diff 1df3716..2d5618e qma_coulomb_hardness_v2.txt
diff --git a/qma_coulomb_hardness_v2.txt b/qma_coulomb_hardness_v2.txt
index 72a9383..67d9c8a 100644
--- a/qma_coulomb_hardness_v2.txt
+++ b/qma_coulomb_hardness_v2.txt
@@ -1,30 +1,12 @@
-Conventions. Atomic units throughout. |A| denotes volume of a set A.
-"p(n,D)" means some polynomial in the named variables whose coefficients
-and exponents are absolute constants; it may be enlarged from line to
-line. E(H) is the smallest eigenvalue (bottom of the spectrum) of a
-self-adjoint operator H. R is the real line and R^3 is three-dimensional
-space.
-
-A single electron near a fixed proton has a lowest energy, its ground
-energy. The same holds for any molecule: fix the nuclei, keep them rigid,
-and ask for the lowest energy the electrons can take. That energy is a
-single real number. Exact computation is impossible in general, and even
-deciding whether it lies below a threshold is extraordinarily hard.
-
-Qma is the class of decision problems whose YES answers admit a quantumly
-checkable proof of size polynomial in the input. It contains NP: a
-classical proof is a quantum proof of a special form. No efficient
-classical or quantum algorithm is known for QMA-hard problems, in the same
-sense none is known for NP-hard problems.
-
-Aharonov, D. and Naveh, T., "Quantum NP - a survey", arXiv:quant-ph/0210077.
-
-Kitaev, A. Yu., Shen, A. H., and Vyalyi, M. N., "Classical and Quantum
-Computation", Graduate Studies in Mathematics 47, AMS, 2002.
-
-The framing that physical simulation is a natural quantum-computational
-task goes back to Feynman (1982) and Lloyd (1996); QMA-hardness asks
-when even this task remains intractable.
+Conventions. Atomic units throughout. |A| is the volume of A. p(n,D) is some polynomial in the named variables with absolute constants; it can be enlarged from line to line. E(H) is the bottom of the spectrum of H.
+
+A single electron near a proton has a lowest energy; any fixed nuclear configuration gives one real number whose exact computation is hard, and even its threshold decision is extraordinarily hard.
+
+QMA is the class of decision problems whose YES instances admit a polynomial-size quantum proof. It contains NP; no efficient algorithm is known for QMA-hard problems.
+
+Aharonov-Naveh 2002, arXiv:quant-ph/0210077. Kitaev-Shen-Vyalyi 2002, AMS Grad Studies 47.
+
+Feynman (1982) and Lloyd (1996) framed physical simulation as a natural quantum-computational task; QMA-hardness is the case where even this remains intractable.
Feynman, R. P., "Simulating physics with computers", Int. J. Theor. Phys.
21 (1982), 467-488. https://doi.org/10.1007/BF02650179
@@ -141,133 +123,15 @@ obeying some rule (the promise). The inputs are described exactly here.
Answer
YES if E_0 <= a. NO if E_0 >= b.
- (Because b-a >= 1/L, these never conflict; one may not care about
- inputs sitting strictly between a and b, but the promise says they do
- not happen.)
-
-Note the promise "b - a >= 1/L" is a statement about how precisely the
-decision is posed; it is not a claim that the molecule has a spectral gap
-between its ground state and first excited state. These are different
-things. The theorem makes no assumption about excitation gaps, binding,
-neutrality, or existence of a ground eigenvector.
-
- +---------------------+-----------------------------------------------+
- | Input / convention | Specification |
- +---------------------+-----------------------------------------------+
- | Nuclear positions | pairwise distinct rational triples R_alpha |
- | Nuclear charges | Z_alpha = 1 for every alpha |
- | Electron number | N >= 1, encoded in unary |
- | Thresholds | rationals a < b |
- | Encoding | signed binary num / positive binary den, low |
- | Input length L | total bit length of the data above |
- | Promise | b - a >= 1/L |
- | Energy convention | electronic energy; nuclear repulsion omitted |
- +---------------------+-----------------------------------------------+
-
-Main theorem.
- The promise problem above is QMA-hard under deterministic classical
- polynomial-time many-one reductions. The reduction produces:
- - polynomially many nuclei (all charge one) at distinct rational
- positions;
- - polynomially many electrons (given in unary);
- - rational thresholds whose bit lengths are polynomial in the source
- length;
- - threshold separation b - a >= 1.
- The only outputs are the nuclear positions, the electron number N, and
- the thresholds.
-
-Consequences stated in the theorem: the absolute charge bound is
-Z_max = 1 and the precision exponent (in the sense of the bounded-charge
-input model) is c = 1. The theorem claims hardness only; membership in QMA
-is not asserted. The strong output gap "b - a >= 1" is stronger than the
-required input gap "b - a >= 1/L", and neither says anything about an
-excitation gap of the molecule.
-
-Why the source promise survives the extra structure. The reduction is a
-chain of energy maps, each correct to within a fraction of the source gap
-gamma (the final assembly). Because gamma is an inverse polynomial in the source
-length, each error is comfortably smaller than the gap, so YES stays YES
-and NO stays NO. The output gap is expanded by the physical scale lambda
-giving b - a >= 1.
-
-The construction builds a single molecule per source spin problem, so that the molecule's
-ground energy equals (up to a known, computable affine map) the ground
-energy of a hard spin Hamiltonian. Six steps:
-
- (S) Source. Start from the field-free Heisenberg energy problem on a
- finite square-lattice graph of maximum degree bounded and with
- signed couplings, which is QMA-hard (Cubitt-Montanaro-Piddock). It
- has polynomially bounded weights and an inverse-polynomial gap.
- Normalize to rational data with a named gap gamma (the source normalization).
-
-Heisenberg, W., "Zur Theorie des Ferromagnetismus", Zs. Phys. 49 (1928),
- 619-636. https://doi.org/10.1007/BF01328601
-
-Kempe, J., Kitaev, A., and Regev, O., "The complexity of the local
-Hamiltonian problem", SIAM J. Comput. 35 (2006), 1070-1097.
-arXiv:quant-ph/0406180
-
- (P) Positive spin. Replace every signed coupling by positive exchange
- using fresh "singlet-pair" mediators. Each original edge becomes a
- central bond plus two bent nine-link paths. Arrange the spins on
- a plane so that the link lengths can be tuned one at a time (the next
- two steps). Result: a positive-exchange Hamiltonian H_+ = sum K_e
- sigma_i . sigma_j with polynomially bounded K_e.
-
- (W) Wells and superexchange. Put one localized spatial orbital (two spin
- states) on each site. With one electron per site, an occupancy
- penalty is positive away from single occupancy. Virtual hopping of an
- electron onto a neighbor and back produces an effective positive spin
- exchange of size t^2/(U - v) (Anderson superexchange). Tune each
- hopping t_e to give the desired K_e. Also keep the direct
- electron-electron Coulomb coefficients v_{ij} exactly, and cancel the
- resulting linear-in-occupancy shift by a one-body counterterm built
- from slightly deepened wells (the well construction). Result: a finite fermion
- Hamiltonian H_d whose ground energy is H_+ up to scale tau^2 and a
- known scalar.
-
-Anderson, P. W., "New approach to the theory of superexchange interactions",
- Phys. Rev. 115 (1959), 2-13. https://doi.org/10.1103/PhysRev.115.2
-
- (D) Charge density. Realize the wells and the transverse confinement as
- the Coulomb field of an honest nonnegative continuous charge density:
- a broad constant-density slab plus the planar wells. Adjust well
- depths so the density stays nonnegative and the one-body counterterm
- appears. Show a fixed one-particle gap outside the orbital space,
- and compute the matrix on the orbitals precisely (the charge-density
- and point-nucleus steps).
-
- (N) Point nuclei. Replace the continuous density by finitely many point
- nuclei of equal mass 1/lambda (in physical units charge 1). Do it by
- a smooth Moser flow that pushes uniform density onto the target
- density, applied to a degree-three Gauss cubature with equal node
- masses, then round the node positions to rationals. Get two error
- bounds: a weak one on all continuum states, a strong one on the
- orbitals.
-
-Moser, J., "On the volume elements on a manifold", Trans. Amer. Math. Soc.
- 120 (1965), 286-294. https://doi.org/10.1090/S0002-9947-1965-0182927-5
-local: S0002-9947-1965-0182927-5.pdf
-
-Dacorogna, B. and Moser, J., "On a partial differential equation involving
- the jacobian determinant", Ann. Inst. H. Poincare C 7 (1990), 1-26.
- local: AIHPC_1990__7_1_1_0.pdf
-
- (M) Many-body. Give the complement of the orbital space a uniform lower
- bound, then complete the square so that coupling between orbitals and
- complement enters only quadratically in the energy. This bounds the
- full continuum ground energy, in every spin sector, by the finite
- fermion ground energy, and a variational trial vector gives the
- reverse inequality. Compare with H_d, undo scalings, and choose
- thresholds (the implementability and closing steps).
-
-Every analytic estimate is made at a fixed polynomial scale lambda, with
-all constants absolute and all smallness "p(n,D) lambda^{-alpha}" for a
-fixed alpha>0. The final assembly verifies that all these smallness conditions can
-be met simultaneously and that the whole reduction is computable in
-polynomial time.
-
-Start from a problem about spins that is already known to be QMA-hard.
+ (The promise b-a >= 1/L makes the two cases disjoint.)
+
+The promise b - a >= 1/L concerns decision precision, not a molecular spectral gap; the theorem assumes no excitation gap, binding, neutrality, or existence of a ground eigenvector.
+
+Energy convention: electronic energy; nuclear repulsion omitted.
+
+Main theorem. The promise problem is QMA-hard under deterministic classical polynomial-time many-one reductions, with output consisting of polynomially many unit-charge rational nuclei at distinct positions, polynomially many unary electrons, and polynomial-length rational thresholds at separation b - a >= 1. Thus Z_max = 1 and c = 1; membership in QMA is not asserted; and neither b - a >= 1 nor the input gap 1/L constrains the molecular excitation gap. Each stage preserves the promise because every energy error is inverse-polynomially small relative to gamma, and gamma itself is inverse-polynomial in source length.
+
+The proof starts from a QMA-hard problem about spins.
The spin source (Cubitt-Montanaro-Piddock, 2018/2019).
Consider a finite square-lattice graph (vertices are cells of Z^2, edges
@@ -281,21 +145,7 @@ The spin source (Cubitt-Montanaro-Piddock, 2018/2019).
sigma_x_L sigma_x_R + sigma_y_L sigma_y_R + sigma_z_L sigma_z_R. There
are no one-site terms.
-The following facts, supplied by the cited theorem, are needed:
-
- (a) QMA-hardness. Deciding whether the ground energy of (2) is below a
- lower threshold or above an upper threshold is QMA-hard, with a
- promise gap that is inverse polynomial in the input size.
- (b) Polynomial geometry. The lattice can be chosen with polynomially
- many vertices and edges and polynomially bounded coupling
- magnitudes. The construction is "spatially sparse": it uses a
- polynomial number of poly-sized gadgets laid out in the plane.
- (c) Polynomial weights. The couplings J_e are real with polynomial
- magnitude bounds; the number of nonzero couplings is polynomial.
-
-The precise theorem numbers in the extended paper are Theorem 48, Lemma
-47, and section 10.1 of Cubitt-Montanaro-Piddock. Facts (a)-(c) are not re-proved;
-the incoming instance is only made to meet a clean rational format.
+The cited result supplies three facts, not re-proved here: deciding the ground-energy threshold of (2) is QMA-hard with inverse-polynomial promise gap; the lattice has polynomially many vertices and edges and polynomially bounded couplings; and the couplings are real with polynomially many nonzero entries. The incoming instance is only reshaped to a clean rational format.
Identity of h^s. On two spins, h^s = sigma . sigma = 2 SWAP - I, where
swap exchanges the two spins. Its eigenvalues are
@@ -347,13 +197,6 @@ there is at least one vertex (add a free spin if the source is empty) and
one may drop couplings of zero. All magnitudes and inverse gaps
encountered later are polynomially bounded in n.
-Output.
-A signed rational Heisenberg Hamiltonian H_in = sum_e J_e h^s_{LR} on a
-finite square-lattice subgraph, with rational thresholds a_s < b_s and gap
-gamma = min(1, b_s - a_s), plus polynomial bounds on size, extent, and
-coupling magnitudes. This is the exact input to the signed-to-positive step and the planar
-layout.
-
The source couplings J_e can be positive or negative. The orbital
construction naturally produces positive exchange (the electron kinetic
energy prefers singlets). So first convert the signed
@@ -361,13 +204,7 @@ source into a positive-exchange spin model. The tool is a "singlet-pair
mediator": a fresh pair of spins forced into a singlet, with weak links to
the original spins. Its effective coupling to the two ends has the sign one chooses.
-This uses the same mechanism as Lemma 5 of Piddock and Montanaro.
-The quantitative proof, including the scalar shifts, appears below.
-
-Needed here is the rule "remove the excited states, keep second order".
-The exact statement follows. It has fewer hypotheses than the usual
-Schrieffer-Wolff theorem and does not require the first-order block
-pbp to vanish.
+This is the mechanism of Piddock and Montanaro Lemma 5. The needed rule: eliminate excited states and keep second order. The quantitative statement, with fewer hypotheses than the usual Schrieffer-Wolff theorem and no requirement that the first-order block pbp vanish, follows.
Schrieffer, J. R. and Wolff, P. A., "Relation between the Anderson and
Kondo Hamiltonians", Phys. Rev. 149 (1966), 491-492.
@@ -396,9 +233,9 @@ projected onto the range of Q, reads
(A|_Q) q + QB(p+q) = E q .
-So q = -( A|_Q + QBQ|_Q - E )^{-1} QBp (provided p != 0; if p = 0 the
-energy is at least g - ||B|| > 0 > -||B||, contradicting |E| <= ||B||, so
-p is indeed nonzero). Because |E| <= ||B|| and ||QBQ|| <= ||B|| <= g/4,
+So q = -( A|_Q + QBQ|_Q - E )^{-1} QBp; indeed p != 0, since p = 0 would
+force energy >= g - ||B|| > -||B||, contradicting |E| <= ||B||. Because
+|E| <= ||B|| and ||QBQ|| <= ||B|| <= g/4,
the inverse (A|_Q + QBQ|_Q - E)^{-1} differs from (A|_Q)^{-1} by O(||B||/g^2)
in operator norm, by the resolvent identity:
@@ -490,10 +327,9 @@ Delta. The total shift is sum_e (3 Delta + (3/2) v_e^2) - (correction).
The singlet-pair gadget produces a positive spinner vertex (the fresh pair)
and two spokes of positive weight sqrt(Delta) v_e, one joining L to A_e and
-one joining C_e to R. But in the orbital construction needs
-the link strengths K_e themselves to be encoded in the distances between
+one joining C_e to R. But the link strengths K_e must themselves be encoded in the distances between
orbitals. Having an intermediate A_e or C_e in the middle of a "spoke" is
-fine, but a spoke of positive weight is still a h^s coupling, and the goal is to
+fine, but a spoke of positive weight is still an h^s coupling, and the goal is
to realize all positive h^s couplings as orbital hoppings.
Subdivide each positive spoke into a path of length three using the same
@@ -502,7 +338,7 @@ its two ends). Each subdivider replaces one h^s coupling K h^s_{XY} by a
chain X -- fresh pair -- Y, with two couplings sqrt(Delta') K h^s on each
side and a central bond Delta' h^s. Lemma 1 leaves the effective X-Y
coupling equal to (K/2) h^s up to a scalar shift; because one may rescale
-the whole target (and absorb the 1/2 into K_e), only care that the
+the whole target (and absorb the 1/2 into K_e), the only care is that the
sign is preserved and the strength is positive.
Two rounds of subdivision (three per spoke, nine per original edge, because
@@ -530,13 +366,9 @@ gadgets and the subdivisions and their second-order corrections) with
Every intermediate weight K_e and its shift are computable (as real
numbers) to inverse-polynomial absolute accuracy in polynomial time.
Square roots need not be exact rationals at this stage; they will be
-absorbed into the final rational approximation in the final assembly.
+absorbed into a final rational approximation.
-Output.
-A positive-exchange Heisenberg Hamiltonian H_+ = sum K_e h^s_{ij} on a
-graph of polynomial size m, with every K_e, 1/K_e polynomially bounded,
-and a scalar shift C_s with (9). Every such K_e is realized as a target
-"hopping" number for the orbital construction of the well construction.
+The positive-exchange Heisenberg Hamiltonian H_+ = sum K_e h^s_{ij} has polynomial size m, every K_e and 1/K_e polynomially bounded, and a scalar shift C_s satisfying (9).
A Hamiltonian H_+ = sum K_e h^s_{ij} is just a weighted graph. In the
well construction each vertex is realized as a localized spatial orbital
@@ -568,7 +400,7 @@ Picture (topology only, not to scale):
off-path sites stay >= 1.4 apart)
=== = the central unit-length bond A_e--B_e
L, R = the two original lattice vertices, 17 apart
- A_e, B_e = the fresh singlet-pair sites from the signed-to-positive step
+ A_e, B_e = the fresh singlet-pair sites from the positive-exchange conversion
The central bond A_e--B_e has length 1 in each case. The two spokes are
nine-link paths of length nine each:
@@ -644,32 +476,14 @@ implement all eighteen independent adjustments at once.
Computability. Each such solve is a fixed-dimensional contraction with
algebraic initial data (rational or nested-square-root). Iterating with
guard precision computes every internal position to any inverse-polynomial
-accuracy in polynomial time. the final assembly takes care of final rational
-rounding.
+accuracy in polynomial time.
-The unit-distance layout above is then rescaled by D = log(lambda) (see
-the well construction for the significance of D). This gives minimum spacing at least
-0.95 D, nonlink spacing at least 1.2 D, and spatial extent poly(n) * O(D).
+The unit-distance layout above is then rescaled by D = log(lambda), giving minimum spacing at least 0.95 D, nonlink spacing at least 1.2 D, and spatial extent poly(n) * O(D).
The orbit centers u_i are placed at the rescaled layout positions.
-Output.
-A planar drawing of the positive-exchange graph of H_+ with:
- - one site per vertex (each tied to a planar position u_i at scale D);
- - one site per fresh spacer vertex (from the signed-to-positive step), all also realized
- as planar positions;
- - each link having an independently adjustable length in
- [D(1 - eps), D(1 + eps)];
- - nonlinks farther apart than 1.2 D;
- - the whole drawing contained in a poly(n) * O(D) box.
-
-the signed-to-positive step gave the target positive-exchange Hamiltonian
-H_+ = sum K_e h^s_{ij} on a graph of m sites. The planar layout gives a
-placement in which independent
-link lengths exist. Now (1) build one smooth localized spatial orbital
-per site, (2) write the finite-fermion Hamiltonian realizing H_+, with an
-occupancy penalty, and (3) calibrate the link hoppings to the desired
-strengths by choosing separations. The next three steps then realize the orbital
-potential of part (1) as the field of real nuclei.
+Every link is independently adjustable in [D(1 - eps), D(1 + eps)], and nonlinks remain farther apart than 1.2 D.
+
+The sign conversion produced the target H_+ = sum K_e h^s_{ij} on m sites; the planar layout gives a placement with independent link lengths. Each site hosts one smooth localized spatial orbital, and the target hoppings are calibrated by choosing separations.
Because the eventual nuclei have charge 1, the physical scale is set by a
large number lambda. Work in dilated coordinates
@@ -735,11 +549,11 @@ The vertical mode. For the z direction use the normalized Gaussian
g(z) = (omega/pi)^{1/4} e^{- omega z^2 / 2}, omega = sqrt(4 pi rho),
the ground mode of -d^2/dz^2/2 + 2 pi rho z^2 with energy omega/2. (rho > 0
-is a large fixed integer chosen in the charge-density step.) The oscillator has a fixed
+is a large fixed integer chosen in the slab-and-wells density construction.) The oscillator has a fixed
gap omega above its ground mode.
The three-dimensional mode. Per site i, centered at planar position u_i
-(from the planar layout, at scale D):
+(from the contact-gadget layout, at scale D):
psi_i(y) = varphi(r - u_i) g(z), e_* = -1/2 + omega/2, (12)
@@ -810,13 +624,7 @@ which is >= 2c. Therefore
and it is uniformly bounded above. So the superexchange denominator
U - v_{ij} below is uniformly positive and bounded.
-In one breath. Two electrons on neighbouring sites with opposite spins
-can momentarily pile onto a single site and then split back. Pauli's
-principle permits the pile-up only when the two spins are antiparallel
-(a singlet); for parallel (triplet) spins the first hop is blocked. So
-the double hop lowers the singlet's energy relative to the triplet's by
-roughly t^2/(U - v), i.e. a favourable spin exchange whose size is set
-by the virtual-excitation cost.
+Superexchange: two electrons on neighbouring sites momentarily pile onto one site, then split back; Pauli blocking restricts this to singlets, lowering their energy by roughly t^2/(U - v).
The fermion model. Consider 2m canonical fermion modes: at each site i, two
spin states sigma in {up,down}. Restrict to exactly m electrons (the
@@ -939,7 +747,7 @@ tail bound exp(-|r|) of phi it satisfies, for some absolute C,
c (1+d)^{-C} e^{-d} <= T(d) <= C (1+d)^{C} e^{-d} . (26)
-The hopping to be supplied by the continuum. The point-nucleus step shows that the real
+The hopping to be supplied by the continuum. The cubature replacement shows that the real
continuum orbital matrix, evaluated on the orbitals centered at u_i, has
hopping entries t_e = lambda T(|u_i - u_j|) in the units of H_d, up to an
error of order p(n,D) lambda^{-0.1}. So to realize the target t_e^tar of
@@ -955,7 +763,7 @@ Define the residual
F_e(d) = lambda T(d) - tau sqrt( K_e ( U - v(d) ) ) . (28)
A root exists. Evaluate at d = (1 - eps/2) D and d = (1 + eps/2) D, both
-inside the adjustable range of the planar layout:
+inside the adjustable range of the contact-gadget layout:
T((1-eps/2)D) ~ exp(-(1-eps/2)D) => lambda T ~ lambda^{eps/2} up to poly(D),
T((1+eps/2)D) ~ exp(-(1+eps/2)D) => lambda T ~ lambda^{-eps/2} up to poly(D).
@@ -971,20 +779,19 @@ accuracy. If the sign is ambiguous at evaluation accuracy, the residual is
already small enough, so stop. Otherwise keep the sign-change bracket, and
continue until bracket width times a Lipschitz bound is within tolerance.
No monotonicity or derivative lower bound is needed: only need some root
-with small residual, and any root works. the final assembly handles the numerical
-evaluation details.
+with small residual, and any root works.
Resulting centers. The calibrated distances d_e, rescaled back, are the
-planar separations |u_i - u_j| for link pairs. the planar layout's contraction then
+planar separations |u_i - u_j| for link pairs. the contact-gadget layout's contraction then
produces internal-site coordinates realizing all these link lengths at once
-with nonlinks separated, and a final rational approximation (the final assembly)
-gives rational centers u_i.
+with nonlinks separated; rational approximation of the resulting positions
+gives the centers u_i.
From now on, H_d means H_d(t) at these final centers, with
t_e = lambda T(|u_i - u_j|). Choose the per-link calibration accuracy so
that replacing t_e^tar by t_e changes the total Hamiltonian norm by at most
gamma tau^2/100. Combining this and the second-order remainder from
-the occupancy-penalty step (which is at most gamma tau^2/100) and the internal error of
+the superexchange calculation (which is at most gamma tau^2/100) and the internal error of
Lemma 1, one gets
| E(H_d) + sum_{i<j} v_{ij} - tau^2 ( E(H_+) - sum_e K_e ) |
@@ -993,51 +800,18 @@ Lemma 1, one gets
Line (29) says the bottom of H_d, shifted by the direct-Coulomb scalar
sum_{i<j} v_{ij}, equals tau^2 times the positive-exchange ground energy
H_+ minus a scalar, up to the stated error. This is the finite-fermion
-target that the remaining steps will realize as a continuum molecule.
-
-Every analytic error produced later has the form p(n,D) lambda^{-alpha}
-for some fixed alpha > 0, with required smallness at worst inverse polynomial
-in n. Now state how lambda, D, and the cubature cell h are chosen so that
-all these conditions can be met at once.
-
-Order of the scale choices:
- 1. The bounded well data (f, phi, W) and the fixed integer rho are fixed
- first, independent of anything else (the charge-density step picks rho using a
- derivative bound in (30)).
- 2. The source cutoff and the three spin-gadget penalty scales Delta,
- Delta', Delta'' are fixed from n and gamma in the source normalization.
- 3. The rational tau is fixed from the uniform Coulomb coercivity constant
- c and the polynomial size bounds (the occupancy-penalty step).
- 4. then choose the physical scale. Take an integer
-
- k = ceil( C n^A ) , h = 1/k , lambda = 8 k^3 / rho , (31)
-
- with A, C fixed absolute positive integers, then large enough.
-
- The relation rho h^3 / 8 = 1/lambda is exactly the statement that each of
- the eight equal nodes of one cubature cell (side h) carries mass 1/lambda.
- This will make each node a unit nucleus in physical coordinates
- The fixed integer rho appears so that this mass relationship
- matches rho h^3.
-
-At this last stage, the target sizes K_e, 1/K_e, the layout extent, and the
-coefficients b_i (the charge-density step) already have fixed polynomial bounds in n, so
-they do not change the choice. Each smallness requirement
-"p(n,D) lambda^{-alpha} <= inverse polynomial in n" is met by a sufficiently
-large fixed A, and the constant C covers all n >= 2 and the fixed
-separation and gap conditions. The polynomials p(n,D) come from fixed-order
-derivatives, moments, fixed-dimensional wells and gadgets, and polynomially
-many matrix entries, so no exponent needs to grow with the input. Thus the
-large box, its volume, and 1/delta_R are all polynomial in n after A, C
-are fixed. the final assembly revisits this and shows every scalar budget can be
-met simultaneously.
+target to be realized as a continuum molecule.
+
+Every analytic error produced below has the form p(n,D) lambda^{-alpha}, with required smallness at worst inverse polynomial in n. The scales lambda, D, and the cubature cell h are then chosen so that all conditions hold at once.
+
+Take k = ceil(C n^A), h = 1/k, lambda = 8 k^3/rho, with A, C fixed positive integers chosen large enough. The relation rho h^3/8 = 1/lambda gives each of the eight equal nodes of one cubature cell (side h) mass 1/lambda, so each node becomes a unit nucleus in physical coordinates.
Finally D = log lambda. Then D is polynomial in n, and "exp(-D) beats
poly(n)" and similar statements hold by taking lambda a large enough power
of n.
Now construct an attractive Coulomb potential whose low-energy matrix
-realizes the modes and hoppings of the well construction. The potential is the Coulomb
+realizes the prescribed modes and hoppings. The potential is the Coulomb
field of an honest nonnegative charge density: a broad constant-density
slab (supplying transverse confinement and a positive background) plus the
planar wells (supplying the orbitals). Adding the wells changes the density
@@ -1068,7 +842,7 @@ like a constant minus a vertical harmonic well centered on z=0, plus small
error (see (34)).
Why we need a well-depth adjustment. The occupancy identity (C) of the
-occupancy-penalty step requires a one-body diagonal term -s_i/lambda on
+superexchange calculation requires a one-body diagonal term -s_i/lambda on
site i, where
s_i = sum_{j!=i} v_{ij}. In the mode basis, the diagonal entry from
-s_i/lambda must be supplied by a small deepening of well i. Because
@@ -1147,8 +921,7 @@ negative at y=0), and |C_H| <= C S H.
Together (36) and (34) say: near the wells the potential is the vertical
harmonic oscillator 2 pi rho z^2 (min energy omega/2, gap omega) plus the
-planar multiwell sum_i W_i, plus perturbations. The rest of the charge-density construction makes
-this precise.
+planar multiwell sum_i W_i, plus perturbations. The precise statements follow.
The potential F_H + V_ell is now expressed as the Coulomb field of a
nonnegative density (36). Before replacing that density by point nuclei,
@@ -1195,7 +968,7 @@ Proposition 1 (continuous one-particle control).
Proof.
-Step 1: lower bound by a separable operator. By (9.6a), (36), and
+Step 1: lower bound by a separable operator. By (34)(a), (36), and
zeta W_i >= W_i (zeta <= 1, W_i <= 0) with the b_i/lambda terms small and
bounded everywhere, L_c is at least the separable operator
@@ -1215,7 +988,7 @@ angle interpolation between 0 and pi/2 on the transition annuli using sine
and cosine, so that sum_i |nabla theta_i|^2 + |nabla theta_ext|^2 <= C/D^2.
The IMS localization identity (product rule with kinetic coefficient 1/2)
then costs O(1/D^2) in the form. On each site term use the isolated ground
-state and gap of -Delta_r/2 + W (the one-particle step); on the exterior term use the
+state and gap of -Delta_r/2 + W; on the exterior term use the
free kinetic lower bound and the fact that no well sits on its support. With
site cutoffs theta_i and varphi_i = varphi(.-u_i), the site term evaluated
on theta_i u is at least
@@ -1346,10 +1119,10 @@ matrix is exactly the one-particle part of H_d, up to p(n,D) lambda^{-0.1}.
Where we are. There is a continuous charge density whose one-particle
operator L_c has (i) the modes as approximate low-energy states with a fixed
complementary gap, and (ii) a compressed matrix equal to the one-particle
-part of H_d to accuracy p(n,D) lambda^{-0.1}. The point-nucleus step replaces the
+part of H_d to accuracy p(n,D) lambda^{-0.1}. The cubature replacement replaces the
density by honest unit point nuclei; the comparison with H_d is then completed.
-The input to the charge-density step was a nonnegative charge density (36). One must now
+The input to the slab-and-wells density construction was a nonnegative charge density (36). One must now
replace its Coulomb field by finitely many point nuclei of equal mass
1/lambda, so that in physical coordinates each nucleus has charge exactly 1.
There are two distinct error requirements:
@@ -1360,7 +1133,7 @@ There are two distinct error requirements:
The comparison with H_d uses energies multiplied by lambda. So the
compressed error must be smaller than lambda^{-1} up to a polynomial margin.
The error coupling the mode space to its complement can be larger: the fixed
-complementary gap of the one-particle-gap step makes its energy contribution quadratic in
+complementary gap makes its energy contribution quadratic in
the coupling, so a weaker bound suffices there.
Cubature side h will be chosen as
@@ -1371,18 +1144,7 @@ A cubature rule exact through polynomial degree three will then give an
O(h^4) compressed error and an O(h^2) form error. Both resulting energy
errors are O(lambda^{-1/3}) in the units of H_d, up to polynomial factors.
-In one breath. To replace a continuous charge smear by point nuclei,
-tile the uniform slab it was built from by an evenly spaced cubature and
-let a smooth vector field push each cubature node toward the thicker
-region. Choose the flow so the pushed nodes' local density equals the
-target smear everywhere; then the discrete Coulomb field mimics the
-continuum field on all smooth test functions, and every node keeps its
-common weight (charge one). This is Moser's construction.
-
-The density interpolation is Moser's deformation method (Moser 1965;
-Dacorogna-Moser 1990). Here the vector field is explicit and compactly
-supported inside the box, and prove the required transport directly from
-the continuity equation.
+The replacement uses Moser's deformation method (Moser 1965; Dacorogna-Moser 1990): tile the uniform slab with an evenly spaced cubature, then push nodes by a smooth compactly-supported flow so that the pushed-node density equals the target smear; the discrete Coulomb field mimics the continuum field on smooth test functions, and every node keeps common weight (charge one).
Define the interpolation path from the uniform slab density rho to the
target density,
@@ -1420,14 +1182,14 @@ measure rho db onto the target density q_1 db. The second line is the crucial
one for cubature: applying F to a transported point G(b) and keeping the
weight rho db reproduces int F q_1.
-The cubature. Because H, S, k are integers (k from the scale choices, h = 1/k),
+The cubature. Because H, S, k are integers (k = ceil(C n^A), h = 1/k),
cubes of side h tile Omega. On each cube Q use the eight tensor products of
two equal-weight Gauss points per coordinate, at the midpoint plus or minus
h/(2 sqrt(3)). This eight-point rule integrates polynomials of total degree
at most three exactly, by matching the moments on each interval. When mapped
these nodes by the flow G, their common weight stays rho h^3 / 8 = 1/lambda
by (52). (The rule's exactness is used in the original cube coordinates,
-on integrands composed with G; see the form and compression step.)
+on integrands composed with G.)
The rational rounding. Each exact mapped point b_alpha (the image under G of
a cubature node) will be approximated within Euclidean error at most
@@ -1444,7 +1206,7 @@ So the data fed to the physical problem are:
x~_alpha = b~_alpha / lambda,
- every one of them a unit nucleus (the physical charge is then exactly 1),
- the electron number N = m,
- - rational thresholds from the implementability step.
+ - rational thresholds from the computability discussion.
The single-particle operator with the actual point nuclei is
@@ -1629,19 +1391,10 @@ fermion operator below. The one-particle kinetic bound (39) gives
This is a genuine finite-dimensional minimum, because Pi projects onto the
m-electron space of 2m one-particle spin modes.
-In one breath. A lower bound is wanted on the full ground energy, but
-only the modes are known well. Split any trial state as w (modes) plus q
-(the rest). The rest costs at least alpha times its H^1 norm; the modes
-part costs at least E_Pi; the only danger is the cross-coupling <q, Hcal w>,
-bounded by beta times ||q|| times ||w||. Completing the square in ||q||:
-alpha ||q||^2 - 2 beta ||w|| ||q|| >= -beta^2/alpha * ||w||^2. Hence
-E_full >= E_Pi - beta^2/alpha, up to polynomial lambda factors.
-
-The complement first receives a uniform lower bound, and its coupling to
-the mode space is bounded. Then a square completion bounds the full infimum.
+The complement receives a uniform lower bound, and its coupling to the mode space enters the energy only quadratically by square completion.
Proposition 3 (eliminating the continuum complement).
- For the unit-nucleus construction with N = m and sufficiently large
+ For the constructed point nuclei with N = m and sufficiently large
polynomial scale lambda, the full antisymmetric m-electron operator in
dilated and shifted units satisfies
@@ -1700,7 +1453,7 @@ vector of the compressed operator (which lies in Ran Pi). The lower-bound
argument applies to every normalized antisymmetric state, in any spin sector,
without any binding or ground-eigenvector assumption. QED
-It remains to evaluate lambda E_Pi and match it to E(H_d). the compressed-matrix step
+It remains to evaluate lambda E_Pi and match it to E(H_d). the compression computation
computed the one-particle matrix M on the orthonormal modes to accuracy
p(n,D) lambda^{-0.1}. Multiplying the compressed replacement error (60) by
lambda costs only p(n,D) lambda^{-1/3}, so the same matrix accuracy
@@ -1730,13 +1483,13 @@ errors costs a polynomial factor over the entire fermion space.
Identifying the retained operator. In this representation, the retained
two-body term is
- (U/2) sum_i n_i (n_i - 1) + sum_{i<j} v_{ij} n_i n_j , (11.10a)
+ (U/2) sum_i n_i (n_i - 1) + sum_{i<j} v_{ij} n_i n_j ,
and the retained one-body diagonal is
- - sum_i ( sum_{j != i} v_{ij} ) n_i = - sum_i s_i n_i . (11.10b)
+ - sum_i ( sum_{j != i} v_{ij} ) n_i = - sum_i s_i n_i .
-At the filling sum_i n_i = m, the identity (C) of the occupancy-penalty step combines these
+At the filling sum_i n_i = m, the identity (C) of the superexchange calculation combines these
two lines into the first line of H_d (20), including its scalar
-sum_{i<j} v_{ij}. The hopping matrix M from (50) gives the remaining line
of the calibrated H_d. So the compressed operator is exactly H_d.
@@ -1754,9 +1507,9 @@ matches by (51), the pair part by (77) and (79). Multiplying (72)
by lambda contributes only p(n,D) lambda^{-1/3} (because lambda times
lambda^{-4/3} is lambda^{-1/3}), which fits within the bound. QED
-The roles of the errors. The table below records that the different
-errors play different roles; in particular the arbitrary-state form error
-cannot be used as a direct energy error after multiplication by lambda.
+Distinct estimates play distinct roles; in particular the arbitrary-state
+form error cannot be used as a direct energy error after multiplication by
+lambda.
+----------------------------------+----------------+---------------------+
| Estimate | Error scale | Role |
@@ -1768,10 +1521,7 @@ cannot be used as a direct energy error after multiplication by lambda.
| Total comparison (78) | lambda^{-0.1} | + other mode errors |
+----------------------------------+----------------+---------------------+
- The first three rows are in the units of Hcal; the last two in the units
- of H_d, up to polynomial factors p(n,D). The O(D^{-2}) and O(S^{-2})
- localization losses only establish a fixed complementary gap and do not
- enter the energy-precision budget.
+ Rows are in the units shown, up to polynomial factors p(n,D). The O(D^{-2}) and O(S^{-2}) localization losses only establish a fixed complementary gap and do not enter the energy-precision budget.
Choose the polynomial scale in lambda large enough that (78) costs at
most tau^2 gamma/100 and that
@@ -1779,7 +1529,7 @@ most tau^2 gamma/100 and that
lambda tau^2 gamma >= 8 . (80)
All earlier smallness, separation, and calibration estimates hold
-simultaneously by the growth choices of the scale choices. Lifting from the
+simultaneously by the growth of k. Lifting from the
dilated coordinates back to physical coordinates gives the physical
electronic ground energy exactly as
@@ -1796,7 +1546,7 @@ lambda tau^2 gamma/8:
| E_0 - F( E(H_in) ) | <= lambda tau^2 gamma / 8 . (83)
-Here is where each piece comes from: (9) contributes 4 gamma/100 in the
+The error decomposition: (9) contributes 4 gamma/100 in the
H_in variable (which maps to at most 4 lambda tau^2 gamma/100 through F);
(29) contributes 2 gamma tau^2/100; and (78) contributes at most
tau^2 gamma/100 by choice of lambda. Summing gives 7/100 < 1/8.
@@ -1805,7 +1555,7 @@ The output thresholds. Take rational numbers a and b approximating
a ~ F( a_s + gamma/4 ) , b ~ F( b_s - gamma/4 ) , (84)
-each within lambda tau^2 gamma/32. The implementability step shows these are computable
+each within lambda tau^2 gamma/32. The computability discussion shows these are computable
and have polynomial rational bit length.
Yes case. Suppose the source is YES: E(H_in) <= a_s. Then
@@ -1834,62 +1584,15 @@ b_s - a_s >= gamma,
using (80) in the last line. So both promises survive rational rounding,
and the reduction meets the required output separation b - a >= 1.
-Why the chosen thresholds preserve the promise. The map F carries the
-source energies (in the small source units) to physical continuum energies
-by scaling with lambda^2, shifting by the slab constant, subtracting the
-direct Coulomb scalar, and scaling the spin part by tau^2. The gaps
-gamma/4 inside the arguments absorb the 7/100 errors, and the outer
-lambda tau^2 gamma/32 slack absorbs rational rounding. The separation
-b - a >= 7/2 then exceeds the required value 1 with room to spare.
-
-A nuclear configuration and thresholds have now been specified. It
-remains to show that they are computable in deterministic polynomial time
-and have polynomial encoding length. The implementability discussion
-also confirms that all the scalar smallness budgets can be met at once, and
-that the output format matches the promise problem.
-
-The scale choices are fixed in a definite order:
-
- 1. bounded well data (f, phi, W) and the fixed integer rho, using the
- derivative bound (30) - depends only on f, phi, W, zeta, 2;
- 2. the source cutoff and the three spin-gadget penalties Delta, Delta',
- Delta'' - from n and gamma by the source normalization;
- 3. the rational tau from the uniform Coulomb coercivity constant c and
- the polynomial size bounds (the occupancy-penalty step);
- 4. the integers k = ceil(C n^A), h = 1/k, lambda = 8 k^3/rho (the
- scale choices), A and C fixed absolute positive integers then large
- enough.
-
-At stage 4, the quantities K_e, 1/K_e, the layout extent, and the b_i already
-have fixed polynomial size bounds in n, so they do not affect the choice of
-k. Every remaining smallness condition has the form
-p(n, log lambda) lambda^{-alpha} <= (inverse polynomial in n) for a fixed
-alpha > 0, and the hopping endpoint brackets of (27) have the same
-fixed-power character. A sufficiently large fixed A meets every asymptotic
-requirement; increasing C covers all n >= 2 and the fixed separation and gap
-conditions.
-
-Uniformity. The polynomials p(n,D) come from fixed-order derivatives and
-moments, fixed-dimensional wells and gadgets, and polynomially many matrix
-entries. The large box is treated separately: its effects are bounded in
-(34), by orbital tails, and by summable fourth derivatives in the cubature
-argument. Its total mass enters only the (much smaller) node-rounding error.
-So no exponent needs to grow with the input, and H, the box volume |Omega|,
-and 1/delta_R are all polynomial in n after A, C are fixed.
-
-All required real quantities can be evaluated to inverse-polynomial absolute
-accuracy in polynomial time.
+The configuration and thresholds are next certified to be computable in deterministic polynomial time with polynomial encoding length. The scale order is: (1) bounded well data (f, phi, W) and fixed integer rho from derivative bound (30); (2) source cutoff and spin-gadget penalties Delta, Delta', Delta'' from n and gamma by the normalization to rationals; (3) rational tau from the uniform Coulomb coercivity constant c and the polynomial size bounds; (4) integers k = ceil(C n^A), h = 1/k, lambda = 8 k^3/rho with A, C fixed absolute positive integers then large enough.
-The large scalar shift. The dominating scalar is C_H times lambda^2 m. Since
-C_H is multiplied by lambda^2 m, evaluating it within
+At stage 4, K_e, 1/K_e, the layout extent, and the b_i are already polynomial in n, so they do not affect k. Every remaining condition is p(n, log lambda) lambda^{-alpha} <= inverse polynomial in n, and the hopping endpoint brackets of (27) sit at the same fixed-power rate; the polynomials p(n,D) come from fixed-order derivatives and fixed-dimensional data, and the large box enters only through (34) and summable cubature derivatives. A large enough A meets every asymptotic requirement; C covers all n >= 2 plus the fixed conditions; hence H, |Omega|, and 1/delta_R are polynomial in n.
+
+The large scalar shift. The dominating scalar is C_H times lambda^2 m; evaluating it within
tau^2 gamma / ( 128 lambda m ) (88)
-uses at most a quarter of the allowed threshold error lambda tau^2 gamma/32.
-The other scalar terms can be allocated the remaining budget the same way,
-because their amplification factors and their number are polynomially
-bounded. The magnitude |C_H| <= C S H therefore causes no demand for
-exponentially fine absolute precision, only inverse-polynomial.
+uses at most a quarter of the allowed threshold error lambda tau^2 gamma/32. The other scalar terms fit the remaining budget the same way, their amplification factors and count being polynomially bounded.
The orbital integrals. The modes come from phi = (-Delta_r + 1)^{-1} f, so
in the Gaussian heat formula, substitute t = s^2 and rescale the Gaussian
@@ -1928,7 +1631,7 @@ evaluate the residual to a small fraction of its per-link tolerance. If the
sign is ambiguous, the residual is already small enough; otherwise keep the
sign-change bracket until its width times the Lipschitz bound is within
tolerance. No derivative lower bound is needed. The fixed layout contractions
-of the planar layout and the rational center approximations can be computed more
+of the contact-gadget layout and the rational center approximations can be computed more
accurately than these tolerances require.
Once the rational centers u_i are fixed, their exact real coefficients b_i
@@ -1953,66 +1656,15 @@ evaluation error, so all numerical and final rational-rounding errors can be
charged to the single displacement budget delta_R without changing the
reference density or any cell mass.
-None of these operations uses a many-electron wavefunction. Everything is
-single-particle quadrature, fixed-dimensional flow integration, and rational
-rounding.
-
-There are 8 |Omega| h^{-3} point nuclei, which is polynomial in n, and each
-has charge exactly one. Their rational physical positions x~_alpha =
-b~_alpha / lambda, the electron number N = m (encoded in unary), and the
-rational thresholds a, b of (84) can all be computed with polynomial
-encoding lengths, by the scale order, numerical evaluation, and flow
-evaluation. Reducing rational outputs to lowest terms gives the input
-format of the promise problem.
-
-To define the reduction on every input, first check the source syntax and the
-fixed polynomial bounds used in the construction; if they fail, output a
-fixed instance (any instance obeys its own promise vacuously, and the
-fallback does not affect either promised implication). Otherwise run the
-deterministic construction above.
-
-This completes the implementability of the reduction: the only outputs are
-nuclear positions, electron number, and thresholds, all polynomial-length and
-deterministic-polynomial-time computable.
+None of these operations uses a many-electron wavefunction: only single-particle quadrature, fixed-dimensional flow integration, and rational rounding. There are 8 |Omega| h^{-3} point nuclei, polynomially many in n, each of charge exactly one. Their rational physical positions x~_alpha = b~_alpha / lambda, the electron number N = m (in unary), and the rational thresholds a, b of (84) all have polynomial encoding length and are computable by the scale order, numerical evaluation, and flow evaluation. Reducing rational outputs to lowest terms gives the input format of the promise problem.
-Now chain all the pieces and prove the Main Theorem of the promise problem.
+On any input, first verify source syntax and the fixed polynomial bounds used in the construction; if they fail, output a fixed instance (which obeys its own promise vacuously and does not affect either promised implication). Otherwise run the deterministic construction above.
-For any source instance (the source normalization), the construction produces a molecule and
-two rational thresholds. The energy relation is a chain, each link within a
-stated error budget expressed as a fraction of gamma tau^2 (or gamma) in the
-relevant variable:
-
- (L0) Source. E(H_in), with signed rational Heisenberg couplings and
- gap gamma = min(1, b_s - a_s) > 0. (Sec 5)
-
- (L1) Positive | E(H_+) - E(H_in) + C_s | <= 4 gamma/100 .
- spin. (9)
-
- (L2) Wells and | E(H_d) + sum_{i<j} v_{ij}
- finite model. - tau^2 ( E(H_+) - sum_e K_e ) | <= 2 gamma tau^2/100 .
- (29)
-
- (L3) Continuum | lambda inf Spec Hcal - E(H_d) | <= p(n,D) lambda^{-0.1}
- comparison. and the complement bound
- E_Pi - p(n,D) lambda^{-4/3} <= inf Spec Hcal <= E_Pi .
- (72),(78)
-
- (L4) Physical. E_0 = lambda^2 ( m(C_H + e_*) + inf Spec Hcal ) , and
- with all errors,
- | E_0 - F( E(H_in) ) | <= lambda tau^2 gamma/8 .
- (83)
-
-Here F is the affine map (82). Combining (L0)-(L4) gives the single
-statement used: for the molecule built from the source instance,
+The affine map F is (82). The estimates (9), (29), (78) and (81) chain to a single bound on the source-instance molecule,
| E_0 - F( E(H_in) ) | <= lambda tau^2 gamma / 8 . (91)
-Every constant entering (91) - lambda, tau, gamma, C_s, sum_e K_e,
-sum_{i<j} v_{ij}, m, C_H, e_* - is either fixed by the source instance or
-fixed by the scale choices; all are computable in
-deterministic polynomial time to the required inverse-polynomial accuracy
-This is a many-one reduction: one source instance yields one
-molecule plus thresholds.
+Each constant entering (91) is fixed by the source instance or by k, A, C, and is computable in deterministic polynomial time to the required inverse-polynomial accuracy. One source instance thus yields one molecule plus thresholds.
The output thresholds are rationals a < b approximating
F(a_s + gamma/4) and F(b_s - gamma/4), each within lambda tau^2 gamma/32.
@@ -2023,10 +1675,7 @@ F(a_s + gamma/4) and F(b_s - gamma/4), each within lambda tau^2 gamma/32.
If the source is NO (E(H_in) >= b_s), then (91) and the definition of b
give E_0 >= b (86). So the molecule is in the NO set.
-The threshold separation satisfies b - a >= 7/2 > 1 (87), so the
-produced instance satisfies the input promise of the promise problem with room to
-spare. The energy errors never exceed the slack built into the thresholds,
-so no YES instance turns into a NO instance or vice versa.
+By (87), b - a >= 7/2 > 1, so the produced instance meets its own input promise with room to spare; every energy error stays inside the built-in slack, so no YES instance becomes a NO instance and no NO becomes a YES.
Theorem (main).
The electronic promise problem for Hamiltonian (1) on the full spinful
@@ -2038,43 +1687,6 @@ Theorem (main).
and threshold separation at least one. The only output data are the
nuclear positions, electron number, and thresholds.
-Proof. Take any instance of the Cubitt-Montanaro-Piddock signed Heisenberg
-energy problem (the source normalization). The following deterministic polynomial-time
-procedure produces a molecule and thresholds:
-
- 1. Normalize to rational couplings and thresholds (the source normalization), obtaining
- H_in with gap gamma = min(1, b_s - a_s) > 0.
-
- 2. Convert signed to positive exchange with singlet-pair mediators and two
- subdivision rounds (the signed-to-positive step). This gives H_+ = sum K_e h^s_{ij} with
- positive polynomially-bounded K_e, plus the scalar shift C_s, to within
- 4 gamma/100 (9).
-
- 3. Build the localized modes and finite fermion model (the well construction), with
- direct Coulomb coefficients v_{ij}, occupancy penalty, and calibrated
- hoppings chosen by (27). This gives H_d related to H_+ by (29).
-
- 4. Construct the nonnegative charge density (the charge-density step), replacing its
- continuous field by unit point nuclei via Moser transport and degree-
- three Gauss cubature (the point-nucleus step), with the form and compression bounds
- (60).
-
- 5. Compare the full continuum infimum with E(H_d) using the complement gap
- and square completion (the complement-control step) and the compressed matrix (70),
- giving (72) and (78).
-
- 6. Compute the physical energy and thresholds through the affine map F
- (82)-(84), verifying (85)-(87) and the threshold gap > 1.
-
-Steps 1-6 are all deterministic polynomial time (the implementability step); all outputs
-are polynomially many rational data plus a unary electron number, and all
-are charge-one nuclei at distinct rational positions. By (85) and
-(86) YES and NO are preserved, so this is a valid reduction. Since the
-source is QMA-hard (the source normalization), the electronic problem is QMA-hard. QED
-
-Remarks. The theorem imposes no neutrality, binding, existence of a ground
-eigenvector, molecular spectral-gap, or lower nuclear-separation promise. It
-supplies no orbital basis, electronic state, magnetic field, or additional
-external potential. Membership in QMA is not asserted. The constructed
-output gap b - a >= 1 is stronger than the input requirement b - a >= L^{-1};
-neither inequality specifies an excitation gap of the molecule.
+Proof. Given any signed Heisenberg instance, the construction above runs in six deterministic polynomial-time stages: (1) normalize to rational couplings and thresholds, obtaining H_in with gap gamma = min(1, b_s - a_s) > 0; (2) convert to positive exchange by singlet-pair mediators, giving H_+ = sum K_e h^s_{ij} with (9); (3) build the localized modes with direct Coulomb coefficients and calibrated hoppings, giving H_d with (29); (4) replace the smooth slab density by unit point nuclei via Moser transport and degree-three cubature, giving the form and compression bounds (60); (5) compare the full continuum infimum to E(H_d) using the complement gap and square completion, giving (72) and (78); (6) compute the physical thresholds through the affine map F, verifying (85)-(87) and the threshold gap > 1.
+
+All six stages run in deterministic polynomial time; the outputs are polynomially many rational data plus a unary electron number, all at charge-one distinct rational positions. By (85) and (86) both promises are preserved, so the source's QMA-hardness transfers and the electronic problem is QMA-hard. QED