that the paper's affine bijection identifies with the $n=3$ case.","marker":"[23]"},{"why":"Defines the support-cardinality problem for discrete Schrödinger equations and motivates the sharp-exponent question addressed here.","marker":"[22]"},{"why":"Provides the whole-space support-dimension result for discrete Schrödinger solutions with which the simplex bound is compared.","marker":"[16]"},{"why":"Supplies the lattice-path determinant lemma used to prove the Pascal submatrix rank lemma behind the one-variable uncertainty bound.","marker":"[24]"},{"why":"Companion path-counting result used with the previous reference to identify the rank of Pascal submatrices as a matching number.","marker":"[14]"}],"fun_headline_variants":["Discrete unique continuation: one value forces R^(n/2) points","Simplex center nonzero implies optimal support lower bound","Optimal support bound from a single lattice point","Pascal uncertainty principle yields sharp simplex bound","Discrete simplex: one nonzero center forces many nonzeros"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire proof rests on the tensorized Pascal uncertainty principle: a single nonzero monomial coefficient forces the total monomial count across all partial unit shifts to be at least the product of the shifted exponents plus two, and if that induction step failed, the boundary lower bound and the final exponent would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Discrete unique continuation: one value forces R^(n/2) points","Simplex center nonzero implies optimal support lower bound","Optimal support bound from a single lattice point","Pascal uncertainty principle yields sharp simplex bound","Discrete simplex: one nonzero center forces many nonzeros"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001275,"raw_usage":{"total_tokens":5260,"prompt_tokens":1034,"completion_tokens":4226,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":650,"completion_tokens_details":{"reasoning_tokens":4148}},"tokens_in":650,"tokens_out":4226,"duration_ms":28924,"temperature":1.0,"reasoning_tokens":4148,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:02:30.515184+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to search for a two-variable polynomial $p$ with $[xy]p\\neq 0$ whose four unit-shifted versions have total support size below $9$; Lemma 5.1 with $\\kappa=(1,1)$ requires at least $9$. A counterexample would disprove the key lemma, and a failure there would remove the foundation of the main theorem's proof.","supporting_citations":[{"cited_title":"Li and L","cited_arxiv_id":null,"evidence_quote":"Supplies the three-variable triangular-lattice unique continuation result that the paper's affine bijection identifies with the $n=3$ case."},{"cited_title":"Li,On support cardinality for the discrete Schr¨ odinger equation, Lett","cited_arxiv_id":null,"evidence_quote":"Defines the support-cardinality problem for discrete Schrödinger equations and motivates the sharp-exponent question addressed here."},{"cited_title":"Lindstr¨ om,On the vector representations of induced matroids, Bull","cited_arxiv_id":null,"evidence_quote":"Supplies the lattice-path determinant lemma used to prove the Pascal submatrix rank lemma behind the one-variable uncertainty bound."},{"cited_title":"Gessel and G","cited_arxiv_id":null,"evidence_quote":"Companion path-counting result used with the previous reference to identify the rank of Pascal submatrices as a matching number."}],"review_version":2}