Venue-neutral preprint manuscript
Author: Josip Fellner
Affiliation: Independent Researcher
ORCID: https://orcid.org/0009-0003-1705-3985
Correspondence: info@apeironfield.com
Version: 1.2, 14 September 2026
Numerical reference: terminal v7.13 Stage-0 HARD-PASS;
Float64 crossing schedule complete and frozen
Follow-up status reference: frozen v7.13 and the
documented AP1 numerical status remain unchanged; this revision adds the
current single-carrier ontology and the separate black-hole reconversion
feasibility result without importing either into the frozen numerical
evidence.
Scope. This document reports a frozen numerical classification and package integrity. It also records, without importing new evidence into the v7.13 result, the confirmed gate status of the separate AP1 follow-up program. Version 1.2 additionally aligns the current-facing Apeiron interpretation to a single-carrier ontology and records the independent black-hole reconversion feasibility branch. Those conceptual additions do not reclassify, recompute or retroactively reinterpret the numerical gates. The document does not report continuum convergence, a unique fundamental covariant action, a completed physical response kernel, observable curves or empirical confirmation.
The stored v7.13 endpoint is a protocol HARD-PASS, not an independent-map reproduction. A later clean-room comparison did not reproduce the stored HARD-PASS classification; independent-map reproduction level V4 remains open. The comparison independently recomputed only selected scalar diagnostics from the frozen state: 7 of 10 scalar rows met the stricter cross-implementation tolerances and 9 of 10 original gates passed. A preregistered no-map precision isolation accounts for the Wronskian discrepancy under frozen forcing but does not close V4.
The frozen homogeneous Apeiron/AME-1 model is defined operationally on a spatially flat expanding FRW background by a noncanonical pressure function, a portal-coupled renormalized mode source and a discrete fixed-point map on 15361 time nodes. At the binary64 endpoint (), corresponding to the exact protocol decimal (0.47878124999999994315658113919198513031005859375), the inherited v7.13 state already satisfied the fixed-point criterion and required no Newton or GMRES update. The recorded fixed-point, Friedmann, continuity, Raychaudhuri, quantum-source and Wronskian diagnostics are (4.6838977141305804^{-11}), (3.818888298257304^{-12}), (3.7214611922998273^{-6}), (1.4051164250399966^{-5}), (3.883856960984437^{-8}) and (7.432742701266731^{-9}), respectively. All preregistered v7.13 gates passed and no finalization failure was recorded, yielding a HARD-PASS under the frozen protocol. The adjacent binary64 endpoint () is an immutable v6.74 NONPASS because its fixed-point and Friedmann gates failed. The classified endpoints are adjacent representable values; the exact protocol midpoint rounds to the NONPASS endpoint, terminating the frozen Float64 schedule. The independently calculated binary64 Wronskian was (2.4628917628444214^{-5}), whereas long-double mode arithmetic in the no-map isolation gave (7.613423383217004^{-9}), consistent with the stored (7.432742701266731^{-9}) under the registered mixed tolerance. The corrected source passed synthetic and static conformance audits without a new physical-map execution. The result establishes compliance with the stated numerical criteria for the frozen discretization, not continuum convergence, empirical confirmation or an Apeiron ontology. The separate AP1 program has advanced only through confirmed, bounded numerical and structural preflights; it remains ORANGE with the G32 three-level pilot open and the physical response kernel blocked. AP2 and AP3 remain locked. No observable curve, data fit, significance or empirical preference is claimed.
For current-facing interpretation, Apeiron is now stated explicitly
as a single fundamental carrier. Ordinary matter,
dark-matter-like gravitational phenomena and dark-energy-like
large-scale behavior are not treated as ontologically separate
fundamental substances inside the Apeiron model; they are hypotheses
about different states, organizations or manifestations of the same
carrier. This is an ontological/model statement, not an empirical result
of the v7.13 computation. A separate black-hole reconversion branch has
reached a structural scientific-feasibility GO: an
organized Apeiron manifestation can be represented as undergoing a
local, energy-momentum-conserving transition toward more fundamental
Apeiron states under extreme curvature. That branch likewise supplies no
observational confirmation.
Keywords: numerical fixed-point classification; binary64 endpoint; gate protocol; reproducibility levels; Pauli-Villars source; FRW background map
Foundational model programs face distinct burdens of evidence. First, their internal equations and numerical decisions must be explicit enough to audit. Second, their computations must satisfy declared criteria without retrospective changes. Third, their physical interpretation must be converted into observables and compared with data. Passing one burden does not discharge the others.
The present work addresses the first two burdens for one frozen homogeneous Apeiron/AME-1 calculation. It asks:
The v7.13 result and its adjacent v6.74 counter-endpoint are treated as immutable. No solver rerun, gate change, parameter adjustment or new physical assumption is introduced in preparing this manuscript.
The paper uses “Apeiron/AME-1” as the name of a frozen homogeneous numerical model. Its publication definition is operational: the variables, parameters, pressure function, portal coupling, quantum-source prescription, background equations, initial data, discretization and active source hashes are supplied in the associated versioned reproducibility archive.
The background is spatially flat FRW with scale factor (a), e-fold variable (N=a), Hubble rate (H>0), natural units (c=) and reduced Planck mass (M_{}=1). The numerical time coordinate is
[ =H_t,=. ]
The registered calculation is homogeneous and background-level. It does not contain:
The lack of a unique action-level completion is not concealed. It limits the article to the exact operational discrete model that was computed.
() is the registered numerical time/control endpoint of the frozen calculation. A unique observational or cosmological mapping beyond (=H_t), including any direct mapping to redshift, remains an AP1 task. The terminal Float64 bracket must therefore not be described as a measured physical constant or exact physical phase boundary.
The current Apeiron model uses one fundamental carrier as its binding ontology. The numerical variables of AME-1 are effective variables of a frozen operational model and are not asserted to enumerate the microscopic constituents of that carrier.
Within this ontology:
This ontology is a model-level hypothesis. It is not established by the v7.13 HARD-PASS, and it does not change any frozen equation, gate, parameter, source hash or numerical classification reported in this paper. If unknown dark-sector particles are discovered experimentally, their existence would not by itself contradict the single-carrier ontology; they could be represented as additional organized manifestations, but that interpretation would require separate physics.
The full trajectory container is
[ y=(,,,,,,H,N). ]
The iterated state uses six columns,
[ u=(,,,,H,N), ]
while the homogeneous () bookkeeping coordinate is fixed to zero. The portal sector enters through renormalized mode expectation values (_q), (p_q) and (^2_q). This six-column numerical state is not claimed to be a count of fundamental degrees of freedom in an unknown covariant completion.
The fixed-point vector is the flattened scaled state
[ x=!( , , , , , ), ]
with 15361 nodes, 92166 Float64 components and a full stored solution array of shape (15361).
Define
[ X=^2,z=, b=b_+b_{}-, ]
[ F(z,b)=14z4-12z2+bz . ]
The registered pressure is
[ P(X,,) =X+ (z-)2X2 -E_cF(z,b)-V_s(), ]
where (V_s()={i=0}{4}c_ii) and the five frozen coefficients are given in the canonical specification. The executable derivatives (P_X), (P{XX}), (P_), (P_) and (P_{X}) are stated there in full.
The AME stress is
[ {}=^2+2XP_X-P, p{}=^2+P. ]
The field-dependent portal mass is
[ m_2=m_{,0}2+ 2(v+)^2!(). ]
The quantum routine uses four Pauli-Villars sectors with coefficients ((1,-3,3,-1)), regulator indices ((0,1,2,3)), 128 resolved momentum nodes on the initial physical support (0k), an analytic order-0 tail and auxiliary adiabatic curvature orders 2 and 4. Mode functions are advanced by a three-substep Yoshida composition of an implicit-midpoint update.
The normalized quantum-source diagnostic represents
[ _q+3H(q+p_q) -m22_q=0. ]
With ({}={}+q) and (p{}=p_{}+p_q), the image map integrates
[ =-3H+P_ -m22_q, ]
[ = , ]
[ H=+, N=H. ]
The Raychaudhuri relation
[ H=- ]
is retained as an independent diagnostic. Hyperbolicity gates require (P_X>0) and (P_X+2XP_{XX}>0).
For the hash-identified image map (G),
[ R(x;)=x-G(x;), {}=|R|. ]
All parameters, source files, gates, tolerances, discretization and solver budgets were frozen before the terminal classification.
The grid spacing is (=_{}/15360). The registered finite-difference step for Jacobian-vector products is (10^{-5}). The allowed nonlinear budget was:
These values describe the frozen allowance. The terminal v7.13 evaluation consumed no Newton update because the inherited state already passed the fixed-point gate.
The v7.13 result JSON, solution NPZ, gate register, protocol, crossing decision, terminal checkpoint state and active runtime sources are individually hashed in the archive manifest. The adjacent v6.74 result, solution and gate register are also included. The v6.74 gate table is preserved for its own classification and is not substituted for the strict v7.13 table.
The read-only verifier supplied with the archive:
It imports no solver module, evaluates no physical map and modifies no file. Its passing terminal message is APEIRON_PUBLICATION_PACKAGE_VERIFY_PASS. This is package-integrity verification, not independent numerical reproduction.
| Level | Meaning | Status |
|---|---|---|
| V0 | narrative and filenames | passed |
| V1 | narrative and structured handoff agree | passed |
| V2 | immutable artifacts resolved and hashes verified | passed |
| V2.5 | stored gates, array forms and crossing identities checked read-only | passed |
| V3 | selected physical diagnostics independently recomputed from the stored state | partial: 7/10 stricter scalar comparisons and 9/10 original gates |
| V4 | independent implementation reproduces the classification | open |
| V5 | observable templates preregistered and tested | not begun; AP1 methods work has not released an observable template |
The C0.29 comparison was a single preregistered passive execution.
The external executor preflight passed 39/39 tests, the internal
comparator preflight passed 58/58 tests, and exactly one executor and
one comparator process ran. No new map, Newton, GMRES, seed or
preconditioner call occurred. Seven of ten scalar rows met the stricter
cross-implementation mixed tolerances, both boolean flags agreed, and
nine of ten original gates agreed with the stored classification.
Fixed-point and Friedmann missed only the stricter cross-implementation
tolerances while still passing their original gates. The independently
calculated binary64 Wronskian, (2.4628917628444214^{-5}), failed both
its cross-implementation tolerance and the original (2^{-8}) gate. The
registered result was therefore
independent_map_does_not_match_stored_hard_pass_classification.
C0.30 traced the Wronskian definition and found no mathematical-definition mismatch in the mode variable, (a^{3/2}) transform, cosmic-time conversion, (H_) scaling, complex sign, sector range, normalization, node range or indexing. C0.32 then executed one separately registered no-map precision isolation using the frozen binary64 forcing. It exactly reproduced the C0.29 binary64 value and obtained (7.613423383217004^{-9}) with long-double mode arithmetic. That value passes the original gate and agrees with the stored v7.13 scalar under the registered mixed tolerance. C0.33 froze the corresponding mode/physical-transform/Wronskian/low-mode-stress source delta and passed 14/14 synthetic tests before freeze and 14/14 after fresh extraction. C0.34 separately promoted the direct order-0 PV quadrature accumulation and passed 8/8 tests before freeze and 8/8 after fresh extraction.
A source-only C0.35 audit found the corrected core quantum-source and
gate implementation statically conformant in all 20 audited items. It
also found one non-gating auxiliary-diagnostic definition deviation
(min_heavy_omega2 sampled at three Yoshida substep
midpoints rather than the global-step midpoint), five computed auxiliary
values not exposed by the final evaluation interface, and an absent
k90_over_Lambda diagnostic. These findings do not alter the
core source arrays or registered gates, but they prevent a claim of
complete diagnostic-surface conformance.
The combined conclusion is deliberately narrow: the arithmetic precision path is causally sufficient for the stored Wronskian discrepancy under frozen forcing, and the corrected core source is statically conformant. Because the corrected complete physical map has not been run on the frozen input, neither v7.13 nor C0.29 has been independently confirmed by a full second implementation.
| Gate | Operator | Threshold | Recorded value | Result |
|---|---|---|---|---|
| fixed-point infinity norm | < | (5.0^{-11}) | (4.6838977141305804^{-11}) | pass |
| Friedmann constraint | < | (1.0^{-10}) | (3.818888298257304^{-12}) | pass |
| continuity residual | < | (1.0^{-5}) | (3.7214611922998273^{-6}) | pass |
| Raychaudhuri residual | < | (2.0^{-5}) | (1.4051164250399966^{-5}) | pass |
| quantum-source identity | < | (1.0^{-7}) | (3.883856960984437^{-8}) | pass |
| Wronskian diagnostic | < | (2.0^{-8}) | (7.432742701266731^{-9}) | pass |
| minimum total density | > | 0 | (2.0729288609194362^{-14}) | pass |
| minimum H | > | 0 | (8.312498343027667^{-8}) | pass |
| minimum (P_X) | > | 0 | (1.0523451240627972) | pass |
| minimum (P_X+2XP_{XX}) | > | 0 | (1.052350229025336) | pass |
| finite | is | true | true | pass |
| finalization failure | is | null | null | pass |
HARD_PASS requires every listed numerical gate to pass and the finalization-failure field to be null. PASS/NONPASS is a deterministic protocol classification. It is not a statistical (p)-value, likelihood statement, model-selection score or empirical verdict.
The terminal exact protocol value is
[ _{}^{} =0.47878124999999994315658113919198513031005859375, ]
stored as binary64
[ _{}^{64}=0.47878124999999994. ]
The terminal record reports one physical-map evaluation, zero Newton steps, no iteration log, no finalization failure and HARD_PASS. The exact diagnostic values and thresholds are listed in Section 6.
The adjacent endpoint
[ _{}^{64}=0.47878125 ]
is classified NONPASS in the immutable v6.74 record. Its fixed-point diagnostic (1.8866305739884126^{-10}) and Friedmann diagnostic (1.7440413453847122^{-10}) failed the gates registered for that result; its remaining documented gates passed. The counter-endpoint is used only for the frozen crossing classification.
| Quantity | Value |
|---|---|
| exact protocol bracket width | (5.684341886080801486968994140625^{-17}) |
| binary64 bracket width | (5.551115123125783^{-17}) |
| next exact protocol midpoint | (0.478781249999999971578290569595992565155029296875) |
| midpoint converted to binary64 | (0.47878125), the NONPASS endpoint |
| endpoint relation | adjacent representable binary64 values |
| frozen decision | schedule complete; further Float64 bisection forbidden |
The bracket is a representation- and protocol-dependent classification result. It is not an exact boundary in the real numbers or a measured physical uncertainty.
The result establishes a narrow technical fact: the stored v7.13 state meets the frozen numerical acceptance criteria, while the adjacent stored v6.74 endpoint is NONPASS. This closes the registered Float64 continuation schedule without reopening the solver.
The result does not demonstrate that the operational equations are unique, fundamental or empirically adequate. Passing positivity and hyperbolicity gates at the terminal trajectory does not prove global stability. Small residuals do not provide observational evidence. The ontology of an Apeiron medium remains a hypothesis outside the reach of the present calculation.
The current single-carrier interpretation therefore must not be mistaken for an additional numerical result. In particular, calling dark-matter-like effects an Apeiron manifestation is a model interpretation, not a claim that existing observations have already distinguished that interpretation from particle dark matter or from ()CDM.
The independent-comparison record adds a concrete numerical limitation. The stored reference is internally self-consistent, but the available release does not contain the complete reference residual, (D_G), the (_q), (p_q) and (^2_q) series, their component series, or mode invariants. Those quantities cannot be reconstructed or inferred from the scalar gate record. The precision isolation resolves the Wronskian mechanism, not the full-map comparison.
Physical plausibility requires separate work: action-level interpretation or another principled completion, perturbation dynamics, regulator and scheme analyses beyond the registered discrete scope, continuum/refinement evidence, comparison with established constraints and ultimately preregistered data tests.
The principal open questions are:
Items 1-7 constrain interpretation and later claims. They do not alter the already completed v7.13 classification. The current AP1 methods work and the locked AP2/AP3 packages are not prerequisites for reporting this narrowly scoped numerical result, and none may be projected backward onto it.
Section 10 summarizes a prospective program, not completed evidence.
AP1 is intended to derive (H(z)), (w_{}(z)), (f_8(z)), ((k,z)), ((k,z)) and ((k,z)) from one frozen parameter core. Its separate workstream has fixed the minimal M-1 matter extension and passed bounded implementation, junction, transport and response-structure preflights through G32. The current atomic state is a methods-preflight PASS for a balanced coupled three-level pilot: the pilot through relative (N=0.08) remains open, any later tolerance freeze and the disjoint (N=0.0850.10) heldout remain conditional, and the physical response kernel is blocked. AP1 therefore remains ORANGE. Before data contact it still requires a unique physical () mapping, a released background, a computed and validated physical response kernel, closed perturbation equations, stability gates, fixed grids, priors, masks, likelihoods and versioned output templates. No such curves are reported here.
AP2 is locked pending formal AP1 release. If unlocked, it would prospectively test a residual lensing profile after reconstruction of galaxies, gas and the declared comparison-model matter distribution:
[ {} =F(r,z,d_{},{}_{}). ]
It must reuse the AP1 parameter core and pass prespecified void and rotated-axis null tests. No filament signal is claimed in this paper.
AP3 is locked pending formal AP1 and AP2 release. If unlocked, it would treat (c_{}=c) as a survival condition and prospectively derive amplitude, polarization or line-of-sight observables from a closed tensor sector. No modified-propagation detection is claimed.
AP1-AP3 must use one preregistered parameter core. Channel-by-channel retuning is forbidden. A conflict with any hard gate produces NONPASS for the affected hypothesis rather than retrospective adjustment of v7.13.
AP1 has executed only bounded, gate-limited methods development and remains unreleased for physical-kernel or observable production. AP2 and AP3 have not entered production work. No template values, likelihoods, curves, fits, significances or observables are reported in this manuscript.
A separate post-freeze branch tests a concept that is not part of the homogeneous v7.13 map: whether an organized Apeiron manifestation can return toward more fundamental Apeiron states in extreme black-hole curvature.
The minimal effective description introduces an organization variable
[ 0s, ]
with (s=1) denoting an organized matter-like manifestation and (s=0) a more fundamental Apeiron state. A local directed state-change law can be written as
[ u^s=-{} C(K)s, _{},C(K)<1, ]
together with one conserved total stress-energy tensor,
[ T{}^{}=0. ]
A local curvature-response variable (K) can be driven by a non-negative strong-curvature diagnostic (W),
[ K u^K=W-K, ]
so that the reconversion variable reacts to a smooth bounded local response rather than to an imposed coordinate radius.
The feasibility result is deliberately limited. It establishes the existence of an effective class in which one carrier, directed de-organization, positivity, local total energy-momentum conservation, causal transport and a compatible hyperbolic evolution structure can coexist. It does not establish a unique microscopic Apeiron end state, a measured reconversion rate, a unique fundamental action or an observational detection.
The separate branch status is therefore
[ ]
at the structural-feasibility level only.
The frozen Apeiron/AME-1 v7.13 Stage-0 state at binary64 () satisfies every exact registered gate and is classified HARD_PASS. The adjacent binary64 endpoint () is an immutable v6.74 NONPASS because its fixed-point and Friedmann gates fail. The exact protocol midpoint converts to the NONPASS binary64 endpoint, so the registered Float64 crossing schedule is complete.
The publication package states the operational homogeneous model, exact v7.13 thresholds, artifact hashes, the partial independent-comparison result, the Wronskian precision diagnosis and a read-only verification path. The terminal v7.13 HARD-PASS remains frozen: the later work neither lifts nor independently reconfirms it. C0.29 is an open independent-reproduction finding, C0.32 isolates a sufficient technical cause for its Wronskian conflict, and C0.33-C0.35 freeze and audit the corrected no-map source. Full independent-map reproduction, continuum convergence, complete semiclassical stability, empirical confirmation and an Apeiron ontology remain open.
This manuscript reports the frozen numerical classification at the stated reproducibility level. AP1 remains a separate ORANGE follow-up workstream, while AP2 and AP3 remain locked. None has produced an empirical result.
Version 1.2 also records, without changing that evidentiary core, the current Apeiron single-carrier ontology and the separate black-hole reconversion feasibility result. Ordinary matter, dark-matter-like and dark-energy-like phenomena are treated as possible organizations/states of one carrier within the model; this remains an unconfirmed physical hypothesis. The black-hole branch is structurally feasible but empirically open.
APEIRON_CANONICAL_MODEL_SPECIFICATION_LATEST.md:
operational model, equations, units, state and source hashes.APEIRON_V7_13_GATE_SPECIFICATION_LATEST.json: exact
thresholds, operators, values and classification.APEIRON_REPRODUCIBILITY_MANIFEST_LATEST.json: SHA-256
manifest.README_REPRODUCIBILITY.md: verification scope and
instructions.verify_apeiron_publication_package.py: read-only
verifier.TECHNICAL_SOURCES/STAGE00_HARD_PASS_RESULT_v7_13.json:
immutable terminal result.TECHNICAL_SOURCES/STAGE00_HARD_PASS_SOLUTION_v7_13.npz:
immutable terminal arrays.TECHNICAL_SOURCES/RECOVERY_GATE_REGISTER_v7_13_STAGE00.json:
immutable terminal gate register.TECHNICAL_SOURCES/RECOVERY_PROTOCOL_v7_13.json:
immutable protocol.TECHNICAL_SOURCES/CROSSING_RESOLUTION_RESULT_v7_13.json:
immutable crossing decision.TECHNICAL_SOURCES/STAGE00_NONPASS_RESULT_v6_74.json:
immutable adjacent NONPASS result.TECHNICAL_SOURCES/STAGE00_NONPASS_SOLUTION_v6_74.npz:
immutable adjacent NONPASS arrays.TECHNICAL_SOURCES/RECOVERY_GATE_REGISTER_v6_74_STAGE00.json:
immutable adjacent NONPASS register.TECHNICAL_SOURCES/STATE_SNAPSHOT: terminal read-only
checkpoint snapshot.TECHNICAL_SOURCES/ACTIVE_RUNTIME: hash-frozen active
source path.All exact hashes are stored in the machine-readable manifest.
From the package root:
python3 verify_apeiron_publication_package.pyExpected terminal line:
APEIRON_PUBLICATION_PACKAGE_VERIFY_PASSData and code supporting the reported v7.13 classification are available in the public, read-only Apeiron/AME-1 Reproducibility Archive v1.0 at https://apeiron-publication-files.info523741.chatgpt.site. The core archive contains the canonical operational specification, exact gate table, immutable result/state/protocol artifacts, active source path, hash manifest and verifier. The same public download surface separately labels later AP1 progress artifacts through the G32 methods preflight; their presence does not incorporate them into the v7.13 evidentiary core or release a physical kernel. The frozen C0.29-C0.35 audit records are available from the author on request; they are not required for the archive’s read-only verification of the stored v7.13 classification. No observational dataset was analyzed.
No statistical fit, parameter estimation, hypothesis-test significance, Bayes factor, information criterion or observational model comparison is reported. PASS/NONPASS denotes only deterministic numerical gate classification.
The status note in Sections 1 and 10 is bound to
APEIRON_WORKSTREAM_REGISTRY_LATEST.json, schema
apeiron-workstream-registry-v3.0, updated 3 September 2026
at 03:32:58 UTC, and to the contemporaneous project safeguard record
Apeiron_Sicherung_aktuell.md. Those records classify AP1 as
ORANGE at the G32 methods-preflight checkpoint, with the balanced
coupled three-level pilot open and the physical kernel blocked. They
also state that no physical kernel has been computed, no production
curve has been released, and AP2 and AP3 remain locked. This paragraph
is documentary status alignment only; it adds no numerical result to the
frozen v7.13 calculation.
| Table | Content | Status |
|---|---|---|
| Table 1 | exact v7.13 gate definitions and outcomes | complete in Section 6 |
| Table 2 | Float64 crossing resolution | complete in Section 7 |
| Table 3 | verification levels | complete in Section 5 |
| Table 4 | controlled file manifest | complete in Section 12 |
| Product | Status |
|---|---|
| terminal result JSON and solution NPZ | included and hash-verified |
| exact v7.13 gate and protocol JSON | included and hash-verified |
| crossing-resolution JSON | included and hash-verified |
| adjacent v6.74 NONPASS artifacts | included and hash-verified |
| terminal checkpoint snapshot | included and hash-verified |
| active runtime source path | included and hash-verified |
| read-only verifier and manifest | included; PASS |
| C0.29 passive comparison | partial scalar diagnostic evidence; not a full reproduction |
| C0.30-C0.35 precision and source audits | internally frozen; no new physical-map run |
| AP1/AP2/AP3 outputs | AP1 methods/preflight records are separate; no observable values exist; AP2/AP3 remain locked |
The retained external sources support the numerical-method context only. No external source is used to fit, calibrate or empirically test the frozen v7.13 state.