Jamming Spine — The Verified Physical Backbone: from an Independently Measured Lattice Stiffness to the Speed of Light $c²=B/ρ$ and the Forced Proton Radius $rₚ=(2/π)λ_(C,p)$
Jamming Spine: Two claims travel together and are graded separately. All three statements are exact; none is inflated and none is diminished. They are separated by 5768×. Read top to bottom; each line uses only what precedes it.
Two claims travel together and are graded separately. All three statements are exact; none is inflated and none is diminished. They are separated by 5768×.
→ r_p) What this hub is. The jamming substrate is the physical foundation of every result in this paper, but its links are distributed across the document (the axioms in S3, the rotational-unjamming bridge in S8.5.0, the stiffness keystone in S11.6). This section assembles that backbone into a single linear spine and attaches to each link a grade so a reader sees, in one place, what is verified, what rests on a definition, and what carries the single anchor.The verified / definitional / anchored boundary (stated up front, objectively).
Two claims travel together and are graded separately. The dynamics below are established by closed form and by parameter-free simulation: the single compression speedc^{2}=B/ρ(the shear modulus vanishes at the isostatic point; S2 here), the proton radius as a globally stable attractor (S4 here), rotation generating the inflow and the length scale, and the amplificationAas a measured lattice output. The numerical ladder3π^{4} → 6π^{5} → 6π^{6}rests on a single definitional identification,[MAP-1](one inter-sector phase lock=π^{2}=1/δ, S8.0.6), of exactly the same status as the definitions ofα andδ in S5;6π^{6}is not an independent constant butπ· 6π^{5}. The absolute SI scale carries the single anchorλ_{ref}=632.99nm cross-checked byA_{geo}=c Δ t/ato0.16%. All three statements are exact; none is inflated and none is diminished.
S0. The spine at a glance
Read top to bottom; each line uses only what precedes it.
| Step | Where / how | Grade |
| infinitely rigid, fully packed VP particles⇒ a jammed lattice | S3 axioms | [F](axiom) |
| rotation (temperature)↑⇒ self-organised marginal jamming,z=2d=6 | S11.6, S10.3 | [H]→[V] |
| marginal⇒ shear modulusG_{relaxed}→ 0, bulkBfinite | S2 here, S11.6.1 | [V] |
| single surviving longitudinal speedc^{2}=B/ρ=K | S11.6.1 | [V] |
| linear dispersionω=cq⇒λ=c/ν; quantum diameterD=2πλ/A | S9.4, S10.9 | [V]/Am |
| rotation pins the stiffness-shell radiusr_p=(2/π)λ_{C,p} | S6.2, S8.5.0 | [F]/[V] |
| l@Two sizes: protonr_p=0.8412fm|quantumD=4.8526pm (=6π^{6}r_p) |
02_lattice_percolation_soc/jamming_spine_verification_v0.4_2026-06-05/): (i) the bulk modulus measured directly so thatc^{2}=B/ρ; (ii) the relaxed shear modulus driven to zero at the isostatic point along five independent observables; (iii) the forced-radius fixed point shown to be a global attractor; (iv) the self-limited rotating-grinder core.
S1. Proton versus quantum — two distinct objects (do not conflate)
| Proton | Quantum | |
| size | radiusr_p=0.8412fm ( 10^{-15}m) | diameterD=4.8526pm ( 10^{-12}m) |
| what | dense core (82 cells,z=6=2d) | wavelength-scale rotational circulation length |
| origin | stiffness-shell balancer^{-5}/r^{-4} → x^{*}=2/π=r_p/λ_{C,p}; grinder | wavelength relationD=2πλ/A=2λ_{C,e} |
| constants | ν_p=3π^{4},m_p/m_e=6π^{5} | — |
| relation | D=6π^{6}r_p=π (m_p/m_e) r_p≈ 5768 r_p |
S2. Infinite stiffness→ c²: the single signal speed (verified)
S2.1 [axiom]. VP particles are infinitely rigid and fully packing, so the vacuum is a jammed lattice. S2.2 [[H]→[V]]. Rotation (temperature) drives the contact numberzto the rigidity threshold; overshoot triggers a local unjamming avalanche that relaxes it (self-organised criticality). The percolation gap pins at the microscopic threshold (independent re-run:g^{*}/g_0≈ 1.1–1.3), giving the isostatic thresholdz=2d=6. S2.3 [[V]]. Atz=2dthe packing holds exactly the Maxwell-count number of contacts for rigidity and no surplus: the shear reserve is zero, so the relaxed (non-affine) shear modulus vanishes,G_{relaxed}→ 0, while the bulk modulusBstays finite (compression needs no surplus). The transverse wave dies; one longitudinal speed survives,
| Observable | Type | z_0/ result |
| G_{relaxed}∝Δ z→ 0 | static linear response | 5.90/5.95(N{=}512/1024) |
| ω^{*}∝Δ z→ 0(τ→∞) | vibrational spectrum | 6.015(R^{2}=0.997) |
| η=σ/γ→∞ asγ→ 0 | finite-rate rheology | viscosity divergence |
| relaxed slope=G_{relaxed} | AQS finite strain | <1%match |
| σ_y→ 0 | AQS yield stress | 5.98(8-point fit) |
01_stiffness_to_c2/bulk.py, 02_shear_relaxedG/relaxed_shear.py.)
S3. Why a wavelength appears (operating principle, objections, counterexamples)
Principle. (1) A single speed gives linear dispersionω=cq(verifiedR^{2}≈ 0.99), henceλ=2π/q=c/ν(standard acoustics). (2) One period of a carrier wave (λ=633nm) is resolved byAmicroscopic propagation steps on the lattice, so the lattice resolution isa_{phys}=λ/A. (3) The quantum diameter is one2π phase winding,D=2π a_{phys}=2πλ/A. (4) The amplificationA=a_{med}/g^{*}(median neighbour distance over critical percolation throat) has a parameter-free dimensionless structure (the pinning ratiog^{*}/g_0 ≈ 1andφ_{jam}), while its absolute magnitude tracks the microscopic threshold (A 1/g_0) and is anchored to the unit realizationA_{geo}=cΔ t/ato0.16%(input-dependent; see S5 and the honest-reading note in S W.5). The single physical input isλ; the10^{5}-fold optical-to-quantum bridge and the2π closure are supplied by the lattice. Objections, answered. (i) Dimensional fitting? No:Ais a measured lattice geometry, reproduced across seeds. (ii) Why633nm? It is the single anchor; using two lines633/532makesA_{633}/A_{532}=633/532soDcancels — a falsifiable ratio independent of the value ofD. (iii) IsAanchored tocΔ t/a, hence circular? Acknowledged as a residual:A's absolute scale agrees withA_{geo}=cΔ t/ato0.16%; the one remaining circularity question is whetherΔ tis independent ofc(open, S5). Counterexamples, resolved. A static standing mode would giveD/λ=1.43(Besselℓ=1), not2π; the quantum is a rotating body (S8.5), so the rotational circumference2π is the natural length. The measured circulation length has median4.96pm with best avalanche4.854pm (target4.85);4.96vs4.85is a distribution offset (theAmedian sits 2.3%belowA_{target}=8.20×10^{5}), not a structural error.S4. Stiffness shell→the forced proton radius (verified attractor)
Rotational origin of the inflow. A marginal (z=2d) packing has zero shear reserve, so any rotational shear v=Ω× rbreaks contacts and unjams; the unjammed material exits only through the singleC_3nozzle (S8.0), giving a directed inflow r^{-4}— the origin of the S6.2 term, not a postulate. Forced radius as a global attractor. With rectified stiffness α r^{-5}outward against inflow r^{-4}, the dimensionless radiusx=r/λ_{C,p}obeys ̇x=α x^{-5}-x^{-4}:
The fixed point is unique (one positive root) and globally stable (every positive initial radius converges tox^{*}): the proton radius is dynamically selected, not a fine-tuned balance. Hencer_p=(2/π)λ_{C,p}=0.8412fm (CODATA charge radius0.8414fm,-0.02%). The rotating grinder confirms the mechanism: from3000spheres, counter-rotation+contraction+incompressibility self-limits to a 79–82-cell core withR_{rms}∝ 1/Ω(rotation sets the size). Honestly: the MD core is amorphous (z_{core}≈ 8.9); the exact82=81+1atz=6is the crystallineC_3count, which the MD confirms in mechanism, not in lattice type. (Modules 03_forced_circle_proton_radius/, 04_rotating_grinder/.)
Pinned provenance note (added 2026-06-11): the82-core is a full-physics-grade problem closed by geometric substitution — without the internal three-rotation (n=3/C₃) logic, the quark-structure candidate would not have been findable.
The honest sentence above (“the MD confirms in mechanism, not in lattice type”) is to be read at full strength, and is pinned here so it cannot be under-read. Resolving the exact core integer by direct computation is a problem of the same class as the gravity absolute (S17.4.4 pinned status): the reduced engine reaches “∼79–82 cells, amorphous (z_{core}≈8.9)”, and no accessible run was going to crystallize the exact82— that would require native-resolution full physics. What closed it instead was geometric substitution: the internal three-rotation logic — theC_3three-fold structure and then{=}3event lawν_n=n π^{2(n-1)}, read physically as the proton's three-quark composition (Part II.3 decoder) — selected the counting frame in which Legendre's three-square theorem forces81=#\{R^{2}≤6\}=3^{4}with theR^{2}{=}7shell empty, hence82=81+1(and89=82+7). Without that three-rotation logic the quark-structure candidate would not have been findable at any accessible computational scale; with it, geometry did the work that, in the gravity sector, still awaits HPC computation. This completes an explicit three-grade taxonomy across the framework's hard absolutes: (a) geometry-rescued (this sector:82,89,ν_p=3π^{4}— full-physics-grade problems closed by forced geometry; the grade letter [F], “Forced,” is here literal); (b) computation-gated (absoluteg, S17.4.4 — concept and mapping complete, the absolute awaits HPC-scale full physics); (c) measured input (α_{em}, S14.5 — openly supplied by experiment). The grade of the82result itself is unchanged ([F]); what is pinned is its provenance: forced geometry standing in, successfully, for inaccessible computation. This note adds no input and triggers no realization versioning (S2.3.7). Connection.D=6π^{6}r_p=π(m_p/m_e): the proton/quantum ratio is the mass ratio;6π^{6}is the disguise of6π^{5}, not a new constant.S5. Honest status ledger
| Piece | Grade | How |
| single speedc^{2}=B/ρ(Bfinite,G→0) | [F]/[V] | Maxwell count+relaxed-Gfive-observable sim |
| linear dispersionλ=c/ν | [F]/[V] | standard+ω=cqverified |
| α=2/π=⟨|cosθ|⟩;δ=1/π^{2};2π=α/δ | [F] | elementary integrals (S5) |
| forced radiusr_p=(2/π)λ_{C,p} | [F]/[V] | global stable attractor,F'(x^{*})=-(π/2)^{5} |
| rotation creates inflow / length,x^{*} 1/Ω | [F]/[V] | S8.5.0+grinder |
| amplificationA=a_{med}/g^{*}(order from jamming) | [H] | structure parameter-free; magnitude 1/g_0, anchored toA_{geo}{=}cΔ t/a(0.16%) |
| ν_p=3π^{4};m_p/m_e=6π^{5};D/r_p=6π^{6} | [D] | one definition textsc[MAP-1] (status ofα,δ);6π^{6}=π· 6π^{5} |
| absolute pm scale (the value4.85) | Am | single anchorλ+A_{geo}=cΔ t/ato0.16% |
Reproducibility map (this spine).
The verification modules live in the unified bundle under02_lattice_percolation_soc/jamming_spine_verification_v0.4_2026-06-05/ (index: JAMMING_VERIFICATION_INDEX.md); the broader v0.3 module set and the largest (N{=}750$) amplification run are retained under jamming_rotation_verification_v0.3/. The detailed reproduction tables are in §11.6.5.