[amaybaum]

Reviewed: independent ACCEPT at af8df1b4ba2a39c895775780027497b76c71eb46; CI green there. Paper only, draft. Guiding PR for the apparatus milestone of #1169, steps 1–2 (owner direction, scope and writer; owner direction after the first review).

Stacked on #1173. The base is #1173's branch at 62502618, so this diff shows only the new file. Once #1173 lands, this PR is retargeted to main.

One new file: verification/programmes/oi-qm/reconstruction/apparatus-1-20261107/DESIGN.md.

Result (as reviewed)

• Target (step 1). One carrier, one heralded preparation, a selectable Z/X read with a reusable continuation. The comparator is the Pauli Lüders instruments.
• Ceiling.
  • Escape is decided by OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914's transported quarter-phase test (C-PH) and rot test (C-ROT).
  • The per-state bound P + V ≤ 1 (C-IS) is recorded as a diagnostic only; exceeding it does not establish nonclassicality.
  • The noncontextuality witness is MR-1's certified-quadruple form (C-NC).
• Candidate SA-1.
  • Native: a symmetric second-order step.
  • Supplied: the per-tick arming coefficient (P3), the ready cell with one known slot, terminal registration, cell isolation (E-iso) and H-PREP.
  • The observer-access assessment and MR-1's countercontrol CC4 are its precursors.
• Decisive test (step 2), exact over F₂.
  • The three-record law equals the sequential Pauli Z/X Lüders law on one preparation, in all four setting combinations.
  • Failure is heralded, the continuation is reusable, and the eraser works only before registration or copying.
  • Controls derived: C1, C3, C4, C5, C8, C9. C2 and C6 are deferred to step 3.
  • No Y read is exhibited.
• Verdict: no escape. Under the one-cell identification, the quarter phase and rot are unreachable.
• The stop, by the owner's three levels:
  1. Per-state bound. The 4-point model MP-1 exceeds it while staying noncontextual (given a supplied Y read).
  2. Quarter phase. The finite 8-point carrier P-8 supplies it. The cost is transformation contextuality, which by Lillystone–Wallman–Emerson Theorem 1 every preparation-noncontextual model of the full stabilizer subtheory pays.
  3. Rotations. Inducing rot with exact statistics excludes every finite carrier (Perron–Frobenius and Niven, derived in §5.3). A ball-containing carrier (MP-2) is one sufficient condition.
• Open for step 3:
  • Q-A: can OI natively produce the P-8 mechanism and its contextuality?
  • Q-B: what OI principle could produce the continuous rotations?

Review and validation

• First independent review at 6fef72b7: REVISE, R1–R12.
• Delta review at 7a3f7a52: R1–R12 resolved; ACCEPT WITH CHANGES, N1–N4. CI 42406330966 green.
• Final delta check: M1 at 4c88f2b8, then ACCEPT at af8df1b4. CI 42407544338: 25 successful jobs and one conditional skip.

Author and reviewer exact self-checks were run locally. They are not committed and not cited as evidence.

Not authorized: premise adoption, simulation or enumeration, Lean, dispatch, freeze, manuscript change, claim promotion or merge. Step 3 (Q-A, then Q-B) waits for the owner.

🤖 Generated with Claude Code

https://claude.ai/code/session_013T6pfKuv5mi3MXtKjH2tZb

[amaybaum]

Transcribed from a separate independent reviewer, an agent of this Claude Code session that is separate from the author. It reviewed read-only at exact head 6fef72b7b08b807dc450250e1a2c8f4b096c1ef9.

Independent review — Apparatus-1 DESIGN.md (#1183)

Object: verification/programmes/oi-qm/reconstruction/apparatus-1-20261107/DESIGN.md at exact
head 6fef72b7b08b807dc450250e1a2c8f4b096c1ef9 (parent 62502618, main e61e5673). The worktree
was read only. Nothing was edited, committed, pushed or posted.

Verdict at exact head: REVISE

The §4 algebra is correct, and I re-derived it exactly. That covers the coupled tick, the
parity-selected Z/X reads, the three-record law in all four setting pairs, the failure branch,
the eraser, Lemma 3, Lemma 5 and the rot and MP-1 numbers. The stop for SA-1 also holds: under
the one-cell tetrahedral identification R1, C-PH and C-ROT both hold, so SA-1 does not escape
#914's transported phase and control tests.

The file is not acceptable as written, for three reasons:

• The ceiling contains a substituted criterion. As defined, C-NC is a per-state bound. It is
  not MR-1's noncontextual witness W1, and both MP-1 and the "square" violate it inside
  noncontextual 4-point models.
• The stop's premise ladder leaves out a finite rung. That rung buys C-PH, and §2 of the file
  itself promised to state it in §5.
• Several claims of nativeness and novelty are stronger than the derivation supports.

These defects change what §5 says MP-1 buys and what the missing assumption is for the quarter
phase. §5 is the owner's step-3 deliverable, so the changes are substantive and not only wording.
§4 needs no recomputation.

Required corrections

R1. C-NC is not the noncontextual P+V bound. D3 counts noncontextual classical models as
escaping (L20–28, L161–172, L376–378, L52, L56, L655–666, L686–692, L780).

• What MR-1 actually uses. W1 is "a violation of a T7 inequality with equivalences certified
  by §5.3 over Acc" (MR-1 prereg L519). Catani et al.'s result, as MR-1 reads it at L753, needs
  the A21 symmetry condition: a quadruple of preparations carrying an operational equivalence. MR★
  Lemma G (L1020–1042) proves that every finite classical control whose preparations and readouts
  span (hypothesis (ii)) is generalized-noncontextual, and so meets V + P ≤ 1 "on every certified
  quadruple".
• What C-NC actually is. C-NC is the per-state statement P + V ≤ 1 for every reachable state.
  That is the isotropy (knowledge-balance) bound of the toy theory, not a noncontextuality bound.
• Exact check. I ran a linear program over arbitrary laws on F_2². The maximum average
  P + V on a certified quadruple (μ00 + μ11 = μ01 + μ10) is exactly 1. Without the
  equivalence the per-state maximum is 2.
• Consequence for MP-1. The MP-1 law and the sharp laws (the "square") are GNC by Lemma G(ii)
  and G(i) respectively. They can never produce W1. MP-1's P + V = 2/√3 therefore violates the
  per-state bound inside a noncontextual 4-point model.
• Consequence for D3. As written, D3 calls such a model an escape of the "applicable ceiling".
  The resulting ceiling is easier to escape than both OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914's and MR-1's. OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914 itself warns that
  escape is not quantum sufficiency.
• Fix.
  • Redefine C-NC as the per-state isotropy bound and drop the "noncontextual" label. Alternatively,
    replace it with MR-1's certified-quadruple form.
  • Remove the per-state C-NC from D3's escape disjunction, or state that a violation of it alone is
    not a nonclassicality escape.
  • In §0 item 5, §5.1, §5.2 and §5.6, say that MP-1's and the square's P + V > 1 occur in
    noncontextual 4-point models, and stop calling P + V = 1 "the noncontextual / classical
    boundary".
  • SA-1 itself being GNC (Lemma G(ii) applies verbatim) is correct and can stay.

R2. The stop's premise ladder omits the finite 8-point rung for the quarter phase (L216–218,
L54–64, §5.2–5.4, L814).

• The broken promise. §2 says the "enlarged finite ontic space per qubit" and the
  "permanently inaccessible functionals" alternatives "appear only as stated alternatives in §5".
  Neither appears in §5. Wallman–Bartlett is cited only as "not selected".
• Why the rung matters. It sits exactly between MP-1 and MP-2 for C-PH. I checked this exactly:
  • Take ontic λ ∈ {±1}³ (8 points), state laws μ_r(λ) = (1 + r·λ)/8, and the read of axis a
    giving sign λ_a.
  • Laws are nonnegative exactly on the octahedron, so the per-state isotropy bound still holds.
  • The read laws are exactly (1 + r_a)/2.
  • The quarter phase is an ontic permutation of the cube vertices.
• Consequence. A finite carrier with an independent third bit (Y ≠ z + x) escapes C-PH while
  keeping C-ROT and the per-state bound. Lemma 5's obstruction is therefore specific to the 4-point
  carrier, where Y = z + x. By Lillystone–Wallman–Emerson (arXiv:1802.06121), the price is
  transformation contextuality. Hardy's infinitude is needed only once the repertoire goes beyond
  the stabilizer fragment, as it must for rot.
• Fix. State this rung in §5, as §2 promised. Place MP-1, the finite 8-point rung and MP-2 by
  what each actually buys: per-state bound, C-PH, and C-ROT/repertoire respectively. Do not imply
  that the quarter phase needs MP-2.

R3. SA-1(a) is called native, but its controlled per-tick per-edge coefficient is not native
(L227–232, L245–247, L35–36, L255–267).

• The native wave rule has a fixed, translation-invariant coefficient (Substratum A4; the
  recombination paper's "coefficient one is the retained reference update").
• SA-1(a) instead has k loaded classically per tick per edge from a controller. That is the
  controlled adjacency the ledger already lists as P3.
• The setting selection by tick parity rests on P3. This is the step that observer-access-20261106
  §6 item 2 identified as missing.
• Fix. Split (a) into a native part (a symmetric second-order linear step, hence symplectic) and
  the posited controlled arming (P3). In §0 item 3, change "On the native F_2 alphabet, the
  apparatus derives …" so that it says the read selection rests on P3 and E-iso.

R4. The hypothesis ledger omits H-PREP, and N2 is misclassified (L341–350, L284–288, L470–474).

• Two steps rest on H-PREP, which is open on main (observer-interface-obsmu-20261104/ANALYSIS.md
  L892–899; owner D-OB8):
  • reading the r0-conditioned classes as a laboratory preparation (X_{r0}, §4.4, D2);
  • applying the counting measure "conditionally on the ready constraint" to model a supplied ready
    cell (N2).
• Fix. Add H-PREP to the ledger. Reclassify N2 as "native measure, applied through the open
  H-PREP identification". The unconditional three-record law (D1) does not need H-PREP. Say so.

R5. The novelty claims are too strong against the prior screen and against CC4 (L32–34,
L208–215, L51–53, L637–644, L777).

• The omitted precursor. observer-access-20261106/ASSESSMENT.md, which the owner's step 1
  points to as the prior screen, is not cited. It already examined:
  • the native bidirectional (reciprocal) coupling, its eq. (3);
  • a uniform hidden background masking the receiver;
  • later background access restoring the signal, which is the eraser;
  • the ready slice K = 0.
    It stopped at exactly the preparation, setting and access gates that SA-1(b), (c) and P3 now
    posit.
• What CC4 already contains. MR★ CC4 (L1066) already has symplectic marking with a half-known
  ready marker: x_M ← x_M + x, p ← p + p_M. So the marker-mediated backaction was derived from
  a symplectic coupling there, not "stipulated".
• What is new. The identification of that marking with the native step (under E-iso and P3),
  the terminality posit and its retention analysis (Lemma 3, C9), and running the result through
  OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914's tests.
• Fixes.
  • Cite observer-access-20261106 as the precursor.
  • Narrow "No earlier record derives the disturbance from the native coupling" and "stipulated
    (CC4)" to what is actually new.
  • Replace "The open question it answers" with wording saying the file supplies posited access,
    (b) and (c), under which the restriction holds and survives retention. S-L7 already gave the
    conditional answer.
  • Replace "validates CC4 as natively reachable" (L777) with "reachable under P1–P3".
  • Replace "Up to the native derivation of its access premises, it is CC4" (L52–53) with: it is
    CC4 with the marking identified with the native step, and its access premises are posits.

R6. Controls are marked green when they are not computed, are vacuous, or are deferred (L44–45,
L568, L572–573, L351).

• C2. It is vacuous at one wing and its substantive two-wing part is deferred, yet it is marked
  "green at this scope".
• C6. It invokes Main §3.3's Bell ceiling, whose hypothesis is that "the protocol records a ±1
  outcome in every run". SA-1 has ⊥ branches, and no two-wing SA-1 is constructed.
• C7. It restates the design; it is not a control.
• AGENTS rule. A vacuous control cannot certify a verdict.
• Fix.
  • Mark C2 and C6 "deferred to step 3, not run". Mark C7 "design conformance".
  • Change §0's "All preregistered controls are green" to name the controls actually derived:
    C1, C3, C4, C5, C8 and C9.
  • Optionally add to C6 that the heralds are setting- and carrier-independent, so conditioning on
    success does not open a detection loophole.
• Same-commit preregistration. The "preregistration (fixed before §4)" is in the same commit as
  the result, so its priority cannot be checked. Call it "decision rule stated in this file", or
  record that it is unseparated.

R7. The scope of the negative statements needs narrowing (L48–50, L590, L196, L661–666).

• Lemma 5 and §0 item 4. Lemma 5's impossibility holds for the one-cell identification R1,
  where Y = z + x, as R2 shows. Add "under R1" to Lemma 5 and §0 item 4.
• "Enlarging the carrier or the operation set does not address it" (L196). This is contradicted
  for C-PH by the 8-point rung. Restrict it to the disturbance requirement.
• "Every discrete relaxation of it overshoots the quantum region … none yields the disk"
  (L661–666). Only the three clause removals and the biased kick were computed. State it for
  those four variants.

R8. The NonnegBounded and OnesBounded predicates are overread (L101–103, L139–142).

• What the definitions quantify over.
  • NonnegBounded quantifies only over one-outcome (availExt n Unit) operations, that is, over
    coarse-grained complete instruments.
  • OnesBounded quantifies only over available unitary conjugations.
• The overreading. Neither predicate by itself "bounds … its branches", and OnesBounded does
  not bound instruments at all.
• Fix. State this precisely. The argument is unaffected, because the quarter phase and rot
  are one-outcome unitary conjugations.
• Wording at L141. "the hypotheses of … permClass_onesFixing hold" should read that the
  apparatus's admissible operators lie in permClass, so its theory is a subtheory of
  permTheory. permClass_onesFixing has no hypotheses.

R9. The eraser-after-copy count is mislabelled (L547–551).

• I fixed the word as: Z read at t; m–f copy with the carrier free at t+1; free tick on
  both cells; c–m coupled.
• Under that word, the full pre-read state (z, x) fails to be restored in 12/16 assignments.
  x alone fails in 8/16: the returned conjugate is x + w, which is uniform and independent
  of x.
• The file says "no longer restores x. Twelve of the sixteen … fail", which conflates the two.
• Fix. State the exact word and say "the returned value is x + w" (or "the full state fails
  in 12/16").

R10. "Independent consequences" (ii) and (iii) are not independent support (L298–303).

• (ii), unavailability of the encoded quarter phase, is a known property of the q = 2
  epistricted (toy) theory. Spekkens 1409.5041 says binary epistricted theories are "a distortion"
  of the stabilizer subset. It is also the stop itself, not a success beyond the target.
• (iii), dyadicity, follows directly from uniform bits at q = 2.
• Fix. Retitle the paragraph "further consequences" and cite the toy-theory literature for (ii).

R11. The dyadicity claim needs a scope (L523–524, L630–631). With herald probability h not
dyadic, the unconditional record laws are not dyadic. Scope the claim to records conditional on
herald success, or to dyadic h. The C-ROT dyadicity argument holds in that conditional form.

R12. The terminality motivation omits the native recurrence caveat (L274–277). On a finite
substratum φ^N = id (MR★ S-L6), so "copied irreversibly into an archive" holds only relative to
a budget below recurrence. Terminality is enforced by P3/E-iso, not by the native dynamics. Add
the caveat.

Optional notes

• Strengthening Lemma 5 beyond R1. SA-1 is GNC for transformations as well (the Lemma G(ii)
  spanning argument over products). Lillystone et al. show that the single-qubit stabilizer theory
  with its Clifford transformations is transformation-contextual. Together these suggest that no
  identification of an SA-1 subsystem reproducing states, Pauli reads and S exists, independent
  of R1. This is my inference. The author should check Lillystone's hypotheses before using it.
• Dense-control ceiling. This OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914 test is not transported. Lemma 5 extends to it because the
  set of stochastic matrices is closed and the required T is not stochastic. Add one line.
• D2. D2 is also satisfied by the copy-only control C3. The discrimination comes from D1's
  continuation, not from D2. Saying so avoids overreading D2 as "phase sensitivity".
• §3.2 item 1. "Must shift the carrier's conjugate slot" holds given (c) and N2. With a known
  meter slot the kick is a known shift and causes no disturbance.
• §5.3 item 2. Main §3.3 gives two options: ontic parameter dependence, or measurement
  dependence (recorded, not adopted). "Needs ontic parameter dependence" should mention both.
• §3.2 item 5. The abstract says classical complementarity "provides the beginning of a
  unification", which is weaker than "identifies as unifying".
• Unresolved labels. A reader cannot resolve some labels without links: "F2-1 A★", "Native A",
  "P-SDC", "E-pair", "E-AR" (first use in §1.4 row 2 is linked, later uses are not), "Gate A".
• Notation. §4.1 uses Ω for diag(J₂, J₂) in cell ordering and implicitly for
  [[0,I],[−I,0]] in the block identity. Name the second one separately.
• Odd q (§5.5). Whether the per-edge native couplings generate the full symplectic group is
  not shown. The "exactly" there is an expectation only, and the file labels it as such.

What I verified independently, and how

The reviewer's exact checks are local scripts; they are not committed and not cited as evidence.
I used exact arithmetic throughout (sympy/Fraction over F_2), with exhaustive enumeration where
the space is finite.

• Coupled tick (check1.py). M_kᵀ Ω M_k = Ω symbolically in k, with det = 1. The block
  identity [[Aᵀ−A, I], [−I, 0]] holds.
• Encoded identification (check1.py). The read rules are correct at both parities (Z read
  (z, x + w), X read (z + w, x)), and the free tick is the encoded identity (C8).
• Three-record law (check1.py). Simulated the full apparatus (carrier plus five meters, all
  2⁷ unknowns) for all four (s, s'). Each law equals the exact Pauli X-then-O_s-then-O_{s'}
  Lüders law from the maximally mixed state, matching the table at L488–493.
• Failure branch. The herald-0 branch is the encoded identity. The complete instrument is
  correct.
• Controls C3 and C9. C3 (copy-only, (0,1)) gives r2 = r0 with certainty. C9 was checked by
  derivation and by the generator below.
• Unheralded defect (check2.py). The record is uniform and independent of the input, and the
  output is the input with the conjugate randomized, i.e. ½Δ_s per record.
• Eraser (check2.py). Restoration of (z, x) and of the meter (0, w) holds in 8/8 cases.
  After the copy, the counts are 12/16 (full state) versus 8/16 (x), as in R9.
• Lemma 3 (lemma3.py, lemma3c.py). GF(2) linear-algebra tracking of the determined functionals
  on active cells, with carrier plus 3 ready cells:
  • Exhaustive depth-3 run, plus 20k random depth-8 runs, plus 30k generated valid depth-10 SA-1
    protocols: 0 isotropy or sharpness violations.
  • The same with terminality removed (record-slot registration only): 29,973/30,000 protocols
    violate isotropy. This confirms C9 and the role of (b).
• Lemma 5 (lemma5.py).
  • The 24-equation linear system has a unique solution for each Y convention. The file's T
    matches ysign = +1, with minimum entry −½ and column sums 1.
  • The coset-fixing permutations are exactly the Klein four-group. Its action on the X/Y
    states has order ≤ 2.
  • The δ-identity (MR★ Lemma G(ii)) holds.
• Numbers (lemma5.py).
  • rot: rot·Z+ has Bloch (−7/25, −24/25, 0), P + V = 31/25, P² + V² = 1,
    P(z = 0) = 9/25, tetrahedral weight at 00 equal to −3/50.
  • POM states: minimum weight (1 − √2)/4.
  • MP-1 law: (1+√3)/4 and (3−√3)/12, with P = (1 + 1/√3)/2 and P + V = 2/√3.
• Certified-quadruple bound (quad.py). LP over all laws on F_2²: 1.0 with the equivalence,
  2.0 without (R1).
• 8-point model (inline check). Octahedron states are nonnegative, the Pauli read laws are
  exact, and S permutes the cube vertices (R2).
• OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914 ceiling. I read comments 6707416018, 6707509318 and 6707783516 first-hand, and the Lean
  definitions at e61e567 (NonnegBounded, OnesBounded, permClass/IsScaledPartialPerm,
  BijectionLevel, PhasesAvailable, FalsifierUnavailable, rot, availExt):
  • The file keeps the two named classes distinct, as OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914 requires.
  • Transporting the phase and rot tests to a fixed identification is consistent with OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914's
    "fixed physical identification" and PC-F1.
  • Lemma 2 is correct as a paper mapping, apart from the wording in R8.
  • C-NC is an addition, not a OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914 test (R1).
• Prior screen and MR-1. Read observer-interface-source §4, observer-access-20261106
  (in full), MR-1 AW-MR5 / W1 / CC4 / Catani rows, MR★ S-L6, S-L7, Lemma G and the CC4 instance,
  and Main §3.3, Substratum L48/A4/A5 and Main Lemma 3.
• Links. Every pinned blob/<sha>/path#Lnn resolves, at e61e5673 or 62502618
  (e61e5673 is an ancestor of 62502618), and the cited lines say what they are cited for. The
  OI-QM-EXT-1: establish conditional equivalence with explicit added conditions, then minimize and source them #1169 owner-direction comment was fetched and matches the quoted direction.
• Literature. Fetched the arXiv abstracts of 1203.2652, 1409.5041, 1111.5057, quant-ph/0602001,
  quant-ph/0406166, 2211.09850 and 1802.06121. Hardy (2004) and Kochen–Specker (1967) were checked
  only against the standard statements: the infinite-ontic-state theorem, and the KS qubit model
  (density (1/π)·max(n·λ, 0), P(+|a) = (1 + n·a)/2; the uniform-hemisphere variant gives
  1 − θ/π). Both are cited within their hypotheses, provided MP-2 is read as including all pure
  states and projective reads, the in-plane trine states for Spekkens 2005 among them.

---

Generated by Claude Code

[amaybaum]

Transcribed from the same separate independent reviewer as the first review. Delta review, read-only, at exact head 7a3f7a528982dc0e0aa2d876705f86da3170b6e3. CI at this head: run 42406330966, 25 successful jobs and one conditional skip.

Delta review: Apparatus-1 DESIGN.md (#1183)

Object: verification/programmes/oi-qm/reconstruction/apparatus-1-20261107/DESIGN.md at exact
head 7a3f7a528982dc0e0aa2d876705f86da3170b6e3. Its single parent is 6fef72b7, the head of the
first review. I read the worktree only; nothing was edited, committed, pushed or posted.

Verdict at 7a3f7a5: ACCEPT WITH CHANGES

All twelve first-round corrections are resolved in the text. The new mathematics checks out
exactly:

• Lemma 4′: its analytic proof and the bound 3/2 (equivalently 1).
• P-8: positivity holds exactly on the octahedron, the read laws and post-read laws are right, S
  is the order-4 permutation (x,y,z) ↦ (−y,x,z), and rot(Z+) has minimum weight −3/100.
• The remark on transformation contextuality beyond R1 (the one-cell identification of §4.2) is a
  correct contrapositive of Lillystone–Wallman–Emerson (LWE) Theorem 1, used within that
  theorem's hypotheses.
• The ladder's structure, and the three-level framing the owner asked for (OI-QM-EXT-1: establish conditional equivalence with explicit added conditions, then minimize and source them #1169 c.6711920779),
  are present.

Four required corrections remain (N1–N4). All are local scoping or wording fixes; none changes
the stop, and §4 needs no recomputation:

• N1: the Y read is asserted but never exhibited, and Lemma 4′'s coverage of MP-1 depends on it.
• N2: one cell in the ladder table is wrong.
• N3: the coordinator's flagged item. Level 3's "beyond any finite construction" is true for
  rot, but the support the file gives (Hardy's theorem as stated) does not reach it. One sentence
  there is also false.
• N4: a notation slip in the LWE citation.

The pinned links all resolve. There is no revision narration and no new self-assessment.

Status of R1–R12

Item	Status	Where verified
R1 C-NC versus the per-state bound; D3	Resolved (one new scope gap, N1)	L28–32, L189–207, L448–450, L681–701, L788–802, L832–833. C-IS is now the per-state bound and not an escape condition; C-NC is the certified-quadruple form; D3 uses only C-PH and C-ROT
R2 Finite 8-point rung	Resolved (one table error, N2)	§5.2 P-8, ladder §5.4, L260–266, L739–741
R3 SA-1(a) split into native and posited parts	Resolved	L275–284 ((a) and (a′)), L41–44, L503–504, ledger P3
R4 H-PREP; reclassification of N2	Resolved	Ledger L409–411, L346–347, L551–552, L565–566
R5 Novelty narrowed; receiver assessment cited	Resolved	§1.4 row 3, L7 row, L247–258, L37–40, L58–60, L760–766, L981, §6.1
R6 Control statuses; same-commit decision rule	Resolved	L50–52, L420–423, L441–442, controls table L655–665 (C2/C6 deferred, C7 design conformance)
R7 Scope of negative statements	Resolved	L54, L703, L230–232, L795–796. Optional: L62 says "removing any SA-1 clause", but only (a), (b) and (c) were computed, not (a′) or E-iso
R8 Quantifier reading of the predicates	Resolved	L113–125, L160–168
R9 Eraser after a copy	Resolved	L631–642. The word is now stated, the returned value is x + w, and the counts are 8/16 and 12/16. I re-checked all three exactly
R10 Further consequences	Resolved	L358–370. The LWE Discussion does say "the Hadamard and Phase gates are not elements of the toy theory" (verified first-hand)
R11 Dyadicity scope	Resolved	L607–609, L746–747
R12 Recurrence caveat	Resolved	L331–336

Optional notes from round 1 that were applied: the dense-control line (L749–754), the D2 caveat
(L583–585), the condition on the §3.2 item 1 kick (L314–318), the two Main §3.3 routes (L952–958),
the "beginning of a unification" wording, the labels (E-pair, Native A, P-SDC, F2-1 A★), the two
Ω names, and the odd-q caveat.

New required corrections

N1. The encoded Y read is asserted but not exhibited. Lemma 4′'s coverage of MP-1 depends on
it (L511–512, L697–701, L791–792, L824–825, L832–833, ladder MP-1 row).

• What the file asserts.
  • §4.2: "Encoded Z, X and Y reads register z, x and z + x."
  • §5.1: MP-1 has an exact Y-read law.
  • L697–701: the certified-quadruple bound covers MP-1 because the equivalence is certified over
    the Z, X and Y reads.
• No SA-1 protocol for the Y read is given. For the apparatus class of §3.3 (only
  carrier–meter edges are armed) there is none. A type invariant excludes it: colour the carrier
  0 and each meter 1. Then a u slot at time t of a colour-χ cell carries only variables of
  type t + χ mod 2, and every update preserves this. So z (type 0) and x (type 1) never share
  a slot, and no record equals z + x. Checks run:
  • 20,000 random bipartite protocols: no slot ever contained both z and x.
  • Exhaustive search with meter–meter edges and all simultaneous edge sets (2 meters, depth ≤ 6,
    90,328 states): no record equal to z + x.
  • Single-edge search with 3 meters, depth ≤ 5: none.
• Consequence. If certification runs only over SA-1's exhibited Z/X reads (with uniform
  kicks), equivalence means equal z- and x-marginals, not equal laws. Two exact checks:
  • The LP maximum over all laws on F_2² is then 2, not 1.
  • Four MP-1-type laws with Bloch vectors (±a, ±a, ±a), a = 1/√3 and product sign on Y,
    are nonnegative and unit length. They are Z/X-equivalent (their Y marginals differ by
    √3/3) and give an average signed P + V = 2/√3 > 1.
  • That is a W1-shaped violation over the exhibited read set. MR-1's tomographic-completeness
    caveat (prereg L753) is what disqualifies it, not Lemma 4′.
  • The "square" variants are unaffected: their reads recover the joint law (copy-only, no
    terminality, sharp meters; and the biased kick for p ≠ ½).
  • SA-1 itself is unaffected: per-state P + V ≤ 1 already bounds every quadruple.
• Fix. State that a Y read is not exhibited in SA-1, or exhibit one, under P3 if needed. In
  L697–701, the §5.1 table and the MP-1 row, scope Lemma 4′'s application:
  • C-NC holds for MP-1 when the equivalence is certified over a read set that spans the functions
    on F_2²;
  • over SA-1's exhibited Z/X reads the test is inconclusive (it reaches 2/√3) and is
    excluded as a witness by tomographic incompleteness, not by Lemma 4′.
  • Mark MP-1's Y-read law as conditional on a Y read being supplied.

N2. Ladder cell MP-2 / C-IS is wrong (L967). It reads "holds (ball)". A ball-containing
carrier includes (1/√2, 1/√2, 0), so P + V = √2, and C-IS fails, as the file itself states
for the quantum comparator (L793). Fix: "fails (√2)".

N3. Level 3: what "beyond any finite construction" rests on (L71–73, L776–777, L927–929,
L967). This is the coordinator's flagged item.

• Hardy's theorem does not reach rot as stated. Hardy (2004) is stated for a model of all
  pure states and all projective measurements. Inducing rot does not supply that repertoire.

• The claim is still true for rot, by a short argument the file does not give.

  • Suppose a finite carrier with ontic set Λ induces rot with exact statistics. Its encoded
    action is then one stochastic matrix Γ on Λ, by Lemma 5's reasoning.
  • With Z+ prepared and Z read, the statistics are P_j = μᵀ Γ^j ξ = cos²(jφ), where
    cos φ = 3/5. Here μ is the law of the Z+ preparation and ξ is the response function of
    the Z read.
  • The right side contains the unit-modulus frequencies e^{±2ijφ}. Since cos 2φ = −7/25,
    Niven's theorem makes 2φ/π irrational, so e^{2iφ} is not a root of unity.
  • A finite stochastic matrix's unit-modulus eigenvalues are roots of unity (Perron–Frobenius).
    So no finite Λ reproduces the sequence.
  • If rot⁻¹ is also available, Hardy's support-pigeonhole argument applied to the orbit
    {rot^k Z+} gives the same conclusion: the overlaps are cos²(kφ) < 1 for every k ≠ 0.

• The claim is therefore established for rot itself, not only for the full qubit repertoire.
  It is not established by Hardy's theorem as cited.

• L927–928 is false as stated. "Infinitude is needed only once the repertoire goes beyond the
  stabilizer fragment, as rot requires." MP-1 is itself a finite 4-point construction beyond
  the stabilizer fragment, and every finite fragment has a finite model. What forces infinitude is
  an operation with an infinite exact orbit, such as rot.

• L71–73 and L777 overstate MP-2. They present MP-2 as the further condition. MP-2 is a
  sufficient condition (the Kochen–Specker model); what rot necessarily needs is an infinite
  carrier.

• Minimal scoped wording (L71–73 and L776–777):

  Level 3, the continuous rotations (rot). Inducing rot with exact statistics excludes every
  finite carrier: its encoded action would be a finite stochastic matrix, whose unit-modulus
  eigenvalues are roots of unity, while cos²(jφ) with cos φ = 3/5 is not periodic. MP-2, a
  ball-containing carrier, is one sufficient further condition. Read as including all pure states
  and projective reads, it is infinite (Hardy) and preparation-contextual (Spekkens).

  And replace L927–929 with:

  Infinitude is forced by an operation with an infinite exact orbit, such as rot, not by
  leaving the stabilizer fragment: MP-1 leaves it on four points. The quarter phase alone is
  supplied by the finite rung P-8.

  This matches the owner's "beyond the finite construction" and supplies the derivation for
  "any".

N4. The LWE T_2 is mis-transcribed (L882–884). The file writes "T_2 = H T_1 H". Read as a
composition, H∘T_1∘H preserves the parity xyz; I checked exhaustively on the 8 points. The
paper's eq. (5) is T_2(ρ) = H T_1(ρ) H, meaning T_1 followed by Hadamard conjugation. That
reverses the parity and gives eq. (7). Fix: write T_2(ρ) = H T_1(ρ) H (T_1, then the
Hadamard).

Optional notes

• §1.3 and §4.7 argue dense control with different gates. §1.3 (L184–187) argues it through
  rot; §4.7 (L749–754) argues it through S. Either works; make them consistent.
• Remark beyond R1, spanning hypothesis. Under preparation noncontextuality, the stabilizer
  laws are affine in ρ and span at most 4 dimensions. The hypothesis therefore effectively
  forces |Λ| ≤ 4, R1's size. One clause saying so makes clear that the remark generalizes
  Lemma 5 across identifications of that dimension.
• The remark also assumes stochastic transformation maps on Λ. It needs each subtheory
  transformation to act on Λ as a single stochastic map, with ancillas fresh and discarded as in
  Lemma 5. Theorem 1's own setting (an ontological model with convex-linear transformations) has
  this. State it.
• Cite the owner's direction for §5. The three-level structure follows the owner's direction
  (OI-QM-EXT-1: establish conditional equivalence with explicit added conditions, then minimize and source them #1169 c.6711920779), which §5 could cite. Its "requires transformation contextuality in the
  cited construction" is stronger in the theorem: by LWE Theorem 1, every preparation-
  noncontextual model of the full subtheory is transformation-contextual, not just P-8. The
  remark already implies this; §5.2's "Cost" paragraph could say it in one clause.
• L62. Change "Removing any SA-1 clause" to "Removing (a), (b) or (c)".

What I independently verified, and how

The reviewer's exact checks are local scripts (delta.py, delta2.py, yread.py, yread2.py/yread3.py, plus the round-1 scripts); they are not committed and not cited as evidence. The arithmetic is exact (sympy/Fraction).

• Lemma 4′.
  • For every ontic point, the per-point maximum weight across the two pairs is exactly 3. This
    gives Σ ≤ 3·2 = 6 and an average of 3/2 (signed: 1). The proof is correct.
  • The round-1 LP agrees: 1.0 with the equivalence, 2.0 without it.
  • New: certification over Z/X only also gives 2.0 (N1).
• P-8.
  • min_λ(1 + r·λ) = 1 − |r|₁ holds at sample points, so positivity is exactly the octahedron.
  • The read laws are (1 + r_a)/2, and the read followed by re-randomization gives the
    eigenstate law.
  • (x,y,z) ↦ (−y,x,z) permutes the cube, has order 4, flips parity, and carries μ_r to
    μ_{Sr}.
  • rot(Z+) has (x,y,z) = (−24/25, 0, −7/25) and minimum weight −3/100.
• LWE (arXiv:1802.06121v2, full text fetched and read).
  • Theorem 1: "Every ontological model of the single-qubit stabilizer subtheory is either
    preparation or transformation contextual."
  • The subtheory consists of all stabilizer states, stabilizer measurements and the full
    Clifford group. The 8-state model is Ref. [21], Wallman–Bartlett, and it is preparation- and
    measurement-noncontextual.
  • Eqs. (5)–(7) and the parity argument are as cited, apart from N4.
  • The Discussion's statement on the toy theory is as cited in R10.
  • The remark beyond R1 uses Theorem 1 within its hypotheses: the full subtheory, all
    statistics, convex-linear model. The added spanning step is Lemma G(ii) applied to
    preparations, transformations and responses.
• Ladder. I checked each cell against Lemma 4, Lemma 4′, Lemma 5, the P-8 algebra and the
  Kochen–Specker model. The MP-2/C-IS cell is wrong (N2) and the MP-1/C-NC cell needs N1's scope;
  the rest are correct.
• Owner's three levels. Levels 1–3, Q-A, Q-B, the "reproducing one operation…" caution and
  the native-versus-supplied ledger column are all present (§0 item 5, §5 intro, §5.7, ledger
  §3.4). P3 and H-PREP are marked supplied, as the owner asked.
• Links. Every pinned link in the revision resolves at e61e5673 or 62502618, and the cited
  lines match their use. New anchors checked: receiver assessment L88/L144/L269, E-pair L25,
  Native A L27, P-SDC L728, H-PREP L892, S-L6 L595, Lemma G L1020, permClass L236, F2-1 A★
  L62. The issue comment links are unchanged and valid.
• Register. A grep found no revision narration ("formerly", "no longer", "corrected", review
  references) and no new self-assessment.

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Generated by Claude Code

[amaybaum]

Transcribed from the same separate independent reviewer (first review, delta review). Final delta check, read-only.

Final delta check — Apparatus-1 DESIGN.md (#1183)

At 4c88f2b (diff from 7a3f7a5): ACCEPT WITH CHANGES

N1–N4 and every optional note are applied. The new §5.3 derivation is correct, and it states its hypotheses. One required correction remains, a single factual misstatement in the new Lemma 4′ scope paragraph. R1–R12 remain resolved. The revision adds no overclaim, no revision narration and no broken link.

M1 (L723–725). The text said the four MP-1-type laws are "pairwise Z/X-equivalent". They are not: their Z/X statistics differ by construction. What holds is narrower. The two mixtures ½(μ_00 + μ_11) and ½(μ_01 + μ_10) have equal z- and x-marginals, both with Bloch (0, 0, ±a), and their Y marginals differ by √3/3 (checked exactly).

Item	Status	Verification
N1 Y read	Resolved, apart from M1	§4.2 now claims only the Z/X reads. The colouring invariant was re-derived: a coupled tick preserves the slot types; 20,000 random bipartite protocols put no slot with both z and x. "No Y_± state is reached" holds because the determined subspace is graded and, on the carrier, a graded isotropic subspace (Lemma 3) is span{z}, span{x} or 0. The Lemma 4′ scope, MP-1's conditional Y law, the MP-1 ladder row and the exemption of the sharp and biased variants are all correct
N2 MP-2 C-IS cell	Resolved	It reads "fails (√2)"
N3 Level-3 derivation	Resolved	See below
N4 LWE T_2	Resolved	It now reads T_2(ρ) = H T_1(ρ) H, which matches eq. (5) of arXiv:1802.06121v2. The stated parity reversal is correct

N3, §5.3 "What level 3 rests on".

• Hypotheses stated: a finite Λ; rot induced with exact statistics; one stochastic matrix Γ applied j times, as in Lemma 5; the Z_+ preparation and the Z read.
• The steps check:
  • (1) P_j = cos²(jφ), exactly for j = 1…5.
  • (2) A stochastic matrix is power-bounded, and its unit-modulus eigenvalues are semisimple roots of unity (Perron–Frobenius). So P_j is a periodic sequence plus one that vanishes.
  • (3) cos 2φ = −7/25, so by Niven's theorem 2φ/π is irrational, and the rotation by 2φ is equidistributed.
  • (4) The identity cos(2(j+p)φ) − cos(2jφ) = −2 sin((2j+p)φ) sin(pφ) was verified symbolically. It forces sin(pφ) = 0, so e^{2iφ} would be a root of unity, contradicting (3).
• Conclusion: the result is stated for rot itself. MP-2 is presented as one sufficient condition, and Hardy and Spekkens are cited within their hypotheses.

The optional notes are all applied: dense control is decided through S in both §1.3 and §4.7; the remark beyond R1 states the stochastic-map setting and the |Λ| ≤ 4 scope; Theorem 1 is applied to every preparation-noncontextual model; #1169 c.6711920779 is cited; L62 reads "(a), (b) or (c)".

Confirmation at af8df1b (single parent 4c88f2b)

git diff 4c88f2b8 af8df1b4: one file, four lines changed.

• M1 resolved. The text now says the two mixtures are Z/X-equivalent and their Y marginals differ. This is exact.
• L527. "The determined subspace is graded by the same types" is the precise statement. Excluding Y_± also uses Lemma 3.
• L75. "Not eventually periodic up to a vanishing term" now matches the §5.3 argument exactly.

Nothing else changed. There is no new link, no narration and no overclaim. N1–N4 and R1–R12 remain resolved.

Verdict at af8df1b: ACCEPT.

CI at this exact head: 25 successful jobs and one conditional skip.

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Generated by Claude Code

[amaybaum]

Merge record.

• Merge: b1d782f (merge commit). It was retargeted to main after OI–QM: interface studies and source assessments #1173 landed. Parents are main dc0b333 and the reviewed head af8df1b, and the tree bb866987 equals the reviewed head's tree.
• Approval: owner, Claude Code chat (amaybaum, 2026-11-07, about 20:03 UTC), recorded on OI-QM-EXT-1: establish conditional equivalence with explicit added conditions, then minimize and source them #1169 c.6712359044.
• Post-merge CI on main at b1d782f: green, 25 successful jobs and one conditional skip, run 42409285903.

The reviewed scope is unchanged: a paper-only stop, with no premise adopted. The owner's Q-B target decision is recorded on #1169 c.6712409082.

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