The Irrotational Shift Corridor

⚑Derivation Flags (45)

• high
  §2 wall floor / equation index #2 — Δ_min = ξ/√(α n_max) is asserted with no free-energy, healing-length, or coherence argument. It is the sole determinant of Δ, hence of R via the aspect rule, hence of every row of Table 1 and of the abstract's claim that the geometry is 'forced' to ≈ 6 μm. Addendum A.4 concedes no substrate Hamiltonian exists from which such a scaling could be obtained. Load-bearing for the paper's central geometric conclusion.

  If wrong: If the scaling has a different power or prefactor, Δ changes, R changes, all of Table 1 changes, and the abstract's central claim that micron coherence 'forces' a ≈ 6 μm on-chip geometry is unsupported.

• high
  §4.2 analog layer; §7 Table 3; gate F2 / equation index #6 — The second-pass speed advantage v_coord > c_s inside T_Γ (fiducial 1.5 c_s) is the paper's central operational claim, but no evolution equation, dispersion relation, or analog metric connects the saturated profile n_★(r), the regulator R(n), or the shift β = ∇ψ to a propagation speed. Table 3 is explicitly a target waveform, not a derivation. Load-bearing for the main conclusion.

  If wrong: If no locked configuration supports super-c_s propagation, gate F2 fails by construction and the reusable-rail claim — the entire actionable content of the spec — collapses to a kinematic stipulation.

• high
  §5 lock rule, definition of T_Γ — The lock bit is imposed on a barred quantity n̄ whose relation to the radial profile n_★(r) (axis value, tube average, or some other functional) is never specified. Every post-lock rule and gates F1–F3 depend on whether T_Γ is nonempty under Case A, which cannot be determined without this definition.

  If wrong: If n̄ cannot reach the lock threshold under frozen Case A, T_Γ is empty and no post-lock behavior, second-pass advantage, or F1–F3 test can be realized as specified.

• high
  Section 4.2 and Section 7, central analog propagation claim — The super-medium-speed mode is stipulated without a medium Hamiltonian, field equation, dispersion relation, or derived analog metric.

  If wrong: If the eventual substrate dynamics do not possess the assumed guided mode, F2 and the paper's main reusable-rail conclusion fail. This is load-bearing.

• high
  Section 4.2, 'Operational analog FTL means v_coord > c_s inside the prepared tube T_Γ' — The definition of operational analog FTL is asserted without derivation from the stated order parameter, regulator, or wall geometry. The paper does not provide an analog metric or evolution equation connecting the saturated order parameter to a coordinate speed exceeding c_s.

  If wrong: If the step is invalid, the central claim of operational analog FTL is unsupported, and the entire experimental program (gates F0–F4) lacks a theoretical foundation.

• high
  Section 5 lock criterion and Table 2, parallel row — The maximum frozen Case A response is below the threshold defining a locked prepared tube, but the table classifies that maximum as lock allowed.

  If wrong: Without a locked T_Γ, the post-lock shift, steering-refusal test, second-pass test, and loop-refusal test cannot operate under frozen Case A.

• high
  Section 5 lock definition and Table 2 — The frozen Case A ceiling does not meet the stated lock criterion, while the table classifies that ceiling as locked.

  If wrong: Under the stated Case A definitions, T_Γ cannot satisfy the lock bit. The post-lock shift rule, target waveform, and F1–F3 lock-dependent tests cannot operate as specified.

• high
  Section 5, lock bit definition R(n̄) ≥ 0.95 vs frozen Case A R(40) ≈ 0.945 — The lock threshold cannot be met by the frozen Case A profile, yet Table 2 labels the n=40 core as 'lock allowed' and later sections assume a locked tube.

  If wrong: The defined prepared tube T_Γ is empty under the supplied definitions, so the post-lock rules and F1–F3 predictions cannot operate as written.

• high
  Section 5, prepared-tube lock definition and Table 2 — The frozen Case A regulator cannot reach the stated 0.95 lock threshold, although the table classifies its maximum value as an allowed lock.

  If wrong: No prepared tube locks under the frozen Case A parameters, so the post-lock shift, second-pass F2 test, steering test F1, and loop-lock test F3 are undefined as written.

• high
  Section 6 and Table 2 — The loop-refusal table substitutes an angle function for a circulation that vanishes under the displayed exact-gradient definition.

  If wrong: The stated mathematical mechanism predicts no loop suppression; F3’s loop-refusal prediction requires a separately specified non-exact one-form, singularity, or phenomenological angle law.

• high
  Sections 2, 3, 5, and Table 2 — Case A lock condition — The submitted Case A maximum saturation is below the submitted lock threshold, yet the maximum-saturation state is subsequently classified as locked.

  If wrong: Under the stated definition, T_Γ is empty in Case A. The post-lock shift, target F2 traversal, steering test F1, loop test F3, and conclusion that this Case A chip is a locked rail are not defined as claimed.

• high
  Sections 4.2, 5, and 7 — analog shift and second-pass claim — No substrate Hamiltonian, evolution equation, analog metric, or constitutive coupling derives the claimed propagation law v_coord=1.5c_s or the inequality T_second<L/c_s from the saturated profile and β=∇ψ.

  If wrong: The principal claim of a reusable analog rail with a post-lock speed advantage is unsupported. The document would remain a proposed profile and falsification plan, but not a mathematical specification predicting F2.

• high
  Sections 4.2, 5, and 7: analog second-pass claim — The claimed post-lock super-c_s propagation is asserted as operational kinematics/target waveform, without a substrate evolution equation or derivation from the order-parameter profile and irrotational shift.

  If wrong: The main analog claim—a reusable locked rail with a second-pass speed advantage—and F2's predicted inequality are unsupported rather than consequences of the submitted model.

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  Sections 4.2, 5, and Table 3/F2 — No analog-medium evolution equation derives v_coord>c_s or the 1.5c_s target from n, R(n), or β=∇ψ.

  If wrong: F2 and the paper’s main reusable-rail conclusion are unsupported; the apparatus may merely be an ordinary guide with no post-lock speed advantage.

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  Table 3, gate F2, Section 4.2 — The target waveform v_coord = 1.5 c_s inside T_Γ is stated as an intended F2 waveform, but no derivation connects the saturated order parameter profile n_★(r), the regulator R(n), or the wall geometry to a coordinate speed exceeding c_s. The paper explicitly labels Table 3 as 'not a measurement' and Addendum A.5 states 'the claim is analog kinematics only.'

  If wrong: If the step is invalid, the central claim of a reusable rail with second-pass speed advantage is unsupported, and gate F2 cannot be interpreted as a test of the model.

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  Vacuum companion Section 4 and Addendum A.3, c-speed density — The displayed density requirement was calculated using the mass-density form of the bound but labeled as an energy density, contradicting the corrected SI convention.

  If wrong: The qualitative conclusion that the required density is enormous remains, but the quoted quantitative no-go density is wrong by about 17 orders of magnitude.

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  Vacuum companion Section 4 and Addendum A.3, Case A maximum speed — The quoted maximum speed is incompatible with the packet's corrected energy-density form of the vacuum bound.

  If wrong: The qualitative propulsion no-go is strengthened, but the stated Case A quantitative bound is invalid under the document's own corrected SI convention.

• high
  Vacuum companion Section 4, matching bound — The specialized thick-wall matching bound is asserted at order of magnitude without a derivation or a precise cited formula with hypotheses.

  If wrong: The quantitative vacuum density and speed bounds do not follow. The separate retarded-build inequality may survive, but the claimed energetic no-go would be unsupported.

• medium
  §6 loop penalty / equation index #8 — The refusal mechanism is self-nullifying as written: the trigger requires a nonvanishing loop integral, and the same section concedes the integral vanishes identically for smooth single-valued t_★. Table 2 then substitutes an unrelated dimensionless angle function for the integral. The argument of the exponential also has no declared units, so λ = 10 cannot be dimensionally checked.

  If wrong: The chronology-penalty section and Table 2 provide no quantitative prediction; gate F3 remains a valid empirical test but the damper model behind it is inoperative as written.

• medium
  Addendum A.3 versus vacuum companion Section 4, Case A speed — The printed maximum speed is inconsistent with the corrected SI equation when the stated energy density is used.

  If wrong: The quantitative Case A vacuum bound is wrong by about eight orders of magnitude, although the qualitative conclusion that it is useless for propulsion remains unchanged.

• medium
  Companion page Section 3 — locked-tube metric — The metric is explicitly schematic and is not derived from supplied IPT equations, a stress-energy model, or junction conditions; nevertheless it is used to motivate vacuum coordinate-FTL statements.

  If wrong: The vacuum-side statements about the geometry and coordinate traversal cannot be treated as consequences of IPT; they remain properties of an assumed metric form. This is secondary to the chip claim because the submission disclaims a realizable vacuum corridor.

• medium
  Companion page Section 4 — Case A vacuum velocity — The printed Case A vacuum maximum speed is inconsistent with the corrected SI bound supplied in Addendum A.

  If wrong: The numerical magnitude quoted for the Case A vacuum bound is invalid, although the corrected value is even smaller and therefore preserves the stated qualitative conclusion that it is useless for propulsion.

• medium
  Companion page Section 4 — vacuum energy bound — The displayed vacuum bound omits c^−2 even though ρ₀ is defined as energy density, making the expression dimensionally inconsistent in SI. Addendum A gives a different, dimensionally valid expression.

  If wrong: The printed companion-page vacuum matching formula cannot support its numerical vacuum speed bound when ρ₀ is an energy density. The qualitative conclusion must be recomputed using the corrected SI convention.

• medium
  Section 2 minimum wall thickness — The wall-floor scaling is asserted without a free-energy functional, substrate Hamiltonian, stability equation, or derivation connecting coherence length to αn_max.

  If wrong: The asserted micron wall scale and derived approximately 6 μm guide radius are unsupported, although one could still impose them phenomenologically.

• medium
  Section 2, minimum wall thickness — The minimum-thickness relation is presented as a universal floor but is not derived from a declared substrate equation.

  If wrong: The numerical on-chip wall scale and resulting radius constraint are unsupported, although the regulator profile itself could still be used as an independently chosen parameterization.

• medium
  Section 2, wall floor and bubble cap — An inequality is used to infer a unique radius, although it supplies only an upper bound unless equality is added as a design choice.

  If wrong: The claimed uniquely forced 6 μm geometry does not follow; it is one boundary-saturating design among all radii R ≤ 5.869 μm.

• medium
  Section 2, wall geometry inference — The numerical Delta_min calculation is reproducible, but the aspect-ratio inequality is converted to an equality without an explicit equality-saturating design assumption.

  If wrong: The exact approximately 6 micrometre radius is not forced by the stated wall rule. The claimed fixed chip geometry and all radius-dependent implementation assertions require an additional stated design choice.

• medium
  Section 2, Δ_min = ξ / √(α n_max) — The wall floor formula is asserted with no supporting free-energy or coherence-length argument, yet it fixes Δ = 0.5869 μm, hence R = 5.869 μm via Δ/R ≥ 0.1, hence every entry in Table 1.

  If wrong: The abstract's central claim that the geometry is 'forced' to ≈ 6 μm is unsupported.

• medium
  Section 3, vacuum companion page, ds² = −c² dt_★² + (ds − v_Γ f_R(n) dt_★)² + dΩ_⊥² — f_R(n) appears in the corridor line element without ever being defined, so the metric cannot be evaluated, its signature checked, or the stated junction conditions actually imposed.

  If wrong: The assertions about timelike riders, local propagation, and coordinate speed are properties intended by the ansatz, not established consequences of the supplied model.

• medium
  Section 4.1 / Vacuum companion page, v_Γ² ≲ 32π G ρ₀ Δ² — The vacuum bound's numerical prefactor 32π is imported without the matching calculation that produces it, and the printed numerical estimate is inconsistent with the corrected SI convention in Addendum A.3.

  If wrong: The one quantitative result of the principle layer is numerically unreliable, though the qualitative conclusion of negligible vacuum effect survives.

• medium
  Section 4.1 vacuum-c energy estimate and Addendum A.3 — The required density was not converted consistently after the SI form was corrected for energy density.

  If wrong: The paper’s vacuum-c density figure is quantitatively invalid, though the conclusion that the required density is extraordinarily large is strengthened rather than reversed.

• medium
  Section 4.1, 'v_Γ² ≲ 32π G ρ₀ Δ²' — The vacuum energy bound is presented as an order-of-magnitude matching condition, but the derivation is not shown. The paper states 'Matching an Alcubierre-like thick wall to a bounded residual still gives, at order of magnitude, v_Γ² ≲ 32π G ρ₀ Δ²' without deriving this from the stated equations.

  If wrong: If the step is invalid, the vacuum energy bound is unsupported, and the claim that a vacuum-c wall at Δ = 0.5869 μm is energetically impossible is not established.

• medium
  Section 5, 'After lock, an irrotational shift β = ∇ψ may be updated along Γ on the regulator slice' — The irrotational shift β = ∇ψ is asserted as the mechanism for the second-pass speed advantage, but no derivation connects β to the coordinate speed v_coord or to the saturated order parameter profile. The paper does not derive the analog metric or the evolution equation for ψ.

  If wrong: If the step is invalid, the mechanism for the second-pass speed advantage is unspecified, and the claim of operational analog FTL is unsupported.

• medium
  Section 6 assumed chronology penalty — The phenomenological loop penalty is explicitly assumed, but its displayed loop functional vanishes for the smooth single-valued scalar field the text itself describes; the table substitutes an unrelated angle proxy without defining the necessary non-exact one-form, singularity, or units.

  If wrong: Table 2 is not a consequence of the stated penalty equation, and F3 does not test a mathematically specified loop-refusal mechanism until the nonzero circulation structure and its units are defined.

• medium
  Section 6, chronology penalty — The exponent's dimensions are unspecified, and the displayed exact-gradient loop integral vanishes under the smooth single-valued assumptions.

  If wrong: Table 2 and F3 are not consequences of the stated differential expression; they are only an external angle-dependent switch. The claimed chronology protection remains an explicit, unvalidated assumption.

• medium
  Section 6, chronology penalty R_eff = R(n) exp(−λ |∮ ∇t_★ · dl|) — The loop-refusal mechanism is written as ∮∇t_★·dl ≠ 0, then immediately conceded to vanish identically for smooth single-valued t_★. Table 2 substitutes an unrelated angle stand-in sin(θ/2).

  If wrong: The chronology penalty as written cannot trigger, so the F3 gate tests an unrelated angle switch rather than the stated mechanism.

• medium
  Vacuum companion §3, corridor line element — The function f_R(n) appears in the metric without any definition anywhere in the packet, so the line element cannot be evaluated, its signature verified, or the stated junction conditions (n and ρ_− continuous, Δ/R ≥ 0.1) imposed. The assertions that the rider stays timelike and local light is not outrun therefore rest on an undefined object.

  If wrong: The vacuum principle-layer claims about timelike riders and coordinate-only FTL become unevaluable; the metric sketch supports no conclusion.

• medium
  Vacuum companion Section 3 metric — The metric contains an undefined function f_R(n), and no conditions are supplied to establish the associated causal claims.

  If wrong: The locked-tube metric, junction behavior, and statements about timelike riders and endpoint coordinate speed cannot be evaluated from the submission.

• medium
  Vacuum companion Section 3, schematic metric — The corridor metric is not developed sufficiently to verify the energy bound, junction conditions, or timelike-rider assertion.

  If wrong: Claims that the rider remains timelike, local light is not outrun, and the proposed stress profile matches this geometry cannot be checked from the submitted metric.

• medium
  Vacuum companion Section 4 energy bound — The numerical prefactor and matching relation are presented without a derivation or a precise cited formula establishing their applicability to the schematic corridor metric.

  If wrong: The quantitative vacuum no-go bound and both derived vacuum estimates lack a demonstrated basis, although this does not directly alter the analog chip budget.

• medium
  Vacuum Principle Layer Section 3 metric — The schematic locked-tube metric contains an undefined function f_R(n) and is not derived from an IPT field equation or a specified analog medium.

  If wrong: Claims concerning the vacuum corridor's local causal structure, junction, and coordinate-speed interpretation remain schematic ansatz statements rather than established consequences of a defined metric model.

• medium
  Vacuum Principle Layer Section 4 and Addendum A.3 — The companion page retains a non-SI energy-density formula and associated numerical estimates, while Addendum A.3 gives the required SI correction. The quantitative claims are mutually inconsistent.

  If wrong: The qualitative no-go conclusion remains, but the printed vacuum speed and required-density values cannot be quoted or used quantitatively under the packet's final SI convention.

• low
  Section 2 wall floor and bubble cap — An inequality is converted into a unique radius without explicitly declaring that the design saturates the bound.

  If wrong: The claimed uniquely forced 5.869 μm radius becomes only a maximum permitted radius; the profile can be retained by declaring equality as a design choice.

• low
  Section 2, Δ / R ≥ 0.1 ⇒ R = 5.869 μm — The inference from the aspect-ratio inequality to a unique value of R replaces an upper bound on R with equality without stating an equality-saturating design choice.

  If wrong: The specific R value is not uniquely forced by the stated inequality, though the order of magnitude is unaffected.

• low
  Section 6, 'R_eff = R(n) exp(−λ |∮ ∇t_★ · dl|), λ = 10' — The loop penalty is explicitly labeled as an assumption ('not derived'), and the paper acknowledges that for a smooth single-valued t_★ the loop integral is identically zero. The angle table uses |∮| ∼ sin(θ/2) as a stand-in, not a topology calculation.

  If wrong: If the step is invalid, the loop refusal mechanism is unsupported, but the paper explicitly labels this as an assumption to be tested by gate F3, so the consequence is limited to the chronology protection claim.

⚑Derivation Flags (45)

• high
  §2 wall floor / equation index #2 — Δ_min = ξ/√(α n_max) is asserted with no free-energy, healing-length, or coherence argument. It is the sole determinant of Δ, hence of R via the aspect rule, hence of every row of Table 1 and of the abstract's claim that the geometry is 'forced' to ≈ 6 μm. Addendum A.4 concedes no substrate Hamiltonian exists from which such a scaling could be obtained. Load-bearing for the paper's central geometric conclusion.

  If wrong: If the scaling has a different power or prefactor, Δ changes, R changes, all of Table 1 changes, and the abstract's central claim that micron coherence 'forces' a ≈ 6 μm on-chip geometry is unsupported.

• high
  §4.2 analog layer; §7 Table 3; gate F2 / equation index #6 — The second-pass speed advantage v_coord > c_s inside T_Γ (fiducial 1.5 c_s) is the paper's central operational claim, but no evolution equation, dispersion relation, or analog metric connects the saturated profile n_★(r), the regulator R(n), or the shift β = ∇ψ to a propagation speed. Table 3 is explicitly a target waveform, not a derivation. Load-bearing for the main conclusion.

  If wrong: If no locked configuration supports super-c_s propagation, gate F2 fails by construction and the reusable-rail claim — the entire actionable content of the spec — collapses to a kinematic stipulation.

• high
  §5 lock rule, definition of T_Γ — The lock bit is imposed on a barred quantity n̄ whose relation to the radial profile n_★(r) (axis value, tube average, or some other functional) is never specified. Every post-lock rule and gates F1–F3 depend on whether T_Γ is nonempty under Case A, which cannot be determined without this definition.

  If wrong: If n̄ cannot reach the lock threshold under frozen Case A, T_Γ is empty and no post-lock behavior, second-pass advantage, or F1–F3 test can be realized as specified.

• high
  Section 4.2 and Section 7, central analog propagation claim — The super-medium-speed mode is stipulated without a medium Hamiltonian, field equation, dispersion relation, or derived analog metric.

  If wrong: If the eventual substrate dynamics do not possess the assumed guided mode, F2 and the paper's main reusable-rail conclusion fail. This is load-bearing.

• high
  Section 4.2, 'Operational analog FTL means v_coord > c_s inside the prepared tube T_Γ' — The definition of operational analog FTL is asserted without derivation from the stated order parameter, regulator, or wall geometry. The paper does not provide an analog metric or evolution equation connecting the saturated order parameter to a coordinate speed exceeding c_s.

  If wrong: If the step is invalid, the central claim of operational analog FTL is unsupported, and the entire experimental program (gates F0–F4) lacks a theoretical foundation.

• high
  Section 5 lock criterion and Table 2, parallel row — The maximum frozen Case A response is below the threshold defining a locked prepared tube, but the table classifies that maximum as lock allowed.

  If wrong: Without a locked T_Γ, the post-lock shift, steering-refusal test, second-pass test, and loop-refusal test cannot operate under frozen Case A.

• high
  Section 5 lock definition and Table 2 — The frozen Case A ceiling does not meet the stated lock criterion, while the table classifies that ceiling as locked.

  If wrong: Under the stated Case A definitions, T_Γ cannot satisfy the lock bit. The post-lock shift rule, target waveform, and F1–F3 lock-dependent tests cannot operate as specified.

• high
  Section 5, lock bit definition R(n̄) ≥ 0.95 vs frozen Case A R(40) ≈ 0.945 — The lock threshold cannot be met by the frozen Case A profile, yet Table 2 labels the n=40 core as 'lock allowed' and later sections assume a locked tube.

  If wrong: The defined prepared tube T_Γ is empty under the supplied definitions, so the post-lock rules and F1–F3 predictions cannot operate as written.

• high
  Section 5, prepared-tube lock definition and Table 2 — The frozen Case A regulator cannot reach the stated 0.95 lock threshold, although the table classifies its maximum value as an allowed lock.

  If wrong: No prepared tube locks under the frozen Case A parameters, so the post-lock shift, second-pass F2 test, steering test F1, and loop-lock test F3 are undefined as written.

• high
  Section 6 and Table 2 — The loop-refusal table substitutes an angle function for a circulation that vanishes under the displayed exact-gradient definition.

  If wrong: The stated mathematical mechanism predicts no loop suppression; F3’s loop-refusal prediction requires a separately specified non-exact one-form, singularity, or phenomenological angle law.

• high
  Sections 2, 3, 5, and Table 2 — Case A lock condition — The submitted Case A maximum saturation is below the submitted lock threshold, yet the maximum-saturation state is subsequently classified as locked.

  If wrong: Under the stated definition, T_Γ is empty in Case A. The post-lock shift, target F2 traversal, steering test F1, loop test F3, and conclusion that this Case A chip is a locked rail are not defined as claimed.

• high
  Sections 4.2, 5, and 7 — analog shift and second-pass claim — No substrate Hamiltonian, evolution equation, analog metric, or constitutive coupling derives the claimed propagation law v_coord=1.5c_s or the inequality T_second<L/c_s from the saturated profile and β=∇ψ.

  If wrong: The principal claim of a reusable analog rail with a post-lock speed advantage is unsupported. The document would remain a proposed profile and falsification plan, but not a mathematical specification predicting F2.

• high
  Sections 4.2, 5, and 7: analog second-pass claim — The claimed post-lock super-c_s propagation is asserted as operational kinematics/target waveform, without a substrate evolution equation or derivation from the order-parameter profile and irrotational shift.

  If wrong: The main analog claim—a reusable locked rail with a second-pass speed advantage—and F2's predicted inequality are unsupported rather than consequences of the submitted model.

• high
  Sections 4.2, 5, and Table 3/F2 — No analog-medium evolution equation derives v_coord>c_s or the 1.5c_s target from n, R(n), or β=∇ψ.

  If wrong: F2 and the paper’s main reusable-rail conclusion are unsupported; the apparatus may merely be an ordinary guide with no post-lock speed advantage.

• high
  Table 3, gate F2, Section 4.2 — The target waveform v_coord = 1.5 c_s inside T_Γ is stated as an intended F2 waveform, but no derivation connects the saturated order parameter profile n_★(r), the regulator R(n), or the wall geometry to a coordinate speed exceeding c_s. The paper explicitly labels Table 3 as 'not a measurement' and Addendum A.5 states 'the claim is analog kinematics only.'

  If wrong: If the step is invalid, the central claim of a reusable rail with second-pass speed advantage is unsupported, and gate F2 cannot be interpreted as a test of the model.

• high
  Vacuum companion Section 4 and Addendum A.3, c-speed density — The displayed density requirement was calculated using the mass-density form of the bound but labeled as an energy density, contradicting the corrected SI convention.

  If wrong: The qualitative conclusion that the required density is enormous remains, but the quoted quantitative no-go density is wrong by about 17 orders of magnitude.

• high
  Vacuum companion Section 4 and Addendum A.3, Case A maximum speed — The quoted maximum speed is incompatible with the packet's corrected energy-density form of the vacuum bound.

  If wrong: The qualitative propulsion no-go is strengthened, but the stated Case A quantitative bound is invalid under the document's own corrected SI convention.

• high
  Vacuum companion Section 4, matching bound — The specialized thick-wall matching bound is asserted at order of magnitude without a derivation or a precise cited formula with hypotheses.

  If wrong: The quantitative vacuum density and speed bounds do not follow. The separate retarded-build inequality may survive, but the claimed energetic no-go would be unsupported.

• medium
  §6 loop penalty / equation index #8 — The refusal mechanism is self-nullifying as written: the trigger requires a nonvanishing loop integral, and the same section concedes the integral vanishes identically for smooth single-valued t_★. Table 2 then substitutes an unrelated dimensionless angle function for the integral. The argument of the exponential also has no declared units, so λ = 10 cannot be dimensionally checked.

  If wrong: The chronology-penalty section and Table 2 provide no quantitative prediction; gate F3 remains a valid empirical test but the damper model behind it is inoperative as written.

• medium
  Addendum A.3 versus vacuum companion Section 4, Case A speed — The printed maximum speed is inconsistent with the corrected SI equation when the stated energy density is used.

  If wrong: The quantitative Case A vacuum bound is wrong by about eight orders of magnitude, although the qualitative conclusion that it is useless for propulsion remains unchanged.

• medium
  Companion page Section 3 — locked-tube metric — The metric is explicitly schematic and is not derived from supplied IPT equations, a stress-energy model, or junction conditions; nevertheless it is used to motivate vacuum coordinate-FTL statements.

  If wrong: The vacuum-side statements about the geometry and coordinate traversal cannot be treated as consequences of IPT; they remain properties of an assumed metric form. This is secondary to the chip claim because the submission disclaims a realizable vacuum corridor.

• medium
  Companion page Section 4 — Case A vacuum velocity — The printed Case A vacuum maximum speed is inconsistent with the corrected SI bound supplied in Addendum A.

  If wrong: The numerical magnitude quoted for the Case A vacuum bound is invalid, although the corrected value is even smaller and therefore preserves the stated qualitative conclusion that it is useless for propulsion.

• medium
  Companion page Section 4 — vacuum energy bound — The displayed vacuum bound omits c^−2 even though ρ₀ is defined as energy density, making the expression dimensionally inconsistent in SI. Addendum A gives a different, dimensionally valid expression.

  If wrong: The printed companion-page vacuum matching formula cannot support its numerical vacuum speed bound when ρ₀ is an energy density. The qualitative conclusion must be recomputed using the corrected SI convention.

• medium
  Section 2 minimum wall thickness — The wall-floor scaling is asserted without a free-energy functional, substrate Hamiltonian, stability equation, or derivation connecting coherence length to αn_max.

  If wrong: The asserted micron wall scale and derived approximately 6 μm guide radius are unsupported, although one could still impose them phenomenologically.

• medium
  Section 2, minimum wall thickness — The minimum-thickness relation is presented as a universal floor but is not derived from a declared substrate equation.

  If wrong: The numerical on-chip wall scale and resulting radius constraint are unsupported, although the regulator profile itself could still be used as an independently chosen parameterization.

• medium
  Section 2, wall floor and bubble cap — An inequality is used to infer a unique radius, although it supplies only an upper bound unless equality is added as a design choice.

  If wrong: The claimed uniquely forced 6 μm geometry does not follow; it is one boundary-saturating design among all radii R ≤ 5.869 μm.

• medium
  Section 2, wall geometry inference — The numerical Delta_min calculation is reproducible, but the aspect-ratio inequality is converted to an equality without an explicit equality-saturating design assumption.

  If wrong: The exact approximately 6 micrometre radius is not forced by the stated wall rule. The claimed fixed chip geometry and all radius-dependent implementation assertions require an additional stated design choice.

• medium
  Section 2, Δ_min = ξ / √(α n_max) — The wall floor formula is asserted with no supporting free-energy or coherence-length argument, yet it fixes Δ = 0.5869 μm, hence R = 5.869 μm via Δ/R ≥ 0.1, hence every entry in Table 1.

  If wrong: The abstract's central claim that the geometry is 'forced' to ≈ 6 μm is unsupported.

• medium
  Section 3, vacuum companion page, ds² = −c² dt_★² + (ds − v_Γ f_R(n) dt_★)² + dΩ_⊥² — f_R(n) appears in the corridor line element without ever being defined, so the metric cannot be evaluated, its signature checked, or the stated junction conditions actually imposed.

  If wrong: The assertions about timelike riders, local propagation, and coordinate speed are properties intended by the ansatz, not established consequences of the supplied model.

• medium
  Section 4.1 / Vacuum companion page, v_Γ² ≲ 32π G ρ₀ Δ² — The vacuum bound's numerical prefactor 32π is imported without the matching calculation that produces it, and the printed numerical estimate is inconsistent with the corrected SI convention in Addendum A.3.

  If wrong: The one quantitative result of the principle layer is numerically unreliable, though the qualitative conclusion of negligible vacuum effect survives.

• medium
  Section 4.1 vacuum-c energy estimate and Addendum A.3 — The required density was not converted consistently after the SI form was corrected for energy density.

  If wrong: The paper’s vacuum-c density figure is quantitatively invalid, though the conclusion that the required density is extraordinarily large is strengthened rather than reversed.

• medium
  Section 4.1, 'v_Γ² ≲ 32π G ρ₀ Δ²' — The vacuum energy bound is presented as an order-of-magnitude matching condition, but the derivation is not shown. The paper states 'Matching an Alcubierre-like thick wall to a bounded residual still gives, at order of magnitude, v_Γ² ≲ 32π G ρ₀ Δ²' without deriving this from the stated equations.

  If wrong: If the step is invalid, the vacuum energy bound is unsupported, and the claim that a vacuum-c wall at Δ = 0.5869 μm is energetically impossible is not established.

• medium
  Section 5, 'After lock, an irrotational shift β = ∇ψ may be updated along Γ on the regulator slice' — The irrotational shift β = ∇ψ is asserted as the mechanism for the second-pass speed advantage, but no derivation connects β to the coordinate speed v_coord or to the saturated order parameter profile. The paper does not derive the analog metric or the evolution equation for ψ.

  If wrong: If the step is invalid, the mechanism for the second-pass speed advantage is unspecified, and the claim of operational analog FTL is unsupported.

• medium
  Section 6 assumed chronology penalty — The phenomenological loop penalty is explicitly assumed, but its displayed loop functional vanishes for the smooth single-valued scalar field the text itself describes; the table substitutes an unrelated angle proxy without defining the necessary non-exact one-form, singularity, or units.

  If wrong: Table 2 is not a consequence of the stated penalty equation, and F3 does not test a mathematically specified loop-refusal mechanism until the nonzero circulation structure and its units are defined.

• medium
  Section 6, chronology penalty — The exponent's dimensions are unspecified, and the displayed exact-gradient loop integral vanishes under the smooth single-valued assumptions.

  If wrong: Table 2 and F3 are not consequences of the stated differential expression; they are only an external angle-dependent switch. The claimed chronology protection remains an explicit, unvalidated assumption.

• medium
  Section 6, chronology penalty R_eff = R(n) exp(−λ |∮ ∇t_★ · dl|) — The loop-refusal mechanism is written as ∮∇t_★·dl ≠ 0, then immediately conceded to vanish identically for smooth single-valued t_★. Table 2 substitutes an unrelated angle stand-in sin(θ/2).

  If wrong: The chronology penalty as written cannot trigger, so the F3 gate tests an unrelated angle switch rather than the stated mechanism.

• medium
  Vacuum companion §3, corridor line element — The function f_R(n) appears in the metric without any definition anywhere in the packet, so the line element cannot be evaluated, its signature verified, or the stated junction conditions (n and ρ_− continuous, Δ/R ≥ 0.1) imposed. The assertions that the rider stays timelike and local light is not outrun therefore rest on an undefined object.

  If wrong: The vacuum principle-layer claims about timelike riders and coordinate-only FTL become unevaluable; the metric sketch supports no conclusion.

• medium
  Vacuum companion Section 3 metric — The metric contains an undefined function f_R(n), and no conditions are supplied to establish the associated causal claims.

  If wrong: The locked-tube metric, junction behavior, and statements about timelike riders and endpoint coordinate speed cannot be evaluated from the submission.

• medium
  Vacuum companion Section 3, schematic metric — The corridor metric is not developed sufficiently to verify the energy bound, junction conditions, or timelike-rider assertion.

  If wrong: Claims that the rider remains timelike, local light is not outrun, and the proposed stress profile matches this geometry cannot be checked from the submitted metric.

• medium
  Vacuum companion Section 4 energy bound — The numerical prefactor and matching relation are presented without a derivation or a precise cited formula establishing their applicability to the schematic corridor metric.

  If wrong: The quantitative vacuum no-go bound and both derived vacuum estimates lack a demonstrated basis, although this does not directly alter the analog chip budget.

• medium
  Vacuum Principle Layer Section 3 metric — The schematic locked-tube metric contains an undefined function f_R(n) and is not derived from an IPT field equation or a specified analog medium.

  If wrong: Claims concerning the vacuum corridor's local causal structure, junction, and coordinate-speed interpretation remain schematic ansatz statements rather than established consequences of a defined metric model.

• medium
  Vacuum Principle Layer Section 4 and Addendum A.3 — The companion page retains a non-SI energy-density formula and associated numerical estimates, while Addendum A.3 gives the required SI correction. The quantitative claims are mutually inconsistent.

  If wrong: The qualitative no-go conclusion remains, but the printed vacuum speed and required-density values cannot be quoted or used quantitatively under the packet's final SI convention.

• low
  Section 2 wall floor and bubble cap — An inequality is converted into a unique radius without explicitly declaring that the design saturates the bound.

  If wrong: The claimed uniquely forced 5.869 μm radius becomes only a maximum permitted radius; the profile can be retained by declaring equality as a design choice.

• low
  Section 2, Δ / R ≥ 0.1 ⇒ R = 5.869 μm — The inference from the aspect-ratio inequality to a unique value of R replaces an upper bound on R with equality without stating an equality-saturating design choice.

  If wrong: The specific R value is not uniquely forced by the stated inequality, though the order of magnitude is unaffected.

• low
  Section 6, 'R_eff = R(n) exp(−λ |∮ ∇t_★ · dl|), λ = 10' — The loop penalty is explicitly labeled as an assumption ('not derived'), and the paper acknowledges that for a smooth single-valued t_★ the loop integral is identically zero. The angle table uses |∮| ∼ sin(θ/2) as a stand-in, not a topology calculation.

  If wrong: If the step is invalid, the loop refusal mechanism is unsupported, but the paper explicitly labels this as an assumption to be tested by gate F3, so the consequence is limited to the chronology protection claim.

⚑Derivation Flags (45)

• high
  §2 wall floor / equation index #2 — Δ_min = ξ/√(α n_max) is asserted with no free-energy, healing-length, or coherence argument. It is the sole determinant of Δ, hence of R via the aspect rule, hence of every row of Table 1 and of the abstract's claim that the geometry is 'forced' to ≈ 6 μm. Addendum A.4 concedes no substrate Hamiltonian exists from which such a scaling could be obtained. Load-bearing for the paper's central geometric conclusion.

  If wrong: If the scaling has a different power or prefactor, Δ changes, R changes, all of Table 1 changes, and the abstract's central claim that micron coherence 'forces' a ≈ 6 μm on-chip geometry is unsupported.

• high
  §4.2 analog layer; §7 Table 3; gate F2 / equation index #6 — The second-pass speed advantage v_coord > c_s inside T_Γ (fiducial 1.5 c_s) is the paper's central operational claim, but no evolution equation, dispersion relation, or analog metric connects the saturated profile n_★(r), the regulator R(n), or the shift β = ∇ψ to a propagation speed. Table 3 is explicitly a target waveform, not a derivation. Load-bearing for the main conclusion.

  If wrong: If no locked configuration supports super-c_s propagation, gate F2 fails by construction and the reusable-rail claim — the entire actionable content of the spec — collapses to a kinematic stipulation.

• high
  §5 lock rule, definition of T_Γ — The lock bit is imposed on a barred quantity n̄ whose relation to the radial profile n_★(r) (axis value, tube average, or some other functional) is never specified. Every post-lock rule and gates F1–F3 depend on whether T_Γ is nonempty under Case A, which cannot be determined without this definition.

  If wrong: If n̄ cannot reach the lock threshold under frozen Case A, T_Γ is empty and no post-lock behavior, second-pass advantage, or F1–F3 test can be realized as specified.

• high
  Section 4.2 and Section 7, central analog propagation claim — The super-medium-speed mode is stipulated without a medium Hamiltonian, field equation, dispersion relation, or derived analog metric.

  If wrong: If the eventual substrate dynamics do not possess the assumed guided mode, F2 and the paper's main reusable-rail conclusion fail. This is load-bearing.

• high
  Section 4.2, 'Operational analog FTL means v_coord > c_s inside the prepared tube T_Γ' — The definition of operational analog FTL is asserted without derivation from the stated order parameter, regulator, or wall geometry. The paper does not provide an analog metric or evolution equation connecting the saturated order parameter to a coordinate speed exceeding c_s.

  If wrong: If the step is invalid, the central claim of operational analog FTL is unsupported, and the entire experimental program (gates F0–F4) lacks a theoretical foundation.

• high
  Section 5 lock criterion and Table 2, parallel row — The maximum frozen Case A response is below the threshold defining a locked prepared tube, but the table classifies that maximum as lock allowed.

  If wrong: Without a locked T_Γ, the post-lock shift, steering-refusal test, second-pass test, and loop-refusal test cannot operate under frozen Case A.

• high
  Section 5 lock definition and Table 2 — The frozen Case A ceiling does not meet the stated lock criterion, while the table classifies that ceiling as locked.

  If wrong: Under the stated Case A definitions, T_Γ cannot satisfy the lock bit. The post-lock shift rule, target waveform, and F1–F3 lock-dependent tests cannot operate as specified.

• high
  Section 5, lock bit definition R(n̄) ≥ 0.95 vs frozen Case A R(40) ≈ 0.945 — The lock threshold cannot be met by the frozen Case A profile, yet Table 2 labels the n=40 core as 'lock allowed' and later sections assume a locked tube.

  If wrong: The defined prepared tube T_Γ is empty under the supplied definitions, so the post-lock rules and F1–F3 predictions cannot operate as written.

• high
  Section 5, prepared-tube lock definition and Table 2 — The frozen Case A regulator cannot reach the stated 0.95 lock threshold, although the table classifies its maximum value as an allowed lock.

  If wrong: No prepared tube locks under the frozen Case A parameters, so the post-lock shift, second-pass F2 test, steering test F1, and loop-lock test F3 are undefined as written.

• high
  Section 6 and Table 2 — The loop-refusal table substitutes an angle function for a circulation that vanishes under the displayed exact-gradient definition.

  If wrong: The stated mathematical mechanism predicts no loop suppression; F3’s loop-refusal prediction requires a separately specified non-exact one-form, singularity, or phenomenological angle law.

• high
  Sections 2, 3, 5, and Table 2 — Case A lock condition — The submitted Case A maximum saturation is below the submitted lock threshold, yet the maximum-saturation state is subsequently classified as locked.

  If wrong: Under the stated definition, T_Γ is empty in Case A. The post-lock shift, target F2 traversal, steering test F1, loop test F3, and conclusion that this Case A chip is a locked rail are not defined as claimed.

• high
  Sections 4.2, 5, and 7 — analog shift and second-pass claim — No substrate Hamiltonian, evolution equation, analog metric, or constitutive coupling derives the claimed propagation law v_coord=1.5c_s or the inequality T_second<L/c_s from the saturated profile and β=∇ψ.

  If wrong: The principal claim of a reusable analog rail with a post-lock speed advantage is unsupported. The document would remain a proposed profile and falsification plan, but not a mathematical specification predicting F2.

• high
  Sections 4.2, 5, and 7: analog second-pass claim — The claimed post-lock super-c_s propagation is asserted as operational kinematics/target waveform, without a substrate evolution equation or derivation from the order-parameter profile and irrotational shift.

  If wrong: The main analog claim—a reusable locked rail with a second-pass speed advantage—and F2's predicted inequality are unsupported rather than consequences of the submitted model.

• high
  Sections 4.2, 5, and Table 3/F2 — No analog-medium evolution equation derives v_coord>c_s or the 1.5c_s target from n, R(n), or β=∇ψ.

  If wrong: F2 and the paper’s main reusable-rail conclusion are unsupported; the apparatus may merely be an ordinary guide with no post-lock speed advantage.

• high
  Table 3, gate F2, Section 4.2 — The target waveform v_coord = 1.5 c_s inside T_Γ is stated as an intended F2 waveform, but no derivation connects the saturated order parameter profile n_★(r), the regulator R(n), or the wall geometry to a coordinate speed exceeding c_s. The paper explicitly labels Table 3 as 'not a measurement' and Addendum A.5 states 'the claim is analog kinematics only.'

  If wrong: If the step is invalid, the central claim of a reusable rail with second-pass speed advantage is unsupported, and gate F2 cannot be interpreted as a test of the model.

• high
  Vacuum companion Section 4 and Addendum A.3, c-speed density — The displayed density requirement was calculated using the mass-density form of the bound but labeled as an energy density, contradicting the corrected SI convention.

  If wrong: The qualitative conclusion that the required density is enormous remains, but the quoted quantitative no-go density is wrong by about 17 orders of magnitude.

• high
  Vacuum companion Section 4 and Addendum A.3, Case A maximum speed — The quoted maximum speed is incompatible with the packet's corrected energy-density form of the vacuum bound.

  If wrong: The qualitative propulsion no-go is strengthened, but the stated Case A quantitative bound is invalid under the document's own corrected SI convention.

• high
  Vacuum companion Section 4, matching bound — The specialized thick-wall matching bound is asserted at order of magnitude without a derivation or a precise cited formula with hypotheses.

  If wrong: The quantitative vacuum density and speed bounds do not follow. The separate retarded-build inequality may survive, but the claimed energetic no-go would be unsupported.

• medium
  §6 loop penalty / equation index #8 — The refusal mechanism is self-nullifying as written: the trigger requires a nonvanishing loop integral, and the same section concedes the integral vanishes identically for smooth single-valued t_★. Table 2 then substitutes an unrelated dimensionless angle function for the integral. The argument of the exponential also has no declared units, so λ = 10 cannot be dimensionally checked.

  If wrong: The chronology-penalty section and Table 2 provide no quantitative prediction; gate F3 remains a valid empirical test but the damper model behind it is inoperative as written.

• medium
  Addendum A.3 versus vacuum companion Section 4, Case A speed — The printed maximum speed is inconsistent with the corrected SI equation when the stated energy density is used.

  If wrong: The quantitative Case A vacuum bound is wrong by about eight orders of magnitude, although the qualitative conclusion that it is useless for propulsion remains unchanged.

• medium
  Companion page Section 3 — locked-tube metric — The metric is explicitly schematic and is not derived from supplied IPT equations, a stress-energy model, or junction conditions; nevertheless it is used to motivate vacuum coordinate-FTL statements.

  If wrong: The vacuum-side statements about the geometry and coordinate traversal cannot be treated as consequences of IPT; they remain properties of an assumed metric form. This is secondary to the chip claim because the submission disclaims a realizable vacuum corridor.

• medium
  Companion page Section 4 — Case A vacuum velocity — The printed Case A vacuum maximum speed is inconsistent with the corrected SI bound supplied in Addendum A.

  If wrong: The numerical magnitude quoted for the Case A vacuum bound is invalid, although the corrected value is even smaller and therefore preserves the stated qualitative conclusion that it is useless for propulsion.

• medium
  Companion page Section 4 — vacuum energy bound — The displayed vacuum bound omits c^−2 even though ρ₀ is defined as energy density, making the expression dimensionally inconsistent in SI. Addendum A gives a different, dimensionally valid expression.

  If wrong: The printed companion-page vacuum matching formula cannot support its numerical vacuum speed bound when ρ₀ is an energy density. The qualitative conclusion must be recomputed using the corrected SI convention.

• medium
  Section 2 minimum wall thickness — The wall-floor scaling is asserted without a free-energy functional, substrate Hamiltonian, stability equation, or derivation connecting coherence length to αn_max.

  If wrong: The asserted micron wall scale and derived approximately 6 μm guide radius are unsupported, although one could still impose them phenomenologically.

• medium
  Section 2, minimum wall thickness — The minimum-thickness relation is presented as a universal floor but is not derived from a declared substrate equation.

  If wrong: The numerical on-chip wall scale and resulting radius constraint are unsupported, although the regulator profile itself could still be used as an independently chosen parameterization.

• medium
  Section 2, wall floor and bubble cap — An inequality is used to infer a unique radius, although it supplies only an upper bound unless equality is added as a design choice.

  If wrong: The claimed uniquely forced 6 μm geometry does not follow; it is one boundary-saturating design among all radii R ≤ 5.869 μm.

• medium
  Section 2, wall geometry inference — The numerical Delta_min calculation is reproducible, but the aspect-ratio inequality is converted to an equality without an explicit equality-saturating design assumption.

  If wrong: The exact approximately 6 micrometre radius is not forced by the stated wall rule. The claimed fixed chip geometry and all radius-dependent implementation assertions require an additional stated design choice.

• medium
  Section 2, Δ_min = ξ / √(α n_max) — The wall floor formula is asserted with no supporting free-energy or coherence-length argument, yet it fixes Δ = 0.5869 μm, hence R = 5.869 μm via Δ/R ≥ 0.1, hence every entry in Table 1.

  If wrong: The abstract's central claim that the geometry is 'forced' to ≈ 6 μm is unsupported.

• medium
  Section 3, vacuum companion page, ds² = −c² dt_★² + (ds − v_Γ f_R(n) dt_★)² + dΩ_⊥² — f_R(n) appears in the corridor line element without ever being defined, so the metric cannot be evaluated, its signature checked, or the stated junction conditions actually imposed.

  If wrong: The assertions about timelike riders, local propagation, and coordinate speed are properties intended by the ansatz, not established consequences of the supplied model.

• medium
  Section 4.1 / Vacuum companion page, v_Γ² ≲ 32π G ρ₀ Δ² — The vacuum bound's numerical prefactor 32π is imported without the matching calculation that produces it, and the printed numerical estimate is inconsistent with the corrected SI convention in Addendum A.3.

  If wrong: The one quantitative result of the principle layer is numerically unreliable, though the qualitative conclusion of negligible vacuum effect survives.

• medium
  Section 4.1 vacuum-c energy estimate and Addendum A.3 — The required density was not converted consistently after the SI form was corrected for energy density.

  If wrong: The paper’s vacuum-c density figure is quantitatively invalid, though the conclusion that the required density is extraordinarily large is strengthened rather than reversed.

• medium
  Section 4.1, 'v_Γ² ≲ 32π G ρ₀ Δ²' — The vacuum energy bound is presented as an order-of-magnitude matching condition, but the derivation is not shown. The paper states 'Matching an Alcubierre-like thick wall to a bounded residual still gives, at order of magnitude, v_Γ² ≲ 32π G ρ₀ Δ²' without deriving this from the stated equations.

  If wrong: If the step is invalid, the vacuum energy bound is unsupported, and the claim that a vacuum-c wall at Δ = 0.5869 μm is energetically impossible is not established.

• medium
  Section 5, 'After lock, an irrotational shift β = ∇ψ may be updated along Γ on the regulator slice' — The irrotational shift β = ∇ψ is asserted as the mechanism for the second-pass speed advantage, but no derivation connects β to the coordinate speed v_coord or to the saturated order parameter profile. The paper does not derive the analog metric or the evolution equation for ψ.

  If wrong: If the step is invalid, the mechanism for the second-pass speed advantage is unspecified, and the claim of operational analog FTL is unsupported.

• medium
  Section 6 assumed chronology penalty — The phenomenological loop penalty is explicitly assumed, but its displayed loop functional vanishes for the smooth single-valued scalar field the text itself describes; the table substitutes an unrelated angle proxy without defining the necessary non-exact one-form, singularity, or units.

  If wrong: Table 2 is not a consequence of the stated penalty equation, and F3 does not test a mathematically specified loop-refusal mechanism until the nonzero circulation structure and its units are defined.

• medium
  Section 6, chronology penalty — The exponent's dimensions are unspecified, and the displayed exact-gradient loop integral vanishes under the smooth single-valued assumptions.

  If wrong: Table 2 and F3 are not consequences of the stated differential expression; they are only an external angle-dependent switch. The claimed chronology protection remains an explicit, unvalidated assumption.

• medium
  Section 6, chronology penalty R_eff = R(n) exp(−λ |∮ ∇t_★ · dl|) — The loop-refusal mechanism is written as ∮∇t_★·dl ≠ 0, then immediately conceded to vanish identically for smooth single-valued t_★. Table 2 substitutes an unrelated angle stand-in sin(θ/2).

  If wrong: The chronology penalty as written cannot trigger, so the F3 gate tests an unrelated angle switch rather than the stated mechanism.

• medium
  Vacuum companion §3, corridor line element — The function f_R(n) appears in the metric without any definition anywhere in the packet, so the line element cannot be evaluated, its signature verified, or the stated junction conditions (n and ρ_− continuous, Δ/R ≥ 0.1) imposed. The assertions that the rider stays timelike and local light is not outrun therefore rest on an undefined object.

  If wrong: The vacuum principle-layer claims about timelike riders and coordinate-only FTL become unevaluable; the metric sketch supports no conclusion.

• medium
  Vacuum companion Section 3 metric — The metric contains an undefined function f_R(n), and no conditions are supplied to establish the associated causal claims.

  If wrong: The locked-tube metric, junction behavior, and statements about timelike riders and endpoint coordinate speed cannot be evaluated from the submission.

• medium
  Vacuum companion Section 3, schematic metric — The corridor metric is not developed sufficiently to verify the energy bound, junction conditions, or timelike-rider assertion.

  If wrong: Claims that the rider remains timelike, local light is not outrun, and the proposed stress profile matches this geometry cannot be checked from the submitted metric.

• medium
  Vacuum companion Section 4 energy bound — The numerical prefactor and matching relation are presented without a derivation or a precise cited formula establishing their applicability to the schematic corridor metric.

  If wrong: The quantitative vacuum no-go bound and both derived vacuum estimates lack a demonstrated basis, although this does not directly alter the analog chip budget.

• medium
  Vacuum Principle Layer Section 3 metric — The schematic locked-tube metric contains an undefined function f_R(n) and is not derived from an IPT field equation or a specified analog medium.

  If wrong: Claims concerning the vacuum corridor's local causal structure, junction, and coordinate-speed interpretation remain schematic ansatz statements rather than established consequences of a defined metric model.

• medium
  Vacuum Principle Layer Section 4 and Addendum A.3 — The companion page retains a non-SI energy-density formula and associated numerical estimates, while Addendum A.3 gives the required SI correction. The quantitative claims are mutually inconsistent.

  If wrong: The qualitative no-go conclusion remains, but the printed vacuum speed and required-density values cannot be quoted or used quantitatively under the packet's final SI convention.

• low
  Section 2 wall floor and bubble cap — An inequality is converted into a unique radius without explicitly declaring that the design saturates the bound.

  If wrong: The claimed uniquely forced 5.869 μm radius becomes only a maximum permitted radius; the profile can be retained by declaring equality as a design choice.

• low
  Section 2, Δ / R ≥ 0.1 ⇒ R = 5.869 μm — The inference from the aspect-ratio inequality to a unique value of R replaces an upper bound on R with equality without stating an equality-saturating design choice.

  If wrong: The specific R value is not uniquely forced by the stated inequality, though the order of magnitude is unaffected.

• low
  Section 6, 'R_eff = R(n) exp(−λ |∮ ∇t_★ · dl|), λ = 10' — The loop penalty is explicitly labeled as an assumption ('not derived'), and the paper acknowledges that for a smooth single-valued t_★ the loop integral is identically zero. The angle table uses |∮| ∼ sin(θ/2) as a stand-in, not a topology calculation.

  If wrong: If the step is invalid, the loop refusal mechanism is unsupported, but the paper explicitly labels this as an assumption to be tested by gate F3, so the consequence is limited to the chronology protection claim.

⚑Derivation Flags (45)

• high
  §2 wall floor / equation index #2 — Δ_min = ξ/√(α n_max) is asserted with no free-energy, healing-length, or coherence argument. It is the sole determinant of Δ, hence of R via the aspect rule, hence of every row of Table 1 and of the abstract's claim that the geometry is 'forced' to ≈ 6 μm. Addendum A.4 concedes no substrate Hamiltonian exists from which such a scaling could be obtained. Load-bearing for the paper's central geometric conclusion.

  If wrong: If the scaling has a different power or prefactor, Δ changes, R changes, all of Table 1 changes, and the abstract's central claim that micron coherence 'forces' a ≈ 6 μm on-chip geometry is unsupported.

• high
  §4.2 analog layer; §7 Table 3; gate F2 / equation index #6 — The second-pass speed advantage v_coord > c_s inside T_Γ (fiducial 1.5 c_s) is the paper's central operational claim, but no evolution equation, dispersion relation, or analog metric connects the saturated profile n_★(r), the regulator R(n), or the shift β = ∇ψ to a propagation speed. Table 3 is explicitly a target waveform, not a derivation. Load-bearing for the main conclusion.

  If wrong: If no locked configuration supports super-c_s propagation, gate F2 fails by construction and the reusable-rail claim — the entire actionable content of the spec — collapses to a kinematic stipulation.

• high
  §5 lock rule, definition of T_Γ — The lock bit is imposed on a barred quantity n̄ whose relation to the radial profile n_★(r) (axis value, tube average, or some other functional) is never specified. Every post-lock rule and gates F1–F3 depend on whether T_Γ is nonempty under Case A, which cannot be determined without this definition.

  If wrong: If n̄ cannot reach the lock threshold under frozen Case A, T_Γ is empty and no post-lock behavior, second-pass advantage, or F1–F3 test can be realized as specified.

• high
  Section 4.2 and Section 7, central analog propagation claim — The super-medium-speed mode is stipulated without a medium Hamiltonian, field equation, dispersion relation, or derived analog metric.

  If wrong: If the eventual substrate dynamics do not possess the assumed guided mode, F2 and the paper's main reusable-rail conclusion fail. This is load-bearing.

• high
  Section 4.2, 'Operational analog FTL means v_coord > c_s inside the prepared tube T_Γ' — The definition of operational analog FTL is asserted without derivation from the stated order parameter, regulator, or wall geometry. The paper does not provide an analog metric or evolution equation connecting the saturated order parameter to a coordinate speed exceeding c_s.

  If wrong: If the step is invalid, the central claim of operational analog FTL is unsupported, and the entire experimental program (gates F0–F4) lacks a theoretical foundation.

• high
  Section 5 lock criterion and Table 2, parallel row — The maximum frozen Case A response is below the threshold defining a locked prepared tube, but the table classifies that maximum as lock allowed.

  If wrong: Without a locked T_Γ, the post-lock shift, steering-refusal test, second-pass test, and loop-refusal test cannot operate under frozen Case A.

• high
  Section 5 lock definition and Table 2 — The frozen Case A ceiling does not meet the stated lock criterion, while the table classifies that ceiling as locked.

  If wrong: Under the stated Case A definitions, T_Γ cannot satisfy the lock bit. The post-lock shift rule, target waveform, and F1–F3 lock-dependent tests cannot operate as specified.

• high
  Section 5, lock bit definition R(n̄) ≥ 0.95 vs frozen Case A R(40) ≈ 0.945 — The lock threshold cannot be met by the frozen Case A profile, yet Table 2 labels the n=40 core as 'lock allowed' and later sections assume a locked tube.

  If wrong: The defined prepared tube T_Γ is empty under the supplied definitions, so the post-lock rules and F1–F3 predictions cannot operate as written.

• high
  Section 5, prepared-tube lock definition and Table 2 — The frozen Case A regulator cannot reach the stated 0.95 lock threshold, although the table classifies its maximum value as an allowed lock.

  If wrong: No prepared tube locks under the frozen Case A parameters, so the post-lock shift, second-pass F2 test, steering test F1, and loop-lock test F3 are undefined as written.

• high
  Section 6 and Table 2 — The loop-refusal table substitutes an angle function for a circulation that vanishes under the displayed exact-gradient definition.

  If wrong: The stated mathematical mechanism predicts no loop suppression; F3’s loop-refusal prediction requires a separately specified non-exact one-form, singularity, or phenomenological angle law.

• high
  Sections 2, 3, 5, and Table 2 — Case A lock condition — The submitted Case A maximum saturation is below the submitted lock threshold, yet the maximum-saturation state is subsequently classified as locked.

  If wrong: Under the stated definition, T_Γ is empty in Case A. The post-lock shift, target F2 traversal, steering test F1, loop test F3, and conclusion that this Case A chip is a locked rail are not defined as claimed.

• high
  Sections 4.2, 5, and 7 — analog shift and second-pass claim — No substrate Hamiltonian, evolution equation, analog metric, or constitutive coupling derives the claimed propagation law v_coord=1.5c_s or the inequality T_second<L/c_s from the saturated profile and β=∇ψ.

  If wrong: The principal claim of a reusable analog rail with a post-lock speed advantage is unsupported. The document would remain a proposed profile and falsification plan, but not a mathematical specification predicting F2.

• high
  Sections 4.2, 5, and 7: analog second-pass claim — The claimed post-lock super-c_s propagation is asserted as operational kinematics/target waveform, without a substrate evolution equation or derivation from the order-parameter profile and irrotational shift.

  If wrong: The main analog claim—a reusable locked rail with a second-pass speed advantage—and F2's predicted inequality are unsupported rather than consequences of the submitted model.

• high
  Sections 4.2, 5, and Table 3/F2 — No analog-medium evolution equation derives v_coord>c_s or the 1.5c_s target from n, R(n), or β=∇ψ.

  If wrong: F2 and the paper’s main reusable-rail conclusion are unsupported; the apparatus may merely be an ordinary guide with no post-lock speed advantage.

• high
  Table 3, gate F2, Section 4.2 — The target waveform v_coord = 1.5 c_s inside T_Γ is stated as an intended F2 waveform, but no derivation connects the saturated order parameter profile n_★(r), the regulator R(n), or the wall geometry to a coordinate speed exceeding c_s. The paper explicitly labels Table 3 as 'not a measurement' and Addendum A.5 states 'the claim is analog kinematics only.'

  If wrong: If the step is invalid, the central claim of a reusable rail with second-pass speed advantage is unsupported, and gate F2 cannot be interpreted as a test of the model.

• high
  Vacuum companion Section 4 and Addendum A.3, c-speed density — The displayed density requirement was calculated using the mass-density form of the bound but labeled as an energy density, contradicting the corrected SI convention.

  If wrong: The qualitative conclusion that the required density is enormous remains, but the quoted quantitative no-go density is wrong by about 17 orders of magnitude.

• high
  Vacuum companion Section 4 and Addendum A.3, Case A maximum speed — The quoted maximum speed is incompatible with the packet's corrected energy-density form of the vacuum bound.

  If wrong: The qualitative propulsion no-go is strengthened, but the stated Case A quantitative bound is invalid under the document's own corrected SI convention.

• high
  Vacuum companion Section 4, matching bound — The specialized thick-wall matching bound is asserted at order of magnitude without a derivation or a precise cited formula with hypotheses.

  If wrong: The quantitative vacuum density and speed bounds do not follow. The separate retarded-build inequality may survive, but the claimed energetic no-go would be unsupported.

• medium
  §6 loop penalty / equation index #8 — The refusal mechanism is self-nullifying as written: the trigger requires a nonvanishing loop integral, and the same section concedes the integral vanishes identically for smooth single-valued t_★. Table 2 then substitutes an unrelated dimensionless angle function for the integral. The argument of the exponential also has no declared units, so λ = 10 cannot be dimensionally checked.

  If wrong: The chronology-penalty section and Table 2 provide no quantitative prediction; gate F3 remains a valid empirical test but the damper model behind it is inoperative as written.

• medium
  Addendum A.3 versus vacuum companion Section 4, Case A speed — The printed maximum speed is inconsistent with the corrected SI equation when the stated energy density is used.

  If wrong: The quantitative Case A vacuum bound is wrong by about eight orders of magnitude, although the qualitative conclusion that it is useless for propulsion remains unchanged.

• medium
  Companion page Section 3 — locked-tube metric — The metric is explicitly schematic and is not derived from supplied IPT equations, a stress-energy model, or junction conditions; nevertheless it is used to motivate vacuum coordinate-FTL statements.

  If wrong: The vacuum-side statements about the geometry and coordinate traversal cannot be treated as consequences of IPT; they remain properties of an assumed metric form. This is secondary to the chip claim because the submission disclaims a realizable vacuum corridor.

• medium
  Companion page Section 4 — Case A vacuum velocity — The printed Case A vacuum maximum speed is inconsistent with the corrected SI bound supplied in Addendum A.

  If wrong: The numerical magnitude quoted for the Case A vacuum bound is invalid, although the corrected value is even smaller and therefore preserves the stated qualitative conclusion that it is useless for propulsion.

• medium
  Companion page Section 4 — vacuum energy bound — The displayed vacuum bound omits c^−2 even though ρ₀ is defined as energy density, making the expression dimensionally inconsistent in SI. Addendum A gives a different, dimensionally valid expression.

  If wrong: The printed companion-page vacuum matching formula cannot support its numerical vacuum speed bound when ρ₀ is an energy density. The qualitative conclusion must be recomputed using the corrected SI convention.

• medium
  Section 2 minimum wall thickness — The wall-floor scaling is asserted without a free-energy functional, substrate Hamiltonian, stability equation, or derivation connecting coherence length to αn_max.

  If wrong: The asserted micron wall scale and derived approximately 6 μm guide radius are unsupported, although one could still impose them phenomenologically.

• medium
  Section 2, minimum wall thickness — The minimum-thickness relation is presented as a universal floor but is not derived from a declared substrate equation.

  If wrong: The numerical on-chip wall scale and resulting radius constraint are unsupported, although the regulator profile itself could still be used as an independently chosen parameterization.

• medium
  Section 2, wall floor and bubble cap — An inequality is used to infer a unique radius, although it supplies only an upper bound unless equality is added as a design choice.

  If wrong: The claimed uniquely forced 6 μm geometry does not follow; it is one boundary-saturating design among all radii R ≤ 5.869 μm.

• medium
  Section 2, wall geometry inference — The numerical Delta_min calculation is reproducible, but the aspect-ratio inequality is converted to an equality without an explicit equality-saturating design assumption.

  If wrong: The exact approximately 6 micrometre radius is not forced by the stated wall rule. The claimed fixed chip geometry and all radius-dependent implementation assertions require an additional stated design choice.

• medium
  Section 2, Δ_min = ξ / √(α n_max) — The wall floor formula is asserted with no supporting free-energy or coherence-length argument, yet it fixes Δ = 0.5869 μm, hence R = 5.869 μm via Δ/R ≥ 0.1, hence every entry in Table 1.

  If wrong: The abstract's central claim that the geometry is 'forced' to ≈ 6 μm is unsupported.

• medium
  Section 3, vacuum companion page, ds² = −c² dt_★² + (ds − v_Γ f_R(n) dt_★)² + dΩ_⊥² — f_R(n) appears in the corridor line element without ever being defined, so the metric cannot be evaluated, its signature checked, or the stated junction conditions actually imposed.

  If wrong: The assertions about timelike riders, local propagation, and coordinate speed are properties intended by the ansatz, not established consequences of the supplied model.

• medium
  Section 4.1 / Vacuum companion page, v_Γ² ≲ 32π G ρ₀ Δ² — The vacuum bound's numerical prefactor 32π is imported without the matching calculation that produces it, and the printed numerical estimate is inconsistent with the corrected SI convention in Addendum A.3.

  If wrong: The one quantitative result of the principle layer is numerically unreliable, though the qualitative conclusion of negligible vacuum effect survives.

• medium
  Section 4.1 vacuum-c energy estimate and Addendum A.3 — The required density was not converted consistently after the SI form was corrected for energy density.

  If wrong: The paper’s vacuum-c density figure is quantitatively invalid, though the conclusion that the required density is extraordinarily large is strengthened rather than reversed.

• medium
  Section 4.1, 'v_Γ² ≲ 32π G ρ₀ Δ²' — The vacuum energy bound is presented as an order-of-magnitude matching condition, but the derivation is not shown. The paper states 'Matching an Alcubierre-like thick wall to a bounded residual still gives, at order of magnitude, v_Γ² ≲ 32π G ρ₀ Δ²' without deriving this from the stated equations.

  If wrong: If the step is invalid, the vacuum energy bound is unsupported, and the claim that a vacuum-c wall at Δ = 0.5869 μm is energetically impossible is not established.

• medium
  Section 5, 'After lock, an irrotational shift β = ∇ψ may be updated along Γ on the regulator slice' — The irrotational shift β = ∇ψ is asserted as the mechanism for the second-pass speed advantage, but no derivation connects β to the coordinate speed v_coord or to the saturated order parameter profile. The paper does not derive the analog metric or the evolution equation for ψ.

  If wrong: If the step is invalid, the mechanism for the second-pass speed advantage is unspecified, and the claim of operational analog FTL is unsupported.

• medium
  Section 6 assumed chronology penalty — The phenomenological loop penalty is explicitly assumed, but its displayed loop functional vanishes for the smooth single-valued scalar field the text itself describes; the table substitutes an unrelated angle proxy without defining the necessary non-exact one-form, singularity, or units.

  If wrong: Table 2 is not a consequence of the stated penalty equation, and F3 does not test a mathematically specified loop-refusal mechanism until the nonzero circulation structure and its units are defined.

• medium
  Section 6, chronology penalty — The exponent's dimensions are unspecified, and the displayed exact-gradient loop integral vanishes under the smooth single-valued assumptions.

  If wrong: Table 2 and F3 are not consequences of the stated differential expression; they are only an external angle-dependent switch. The claimed chronology protection remains an explicit, unvalidated assumption.

• medium
  Section 6, chronology penalty R_eff = R(n) exp(−λ |∮ ∇t_★ · dl|) — The loop-refusal mechanism is written as ∮∇t_★·dl ≠ 0, then immediately conceded to vanish identically for smooth single-valued t_★. Table 2 substitutes an unrelated angle stand-in sin(θ/2).

  If wrong: The chronology penalty as written cannot trigger, so the F3 gate tests an unrelated angle switch rather than the stated mechanism.

• medium
  Vacuum companion §3, corridor line element — The function f_R(n) appears in the metric without any definition anywhere in the packet, so the line element cannot be evaluated, its signature verified, or the stated junction conditions (n and ρ_− continuous, Δ/R ≥ 0.1) imposed. The assertions that the rider stays timelike and local light is not outrun therefore rest on an undefined object.

  If wrong: The vacuum principle-layer claims about timelike riders and coordinate-only FTL become unevaluable; the metric sketch supports no conclusion.

• medium
  Vacuum companion Section 3 metric — The metric contains an undefined function f_R(n), and no conditions are supplied to establish the associated causal claims.

  If wrong: The locked-tube metric, junction behavior, and statements about timelike riders and endpoint coordinate speed cannot be evaluated from the submission.

• medium
  Vacuum companion Section 3, schematic metric — The corridor metric is not developed sufficiently to verify the energy bound, junction conditions, or timelike-rider assertion.

  If wrong: Claims that the rider remains timelike, local light is not outrun, and the proposed stress profile matches this geometry cannot be checked from the submitted metric.

• medium
  Vacuum companion Section 4 energy bound — The numerical prefactor and matching relation are presented without a derivation or a precise cited formula establishing their applicability to the schematic corridor metric.

  If wrong: The quantitative vacuum no-go bound and both derived vacuum estimates lack a demonstrated basis, although this does not directly alter the analog chip budget.

• medium
  Vacuum Principle Layer Section 3 metric — The schematic locked-tube metric contains an undefined function f_R(n) and is not derived from an IPT field equation or a specified analog medium.

  If wrong: Claims concerning the vacuum corridor's local causal structure, junction, and coordinate-speed interpretation remain schematic ansatz statements rather than established consequences of a defined metric model.

• medium
  Vacuum Principle Layer Section 4 and Addendum A.3 — The companion page retains a non-SI energy-density formula and associated numerical estimates, while Addendum A.3 gives the required SI correction. The quantitative claims are mutually inconsistent.

  If wrong: The qualitative no-go conclusion remains, but the printed vacuum speed and required-density values cannot be quoted or used quantitatively under the packet's final SI convention.

• low
  Section 2 wall floor and bubble cap — An inequality is converted into a unique radius without explicitly declaring that the design saturates the bound.

  If wrong: The claimed uniquely forced 5.869 μm radius becomes only a maximum permitted radius; the profile can be retained by declaring equality as a design choice.

• low
  Section 2, Δ / R ≥ 0.1 ⇒ R = 5.869 μm — The inference from the aspect-ratio inequality to a unique value of R replaces an upper bound on R with equality without stating an equality-saturating design choice.

  If wrong: The specific R value is not uniquely forced by the stated inequality, though the order of magnitude is unaffected.

• low
  Section 6, 'R_eff = R(n) exp(−λ |∮ ∇t_★ · dl|), λ = 10' — The loop penalty is explicitly labeled as an assumption ('not derived'), and the paper acknowledges that for a smooth single-valued t_★ the loop integral is identically zero. The angle table uses |∮| ∼ sin(θ/2) as a stand-in, not a topology calculation.

  If wrong: If the step is invalid, the loop refusal mechanism is unsupported, but the paper explicitly labels this as an assumption to be tested by gate F3, so the consequence is limited to the chronology protection claim.