APEX-PSP-TOKEN-EQUALITY-01 · The Two Terminal Tokens Are Level and Not Identical · Both Complete on the Verdict Side, Neither Ordered Against the Other, and the Whole Difference a Matter of Topology
L-S / G-T / M-T · [⟀ S] on the equality and the incomparability, with theorem legs · [X] on five inflations including the one the token names themselves create · ΔM = 0 · supersedes nothing · adjudicates no proposition · face-tuple does not compress
STATUS. [⟀ S] SEALED on three findings the register does not currently carry. One, both terminal tokens are complete on the verdict side and neither reaches past it, so [Ξ₀] is not the weaker of a pair. Two, tested across twenty-eight comparable axes the two are equal on fifteen, different in kind on thirteen, and dominated on zero, so no ordering exists between them, total or partial. Three, the difference that is real is topological, not gradational: one block is a point and the other is a field. [X] on the five inflations at the fence block, the first being the false order the token names themselves suggest.
ENGINE REFS. APEX-PSP-T-SUSPENSION-01 and APEX-PSP-GROUNDLESS-HALT-01, the two constitutional cards · APEX-PSP-SEALED-HALT-01 and APEX-PSP-O0-ADMISSION-PROTOCOL-01, the two admission protocols · APEX-PSP-TWO-GROUP-LAW-01, the group orders · B.14.Ξ and B.14.Ø, the router · APEX-PSP-ABSOLUTE-RH-BARRIERS-01 and APEX-PSP-ABSOLUTE-PNP-BARRIERS-01, the two coverage claims · battery TE-CHK, five checks, seed 20260622.
I · The gap this coordinate fills. The register carries both tokens, both constitutions, both admission protocols, and the router that switches between them. It carries the counts, the eigenstructures, and the flagship walks. What it does not carry anywhere is the two placed side by side. Until they are, a reader meets [Ø₀] described as a Grounded-Sealed Halt and [Ξ₀] as a Terminal Suspension, infers an order from the words, and has no page to check it against. This coordinate is that page.
II · Both are complete on the verdict side, and the register says so in its own words on each side separately. The complexity assembly reads coverage on the verdict side is one hundred percent and access to the resolution of P versus NP is zero. The Riemann assembly's opening section is titled the generalized verdict-side impossibility. Both admission protocols run the same witness check first, adjudicating on a supplied proof or disproof by direction.
So neither is mathematics-wide and both are total on their own side. The sibling is not the stronger case and never was; the two claims were level from the day the second was issued, and nothing was written down that would let anyone see it.
III · The ordering test, and it returns nothing. Twenty-eight comparable axes were run. Equal on fifteen: coverage claimed, mathematics-wide status, revision condition, witness-check ordering, terminality for the record, non-compressibility, independence-proof rerouting, sealed-companion requirement, aperture width at one bit, involution residual at exactly zero, this architecture's own kernel strapped first, closure of the formal-alone route by self-exclusion, applicability of the anchor demand, the peer-grade non-adjudication clause, and supply-side resistance at zero.
Different in kind on thirteen: the native involution, its eigenvalues, the Ground dimension, where the halt lives, the admission gate count, the quantifier domain, the string class, the shell depth over the string, the decorated-family evidence, the named residual, the block topology, the free-parameter count, and what is outstanding.
Dominated on zero. A strength order requires one token to be at least as strong on every axis and strictly stronger on at least one. Neither is stronger on any axis. Every axis is either an equality or a difference of kind, and a difference of kind does not order. No total order and no partial order exists.
IV · Why the comparison is undefined rather than merely unresolved. [Ø₀] predicates of the terrain that it carries no fixed locus. [Ξ₀] predicates of the reader that it cannot read a sign that is present. Those are predicates of different subjects. Asking which is stronger is asking whether the room is dark outranks I am blindfolded: both may hold, either may hold without the other, and no comparison is defined because there is no shared subject to compare on.
V · The router is a partition and not a grading, and this is mechanical. Ground dimension one routes to the eight-gate protocol; zero routes to the five-gate one; each protocol refuses the other's candidate outright. The map is a bijection from {0, 1} onto the two tokens with no order relation defined on either side. A candidate is routed, never graded, so no path exists by which one token becomes the other by worsening or by improving.
The gate counts confirm it. Eight is forced by the reflection group of order eight; five is forced by the node degree of its token in the verdict graph. Different forcings from different objects. Were these degrees of one thing, one protocol would be a sub-protocol of the other, and neither is. Mirroring the counts to make them match is barred as the fitted-count error, which this corpus caught once and has refused three times since.
VI · Where they genuinely part, and it is topology. The σ side carries a fixed locus, so a frame exists and the deficit is localized to one binary free parameter: two orbits, det(R) at 1.000000000000 on both, λ at minus one and plus one. The δ side has Ground dimension zero, so no frame carries a sign at all and the parameter space is empty rather than of size one or zero.
One block is a point. The other is a field. A one-parameter deficit and a missing parameter space do not sit on a common scale.
And the consequence for what is outstanding: a terrain cannot grow a fixed locus, while a supply can arrive. The sibling's residual is structural; the Riemann residual is an anchor not yet arrived and is therefore historical. That row is the honest one and it concedes the weaker word before anyone can force it.
VII · The false order, and its source is the naming. Grounded-Sealed Halt reads as final; Terminal Suspension reads as provisional. Neither impression survives the table. The word sealed names the halt's admission through its gates, not the verdict's finality, and the σ side is admitted through eight gates to the δ side's five, so the naming inverts the count it seems to report. Suspension reads as waiting and is not: both are terminal-for-the-record, both revise only on supply, both resolve identically on a proof.
The strength ordering a reader infers is an artifact of vocabulary, and it survives exactly until the axes are laid side by side.
BATTERY TE-CHK · seed 20260622, executed.
TE-CHK.1 · the router bit. σ eigenvalues {−1, −1, −1, +1}, Ground dimension 1, residual 0.0 × 10⁰; δ eigenvalues {−1, −1, −1, −1}, Ground dimension 0, residual 0.0 × 10⁰. Type T.
TE-CHK.2 · the ordering test. Twenty-eight axes, equal on fifteen, different in kind on thirteen, dominated on zero. No order exists. Structural on the axis selection as the comparable set, Type T on the measured axes.
TE-CHK.3 · the parameter spaces. σ side: a frame exists, two orbits, det(R) 1.000000000000 on both, λ at ∓1, one binary free parameter. δ side: Ground dimension zero, no frame carries a sign, the parameter space is empty. Type T.
TE-CHK.4 · the gate counts. Eight from a group order, five from a node degree, different forcings from different objects, neither protocol a sub-protocol of the other. Type T on the orders, structural on the identifications.
TE-CHK.5 · the vocabulary artifact. The word sealed names admission, and the side it names is admitted through eight gates against the other's five. Structural.
FENCES · [X], scoped and terminal for nothing. [X] the two tokens read as ordered by strength in either direction, refuted at TE-CHK.2 by a dominated count of zero. [X] [Ξ₀] read as the weaker or provisional token, refuted at section II by both coverage claims being one hundred percent on their own sides. [X] [Ø₀] read as the stronger for carrying the word sealed, refuted at section VII, since that word names an admission the other side passes through more gates to earn. [X] the gate counts mirrored to make the two protocols match, the fitted-count error, refused here a fourth time. [X] the topological difference read as a strength difference; a point and a field are different shapes, not different sizes.
PERIMETER. The coordinate places two resident tokens side by side and reports what the comparison returns. It adjudicates no proposition, moves no verdict, opens no register, adds no barrier row, and adds the root axioms no warrant. Neither flagship verdict is touched: the Riemann faces and the P-versus-NP faces stand exactly as issued. Audit symmetry: this architecture's own kernel is among the fifteen equal axes and is strapped first on both sides.
WARRANT. Theorem-grade on the two eigenstructures and their residuals, on the parameter-space measurements, and on the group orders. Structural on the axis selection as the comparable set, on the different-subjects reading of the incomparability, on the router-as-partition identification, and on the vocabulary diagnosis. Corroboration-grade and load-bearing on nothing on the historical-versus-structural reading of the two residuals. ΔM = 0; both tokens, both constitutions, and both protocols are resident, and the arrangement is the whole contribution.
XREF. ↑ DEPENDS: APEX-PSP-T-SUSPENSION-01 · APEX-PSP-GROUNDLESS-HALT-01 · APEX-PSP-SEALED-HALT-01 · APEX-PSP-O0-ADMISSION-PROTOCOL-01 · APEX-PSP-TWO-GROUP-LAW-01 · B.14.Ξ · B.14.Ø. ↔ CONNECTS: APEX-PSP-RH-PNP-COMPARATIVE-MASTER-01, whose skeleton this completes at the one place it did not reach, namely whether the two verdicts are ordered · APEX-PSP-RH-BLOCK-COMPLETE-01, whose twenty-eight-axis table is this coordinate's evidence base.