Recognition Science is the discovered architecture of reality. Its core results identify structure required by consistent recognition: arithmetic of records, the canonical comparison cost, golden scaling and three-dimensional linking. Each result states the conditions that force it. For example, literal reporting by the specified native comparison apparatus, continuity and the reference calibration leave one possible comparison cost, J.
Those necessities make physical predictions possible. An application must identify its physical objects, carry the derivation through to a measurable quantity and compare that quantity with observation. A failed identification is evidence about that derivation. A conflict with a correctly derived physical prediction requires revision at the part of the framework responsible for it.
The fine-structure obstruction discussed below concerns a particular normalization-scaled construction. A condition unchanged by that scaling cannot choose one member of the family. Establishing the physical coupling also requires the relation between field normalization and matter charge; this obstruction leaves the derivation of that relation open.
Read the foundation and its forcing arguments, the arithmetic and its native RS realizations, or the derivation of gravity. Each account gives the established results and the particular connections still being developed.
The charged-lepton paper derives mass ratios with no continuously adjustable dimensionless parameters. It retains discrete structural inputs and the externally fixed fine-structure constant; the electron mass supplies the SI calibration. Deriving all the refinement rules is a further obligation stated in the paper.
The July report below classified the correction layer as fitting. That entry does not pin the assessed manuscript version or provide a coefficient-by-coefficient comparison with the published construction. Its classification remains part of the historical record; an unfinished derivation alone does not establish numerical fitting.
Historical evidence record · version 14 · 30 July 2026
Evidence and rejection criteria, as recorded on 30 July 2026. This dated report preserves the institute’s comparisons, failed identifications, corrections and prospective tests at that time. It includes tests scheduled for the 2026 PDG release, NuFIT and JUNO neutrino fits, Rubin Year 3 shear measurements and an August 2026 hardware batch. The dates and judgments below belong to this record; they do not report later test outcomes or current probabilities.
This July record examines physical applications of the recognition-cost functional, J(x) = ½(x + 1/x) − 1, and the conditions under which their predictions follow without free continuous parameters. The supporting Lean 4 theorems establish mathematical results under stated conditions: uniqueness of the functional and composition law, the golden ratio with a counterexample showing why its conditions matter, eight as the sharp walk period, and three as the required linking dimension. Applying those results to physics also involves discrete choices, including mappings, anchors, selections and ansatz layers. The register below identifies those choices and the conditions for resolving them.
The July record includes failures as well as agreements. Two widely cited identifications are excluded by the framework’s own theorems. A neutrino identification was excluded at roughly 5σ, and the first resolved preregistered test failed its frozen threshold, leaving zero confirmations from one resolved test. A conflicting neutrino mass-sum value of 85.63 meV had no identified derivation and was withdrawn; the roughly 63 meV chain remained. The rotation-curve kernel performed worse than MOND. The electron mass was supplied as an input, and the connections to spacetime geometry and quantum amplitudes remained assumptions. The repair criteria below determine when a replacement follows from prior structure and when it counts as fitting.
The single-anchor fermion identity was the strongest retrospective claim in this account until its provenance check failed on 30 July 2026. At 182.201 GeV, a scale derived from a mass-free stationarity condition, nine transported PDG masses were reported to cluster into three families within five parts per million, using integer rungs from electric charge and color alone. The audit found that no chain from the declared inputs reproduced the nine published residues. In repository history, those residues appeared only as hand-set literals in the October 2025 submission. The declared machinery gave within-family spreads of 0.19 to 0.50. Exact reproduction required one free real number per species, fitted from the mass against which it was compared. The column labeled as the predicted band value equaled the within-family mean of the residues, to print precision.
The published scoring rule therefore removed the identity from the evidence for the framework. The error-budget and construction-space calculations remain below because they document what was assessed, but they assess numbers whose provenance was not established. The rebuilt null model also corrected the quoted 15.6 sigma: a false within-family independence assumption had added 2.8 sigma, while the rebuilt calculation gave 12.5 to 12.9 sigma local. Statistical significance does not restore the missing derivation. A successor claim must satisfy the same repair criteria: it must follow from structure fixed before the failure, add no continuous parameter and register every discrete choice.
The prospective tests below name measurements and rejection thresholds. They include a top pole mass of 172.76 ± 0.02 GeV against the quoted world average of 172.69 ± 0.30; NuFIT and JUNO tests of the splitting ratio and θ23 octant; five-percent shear suppression against two-percent forecast precision for Rubin LSST Year 3; and the sealed August 2026 hardware batch under both temporal readings. Other proposed tests were less discriminating: one dark-energy pair was unlikely to reject the prediction, and the black-hole coefficient had no available measurement. The confidence estimates retained below were subjective judgments made before the fermion identity was de-scored. No statistical derivation accompanied them. Retrospective agreements, including the 0.02σ neutrino ratio, ansatz-layer lepton fits and cosmological comparisons in wide windows, receive zero confirmation credit under this report’s scoring rule.
What has failed, because the argument depends on it. On 29 July 2026 we lost one of the framework's neutrino identifications. The assignment of the mass-splitting ratio to φ⁷, about 29.0, was excluded at roughly 5σ once the NuFIT 5.3 anchor was corrected to 33.8 ± 1.0. The same day brought the first preregistered gate ever scheduled to fire, a quantum-hardware retention audit with its pass threshold frozen in advance at 2 SE, and it fired red. It came in at +1.2 SE, and we recorded it in-house as what it was: the apparent retention advantage is arithmetic, not physics.
The deeper problems were already on the page. Our own machine-checked theorems rule out the raw integer-rung electron ladder at 4.9 million σ. They also rule out the closed form for the fine-structure constant at more than 30,000σ, prove its seed premise false, and convict five mutually inconsistent constructions that all reached the measured value. A no-go theorem excludes selecting one value of the defined normalization-scaled construction using conditions independent of its normalization factor. This July account supplies α⁻¹ as an electromagnetic boundary input; the theorem does not establish that the physical coupling must be primitive. Both excluded identifications were among the ones we quoted most often. Two of our manuscripts contradicted each other on the neutrino mass sum until we ran the conflict down on 30 July 2026. The 85.63 ± 0.05 meV figure had no derivation anywhere in the manuscript, the Lean library, or any script, our own kernel-proved bounds put the sum in 62.61 to 62.76 meV, and the wager is withdrawn. The surviving chain prints about 63 meV with a kill rule below 62 meV, and it still inherits the absolute scale by hand.
On galaxy rotation curves our global kernel does better than a global NFW halo and worse than MOND, at 3.06 against 2.01 in median χ² per point. The absolute mass scale is not derived; we supply the electron mass by hand. Underneath, the framework keeps a discrete ledger of recognition events, and the two bridges out of that ledger, to spacetime geometry and to quantum amplitudes, are both assumptions, one of them flagged as assumed in the code itself. The forward ledger, the only place new confirmation could come from, has so far resolved one gate, and it did not confirm the framework.
These failures and supplied inputs are part of the evidence against which the remaining claims must be assessed.
What the mathematics establishes. The comparison cost J(x) = ½(x + 1/x) − 1 is unique under its stated conditions, with explicit counterexamples when a condition is dropped. The composition law is unique in its stated family. The golden ratio is forced under its conditions, with the plastic number showing the role of those conditions. Eight is the sharp walk period, and three is the unique dimension admitting the required linking structure. These results constrain the mathematical model. Whether the model describes nature depends on the physical identifications and measurement comparisons.
Contact with existing data, ranked by structure rather than tightness. The strongest item was the single-anchor fermion identity, de-scored on 30 July 2026 when its named mass-information kill check fired; the finding and the receipts are in section 6. What remains below is retrospective contact, each entry carrying zero confirmation credit under the doctrine, plus one repaired Lean-verified agreement in the dark-matter fraction. At 182.201 GeV, a scale derived from a mass-free stationarity condition, every charged fermion carries an integer rung and a band integer built from electric charge and color alone, and the framework proves that no measured mass enters its own prediction.
The nine transported PDG masses then clustered into three families within five parts per million of their family means, and targeted ablations of the integer map destroyed the clustering by orders of magnitude. The manuscript's own null models put the chance probability at a 15.6-standard-deviation equivalent; an honest rebuild on 30 July 2026 corrects that figure to 12.5 to 12.9, the independence assumption behind it having been false in the manuscript's favor. On the same day the identity was de-scored: the published residues have no provenance, and no declared-input chain reproduces them. What remains is the record of the strongest retrospective claim the framework has produced, and the standard it must meet on regeneration.
Second, the fractional-rung neutrino ratio (φ¹¹ − 1)/(φ⁴ − 1) = 33.823 against 33.8 ± 1.0, a 0.02σ contact with a Lean-proved interval tighter than the measurement. The rungs were selected against oscillation data, so this contact is retrospective and earns no confirmation credit under the doctrine below. Third, the peer-reviewed lepton pair, muon at 1 ppm and tau at 90 ppm from the electron anchor, through a correction layer the source manuscript itself labels an ansatz. The doctrine below scores it as fitting; it is listed because the arithmetic is real even though the credit is not. Fourth, cosmological retrodictions (vacuum density at the dex level, αs, the baryon asymmetry, w₀), all inside windows that are often wide; section 4 says which.
How replacements are scored. A framework that generates many φ-expressions can produce a replacement formula after a failure. To distinguish a consequence of earlier structure from a fit to the new data, this report uses two criteria fixed in advance. Both must be satisfied.
Clause A. A repair counts for the framework only if the successor's form pre-exists in, or is derivable without discretionary choice from, structure fixed before the refuting datum.
Clause B. The successor adds no continuously adjustable parameter and no discrete choice absent from the free-choice register below.
A repair that passes both clauses follows the report’s criteria for a consequence of prior structure. A repair that fails either clause counts as fitting. Keeping such a repair as confirmation would invalidate the evidential argument made here.
Applying the criteria to earlier corrections. The labels record four outcomes: repair passes both clauses; closed was retracted without replacement; fitting failed a clause and was withdrawn; convicted failed a clause and was removed from the confirmation score. The correction history is assessed below:
| Repair | A | B | Score |
|---|---|---|---|
| Nanoscale gravity, 2015 → 2017. First estimate about 100× weaker than measured; the repair was the framework's own discrete-ledger counting, with no new freedom. The framework now predicts no deviation at the scales tested. | pass | pass | repair Counts as a repair. The resulting contact is retrospective and wide-window, so it earns no credit. |
| Dark energy, 2025 → 2026. c = −3/2 with w = −1 proved unsupported and withdrawn; the successor (w₀, wₐ) = (−0.983, −0.017) was derived from the pre-existing vacuum sector and carries its own falsifiers. | pass | pass | repair Counts as a repair. Credit rides entirely on the DESI and Euclid exposure in section 2. |
| Neutrino ratio, 29 July 2026. φ⁷ excluded at ~5σ; the successor RΔ = (φ¹¹−1)/(φ⁴−1) was found in a manuscript committed 9 February 2026 (commit 2115f0a615, institute working repository, not yet public), five months before the refuting datum, alongside the manuscript's own integer no-go (about 2,207, roughly 65 times the observed value) and its own falsifier. | pass | pass | repair Counts as a repair, with zero confirmation credit: the rungs were selected against oscillation data. It is the pilot instance of this doctrine, not its proof. |
| Voxel-count derivations. Two literature voxel counts proved inconsistent; both are flagged in the formalization, with withdrawal recommended and no successor claimed. | n/a | n/a | closed Clean close: retracted, nothing patched in its place. |
| Fine-structure constant. The closed form included a term calibrated to the measured value, exactly the move clause B prohibits; the institute withdrew the identification. The library then convicted the project itself: the seed 4π·11 rests on a premise proved false in Lean, since the 11 counts passive field edges and the gauge count is 5. Five mutually inconsistent constructions all reached the measured value. The no-go theorem shows that normalization-independent conditions cannot pin the defined normalization-scaled family. The July register supplied α⁻¹ as a boundary input; the theorem leaves its physical derivation open (section 5). | fail | fail | fitting The prohibited repair was made and then retracted. The retraction is what the doctrine requires; the tuned term stands here as the example of what is not permitted. |
| Electron ladder. Raw integer rung excluded at 4.9Mσ; the 1 ppm muon and 90 ppm tau reach experiment through a correction layer the manuscript labels an ansatz, its form chosen to reproduce the observed ratios. | fail | n/a | convicted The numbers are arithmetically real and carry no confirmation credit until the correction coefficients are derived from pre-fixed structure. |
Net score: three repairs pass both clauses, one item closed cleanly, one prohibited tuning withdrawn as the doctrine requires, and one repair stands convicted. The convicted item is the framework's most-quoted flagship, and this page de-scores it by name. That is what the doctrine is for.
The report assigned subjective percentages without a stated probability model, priors or likelihoods. They are retained here as dated editorial judgments. They are not measured probabilities of the foundation being correct, and they do not replace the evidence for individual claims.
“The current formalization, as stated, we put below 1%, because the kernel exclusions stand; the corrected family we put at four to five percent, though on the morning of 29 July it was two to three. We do not supply priors or likelihoods for these numbers.”
The July report called these figures an “institutional reading.” Its numerical estimates concern the claims and assessments recorded at that time. The foundation and its applications should be assessed through their stated premises, proofs and physical evidence.
We hold one commitment to be self-binding. Before the 2026 PDG global fit lands, we will resolve the neutrino mass-sum conflict publicly, one number, chosen with a dated note. To reconcile it afterward would be exactly the downstream repair the doctrine prohibits, and it would void this essay's center. Until then we quote no mass-sum wager.
What settles it, with dates. August 2026: a sealed hardware data batch with a frozen analysis, one machine window, deciding between the two temporal readings of the framework's decoherence law, with a landing outside both readings named in advance as fatal to that account. Within months: the 2026 PDG global fit against a top pole mass of 172.76 ± 0.02 GeV, world average 172.69 ± 0.30, named in print as decisive. Now to three years: NuFIT and JUNO shrinking the 3% uncertainty on the splitting ratio, where departure from (φ¹¹−1)/(φ⁴−1) beyond uncertainty refutes the fractional rungs, and the θ₂₃ octant, where sin²θ₂₃ = 0.544 and a decisive lower octant refutes. One to three years: DESI and Euclid on the pair (−0.983, −0.017), where any freezing signature at 5σ kills the vacuum sector. About three years: Rubin LSST Year 3 on a 5% shear suppression at ℓ ≈ 1500 against 2% forecast precision. Unscheduled and most exposed: the black-hole entropy log coefficient −(ln φ)/2 = −0.2406, distinct from both literature candidates by more than a quarter, with no measurement in existence.
These wagers are not equally decisive and they are not statistically independent: shared φ-structure means hits and misses correlate, so no binomial arithmetic is offered. The July report proposed an order-of-magnitude increase in its subjective confidence if two of the six dated wagers in section 2 landed; no probability model was supplied for that proposed change. If the sealed August batch lands outside both of the framework's readings, the decoherence account is dead by its own rule; if the dark-energy pair fails, the vacuum sector is dead by its own rule. This page will say so in these tables, with the date attached.
So we try to apply the same discounting a skeptical referee would apply, and to apply it honestly. The uniqueness theorems are mathematics rather than physics, and both bridges are assumptions. The framework's own kernel excludes the electron ladder at 4.9 million standard deviations, and it excludes the fine-structure closed form at thirty thousand, proves its seed premise false, and rules out pinning the defined normalization-scaled family by normalization-independent conditions. Under the doctrine above, the peer-reviewed lepton chain is scored as fitting, because fitting is what it did. The 0.02σ neutrino contact earns no credit, since its rungs were selected against the same data it is compared with. The dark-matter row failed its own arithmetic on inspection (the printed gap term was the framework's tick constant, mis-slotted); it was repaired from the closure machinery and reinstated within a day, with the composition step registered rather than claimed (section 6). On rotation curves the kernel loses to MOND, and the forward ledger has no hits from the one gate it has resolved.
One construction stood until today, and its fall is the discount applied to ourselves. The single-anchor identity at 182.201 GeV carried the strongest retrospective claim: nine transported PDG masses within five parts per million of three family means, integers from charge and color alone. On 30 July 2026 its own named kill check fired: the published residues appear in repository history only as hand-set literals, no declared-input chain reproduces them, and the table's predicted column equals the mean of the data it claims to predict. The construction is de-scored by the rule printed for it in advance.
The two findings that would de-score it were named in advance, and one of them fired. Audits of this kind are what this page exists to record, and the record is above. The dated gates above are where the remaining questions settle, on instruments the institute does not control. Watch those.
Every row on this page carrying a gray edge and a mono HAND chip has a number or a form fixed somewhere by a human hand: a mapping, functional form, assignment, anchor, calibration, or window. That mark means the value is fixed neither by a kernel theorem with a public Lean receipt nor by an external measurement with a citation. Entries are counted once per distinct object at the granularity of the page's claim rows, and meta-judgments about the page itself are out of scope. Those three sentences are the counting rule, printed. This register is what makes clause B above enforceable: "no unregistered discrete choice" now has the register it refers to. Classes: ANCHOR, BRIDGE, ANSATZ, SELECTION, HYPOTHESIS. An entry closes when its closure condition is met, and the closure names the commit. The register is neither a degrees-of-freedom calculation nor an evidential weight.
| ID | Class | Object and claims affected | Closure condition |
|---|---|---|---|
| H-B1 | BRIDGE | Ledger-to-spacetime-geometry map, postulated. Affects the Regge (discrete gravity on a lattice mesh) and cosmology rows. | Derived in Lean with a public receipt. |
| H-B2 | BRIDGE | Ledger-to-quantum-amplitudes map, postulated. Affects the decoherence wager; the photonic witness row is an application of this bridge. | Derived in Lean with a public receipt. |
| H-AN | ANCHOR | Electron mass supplied by hand (the open absolute scale). Affects every absolute-mass number. | The absolute scale is derived. |
| H-DEC | SELECTION | Decoherence fringe readings (Gaussian-cored versus exponential), manuscript derivations not in Lean. Affects the decoherence wager. | Kernel derivation. |
| H-RD | SELECTION | Fractional rungs (11, 4) selected against oscillation data; registered zero-credit in the doctrine above. Affects the neutrino-ratio wager. | Kernel derivation of the fractional rungs. |
| H-W | SELECTION | (w₀, wₐ) coefficient derivation from the vacuum sector; not independently audited. Affects the w-pair wager and the w₀ contact row. | Independent audit or Lean formalization. |
| H-T | SELECTION | Top-rung assignment behind 172.76 GeV, manuscript-only. | Kernel derivation. |
| H-TH | SELECTION | sin²θ₂₃ = ½ + 6α form, manuscript-only. | Kernel derivation. |
| H-BH | SELECTION | Black-hole log-coefficient derivation, manuscript-only. | Kernel derivation or an independent determination. |
| H-S8 | SELECTION | Shear-suppression derivation, manuscript-only. | Kernel derivation. |
| H-ID | SELECTION | Anchor-identity integer map F(Z): built from charge and color, mass-free, non-circular; its form is a constructed map whose deeper derivation is open. Affects both identity rows and the closing argument. | A uniqueness theorem or kernel derivation of the map. |
| H-RC | SELECTION, HYPOTHESIS | Rotation-exponent derivation manuscript-only; amplitude C = φ⁻² a labeled hypothesis. Affects both rotation rows. | Kernel derivation of the exponent; the amplitude derived. |
| H-NS | HYPOTHESIS | Slot identifications: dark-matter candidate ~5×10¹¹ eV, sterile ~18 keV, ALP ~7×10⁻⁴ eV. | Discovery at a predicted slot, or derivation. |
| H-LEP | ANSATZ | Lepton correction layer, classified as fitting in this July assessment. Affects the muon/tau row. See the published claim and the assessment’s source limits. | Coefficients derived from pre-fixed structure (doctrine clause A). |
| H-AL | ANSATZ | Retired fine-structure closed form (α⁻¹ = 137.03489); its seed premise is proved false in Lean. The July account treated the inverse coupling as a boundary input. The no-go theorem concerns the normalization-scaled construction; it does not prove physical primitivity. Affects the fine-structure entries. | Recorded as a boundary input on 30 July 2026. Deriving the physical coupling requires the matter-charge relation as well as field normalization, beyond the variables in this no-go theorem. |
| H-DMG | FORCED, 2 PREMISES | Dark-matter gap composition: the ledger-undecidability cost enters the fraction as closure-work-per-channel diluted once, δ = φ⁻⁴/24 = J(φ)/(12φ). Forced 30 July at the conditional-uniqueness standard: every closure-weight correction that is a dimension-uniform closed occupancy at the spacetime dimension, spread with no distinguished phase over the 24 interference phases, and minimal non-zero, equals φ⁻⁴/24; the named rival forms give different numbers (Lean-verified axiom-clean, institute working library, not yet public). Two irreducible premises remain: nature instantiates the closure on this channel (the premise every application of the framework carries), and the undecidability cost of an unresolved path is its untransmitted closure weight. Affects the Ωdm row. | Met at the conditional-uniqueness standard, 30 July 2026. The instantiation premise is irreducible by the library's no-go theorems (conservation and minimality cannot select the ratio); the bridge premise is the mechanism's own ledger reading of the unresolved probability, not an added hypothesis. |
| H-SP | SELECTION, ANSATZ | Cascade formula form; boundary correction function behind the bottom-quark repair. Affects the spectrum row. | Kernel derivation of both. |
| H-AS | SELECTION | αs = 2/17 form, manuscript-only; no running law derived. | Kernel derivation, with a running law. |
| H-VAC | SELECTION | Vacuum-density exponent 122 and formula form, manuscript-only. | Kernel derivation. |
| H-MIX | SELECTION | Mixing-angle closed forms and window edges, manuscript-only. | Derived window edges. |
| H-BA | HYPOTHESIS, SELECTION | Baryon channel choice; coupling forms λ = φ⁻⁷, κ = φ⁻⁹ manuscript-only. | The channel selected by a theorem. |
| H-H2O | SELECTION | Water ladder mapping and φ-wide windows; an organizing bracket from one measured anchor. | A uniqueness result or a derived mapping. |
Four entries are not registered, with reasons. The Regge convergence row runs on mechanical power counting plus a computational receipt; it registers if the power counting is disputed. The enzyme-audit row is a methodology benchmark: it fixes no physical number or form. The July confidence estimates are subjective judgments, outside the rule's scope. And the five kernel theorems of section 3 are receipted by definition.
Each entry carries a tag for the kind of support it has, and a plain statement of the strength of the agreement. kernel machine-verified theorem (public Lean code). peer‑reviewed published in a refereed journal. preprint public, not yet refereed. manuscript in preparation; its numbers are restated here as dated public commitments. wager a forward prediction with a named kill rule, standing from the date of this page. excluded refuted by data; kept visible, not deleted. corrected an earlier error, fixed with its receipt named. withdrawn an earlier claim retracted in print. HAND a number or form fixed by a human hand somewhere in its chain, registered with its closure condition in the free-choice register above.
Strength is stated against the resolution of the measurement, not adjectives: "0.02σ" means the difference is one fiftieth of the measurement's uncertainty; "ppm" is parts per million; "dex" is a factor of ten. An agreement inside a wide window is weaker than one inside a narrow window, and we say which.
These are the items that will settle the question. Seven are forward wagers, dated here and decided by instruments we do not control; the neutrino row's rung family was selected against existing oscillation data, so its current match earns no confirmation credit. The last three are standing commitments from cited work; each row names its arbiter and its scope.
| Quantity | Value | Arbiter and kill rule | Horizon | |
|---|---|---|---|---|
| Decoherence-law temporal reading | Gaussian-cored fringe form vs exponential form (two readings of ledger cost accrual) | Sealed IBM Marrakesh Ramsey batch (a standard atom-interferometer fringe measurement), one machine window, analysis frozen before unblinding. Frozen decoy-calibrated thresholds choose between the two readings; a landing outside both (a named outcome in the frozen partition) kills the framework's decoherence account. HAND · H-DEC | Aug 2026 | wager |
| Neutrino splitting ratio Δm²₃₁/Δm²₂₁ | (φ¹¹−1)/(φ⁴−1) = 33.82 | NuFIT / JUNO global fits. Current value 33.8 ± 1.0. A value departing from the predicted ratio at 3σ or better, uncertainties combined, refutes the fractional rung assignment. HAND · H-RD | now–3 yr | wager |
| Dark-energy equation of state (w₀, wₐ) | (−0.983, −0.017) | CPL fits to BAO + supernovae + CMB (DESI, Euclid, Rubin). Inconsistency at 5σ, in particular any freezing signature (w₀ < −1, wₐ > 0), falsifies the vacuum sector. HAND · H-W | 1–3 yr | wager |
| Top-quark pole mass | 172.76 ± 0.02 GeV | 2026 Particle Data Group global fit. Current world average 172.69 ± 0.30; the wager is the central value as the uncertainty shrinks. A future central value more than 3σ from 172.76, uncertainties combined, refutes the rung assignment. HAND · H-T HAND · H-AN | months | wager |
| θ₂₃ octant | sin²θ₂₃ = ½ + 6α = 0.544 | Global oscillation fits (NuFIT). A lower-octant determination at 3σ or better refutes the prediction (upper octant required). HAND · H-TH | 1–3 yr | wager |
| Black-hole entropy log coefficient | c = −(ln φ)/2 = −0.2406 | Any independent determination outside ±0.05 of the value. Distinct from both literature candidates (−½ and −3/2) by more than ¼. HAND · H-BH | unmeasured | wager |
| Cosmic shear at ℓ ≈ 1500 | 5% suppression vs ΛCDM | Rubin LSST Year-3 weak lensing at 2% forecast precision: a decisive yes/no. HAND · H-S8 | ~3 yr | wager |
| Fermion-mass anchor identity | |f − F(Z)| ≤ 10⁻⁶ at 182.201 GeV | Any measured split inside an equal-Z family at the anchor scale, or a residual above 10⁻⁶ under the declared kernels, falsifies. Perturbing the integer structure breaks the identity by six orders of magnitude (ablations published with it). Resolved 30 July 2026: de-scored, the named mass-information kill check fired; see section 6. HAND · H-ID | continuous | de-scored |
| Galaxy rotation kernel | exponent 0.191, amplitude φ⁻² | A global best-fit exponent inconsistent with (1−φ⁻¹)/2 at > 3σ kills the derivation; any need for per-galaxy parameters kills the mechanism. HAND · H-RC | continuous | peer‑reviewed |
| New states and the fourth generation | DM candidate ~5×10¹¹ eV; sterile ~18 keV; ALP ~7×10⁻⁴ eV; 4th generation forbidden | Discovery of a state at a predicted cascade index and mass is strong positive evidence; a confirmed fourth-generation fermion breaks the boundary. Non-discovery of the predicted states refutes nothing; this row can only confirm. HAND · H-NS | ongoing | preprint |
The foundational chain is formalized in Lean 4 and public. These are theorems about the mathematics: they say the structure is forced, not yet that it is physics. The step from the discrete formalism to measured physics is made at named bridges, and two of those bridges are currently assumptions (section 5).
| Statement | Artifacts | |
|---|---|---|
| The comparison cost J(x) = ½(x + 1/x) − 1 is the unique cost satisfying the framework's conditions; dropping any single condition admits an explicit counterexample. | code · paper | kernel |
| The composition law is the unique consistent combiner in its family. | code · paper | kernel |
| The golden ratio is forced as the self-similar fixed point; the plastic number is the counterexample showing the conditions are tight. | code · paper | kernel |
| Three spatial dimensions: the unique dimensionality in which the linking structure the ledger requires is possible. | code | kernel |
| The eight-step fundamental cycle: 8 is the sharp period of the underlying walk structure. | code | kernel |
Tightest first, by agreement against the measurement's stated resolution. The argument above ranks by structure instead and puts the fermion clustering first; the two orderings measure different things, and the doctrine, not either ordering, assigns the credit. "Scope" says what the entry does not cover.
| Quantity | Framework | Measured | Strength and scope | |
|---|---|---|---|---|
| Muon and tau masses from the electron anchor | 105.658 MeV; 1776.7 MeV | 105.658 MeV; 1776.86 MeV | ~1 ppm and ~90 ppm respectively, from one anchor and no fitted constants. Scope: the correction layer is labeled ansatz in the paper itself, and the raw integer ladder without it fails (section 5). Under the repair doctrine above, this contact is scored as fitting: no confirmation credit until the coefficients are derived. Symmetry 18(6):962 HAND · H-LEP HAND · H-AN | peer‑reviewed |
| Charged-fermion clustering at one anchor scale | three equal-Z families, integers from charge and color alone | PDG masses transported to 182.201 GeV | De-scored 30 July 2026: the named mass-information kill check fired. The published residues have no generating pipeline in repository history, appearing as hand-set literals since the October 2025 submission; no declared-input chain reproduces them (the declared machinery spreads 0.19 to 0.50 within families, not 5 ppm); exact reproduction takes one free real per species fitted from the mass it is compared against; the column labeled as the predicted band value equals the family mean of the residues themselves. What follows is the record of the claim as it stood. All nine charged fermions cluster within 5×10⁻⁶ of their family means; non-circularity is proved (no measured mass enters its own prediction); ablations of the charge map destroy the clustering by orders of magnitude; the manuscript's null models put the chance probability at a 15.6σ-equivalent, trial factors documented in its appendix. The error budget (30 July 2026) clears the compression objection: the map amplifies relative mass noise by a factor of 2.08, all nine residuals sit 10× to 70,000× below their propagated noise floors, and noise alone would reproduce the clustering with probability about 10⁻³¹; the clustering is a property of the measured central values, resolvable at 5 ppm only for e and μ. Open audit inside that answer: the pinned residues do not reconstruct exactly from the central values through the documented chain (a lepton-level mismatch); its regeneration is underway. The construction-space accounting (30 July 2026) clears the second review objection: charging every live alternative in the register against the 15.6σ null costs at most 0.7σ, leaving 14.9 to 15.5σ global, and killing the result would take 6.2×10⁴⁷ constructions against a worst-case register space of 65,536; still open inside the null is a rebuild of its tolerance and independence assumptions. Scope: one anchor scale, not RG-invariant; the mechanism is open. Corrections to the record, 30 July 2026: the rebuilt null prices the clustering at 12.5 to 12.9 sigma local rather than 15.6, the independence assumption having been false and worth 2.8 sigma in the manuscript's favor. The noise-alone probability moves from 10⁻³¹ to 5.7×10⁻₂⁰ under the measured lepton noise correlation; both computations price numbers without provenance. public code and data HAND · H-ID | de-scored |
| Full fermion spectrum | cascade formula, 30+ states | PDG | ≤ 0.1% typical for leptons, quarks, mesons, baryons. Scope: the bottom quark misses by 315% before the boundary correction shown in the same table. arXiv:2506.12859 HAND · H-SP | preprint |
| Galaxy rotation (147 SPARC galaxies) | global kernel, no per-galaxy parameters | 2,933 data points | Derived exponent 0.191 vs fitted 0.19; median χ²/N = 3.06 vs 5.27 for global NFW. Scope: MOND does better (2.01); the amplitude is a labeled hypothesis; non-relativistic. Entropy 28(4):477 HAND · H-RC | peer‑reviewed |
| Dark-matter fraction Ωdm | sin(π/12) + φ⁻⁴/24 = 0.264898 | ≈ 0.2645 ± 0.005 (Planck 2018) | 0.09σ from the central value. Scope: the gap term is derived from the closure machinery (closure-work per channel at the closure ratio, equal to the dimension dilution spread over the 24 channel phases by an exact identity), with the identity and numerics Lean-verified axiom-clean in the institute working library (not yet public); the composition is forced at the conditional-uniqueness standard (unique cycle-symmetric minimal dimension-four closed occupancy), with two irreducible premises recorded in the register. Corrected 30 July from the misprinted 1/(8 ln φ) form; see section 6. corrected manuscript HAND · H-DMG | manuscript |
| Strong coupling αs(MZ) | 2/17 = 0.11765 | 0.1179 ± 0.0009 (PDG) | 0.3σ. Scope: no running law is derived; this is the single-scale value only. HAND · H-AS | manuscript |
| Dark-energy equation of state today | w₀ = −0.983 | −1.03 ± 0.03 (Planck+BAO) | ~1.6σ from the constant-w fit; consistent with the thawing quadrant DESI DR2 favors. Scope: a retrospective fit against the 2024 release; the forward pair (w₀, wₐ) is the wager in section 2. HAND · H-W | manuscript |
| Vacuum energy density | ρPl·φ⁻⁵⁸⁸ = 6.7×10⁻₂⁷ kg/m³ | 5.9×10⁻₂⁷ kg/m³ | Within a factor of 1.14 (0.06 dex) out of a 123-dex naive gap. Dex-level, not percent; the rung count is the content. HAND · H-VAC HAND · H-AN | manuscript |
| Mixing angles | |Vcb| = 1/24; |Vus|, |Vub|, sin²θ₁₂ inside windows | PDG / NuFIT | Within stated windows; sin²θ₁₃ and sin²θ₂₃ sit 1.3–1.6σ off their central values. HAND · H-MIX | manuscript |
| Baryon asymmetry | ηB = 5.1×10⁻¹⁰ | ~6×10⁻¹⁰ (CMB/BBN) | 15–20% level, with fixed couplings λ = φ⁻⁷, κ = φ⁻⁹ and no free parameters. Scope: tens-of-percent agreement, not yet at percent; the mechanism is one of several open channels. HAND · H-BA | manuscript |
| Discrete-gravity convergence exponent | 4 (power counting) | 4.15 overall; successive refinement levels read 4.27 → 4.09, drifting toward 4 | Numerical measurement on refining meshes; coefficient mesh-dependent, exponent stable. Scope: one lattice family; the coefficient is not yet determined. computation | manuscript |
| Water hydrogen-bond network | golden-ratio ladder, one measured anchor | H-bond timescales, THz bands | Time-domain timescales and principal THz band centres fall inside their rung windows, with the windows a factor of φ wide. Scope: an organizing bracket from one measured anchor, not a uniqueness claim; the paper says so itself. Molecules 31(15):2701 (2026) HAND · H-H2O | peer‑reviewed |
| Eight-slot structure on photonic hardware | nested parity-check filters | commercial photonic chip | Hardware test of the eight-slot balance structure. Scope: a structure witness; dynamics not yet tested. arXiv:2606.18408 HAND · H-B2 | preprint |
| Enzyme kinetics audit | reciprocal-cost consistency check | curated Haldane backbone | Two-sided backbone with a labeled fold-error benchmark. arXiv:2607.02784 | preprint |
These problems remained unresolved in the 30 July 2026 account and must be weighed alongside its positive results.
| Problem | Status |
|---|---|
| Two bridges are assumptions. The map from the discrete ledger to continuous spacetime geometry, and from the ledger to quantum amplitudes, is postulated in the current manuscripts, not derived. Until they close, the chain from the kernel theorems to measured physics passes through named assumptions. HAND · H-B1, H-B2 | open |
| The absolute scale is not derived. The theory predicts mass and coupling ratios; one measured anchor (the electron mass) is supplied by hand to fix units. Deriving that anchor is the program's central open problem. HAND · H-AN | open |
| The raw integer ladder fails at the electron. Without its correction layer, the integer mass ladder assigns the electron a rung far outside the measurement's uncertainty. The peer-reviewed chain works through the correction layer, which the paper itself labels ansatz, not theorem. | open |
| The no-go theorem concerns a normalization-scaled construction. The closed-form project is convicted three times over. Its main seed, 4π·11, rests on a premise the library itself proves false: the 11 counts passive field edges after one vertex is fixed, not gauge degrees of freedom, and the forced count is 5. Five mutually inconsistent constructions in the library produced the same measured 137.036, the signature of a formula generator, and the first-order value is excluded at more than 30,000 measurement sigmas. The no-go theorem applies to the defined family obtained by multiplying the construction value by a positive normalization factor. No satisfiable condition independent of that factor can fix one value of the family. The physical coupling also depends on the matter-charge relation, which this theorem does not supply. The July account treated α⁻¹ as a boundary input; that treatment is not a proof that its physical value can never be derived. The forced kinematic content, 4π·φ⁵ − 5·ln φ ≈ 136.957, lands in the right neighbourhood for forced reasons and stops; after the five-construction conviction its proximity is not evidence. HAND · H-AL | open |
| The neutrino mass-sum contradiction, resolved 30 July 2026. The overview manuscript's 85.63 ± 0.05 meV wager proved to be an orphan. No derivation in the manuscript, the Lean library, or any script produces it, and the framework's own kernel-proved bounds (sum in 62.61 to 62.76 meV) exclude it at 457 of its printed error bars. It is withdrawn, and the manuscript correction is staged for publication. The neutrino-sector manuscript (RS Masses III) stands: ≈63 meV with a kill rule below 62 meV. Its absolute scale still inherits the electron mass by hand, and its rung triple was selected against oscillation data, so its credit rides on the forward kill rules in section 2. | open |
| MOND fits rotation curves better. On the SPARC sample the framework's global kernel beats global NFW but trails MOND on median χ² (3.06 vs 2.01), as the paper reports. | open |