A Memory-Magic Exchange Law in Streaming Clifford+T Compilation

• The unconditional rate-two theorem gives an explicit-constant lower bound after an initial allowance of A2 bits per coordinate.The bound is stated as Tpre ≥ 2(r−1)(m log2 K − S − mA2), up to O(δ) corrections.Source: §I · p. 2
• A determinant method, sharpened by a height dichotomy and rational-projection fibration, proves the general bounds uniformly over frames without the spectral method’s square-root barrier.Its continuum case analysis is checked exactly by an SMT solver, and every rate below 5/2 is reached.Source: §I · p. 2
• At Clifford-framed cosets, all but a vanishing fraction of z-rotations require T-count 3L − O(L/log L), unconditionally.The volume law holds up to subexponential factors at these cosets.Source: §I · p. 3
• With clean ancillas and a phase-gradient catalyst, batching table lookups across coordinates removes the constant-rate law.The committed cost per share drops to O(L/log L).Source: §I · pp. 2–3
• Three committed T gates per bit are achievable along the fractional-passthrough family when the grid-mean synthesis cost is (3 + o(1)) log2 K.The stated synthesis condition is supported for typical angles by Ross–Selinger synthesis.Source: §I · pp. 3–4
• Under the stated equidistribution conjecture, the general exchange rate is 3, and a committed word carries only O(log log(1/ε)) coordinate bits unless it costs at least 2 log2(1/ε) − O(1) gates.The passage reports 7 bits at ε = 10^-10, compared with 33 bits carried by a coordinate.Source: §I · p. 3
• When downstream side information is available, the bound replaces m log2 K with the conditional entropy H(X | other rounds, Y).A one-shot version uses smooth entropies.Source: §I · p. 3
• The general exchange-rate lower bound reaches every rate below 5/2 asymptotically, while rotation-segmented processes have an unconditional lower bound of 3 −o(1).The 3-rate bound applies when each committed segment is near a Clifford-framed z-rotation.Source: §I · pp. 3–4

• The tuned grid has K = Q−1 phase values and log2 Kε = L − 0.35 + o(1), with spacing chosen to make nearest-grid decoding unambiguous.The accuracy condition is 0 < ε < sin(π/2Q).Source: §II · p. 4
• An r-round dispersed stream consists of shares whose coordinatewise sums modulo Q equal the aggregate phase vector.The canonical sharing makes the first r−1 rounds independent uniform shares.Source: §II · p. 4
• The coordinatewise ancilla-free Clifford+T model emits a write-once list of H, S, and T records, with T-count equal to the number of T records.Each record is assigned to one coordinate, and correctness requires each implemented coordinate rotation to be within projective error ε of its target.Source: §II · p. 4
• The compiler reads the stream once, may use RAM and a random tape, and is charged for the worst-input expected size S of its restart snapshot.It must be correct with probability at least 1−δ for every aggregate and valid stream.Source: §II · p. 4

• A single-qubit Clifford+T unitary of minimal T-count t has 36 · 2^t possibilities when t is positive, while there are 24 at t = 0.These counts are modulo global phase.Source: §III · p. 4
• The cumulative number of single-qubit words of minimal T-count at most t is 72 · 2^t − 48.This follows by summing the exact-count values.Source: §III · p. 4
• For a random m-tuple of single-qubit Clifford+T unitaries, entropy is bounded by expected total T-count plus an overhead depending on m and that count.The stated bound is H(M) ≤ T̄ + m log2 36 + ρm(T̄).Source: §III · p. 4
• The exact word count is the source of the exchange-rate argument: a T gate carries at most one bit.The passage attributes the one-bit bound to log2 2 = 1.Source: §III · p. 4

Proof strategy

• Conditioning on the other rounds lets the proof recover a share from a successful coordinate output, so its information must fit in the snapshot and committed message.The proof applies nearest-grid decoding and Fano’s inequality, then bounds the message size using snapshot entropy and the number of committed T records.Source: §IV · p. 5

Scope

• The factor r − 1 is specific to one-pass processing: a process that rereads earlier rounds after seeing the final round need not commit during those rounds.For multiple forward passes, the argument instead uses crossing snapshots and committed unitary tuples in an alternating-simulation transcript.Source: §IV · p. 5

Cancellation

• The lower bounds still hold when the aggregate is zero and the target unitary is identity; a universally correct process has already committed the same expected magic.A process promised in advance that the aggregate is zero can emit nothing; the claim concerns a stream that turns out to cancel.Source: §IV · p. 5

Elementary tube bound

• Theorem 4 bounds the number of words near any rotation coset with exponent 0.4513, improving the spectral remainder exponent 1/2 uniformly over frames.The determinant method lifts words to quaternion numerators on spheres in both real embeddings; successive determinants reduce points in boxes to sphere sections and then circles.Source: §V · p. 6

Rate eleven fifths

• Rate 11/5 follows from the elementary tube bound for every process and frame, once L is sufficiently large.The stated threshold is L0 = 110; the result raises the rate without assumption (R).Source: §V · p. 7

Height dichotomy

• Height splitting improves the tube exponent to 0.41114 and yields rate 17/7 by treating high- and low-height sphere sections differently.High-height sections use a dual-lattice fibration; low-height sections are counted through their normals and rational spans.Source: §V · p. 7
• The rate-17/7 result is asymptotic only: its small exponent margin makes the implied constants and threshold for a nontrivial bound astronomically large.The numerical limit of the stated lemmas is near 2.435; 17/7 is essentially their limit, not the full method’s limit.Source: §V · p. 7

Rational projections and rate five halves

• Further rational-plane and projection arguments give the exact-certificate rate 249/100 before the continuum analysis extends the result to every α < 5/2.The added cases include annular fibration, projections of boxes, and rank-one configurations.Source: §V · pp. 7–8
• Every rate α < 5/2 holds unconditionally and uniformly over frames, using a continuum case analysis checked exactly by an SMT solver.The method approaches 5/2 but does not reach it.Source: §V · p. 8

Method boundary

• The determinant method’s five-point step forces points in a δ-cube onto a hyperplane only for α < 5/2; beyond that, the method needs additional sparsity evidence.The paper notes that a dense-tube construction would prevent improving Theorem 8 along the relevant sequence.Source: §V · p. 8

Construction

• The family stores q coordinate sums and commits shares of the other m − q coordinates as they arrive, with S ≤⌈q log2 Q⌉+ O(log(mQr)).Its committed cost per round is bounded by (m − q)τ(ε/r), and the coordinatewise error is at most ε.Source: §VI · p. 8

Achievable slope

• When the tuned grid’s mean single-rotation cost is (3 + o(1)) log2 K, fractional passthrough achieves asymptotic slope 3.Ross–Selinger synthesis gives typical cost τ = 3L + O(log L); the worst case is 4L + O(1).Source: §VI · pp. 8–9

Initial tradeoff

• The first bits can be cheaper than the passthrough slope: when 8 divides Q, three bits per coordinate can be shed for an average cost of one-half T gate.The construction depends on grid divisibility; for general Q, the saving is log2 gcd(Q, 8) bits.Source: §VI · p. 9

Conjecture 1

• The general rate-three conclusion depends on Conjecture 1, whose tube-count bound includes a linear term forced by powers of infinite-order words.The conjectured count is stated for all cosets and costs; the enumeration constrains constants from below but does not determine them.Source: §VII · pp. 9–10

Conjecture 1 and Theorem 10

• Under Conjecture 1, the exchange rate is 3, and a committed word carrying more than O(log L) bits about its coordinate costs at least 2L − O(1) gates.At ε = 10^-10, the stated low-cost allowance is 7.0 bits, versus 32.9 bits carried by a coordinate.Source: §VII · p. 10

The quantum of commitment

• The commitment is quantized: words of cost at most about 2L hold only O(L) possibilities, so bits beyond their O(log L) allowance require cost at least 2L − O(1) per word.The argument permits a mixture of cheap words and rare expensive ones; fractional passthrough pays by committing whole coordinates.Source: §VII · p. 10

The entry fee

• At the identity coset, every word of T-count at most 2L − 8 within ε of a z-rotation is a power of T, giving an unconditional 2L − O(1) entry fee.Enumeration places the first non-power examples at T-counts between 2L and 2L + 3 for the listed accuracies.Source: §VII · pp. 10–11

Clifford-framed cosets

• At Clifford-framed cosets, the volume law holds up to a subexponential factor, yielding an unconditional 3L − O(L / log L) lower bound for all but a vanishing fraction of z-rotations.The result also holds for almost every angle at all sufficiently small dyadic accuracies.Source: §VII · p. 11

Rotation-segmented processes

• Rotation-segmented processes with segments within ε of Clifford-framed z-rotations have exchange rate at least 3 − O(1/L), and fractional passthrough attains rate 3 to leading order under the synthesis hypothesis.The lower bound is unconditional; the matching upper bound assumes the synthesis hypothesis.Source: §VII · p. 11

Side information

• With side information Y, the resource bound uses conditional entropy H(X | other rounds, Y) rather than the full aggregate entropy.Theorem 13 gives a Shannon form, while Theorem 14 gives a one-shot form.Source: §VIII · p. 12

Examples and consequences

• If Y determines the earlier shares, the bound is zero; if those shares encode an unknown λ-bit seed, the bound degrades from m log2 K to λ.The seed example identifies the entropy missing to the process, rather than aggregate size, as the relevant quantity.Source: §VIII · p. 12
• A seed spent in earlier rounds and canceled later does not provide a general lower bound, although the upper bound λ survives for general r.The passage’s counterexample uses three rounds, with the first two shares canceling and the final share equal to X.Source: §VIII · p. 12
• For non-product sources, the stated bounds replace the product-source entropy term with conditional entropies and add a total-correlation term to the snapshot cost.The passage specifies the adjustment as Σ_j H(A_j | Z) − m(h2(δ) + δ log2 K) and S + TC(A | Z).Source: §VIII · p. 12

One-shot form

• The Shannon form applies to expected resources and the one-shot form to worst-case resources; the two forms should not be mixed.The discussion states the bound using Hδ^0 and notes a further smooth max-entropy relaxation with a different smoothing parameter.Source: §VIII · p. 12

Conjecture 1 (Fig. 1)

• Across 30 well-populated Haar-pair cells, N divided by its exact prediction lies in [0.98, 1.04] for grid counts and [0.98, 1.01] for tube counts.At T-count 22, the pooled Haar-pair ratios also track the predictions across tested accuracies.Source: §IX · pp. 12–13

Arithmetic cosets

• Arithmetic-frame fits give log2 N = 1.006t − 2.003L for grid counts and 1.002t − 2.009L for tube counts, close to Haar-pair exponents.Clifford frames reproduce identity-coset counts, while arithmetic and Haar-random frames have at most three low-cost hits in the reported comparison.Source: §IX · p. 13

Axis cosets and the size of c

• A finite-cost identity-coset step requires c ≥163.9 at τ = 11, despite the dyadic scan's maximum required c of 53.5.The scans miss the larger step because it occurs at a non-dyadic accuracy.Source: §IX · p. 14

Exact optimal synthesis and the frontier (Fig. 2)

• At L = 5 and 6, the certified ε/2 fractional-passthrough slopes are 3.333 and 3.334, while ε-calibration slopes are 2.478 and 2.721.The ε/2 sequence is the r = 2 family bounded by Theorem 9; the ε sequence is a single-rotation calibration.Source: §IX · p. 15
• The exact-optimal partial-commitment variant costs 4.8, 3.2, 2.9, and 2.7 T gates per bit shed for ℓ = 1,…,4 at L = 5, worse than passthrough.At chemistry scale the quantum of commitment persists: partial commitment costs 61 to 14 gates per bit shed for ℓ = 1,…,8.Source: §IX · p. 15

Chemistry scale (Fig. 2b)

• At ε = 10−10, 20,000 sampled rotations give Eτ = 105.34 at the certified ε/2 share accuracy, or 3.205 gates per bit.The single-rotation ε calibration is Eτ = 102.32 and is not the certified full-stream point.Source: §IX · p. 15

The bounds at chemistry scale (Table III)

• At ε = 10−10 and S = 0, Theorem 10 gives 52.9m committed gates versus 105.34m for achievable passthrough, conditional on Conjecture 1.Theorem 3 gives 25.72m and Theorem 1 gives 21.75m; these unconditional bounds are within factors 4.1 and 4.8 of achievability, respectively.Source: §IX · p. 15

Fractional passthrough versus lower bounds

• At ε = 10−10, the certified ε/2 fractional-passthrough point is 105.34 gates per coordinate per round at S = 0, against a 52.9m conditional bound.The bound uses Theorem 10 and depends on Conjecture 1.Source: Figure 2 · p. 16, Table III · p. 16

Phase-gradient model

• The phase-gradient catalyst removes the phase-gradient model's entry fee, while the batching construction prepares its register once and returns it unchanged.At ε = 10−10, batching costs 80 T gates per share for m = 10^4, r = 2, and total error ε.Source: §X · pp. 16, 18

Mixed-channel models

• At small angles, mixed-channel synthesis can realize rotations with average cost Õ(θ²/δ); mixing rescales the exchange law by one half rather than removing it.The low bits of a share are the small angles that can be shed cheaply.Source: §X · p. 16

Batched commitment

• With clean ancillas, a catalytic phase-gradient register, and batching across coordinates, the committed cost is (4 + o(1))mL/log2 L and the exchange rate is O(1/log log(1/ε)).The construction commits each share on arrival and leaves only a program counter in the snapshot.Source: §X · pp. 17–18

Model scope

• In the fully general n-qubit model, ancilla-assisted synthesis across coordinates can cost far less than summing single-qubit rotation costs, so no constant-rate law survives.The paper locates its lower bounds in coordinatewise synthesis, with ancillas bounded per coordinate.Source: §X · pp. 16, 18

Batched commitment

• At ε = 10−10, batched synthesis uses 64 T gates per share under the coordinatewise criterion, versus 105.34 for ancilla-free passthrough.Asymptotically, the ratio tends to zero like 1/log L; the paper notes that the constant-rate lower-bound side remains open for ancilla-assisted cross-coordinate synthesis.Source: §X · p. 18

The law under mixing

• A non-held coordinate costs at least L−O(log L) committed T gates per round under mixing, half the deterministic 2L floor, while chargeable bits still cost two gates per bit.Here L = log2(1/ε); mixing lets a committed word be accurate to √ε, so only the top half of the bits are charged.Source: A. The law under mixing · pp. 18–19
• The cited passage says obtaining the average 3/2 L requires an assumption in place of Corollary 3, and defines fax as the worst-case deterministic-synthesizer cost at that accuracy over axial targets.Source: A. The law under mixing · p. 20
• The achievable cost is O(1) for low share bits, rises over the next 1/2 log2 L bits, and reaches 3/2 L near the share’s midpoint; the exact transition remains unknown.The construction makes remembering the high half of a coordinate’s bits worthwhile for O(1) committed gates, whereas remembering the low half saves nothing; the boundary estimates agree only within O(log L) bits.Source: A. The law under mixing · pp. 20–21
• Under Conjecture 1, the lower bound approaches 3/2(r−1)(m log2 K−2S), matching the achievable leading term at S = 0 but not its dependence on S.The bound’s slope in S is 3(r−1), while the whole-coordinate passthrough family’s slope is 3/2(r−1); these are rates per different units and should not be treated as one exchange rate away from S = 0.Source: A. The law under mixing · p. 21
• With bounded-ancilla measurement adaptivity, the floor remains order L but falls by a factor depending on ancilla count; with one ancilla it is L/12 gates plus measurements per round.The passage compares this lower bound with mixed fallback at approximately 0.53L achievable cost; the bound’s constant is poor because it counts transcripts crudely.Source: A. The law under mixing · p. 22
• Across deterministic and mixed models, mixing halves quantities linear in L but leaves the rigorous-to-conjectural-to-achievable ratio at 2:3:3 and does not change the exchange rate.The mixed achievable costs cited in this comparison are conditional on synthesis hypotheses, and the lower bounds are asymptotically vacuous at ε = 10^-10.Source: A. The law under mixing · p. 22

A. The law under mixing

• At ε=10^-10, γ=1/4, and S=0, (29) gives T_t ≥4.1m, while (30) is vacuous until L≥56 (L≥45 under bounded support).Theorem 15 is asymptotic in the same sense as Theorems 1 and 3.Source: A. The law under mixing · p. 22

XI. DISCUSSION

• For Clifford-framed rotations, the lower bound is 3 − o(1), and a passthrough family attains three when grid-mean synthesis costs (3 + o(1))L.The attainment is asymptotic in L and applies after the first O(log L) bits of each coordinate.Source: §XI · p. 22
• Every α < 5/2 is proved unconditionally and uniformly over frames; 5/2 is the stopping point of the elementary method.Closing the gap from 5/2 to 1+β for arbitrary frames remains open.Source: §XI · pp. 22–23
• The volume rate is three because a rotation coset has codimension two, while the spectral method yields two through its square-root remainder.The elementary geometry-of-numbers method crosses that spectral barrier but stops at 5/2 as configured.Source: §XI · p. 22
• Under Conjecture 1, the general rate is three, but whether the true rate is three is not established here.The enumeration supports the conjecture, and the Clifford-coset case is proved up to a subexponential factor.Source: §XI · p. 22
• For deterministic words, memory shedding is quantized in whole rotations beyond the initial O(log L) bits each coordinate can absorb.The discussion reports a 4.5-to-20-times passthrough-slope penalty for shedding low bits under Ross–Selinger synthesis at ε = 10^-10.Source: §XI · p. 22
• The stated bounds do not apply to quantum-generated phase coefficients, since the proof uses Fano’s inequality on a classical stream.No reduction from that setting to the classical-input model is provided.Source: §XI · p. 22
• Probabilistic mixing makes low bits of every share free while coarse bits retain the same total rate; batched phase-gradient lookups remove the constant-rate law.With clean ancillas, a phase-gradient catalyst, and batched table lookups, committed cost per share drops to O(L/log L).Source: §XI · p. 23

DATA AVAILABILITY

• Scripts, data, and exact certificates for the paper’s tables, figures, and theorems are available in the cited repository.The repository README maps each table, figure, and theorem to its producing script and data file.Source: DATA AVAILABILITY · p. 24

Appendix A: The streaming model

• A charged one-pass process emits a snapshot at each round cut, and later output must depend only on that snapshot and the remaining stream.The restart rule models an append-only instruction stream: what has not been written must be remembered.Source: Appendix A · p. 24
• For 0 < ε ≤ 1/8, the tuned modulus ensures the stated accuracy condition and gives log2 Kε = L − 0.35 + o(1).The appendix obtains this from Qε = ⌊π/(2 arcsin 2ε)⌋.Source: Appendix A · p. 24

Appendix B: Proofs of the lemmas

• Projective distance between equal-size unitaries is determined by the shortest spectral arc: dproj(A, B) = 2 sin(Θ/4).Here Θ is the length of the shortest closed arc containing the spectrum of B†A.Source: Appendix A · p. 24

Appendix A: The streaming model

• The block compiler stores partial sums for q designated coordinates in at most ⌈q log2 Q⌉ + O(log(mQr)) bits and recovers their exact aggregates.It updates each partial sum modulo Q as shares arrive.Source: Appendix A · pp. 24–25

Appendix B: Proofs of the lemmas

• The appendix uses a unique T-optimal normal form for Clifford+T unitaries, with 24 Clifford choices modulo phase.This form supports the subsequent count of words by T-count.Source: Appendix B · p. 25
• For a unitary close to diagonal, the norm equations force its off-diagonal term to vanish, yielding a diagonal Clifford up to phase.The proof uses the two real embeddings of the relevant norm equation.Source: Appendix B · p. 25

Appendix C: Proof of Theorem 2

• The Hopf map sends the rotation coset problem to spherical caps on S2, with cap measure ε^2 for the relevant ε-neighborhood.The cap is centered at the projection of the coset, and its angular radius satisfies sin^2(θε/2) = ε^2.Source: Appendix C · p. 25
• A Poisson-kernel majorant covers the cap’s preimage while retaining geometrically decaying spherical-harmonic coefficients at scale 1/ε.The construction needs no smoothing or truncation.Source: Appendix C · p. 25
• The proof averages over the octahedral group and uses rank-one harmonic coefficients plus the Ramanujan bound to control the word-count contribution.The group averaging makes the function left-invariant, while the harmonic estimates bound its components.Source: Appendix C · pp. 25–26

Appendix D: Proofs of Theorems 3 and 10

• The cost-weighted mutual-information lemma bounds information about a finite-support variable using the output cost and the number of outputs available below each cost threshold.The proof splits outputs at a threshold and applies an entropy bound to the lower-cost portion.Source: Appendix D · p. 26
• The cost-weighted entropy lemma assumes a word-count profile with a power-law term and requires ε ≤ (2c)^-1/β.Under that condition it separates a finite low-cost alphabet from the remaining words.Source: Appendix D · p. 26
• Theorem 3 applies the information bound coordinatewise, then uses conditional independence and the snapshot entropy budget to lower-bound emitted T-cost.The per-coordinate decoding step uses Fano’s inequality, and the summed snapshot information is at most S.Source: Appendix D · pp. 26–27
• Theorem 10 conditions its cost-profile argument on Conjecture 1, separating low-cost words from the high-cost tail.The proof controls the low-cost alphabet size and bounds the information contribution from the high-cost event.Source: Appendix D · p. 27

Numerators

• Quaternion numerators lie on spheres in both real embeddings, and proximity to a rotation coset becomes a tube-counting problem around a great circle.The two-to-one map from numerators to unitaries and the tube radius reduce the target count to points of X_t.Source: Appendix E · p. 27

Step 1: boxes

• The determinant argument places points in each box in an affine hyperplane, using small determinants in one embedding and the nonzero norm lower bound across embeddings.The box conditions are 2a + 3b > 8 and a0 + a ≥ 2b.Source: Appendix E · pp. 27–28

Patches and cells

• Within a sphere section, patches constrain core parameters and offsets; finer cells then place the points on a circle when the cell-exponent condition holds.The cell argument uses Lemma 16’s hypotheses and inequality (E1).Source: Appendix E · p. 28

Assembly and certificate

• The circle bound and cell-count estimate feed the assembly, while numerical optimization finds these lemmas cannot give a rate above about 2.234.The stated 2.23 rate follows from the alternative parameter computation in the proof.Source: Appendix E · pp. 28–29
• log_R′ #X_t ≤ 1.805001, yielding #X_t ≤ 2^(0.4513t)+O(t/log t) after summing over T-counts.The exact rational certificate checks cell choices over sphere radii and gives the bound in (9).Source: Appendix E · p. 29

Appendix F: Proof of Theorem 6

• For a0 = 28/17, a = a0, b = 157/100, and E* = 11183/6800, the proof establishes #Xt ≤ R′^E*+o(1).For t ≤ (17/7)(L+2), summing the bound over T-counts ≤ t gives (11).Source: Appendix F · p. 29
• The tube is covered by O(R′^b) boxes; points spanning a hyperplane lie on its sphere section, while planar cases are bounded separately.The section is classified by its radius r = R′^(1−g), with g its class.Source: Appendix F · p. 29
• Sphere-section patches have bounded convex-hull volume, and sections of sufficient height intersect small tube regions in few circles and points.For height at least R′^y and patch volume at most R′^v, the tube points lie on at most 1 + C R′^(1+(v−y)/3) circles.Source: Appendix F · p. 30
• Low-height sections are controlled by lattice minima and wedge-height bounds, followed by a fibration dichotomy that bounds their total contribution.The dichotomy compares H_V R′^4ς^2 with R′^E*; in the complementary case, the sector count is at most C R′^max(y, 2y−X−max(0,ϑ+γ)).Source: Appendix F · pp. 31–32
• The assembly divides sections into tiny, large, and intermediate classes, then counts low-height layers using tangent or crossing offsets and per-section bounds.High sections are covered by a cost threshold, while low layers are counted globally rather than box by box.Source: Appendix F · p. 32
• 1.64453 < E*=1.64456, and the exact certificate proves (11) with η(t) = O(t/log t).The rational computation checks every case; the largest checked exponent is 67097/40800 = 1.64453.Source: Appendix F · p. 33

Appendix G: Rational projections

• The annular fibration counts projected lattice values rather than points on the sphere, since points sharing a projection lie on a circle.The projected values lie in a rank-4 lattice and their first embedding lies in an ellipse, which is covered by rectangles for the count.Source: Appendix G · p. 33
• When at most one successive minimum is small, the number of normals is bounded; box projections also reduce sphere-point counts to lattice counts or circles.For hyperplane normals lying in a K-plane, projection gives at most 1 + ēR′HV/h(nB) values, each with a circular fibre.Source: Appendix G · pp. 33–34
• The assembly replaces G with G3, removes y from nor(y), and retains a height factor in the rank-three case.These changes combine the new projection bounds with the prior layer-counting argument.Source: Appendix G · p. 34
• Theorem 7’s finite instance gives rate 249/100; Appendix H replaces this finite instance with the continuum form of the same assembly.Source: Appendix G · p. 34

Appendix H: Proof of Theorem 8

• For 0 < µ ≤ 1/10, z3 verifies the continuum claim with T = E1 + (10/33)µ, yielding the stated rate for Theorem 8.The proof reduces the height split to a quantifier-free linear real arithmetic check; near the limit, the method gives exponent 8/5 − (4/11)(a0 − 8/5), and nothing better.Source: Appendix H · pp. 34–35

Appendix I: Proof of Theorem 11

• Theorem 11’s count reduces words near a Clifford-framed z-rotation torus to quaternion coordinates, bounding the (c, d) choices by C(1 + ε^2 2^τ).For each (c, d), the remaining coordinate pair is bounded using ideal divisors in Z[ω]; the proof sums over costs up to τ.Source: Appendix I · p. 35

Appendix J: Proof of Theorem 14

• For Theorem 15(a), list decoding and Fano’s inequality lower-bound information about the phase from the message and snapshot, while their entropy upper-bounds the same information.The decoder forms a list of size at most (2w + 1)^m containing the phase with probability at least 1 − δ.Source: Appendix J · p. 36
• Frame clustering shows that when two unitary distributions have average infidelity at most η^2, each has a set of at least 3/4 mass whose elements are mutually within 2η.In the share-independent-frame argument, the proof anchors clustering at a value determined by the snapshot and side information, then applies it to the actual share.Source: Appendix J · pp. 36–37
• Coarse cell labels preserve conditional entropy and identify each angular window up to two possibilities, enabling the entropy argument that yields (30).The cells have sizes between h and 2h, where h = 2w′ + 1.Source: Appendix J · p. 37
• Discretizing the rotation angle into 9Q1 candidates places every W in one of nine sets counted by Conjecture 1, giving the union-bound estimate (34).The proof bounds the distance to a candidate angle by ε, so the resulting set uses accuracy 2ε.Source: Appendix J · p. 37
• A branch is encoded by T-gate and measurement tokens followed by a Clifford, yielding an entropy bound for transcripts with a given total token count.For N total tokens, the transcript count is bounded using the Clifford-group size and the token alphabet.Source: Appendix J · p. 38
• Each branch induces a unitary on the coordinate qubit with input-independent probability, so the implemented operation is a mixed-unitary channel.The ancillas are returned to |0⟩ or measured, and do not carry additional information forward.Source: Appendix J · p. 38